Julie says that the triangles are congruent because all the corresponding angles have the same measure.
Ramiro says that the triangles are similar because all the corresponding angles have the same measure.
Is either student correct? Explain your reasoning.
Hint: Find
Answers
congruence means, the figures are a duplicate or an exact twin of the other, that is angles as well as sides are the same, well, clearly ABC is larger, so they're not congruent.
That said, we could have a figure with same angles, and another with the same angles, but their side are not restricted due to the angle, the sides can easily extend or shrink, whilst the angles are being retained all along, and thus the figures being similiar, but never congruent.
Related Questions
What is the value of n if 5.69×10n=5690000?
Answers
Answer:
n=6
Step-by-step explanation:
If you see, you need to move the decimal 6 places to the right to get to 5690000
Let (X,Y) be uniformly distributed on the triangleD with vertices (1,0), (2,0) and (0,1), as in Example 10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You might first deduce the answer from Figure 10.2 and then check your intuition with calculation. (b) Verify the averaging identity for P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:[infinity] −[infinity] P(X ≤ 1 2|Y =y)fY(y)dy.
Answers
suppose f(x) is continuous on [2,7] and −4≤f′(x)≤2 for all x in (2,7). use the mean value theorem to estimate f(7)−f(2).
Answers
We find that -20 ≤ f(7) - f(2) ≤ 10. So, using the Mean Value Theorem, we can estimate the difference between f(7) and f(2) to be in the range of [-20, 10].
We are given that f(x) is continuous on [2,7] and its derivative, f'(x), is between -4 and 2 for all x in (2,7). We are asked to use the Mean Value Theorem (MVT) to estimate f(7) - f(2).
First, let's recall the Mean Value Theorem. If a function is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
Now, let's apply the MVT to our problem. We have:
f'(c) = (f(7) - f(2)) / (7 - 2)
We know that -4 ≤ f'(x) ≤ 2 for all x in (2,7). Therefore, -4 ≤ f'(c) ≤ 2 for some c in (2,7). Using this inequality, we get two separate inequalities:
-4 ≤ (f(7) - f(2)) / 5
2 ≥ (f(7) - f(2)) / 5
Now, we can multiply both sides of each inequality by 5:
-20 ≤ f(7) - f(2)
10 ≥ f(7) - f(2)
Thus, we find that -20 ≤ f(7) - f(2) ≤ 10. So, using the Mean Value Theorem, we can estimate the difference between f(7) and f(2) to be in the range of [-20, 10].
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Suppose n is a vector normal to the tangent plane of the surface F(x,y,z) = 0 at a point. How is n related to thegradient of F at that point?Choose the correct answer below..A. The gradient of F is a multiple of nB. The gradient of F is equal to n.C. The gradient of F is orthogonal to nD. The gradient of F is not related to n
Answers
If n is a vector normal to the tangent plane of the surface F(x,y,z) = 0 at a point, then option (C) the gradient of F is orthogonal to n.
The gradient of F at a point (x₀, y₀, z₀) is defined as the vector (∂F/∂x, ∂F/∂y, ∂F/∂z) evaluated at that point. This gradient vector is perpendicular (or orthogonal) to the level surface of F passing through that point.
The tangent plane to the surface F(x, y, z) = 0 at a point (x₀, y₀, z₀) is defined as the plane that touches the surface at that point and is perpendicular to the normal vector at that point.
Thus, if n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point (x₀, y₀, z₀), then n is perpendicular to the tangent plane. Since the gradient vector of F at the point (x₀, y₀, z₀) is perpendicular to the tangent plane, it is also perpendicular to n.
Therefore, the correct answer is (C) The gradient of F is orthogonal to n.
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4. True or False? Use your knowledge of the indirect proof method to determine whether each statement is true or false. Check all that apply You cannot use an indirect proof sequence within the scope of another indented sequence. You can obtain only conditional statements with the indirect proof method. The explicit contradiction on the last line of an indirect proof sequence must contain the same proposition letters as those in the assumption beginning the sequence. If you can derive an explicit contradiction from an assumption beginning an indented sequence, then the assumption must be false. You should justify the assumption beginning an indirect proof sequence with the abbreviation "AIP" without citing any line numbers. The proposition (29 F). (ZF) could serve as the last indented line of an indirect proof sequence. You can correctly use line numbers from an indirect proof sequence that has been discharged as justification for subsequent lines. You cannot cite line numbers from an indented indirect or conditional proof sequence that has already been discharged as justification for any subsequent lines. The proposition (ZV F) (ZV F) cannot serve as the last indented line of an indirect proof sequence. No proof can end with an indented line. U .
Answers
The following statements are true regarding the indirect proof method:
You cannot use an indirect proof sequence within the scope of another indented sequence.
The explicit contradiction on the last line of an indirect proof sequence must contain the same proposition letters as those in the assumption beginning the sequence.
If you can derive an explicit contradiction from an assumption beginning an indented sequence, then the assumption must be false.
You should justify the assumption beginning an indirect proof sequence with the abbreviation "AIP" without citing any line numbers.
You can correctly use line numbers from an indirect proof sequence that has been discharged as justification for subsequent lines.
You cannot cite line numbers from an indented indirect or conditional proof sequence that has already been discharged as justification for any subsequent lines.
The proposition (ZV F) (ZV F) cannot serve as the last indented line of an indirect proof sequence.
No proof can end with an indented line.
An indirect proof sequence is a separate proof method that cannot be used within the scope of another indented sequence. It must be its own standalone proof.The explicit contradiction on the last line of an indirect proof sequence must contain the same proposition letters as those in the assumption beginning the sequence. This is because the assumption is being negated to arrive at a contradiction, which proves the original proposition.If you can derive an explicit contradiction from an assumption beginning an indented sequence, then the assumption must be false. This is because a contradiction cannot be true, so the assumption that led to the contradiction must be false.The assumption beginning an indirect proof sequence should be justified with the abbreviation "AIP" without citing any line numbers. This is a standard way to indicate that an assumption is being made without referring to specific line numbers in the proof.Line numbers from an indirect proof sequence that has been discharged, meaning the proof has been completed, can be correctly used as justification for subsequent lines. This is because the proof has been established and the lines from the discharged sequence can be referenced.Line numbers from an indented indirect or conditional proof sequence that has already been discharged cannot be cited as justification for any subsequent lines. This is because the indented sequence has already been completed and the lines from it cannot be referenced anymore.The proposition (ZV F) (ZV F) cannot serve as the last indented line of an indirect proof sequence. This is because it is a tautology, meaning it is always true, and it does not lead to a contradiction which is required for an indirect proof.No proof can end with an indented line. An indented line is used to indicate a continuation of a proof within an assumption, but the proof itself must be completed with an explicit contradiction or a valid conclusion outside of the indented lines.Therefore, the above statements are true based on the principles of indirect proof method.
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Which graph shows the solution to y > x – 8?
Answers
Answer: Your answer is B.
Step-by-step explanation: y > x means y is bigger than x and also -8 means that you subtract 8 from x so its B
Susan went to the supermarket to buy some itemsShe bought 5 pounds of meat at $14 per pound and a packets of sodas at $9 per pack which equation can be used to determine the total amount y that Susan paid A y=9x+70 B y=14x+9
C y=9x-14 D Y=70x+9
PLEASE HELPP
Answers
The equation that can be used to determine the total amount y that Susan paid is [tex]y=14x+9[/tex]. The Option B is correct.
What equation can be used to calculate Susan's total purchase cost?To calculate the total amount paid by Susan at the supermarket, we need to add the cost of all the items she bought.
From the given information, we know that:
She bought 5 pounds of meat at $14 per pound She bought pack of sodas at $9 per pack.We can use the equation y = $14x + $9, where y represents the total amount paid by Susan and x represents the pounds of meat which can be multiplied.
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The probability density function f(x) for a uniform random variable X defined over the interval [2, 10] is
a. 4 b. 8 c. 0.20 d. None of these choices.
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The probability density function f(x) for a uniform random variable X defined over the interval [2, 10] is:
d. None of these choices.
Step 1: Identify the interval limits, a and b.
a = 2, b = 10
Step 2: Calculate the width of the interval.
Width = b - a = 10 - 2 = 8
Step 3: Determine the probability density function for a uniform distribution.
f(x) = 1 / (b - a)
Step 4: Substitute the values of a and b in the formula.
f(x) = 1 / (10 - 2)
Step 5: Simplify the expression.
f(x) = 1 / 8 = 0.125
So, the correct answer is none of these choices (d).
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Given the four rational numbers below, come up with the greatest sum, difference, product, and quotient, using two of the numbers for each operation. Numbers may be used more than once. Show your work.
Answers
Answer:
[tex]\textsf{Greatest sum}=25.1=25\frac{1}{10}[/tex]
[tex]\textsf{Greatest difference}=30.9=30\frac{9}{10}[/tex]
[tex]\textsf{Greatest product}=122.1=122\frac{1}{10}[/tex]
[tex]\textsf{Greatest quotient}=2.8\overline{03}=2\frac{53}{66}[/tex]
Step-by-step explanation:
Method 1First, rewrite each number as an improper fraction with the common denominator of 10.
[tex]6.6=\dfrac{66}{10}[/tex]
[tex]-4\frac{3}{5}=-\dfrac{4 \cdot 5+3}{5}=-\dfrac{23}{5}=-\dfrac{23 \cdot 2}{5 \cdot 2}=-\dfrac{46}{10}[/tex]
[tex]18\frac{1}{2}=\dfrac{18 \cdot 2+1}{2}=\dfrac{37}{2}=\dfrac{37\cdot 5}{2\cdot 5}=\dfrac{185}{10}[/tex]
[tex]-12.4=-\dfrac{124}{10}[/tex]
Now order the improper fractions from smallest to largest:
[tex]-\dfrac{124}{10},\;\;-\dfrac{46}{10},\;\;\dfrac{66}{10},\;\;\dfrac{185}{10}[/tex]
The greatest sum can be found by adding the largest two numbers:
[tex]\implies \dfrac{66}{10}+\dfrac{185}{10}=\dfrac{66+185}{10}=\dfrac{251}{10}=25.1=25\frac{1}{10}[/tex]
The greatest difference can be found by subtracting the smaller number from the largest number:
[tex]\implies \dfrac{185}{10}-\left(-\dfrac{124}{10}\right)=\dfrac{185+124}{10}=\dfrac{309}{10}=30.9=30\frac{9}{10}[/tex]
The greatest product can be found by multiplying the largest two numbers:
[tex]\implies \dfrac{66}{10}\cdot \dfrac{185}{10}=\dfrac{66\cdot 185}{10 \cdot 10}=\dfrac{12210}{100}=122.1=122\frac{1}{10}[/tex]
The greatest quotient can be found by dividing the largest number by the smallest number, given the two numbers have the same sign.
[tex]\implies \dfrac{185}{10} \div \dfrac{66}{10}= \dfrac{185}{10} \cdot \dfrac{10}{66}=\dfrac{185}{66}=2\frac{53}{66}[/tex]
[tex]\hrulefill[/tex]
Method 2Rewrite all the numbers as decimals:
[tex]6.6[/tex]
[tex]-4\frac{3}{5}=-4.6[/tex]
[tex]18\frac{1}{2}=18.5[/tex]
[tex]-12.4[/tex]
Now order the decimals from smallest to largest:
[tex]-12.4, \;\; -4.6, \;\;6.6,\;\;18.5[/tex]
The greatest sum can be found by adding the largest two numbers:
[tex]\begin{array}{rr}&6.6\\+&18.5\\\cline{2-2} &25.1\\ \cline{2-2}&^1^1\;\;\;\end{array}[/tex]
The greatest difference can be found by subtracting the smallest number from the largest number:
[tex]18.5-(-12.4)=18.5+12.4[/tex]
[tex]\begin{array}{rr}&18.5\\+&12.4\\\cline{2-2} &30.9\\ \cline{2-2}&^1\;\;\;\;\end{array}[/tex]
The greatest product can be found by multiplying the largest two numbers:
[tex]\begin{array}{rr}&18.5\\\times&6.6\\\cline{2-2} &11.10\\+&111.00\\ \cline{2-2}&122.10\end{array}[/tex]
The greatest quotient can be found by dividing the largest number by the smallest number, given the two numbers have the same sign.
[tex]\implies \dfrac{18.5}{6.6}=\dfrac{185}{66}=2.8\overline{03}[/tex]
2. SIGNS A sign is in the shape of an ellipse. The eccentricity is 0.60 and the length is 48 inches.
a. Write an equation for the ellipse if the center of the sign is at the origin and the major axis is horizontal.
b. What is the maximum height of the sign?
Answers
a. The standard equation for an ellipse with center at the origin and major axis horizontal is:
x^2/a^2 + y^2/b^2 = 1
where a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity e is related to a and b by the equation:
e = √(a^2 - b^2)/a
We are given that the eccentricity e is 0.60 and the length of the major axis is 48 inches. Since the major axis is horizontal, a is half of the length of the major axis, so a = 24. We can solve for b using the equation for eccentricity:
0.6 = √(24^2 - b^2)/24
0.6 * 24 = √(24^2 - b^2)
14.4^2 = 24^2 - b^2
b^2 = 24^2 - 14.4^2
b ≈ 16.44
Therefore, the equation of the ellipse is:
x^2/24^2 + y^2/16.44^2 = 1
b. To find the maximum height of the sign, we need to find the length of the semi-minor axis, which is the distance from the center of the ellipse to the top or bottom edge of the sign. We can use the equation for the ellipse to solve for y when x = 0:
0^2/24^2 + y^2/16.44^2 = 1
y^2 = 16.44^2 - 16.44^2 * (0/24)^2
y ≈ 13.26
Therefore, the maximum height of the sign is approximately 26.52 inches (twice the length of the semi-minor axis).
Use polar coordinates to calculate the area of the region. R = {(x, y) | x^2 + y^2 ≤ 100, x ≥ 6}
Answers
The area of the region with polar coordinates R = {(x, y) | x² + y² ≤ 100, x ≥ 6} is approximately 197.39 square units.
To calculate the area, first, rewrite the given inequalities in polar coordinates: r² ≤ 100 and rcos(θ) ≥ 6. Next, find the bounds for r and θ. Since r² ≤ 100, r must be between 0 and 10.
For the second inequality, rcos(θ) ≥ 6, divide by r (assuming r ≠ 0) to get cos(θ) ≥ 6/r. To satisfy this inequality, θ must be between arccos(6/r) and π for r in [6, 10].
Now, integrate the area using polar coordinates with the following formula: A = 0.5 * ∫(from 6 to 10) ∫(from arccos(6/r) to π) (r^2) dθ dr. After evaluating the integral, you get the area A ≈ 197.39 square units.
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find a 95 confidence interval for the mean surgery time for this procedure. round the answers to two decimal places. The 95% confidence interval is _____ , ______
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To find a 95% confidence interval for the mean surgery time for this procedure, you need to use the following formula:
CI = X-bar ± (t * (s / √n))
where:
- CI is the confidence interval
- X-bar is the sample mean
- t is the t-score, which corresponds to the desired confidence level (95%) and the degrees of freedom (n-1)
- s is the sample standard deviation
- n is the sample size
1. Calculate the sample mean (X-bar), sample standard deviation (s), and sample size (n).
2. Find the appropriate t-score using a t-table or calculator for 95% confidence level and (n-1) degrees of freedom.
3. Plug the values into the formula and calculate the interval.
4. Round the answers to two decimal places.
The 95% confidence interval for the mean surgery time is (lower limit, upper limit). Make sure to provide the specific values for your data in the formula.
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complete question:
find a 95 confidence interval for the mean surgery time for this procedure. what quantities do you need to calculate the 95 confidence interval ?
show how you can factor n = pq given the quantity (p − 1)(q − 1).
Answers
If we are given the quantity (p-1)(q-1), we can use it to factor n=pq. This is because of the relationship between Euler's totient function and the prime factors of n.
Specifically, if we know (p-1)(q-1), we can calculate the value of Euler's totient function for n as follows:
φ(n) = (p-1)(q-1)
We can then use this value to find the prime factors of n. One way to do this is to use the fact that for any prime factor p of n, we have:
n = p^k * m, where m is not divisible by p.
Then, we can use Euler's totient function to calculate the value of φ(n) as:
φ(n) = (p-1) * p^(k-1) * φ(m)
By substituting the value we calculated earlier for φ(n), we can solve for p and q:
φ(n) = (p-1)(q-1) = pq - (p+q) + 1
Solving for p and q using the quadratic formula gives:
p = (p+q) / 2 + sqrt((p+q)^2 / 4 - n)
q = (p+q) / 2 - sqrt((p+q)^2 / 4 - n)
Therefore, if we are given the quantity (p-1)(q-1), we can use it to calculate the value of Euler's totient function for n, and then use that to find the prime factors p and q of n.
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Arcs and Angle Relationships in circles , help fast pls
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The value of x in the quadrilateral is 4.
We are given that;
Four angles of quadrilateral 5x, 102, 4y-12, 3x+8
Now,
The sum of all interior angles of quadrilateral is 360degree
So, 5x + 102 + 4y-12 + 3x+8 = 360
Also opposite angles are equal
5x= 3x+8
2x=8
x=4
Therefore, by the properties of quadrilateral the answer will be 4.
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Could use some help on this proof. I can't figure it out.Use induction on the size of S to show that if S is a finite set, then |2s| = 2|S|*Note: Here, |S| means the cardinality of S.
Answers
We have shown that |2S| = 2|S| for all finite sets S, by induction on the size of S.
To prove that if S is a finite set, then |2S| = 2|S| using mathematical induction, we need to show that the statement is true for a base case and then show that it holds for all possible cases.Base Case:When S has only one element, say {x}, then the power set of S, 2S, contains two elements: {} and {x}. Hence, |2S| = 2 = 2|S|.Inductive Hypothesis:Assume that for any finite set S of size k, the statement |2S| = 2|S| holds true.Inductive Step:Consider a set S' of size k+1. Let x be any element in S', and let S = S' \ {x} be the set obtained by removing x from S'. By the inductive hypothesis, we know that |2S| = 2|S|.Now consider 2S' = {A : A ⊆ S'} to be the power set of S'. For any set A in 2S', there are two possibilities:A does not contain x, in which case A is a subset of S and there are 2|S| possible choices for A.A contains x, in which case we can write A as A = {x} ∪ B for some subset B of S. There are 2|S| possible choices for B, so there are 2|S| possible choices for A.Therefore, |2S'| = 2|S| + 2|S| = 2(2|S|) = 2|S'|, which completes the induction step.Thus, we have shown that |2S| = 2|S| for all finite sets S, by induction on the size of S.For more such question on induction
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The number of students in tumbling classes this week is represented in the list.
8, 6, 12, 5, 15, 12, 3, 10, 9
What is the value of the mode for this data set?
15
12
9
3
Answers
Answer:
12
Step-by-step explanation:
Mode is the number that shows up the most in a data set.
The numbers: 8, 6, 12, 5, 15, 12, 3, 10, 9
Number 12 shows up the most in the list, so the mode is 12.
calculate the double integral of (,)=3−8 over the triangle with vertices =(0,0),=(2,5),=(6,5). (use symbolic notation and fractions where needed.)
Answers
The double integral of f(x,y) = 3 - 8 over the given triangle with vertices (0,0), (2,5), and (6,5) is equal to -24.
To solve the double integral of f(x,y) over the given triangle, we need to set up the limits of integration. Let's first sketch the triangle:
(0,0) (2,5)
*--------*
| / \
| / \
| / \
| / \
| / \
| / \
|/_______\
(6,5)
We can see that the triangle is bounded by the lines y = 0, y = 5, and the line connecting (2,5) and (6,5), which has the equation y = -5/4 x + 15/2. We can find the limits of integration as follows:
y = 0 to y = 5
x = 0 to x = (y-5)/(-5/4)
Thus, the double integral can be written as:
∬(triangle) f(x,y) dA
= ∫(0 to 5) ∫(0 to (y-5)/(-5/4)) (3 - 8) dx dy
= ∫(0 to 5) [(3 - 8) * (y-5)/(-5/4)] dy
= ∫(0 to 5) (-3.2y + 16) dy
= [-1.6y^2 + 16y] from 0 to 5
= -24
Therefore, the double integral of f(x,y) over the given triangle with vertices (0,0), (2,5), and (6,5) is equal to -24.
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A unicycle wheel has a diameter of 25 inches and a radius of 12.5 inches.
How many inches will the unicycle travel in 4 revolutions?
Use π = 3.14 and round your answer to the nearest hundredth of an inch.
Answers
Answer:
314.00 inches
Step-by-step explanation:
r=12.5
every revolution = circumference = 2π r = 25π
4 times that = 100π =314
I NEED HELP ON THIS ASAP!!
Answers
The answers are as follows
A: [tex]-2.3^{x-1} = a^{2} (r^{x-1} )\\[/tex]
∴[tex]y = a(-2.3)^{x}[/tex]
The constant ratio is [tex]-2,3[/tex] and the y-intercept is [tex](0,a)[/tex].
B: [tex]45.2^{x-1} = a(r^{x-1} )[/tex]
∴[tex]y = a(45.2)^{x}[/tex]
The constant ratio is [tex]45.2[/tex] and the y-intercept is [tex](0,a)[/tex].
C: [tex]1234-0.1^{x-1}[/tex] [tex]= a r^{x-1}[/tex]
∴ [tex]y = a(0.1)^{x} + 1234[/tex]
The constant ratio is [tex]0.1[/tex] and the y-intercept is[tex](0,a +1234)[/tex].
D: [tex]-5(1/2)^{x-1}[/tex] is not a geometric sequence as there is no common ratio between consecutive terms.
The constant term is [tex]-5[/tex] and the y-intercept is [tex](0,-5)[/tex].
What is exponential function?An exponential function is a mathematical function in the form [tex]f(x) = a^{x}[/tex], where a is a positive constant called the base, and x is the variable. These functions have a constant ratio between consecutive outputs.
An explicit formula for a geometric sequence as an exponential function, we can write the nth term as[tex]a(r^{n-1})[/tex], where a is the first term and r is the common ratio.
This is equivalent to the general form of an exponential function, [tex]y = ab^{x}[/tex], where a is the initial value and b is the base. The constant ratio between consecutive terms is equal to the base of the exponential function.
Therefore,
A: [tex]-2.3^{x-1} = a^{2} (r^{x-1} )\\[/tex]
∴[tex]y = a(-2.3)^{x}[/tex]
The constant ratio is [tex]-2,3[/tex] and the y-intercept is [tex](0,a)[/tex].
B: [tex]45.2^{x-1} = a(r^{x-1} )[/tex]
∴[tex]y = a(45.2)^{x}[/tex]
The constant ratio is [tex]45.2[/tex] and the y-intercept is [tex](0,a)[/tex].
C: [tex]1234-0.1^{x-1}[/tex] [tex]= a r^{x-1}[/tex]
∴ [tex]y = a(0.1)^{x} + 1234[/tex]
The constant ratio is [tex]0.1[/tex] and the y-intercept is[tex](0,a +1234)[/tex].
D: [tex]-5(1/2)^{x-1}[/tex] is not a geometric sequence as there is no common ratio between consecutive terms.
The constant term is [tex]-5[/tex] and the y-intercept is [tex](0,-5)[/tex].
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Louise is buying wallpaper. It costs $7.98 per meter. She needs 150 feet. How much
will the wallpaper cost? Round to the nearest half dollar.
$365.00
$364.00
$364.94
$364.40
Answers
Answer:
$365.00.
Step-by-step explanation:
Two joggers run 6 miles south and then 5 miles east. What is the shortest distance they must travel to return to their starting point?
Answers
The shortest distance the joggers must travel to return to their starting point is 7.81 miles.
To find the shortest distance the joggers must travel to return to their starting point, we can use the Pythagorean theorem, as the southward and eastward distances form a right triangle. The theorem states that the square of the length of the hypotenuse (the shortest distance, in this case) is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
Here, a is the southward distance (6 miles), and b is the eastward distance (5 miles). We need to find c, the hypotenuse.
(6 miles)^2 + (5 miles)^2 = c^2
36 + 25 = c^2
61 = c^2
Now, take the square root of both sides to find c:
c = √61
c ≈ 7.81 miles
The two joggers form a right triangle with the starting point as the right angle. They have run 6 miles south and 5 miles east, so the legs of the right triangle have lengths 6 and 5.
To find the shortest distance they must travel to return to their starting point, we need to find the length of the hypotenuse of the right triangle. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of the legs.
Plugging in the values we have, we get:
c^2 = 6^2 + 5^2
c^2 = 36 + 25
c^2 = 61
Taking the square root of both sides, we get:
c = sqrt(61)
Therefore, the shortest distance the joggers must travel to return to their starting point is approximately 7.81 miles.
use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum? 4−12z 36z2−108z3 ⋯⋯
sum =
domain =
(Give your domain as an interval or comma separated list of intervals; for example, to enter the region x<−1 and 2
Answers
The sum of the given geometric series is 4/(1+3z) and the domain of convergence is (-1/3, 1/3).
The domain in which the series converges is (-1/3, 1/3) because for the series to converge, the absolute value of the common ratio (r) must be less than 1.
In this case, r = -3z, so the absolute value of r is less than 1 if and only if |z| < 1/3. Therefore, the domain of convergence is (-1/3, 1/3).
A geometric series is a series in which each term is a constant multiple of the preceding term. In this case, the first term is 4 and each subsequent term is obtained by multiplying the preceding term by -3z/3. Therefore, the series can be written as:
4 - 12z + 36z² - 108z³ + ...
To find the sum of the series, we use the formula for the sum of an infinite geometric series:
S = a/(1-r)
where a is the first term and r is the common ratio. In this case, a = 4 and r = -3z/3 = -z. Therefore, the sum of the series is:
S = 4/(1+z)
To determine the domain of convergence, we need to find the values of z for which the absolute value of r is less than 1. That is, we need to find the values of z for which |z| < 1/3. Therefore, the domain of convergence is (-1/3, 1/3).
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if the distribution funciton of x is given by f(b) = 0 b < 0 = 1/2 0 ≤ b < 1 = 3/5 1 ≤ b < 2 = 4/5 2 ≤ b < 3 = 9/10 3 ≤ b < 3.5 = 1 b ≤ 3.5 find the probability distribution
Answers
The probability distribution is:
P(x < 0) = 0
P(0 ≤ x < 1) = 1/2
P(1 ≤ x < 2) = 3/5
P(2 ≤ x < 3) = 4/5
P(3 ≤ x < 3.5) = 9/10
P(x ≥ 3.5) = 0
To find the probability distribution, we need to calculate the probability of each value of x. We can do this by looking at the ranges defined by the distribution function and calculating the area under the curve for each range.
For x < 0, the probability is 0 since there is no area under the curve in this range.
For 0 ≤ x < 1, the probability is 1/2 since the area under the curve in this range is equal to the height (which is 1/2) multiplied by the width of the range (which is 1).
For 1 ≤ x < 2, the probability is 3/5 since the area under the curve in this range is equal to the height (which is 3/5) multiplied by the width of the range (which is 1).
For 2 ≤ x < 3, the probability is 4/5 since the area under the curve in this range is equal to the height (which is 4/5) multiplied by the width of the range (which is 1).
For 3 ≤ x < 3.5, the probability is 9/10 since the area under the curve in this range is equal to the height (which is 9/10) multiplied by the width of the range (which is 0.5).
Therefore, the probability distribution is:
P(x < 0) = 0
P(0 ≤ x < 1) = 1/2
P(1 ≤ x < 2) = 3/5
P(2 ≤ x < 3) = 4/5
P(3 ≤ x < 3.5) = 9/10
P(x ≥ 3.5) = 0
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4
Find the area of the shaded sector in the diagram below. (Round answers to the nearest hundredth)
Area of sector =
Answers
The area of the shaded sector is 9.5 square units.
We know that the formula for the area of sector of circle is:
A = (θ/360°) × π × r²
where the central angle θ is measured in degrees
and r is the radius of the circle
Here, the central angle is θ = 120° and the radius of the circle is r = 3 units
Using above formula the area of the shaded sector would be,
A = (θ/360°) × π × r²
A = (120°/360°) × π × 3²
A = 1/3 × π × 9
A = 3 × π
A = 9.5 sq. units
Thus the required area is 9.5 sq.units
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A random sample of size n = 36 is taken from a population with mean μ = -6.8 and standard deviation σ = 3.
a. Calculate the expected value and the standard error for the sampling distribution of the sample mean. (Negative values should be indicated by a minus sign. Round "expected value" to 1 decimal place and "standard error" to 4 decimal places.)
b. What is the probability that the sample mean is less than -7? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.)
c. What is the probability that the sample mean falls between -7 and -6? (Do not round intermediate calculations. Round "z" value to 2 decimal places and final answer to 4 decimal places.)
Answers
a) The expected value is -6.8 and the standard error is 0.5.
b) The probability that the sample mean is less than -7 is 0.0548.
c) The probability that the sample mean falls between -7 and -6 is 0.8904.
population mean, which is -6.8.
The following formula may be used to get the standard error of the sampling distribution of the sample mean:
SE = σ/√n
Substituting the given values, we get:
SE = 3/√36 = 0.5
Therefore, the expected value is -6.8 and the standard error is 0.5.
b. To find the probability that the sample mean is less than -7, we need to standardize the sample mean using the formula:
z = (X- μ) / (σ / √n)
where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Substituting the given values, we get:
z = (-7 - (-6.8)) / (3 / √36) = -0.8 / 0.5 = -1.6
Using a standard normal distribution table or calculator, we find that the probability of getting a z-score less than -1.6 is 0.0548.
The probability that the sample mean is less than -7 is 0.0548.
c. To find the probability that the sample mean falls between -7 and -6, we need to standardize both values using the same formula as above and subtract the probabilities:
z1 = (-7 - (-6.8)) / (3 / √36) = -1.6
z2 = (-6 - (-6.8)) / (3 / √36) = 1.6
Using a standard normal distribution table or calculator, we find that the probability of getting a z-score less than -1.6 is 0.0548 and the probability of getting a z-score less than 1.6 is 0.9452. Therefore, the probability of getting a z-score between -1.6 and 1.6 is:
P(-1.6 < z < 1.6) = 0.9452 - 0.0548 = 0.8904
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A triangular prism is 38 inches long and has a triangular face with a base of 42 inches and a height of 28 inches. The other two sides of the triangle are each 35 inches. What is the surface area of the triangular prism?
Answers
The surface area of the triangular prism is 5166 square inches.
We need to find the area of the triangular face. The area of a triangle is given by the formula:
Area = (1/2) × base × height
Area = (1/2) × 42 × 28
= 588 square inches
The total surface area of those two faces is 2 times the area we just calculated:
Total area of the two triangular faces = 2 × 588
= 1176 square inches
Area of each rectangular face = length× width
= 38 × 35
= 1330 square inches
There are 3 rectangular faces
The total area of those three faces is 3 times the area
Total area of the three rectangular faces = 3 × 1330
= 3990 square inches
Total surface area of the triangular prism by adding together the areas of the two triangular faces and the three rectangular faces:
Total surface area = Total area of the two triangular faces + Total area of the three rectangular faces
Total surface area = 1176 + 3990 = 5166 square inches
Therefore, the surface area of the triangular prism is 5166 square inches.
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Find the value of x
A. 135
B. 40
C. 50
D. 45
Answers
Hope this helps :)
You just need to complete the right bottom angle, you get 45. Every triangle has 180º in total,you already have 95 and 45, the get 180 you need 40.
(b) suppose = 148. what is the probability that x is at most 200? less than 200? at least 200? (round your answer to four decimal places.)
Answers
We need to determine the probabilities for the given scenarios. Probability is a branch of mathematics that deals with the study of random events or phenomena.
Here's the step-by-step explanation:
1. At most 200: This means x ≤ 200, including x = 200. Since x has 148 possible values, and all of them are less than or equal to 200, the probability, in this case, is 1 (all possible outcomes are included).
2. Less than 200: This means x < 200, not including x = 200. Since x has 148 possible values and all of them are less than 200, the probability is also 1 (all possible outcomes are included).
3. At least 200: This means x ≥ 200, including x = 200. Since there are no values of x in the given range that are greater than or equal to 200, the probability, in this case, is 0 (no possible outcomes are included).
So, the probabilities for the given scenarios are:
- At most 200: 1.0000 (rounded to four decimal places)
- Less than 200: 1.0000 (rounded to four decimal places)
- At least 200: 0.0000 (rounded to four decimal places)
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the 2012 National health and nutrition examination survey reports a 95% confidence interval of 99.8 to 102.0 centimeters for the mean waist circumference of adult women in the United State. a) what is captured by the confidence interval? b) Express this confidence interval as a sequence written in the context of this problem c) what is the margin of error for this confidence interval? Express this interval in the format "estimate plus or minus margin of error" d) would a 99% confidence interval based on the same data be larger or smaller? Explain
Answers
The interval would be wider, with more plausible values, and the margin of error would be larger.
What is confidence interval ?
A confidence interval is a statistical range of values that is used to estimate an unknown population parameter, such as the mean or standard deviation of a distribution, based on a sample of data.
a) The confidence interval captures the plausible range of values for the population mean waist circumference of adult women in the United States. More specifically, it is the range of values that is likely to contain the true population mean with 95% confidence.
b) The confidence interval can be expressed as: "We are 95% confident that the true population mean waist circumference of adult women in the United States falls between 99.8 and 102.0 centimeters."
c) The margin of error for this confidence interval can be calculated by taking half the width of the interval. Therefore, the margin of error is (102.0 - 99.8) : 2 = 1.1. Thus, we can express the confidence interval as "The estimate is 100.9 centimeters plus or minus 1.1 centimeters."
d) A 99% confidence interval based on the same data would be larger than the 95% confidence interval. This is because a 99% confidence level requires a wider interval to capture the true population mean with 99% confidence.
Therefore, the interval would be wider, with more plausible values, and the margin of error would be larger.
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The discrete random variable X is the number of students that show up for Professor Smith's office hours on Monday afternoons. The table below shows the probability distribution for X. What is the expected value E(X) for this distribution?A. 1.2B. 1.0C. 1.5D. 2.0
Answers
The required answer is E(X) = 1.4
The expected value E(X) for this distribution can be calculated by multiplying each possible value of X by its probability, and then adding up these products. Using the table provided, we have:
E(X) = (0)(0.2) + (1)(0.3) + (2)(0.4) + (3)(0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4
Therefore, the closest option to our calculated expected value is option A, 1.2. However, none of the given options match exactly with our calculation.
To find the expected value E(X) of the discrete random variable X, we need the probability distribution table for X. However, the table is not provided in the question. Please provide the table with the probabilities for each possible value of X, and I will be happy to help you calculate the expected value E(X).
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