a rectangle has a perimeter of 48 in. the length and width are scaled by a factor of 2.5. what is the perimeter of the resulting rectangle?
The calculated value of the perimeter of the scaled rectangle is 120 inches
Calculating the perimeter of the scaled rectangleFrom the question, we have the following parameters that can be used in our computation:
Perimeter = 48 in
Scale factor = 2.5
Using the above as a guide, we have the following:
Perimeter of the scaled rectangle = Perimeter * Scale factor
Substitute the known values in the above equation
Perimeter of the scaled rectangle = 48 * 2.5
Evaluate
Perimeter of the scaled rectangle = 120
Hence, the perimeter of the scaled rectangle is 120
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find inverse f(x)=x-3/x+4, g(x)=4x+3/1-x
Answer:
1984
________________________________________________________
Given:
To find the inverse of f(x), we first switch x and y and then solve for y. So, x = y-3/y+4, which we can rewrite as x(y+4) = y-3. Simplifying, we get xy + 4x = y-3, and then we can isolate y on one side: y-xy = 4x-3. Factoring out y on the left side, we get y(1-x) = 4x-3, and then we can divide both sides by (1-x) to get y = (4x-3)/(1-x). This is our inverse function.
Find:
To find the inverse of g(x), we follow the same process of switching x and y and solving for y. So, x = 4y+3/1-y, which we can rewrite as x(1-y) = 4y+3. Simplifying, we get -xy + y = 4x+3, and then we can isolate y on one side: y(-x+1) = 4x+3. Dividing both sides by (-x+1), we get y = (4x+3)/(-x+1). This is our inverse function.
Solve:
As for the given set of values, we have 187, 191, 202, 209, 218, and 1984. The outlier is obviously 1984, and its presence will not affect the range because the range is simply the difference between the largest and smallest values, which will be the same regardless of the presence of an outlier. However, the outlier will greatly affect the interquartile range, which is the difference between the upper and lower quartiles. This is because the upper and lower quartiles are the median of the upper half and lower half of the data, respectively, and including an outlier in one of these halves can greatly skew the median and thus the interquartile range.
What is the end behavior of the function f of x equals negative 2 times the cube root of x?
What is the end behavior of the function f of x equals negative 2 times the cube root of x?
As x → –∞, f(x) → 0, and as x → ∞, f(x) → 0.
As x → 0, f(x) → –∞, and as x → ∞, f(x) → 0.
As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞.
As x → –∞, f(x) → ∞, and as x → ∞, f(x) → –∞.
The end behavior of the given function is As x → –∞, f(x) → –∞, and as x → ∞, f(x) → –∞.
What is end behavior of function?The term "end behavior" describes the behaviour or trend of a function's graph as its x-values go closer to positive or negative infinity. It explains how the function "ends" or acts when x is extremely large (positive infinity) or extremely small (negative infinity). The degree and leading coefficient of the polynomial formulation of the function can be used to predict the final behaviour. For instance, the left and right ends of the graph will both point upward for a function with an odd-degree polynomial and a positive leading coefficient (towards positive infinity).
For the function f(x) = -2∛x as x approaches negative infinity because the function has a negative coefficient in front of the cube root, which means that as x decreases significantly, the absolute value of the cube root increases. The function also approaches negative infinity as x approaches very big (positive) values and the cube root approaches very small absolute values. As a result, the function's final behaviour is that it approaches negative infinity as x gets closer to both negative and positive infinity.
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The end behavior of the given function is As x → –∞, f(x) → –∞, and as x → ∞, f(x) → –∞.
What is end behavior of function?The term "end behavior" describes the behaviour or trend of a function's graph as its x-values go closer to positive or negative infinity. It explains how the function "ends" or acts when x is extremely large (positive infinity) or extremely small (negative infinity). The degree and leading coefficient of the polynomial formulation of the function can be used to predict the final behaviour. For instance, the left and right ends of the graph will both point upward for a function with an odd-degree polynomial and a positive leading coefficient (towards positive infinity).
For the function f(x) = -2∛x as x approaches negative infinity because the function has a negative coefficient in front of the cube root, which means that as x decreases significantly, the absolute value of the cube root increases. The function also approaches negative infinity as x approaches very big (positive) values and the cube root approaches very small absolute values. As a result, the function's final behaviour is that it approaches negative infinity as x gets closer to both negative and positive infinity.
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Answer:
The function f(x) = -2∛x represents a cube root function that is multiplied by -2.
As x approaches negative infinity, the cube root of x becomes more negative, and when it is multiplied by -2, it becomes more positive. Therefore, as x → –∞, f(x) → ∞.
As x approaches positive infinity, the cube root of x becomes more positive, and when it is multiplied by -2, it becomes more negative. Therefore, as x → ∞, f(x) → –∞.
Therefore, the correct answer is: As x → –∞, f(x) → ∞, and as x → ∞, f(x) → –∞. D.
Answer:
The function f(x) = -2∛x represents a cube root function that is multiplied by -2.
As x approaches negative infinity, the cube root of x becomes more negative, and when it is multiplied by -2, it becomes more positive. Therefore, as x → –∞, f(x) → ∞.
As x approaches positive infinity, the cube root of x becomes more positive, and when it is multiplied by -2, it becomes more negative. Therefore, as x → ∞, f(x) → –∞.
Therefore, the correct answer is: As x → –∞, f(x) → ∞, and as x → ∞, f(x) → –∞. D.
Need help please !!
Geometry
Answer:
complementary since the angles sum up to 90
You have a choice of receiving a wage of $34,000 per year, $2840 per month, $650 per week, or $18 per hour. Which pay choice would you take? Assume a 40-hour work week with 52 weeks per year.
Based on calculation, the highest hourly wage is $2,840 per month which gives us $17.75 per hour.
Which pay choice is the best choice take?For purpose of making a comparison, we can calculate the hourly wage for each option.
Hourly wage for $34,000 per year:
= $34,000 / (40 hours per week * 52 weeks per year)
= 16.3461538
= $16.35 per hour.
Hourly wage for $2,840 per month:
= $2,840 / (4 weeks per month * 40 hours per week)
= $17.75 per hour
Hourly wage for $650 per week:
= $650 / 40 hours per week
= $16.25 per hour
Hourly wage for $18 per hour.
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what us the rate of change in the value of one share of company’s stock, in dollars per day, from Day 0 to Day 15
Check the picture below.
the slope goes by several names
• average rate of change
• rate of change
• deltaY over deltaX
• Δy over Δx
• rise over run
• gradient
• constant of proportionality
however, is the same cat wearing different costumes.
[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=0\\ x_2=15 \end{cases}\implies \cfrac{f(15)-f(0)}{15 - 0}\implies \cfrac{11-5}{15}\implies \cfrac{6}{15}\implies \cfrac{2}{5}[/tex]
(b) How many ways are there to distribute two balls into four boxes if each box must have at most one ball in it if (i) both the balls and boxes are labeled? (ii) the balls are labeled, but the boxes are unlabeled? (iii) the balls are unlabeled, but the boxes are labeled? (iv) both the balls and boxes are unlabeled?
(i) distributed in [tex]4 \times 3 = 12[/tex] different ways. (ii)statistical distribution of the balls is done in one of three ways. [tex](1 + 2 = 3)[/tex] . (iii) there are 6 ways to divide the balls. [tex](4 + 2 \times 1).[/tex] (iv) to distribute the balls [tex](1+2+1)[/tex].
What is the statistical distribution?(i) Both balls and boxes are labeled:
Since one box already contains a ball, there are four options for where to place the first ball, and only three options remain for the second ball. The two balls can be distributed in [tex]4 \times 3 = 12[/tex] different ways.
(ii) Balls are labeled, but boxes are unlabeled:
There is just one method to arrange the balls if there is only one ball in each box. There are two ways to determine which box receives both balls if only one box possesses them. The statistical distribution of the balls is done in one of three ways.
(1 + 2 = 3).
(iii) Balls are unlabeled, but boxes are labeled:
There are four alternatives available to the box if they are both in the same box. If they are in different boxes, the first ball's box has two options, leaving the second ball with just one. Accordingly, there are 6 ways to divide the balls.
(4 + 2 * 1).
(iv) Both balls and boxes are unlabeled:
There is only one method to arrange the balls if there is one ball in each box. There are two methods to decide which box gets both balls if only one box has both balls.
There is just one method to arrange the balls if neither box incorporates a ball. There are therefore a total of 4 options to distribute the balls (1+2+1).
Therefore,(i) distributed in [tex]4 \times 3 = 12[/tex] different ways. (ii)statistical distribution of the balls is done in one of three ways. [tex](1 + 2 = 3)[/tex] . (iii) there are 6 ways to divide the balls. [tex](4 + 2 \times 1).[/tex] (iv) to distribute the balls [tex](1+2+1)[/tex].
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A pool measuring 18 meters by 30 meters is surrounded by a path of uniform width, as shown in the
figure. If the area of the pool and the path combined is 1900 square meters, what is the width of
the path?
Answer: 3 meters
Step-by-step explanation:
The area of the rectangular region is given by:
(18 + 2x) * (30 + 2x) = 18*30 + 36x + 60x + 4x^2
Simplifying this expression, we get:
4x^2 + 96x - 1300 = 0
Dividing both sides by 4, we get:
x^2 + 24x - 325 = 0
Now we can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 24, and c = -325.
Plugging in these values, we get:
x = (-24 ± sqrt(24^2 - 4(1)(-325))) / 2(1)
x = (-24 ± sqrt(936)) / 2
x = (-24 ± 30) / 2
x = 3 or -27
Since the width of the path cannot be negative, the only possible solution is x = 3 meters. Therefore, the width of the path surrounding the pool is 3 meters.
Determine whether the equation has no solution, one solution, or infinitely many solutions. -2 ( 11 - 12x )= -4 ( 1 - 6x) Explain your conclusion. Enter your answer, your work, and your explanation in the box provided.
Answer:
No solution
Step-by-step explanation:
-2(11-12x) = -4(1-6x)
Distribute the -2 and -4:
-22 + 24x = -4 + 24x
Subtract 24x from both sides:
-22 = -4
Since this is a contradiction, there is no solution. The equation is inconsistent.
Answer:
No Solution.
Explanation:
When solving for x, the two values do not equate to each other.
URGENT HELP PLEASE PLEASE CAN YOU HELP
Answer:
1/2
Step-by-step explanation:
Let's check if this is a positive of negative slope. Remember that when checking slope, you need to read the graph from left to right.
So let's see: When we look at the graph from left to right, is it moving up or down?
It is moving up, correct!
Therfore, this must be a positive slope.
Ok, now we need to pick 2 points.
I will choose: ( 6 , 0 ) & ( 2 , -2 )
To move between those pouints, we need to move up 2 and 4 units to the right.
Slope is defined as:
Change is y over change in x
Change in y is positive 2
Change in x is positive 2
Therfore, slope is [tex]\frac{2}{4}[/tex]
or, 1/2
A sprinkler set in the middle of a lawn sprays in a circular pattern. The area of the lawn that gets sprayed by the sprinkler
can be described by the equation (z+6)2 + (y-9)² = 196.
What is the greatest distance, in feet, that a person could be from the sprinkler and get sprayed by it?
Answer: The greatest distance a person could be from the sprinkler and get sprayed by it is 14 feet.
Step-by-step explanation:
The equation given describes a circle with the general equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
In the provided equation, (z + 6)^2 + (y - 9)^2 = 196, we can identify the values for h, k, and r^2:
1. h = -6 (from z + 6)
2. k = 9 (from y - 9)
3. r^2 = 196
Now, we need to find the radius r, which is the greatest distance a person could be from the sprinkler and still get sprayed by it. To do this, we take the square root of r^2:
r = √196
r = 14
Convert 8 5/8 to a decimal. Show your work.
Answer: 8.625
Step-by-step explanation:The eight stays an 8 snd 5:8 is .625. Add these together and you get 8.625.
Calculator The diameter of a ball is 9 in. What is the volume of the ball? Use 3.14 for pi. Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth. in³
To find the volume of a ball, we use the formula V = (4/3)πr³, where r is the radius of the ball. We are given the diameter of the ball, which is 9 inches.
The radius of the ball is half of the diameter, so
r = 9/2 = 4.5 inches.
Substituting this value of r in the formula, we get:
V = (4/3)π(4.5)³V = (4/3)π(91.125)V = 4.1888 × 91.125V ≈ 381.71
Therefore, the volume of the ball is approximately 381.71 cubic inches.
When rounded to the nearest hundredth, the answer is 381.71.
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In a particular class of 29 students, 11 are men. What fraction of the students in the class are women?
Answer:
Step-by-step explanation:
Q:
In a particular class of 29 students, 11 are men. What fraction of the students in the class are women?
A:
29-11=18
18/29 are women
How many minutes does it take to travel 11 miles if your average speed is 55 miles per hour?
(Remember to convert 1 hour = 60 minutes)
Answer:
12 minutes
Step-by-step explanation:
11/x = 55/60
11 x 5 = 55
60/5=12
12.01 minutes
time = distance / speed
where distance is the distance traveled, speed is the average speed of the travel, and time is the time taken to cover the distance.
We are given that the distance traveled is 11 miles and the average speed is 55 miles per hour. To convert the speed to miles per minute, we can divide it by 60 since 1 hour is equal to 60 minutes:
55 miles per hour = 55/60 miles per minute
= 0.9167 miles per minute (rounded to four decimal places)
Now we can substitute the values into the formula:
time = distance / speed
= 11 miles / 0.9167 miles per minute
≈ 12.01 minutes (rounded to two decimal places)
Therefore, it would take approximately 12.01 minutes to travel 11 miles at an average speed of 55 miles per hour.
*IG:whis.sama_ent
8) Elana's Math Class has 24 students. She miscounted the class total and
recorded it as 21 students. What is her percent error?
a) Find the mean and median of the following gasoline prices per gallon in California:
regular:
$
3.14
$3.14, mid-grade:
$
3.21
$3.21, premium:
$
3.28
$3.28, diesel:
$
3.53
$3.53. Round to the nearest cent.
The calculated median and the mean gasoline prices per gallon in California are 3.25 and 4.49, respectively
Finding the median and the mean gasoline prices per gallon in California:From the question, we have the following parameters that can be used in our computation:
Prices = 3.14, 3.21, 3.28, 3.53
Start by sorting the number of prices in ascending order
So, we have
3.14, 3.21, 3.28, 3.53
As a general rule.
The median is the middle number
Using the above as a guide, we have the following:
Median = middle number = 1/2 *(3.21 + 3.28)
Evaluate
Median = middle number = 3.25
For the mean, we have
Mean = (3.14 + 3.21 + 3.28 + 3.53)/3
Evaluate
Mean = 4.49
Hence, the value of the median is 3.25
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Quadrilateral CDEF is inscribed in circle A.
If m∠CFE = (2x + 6)° and m∠CDE = (2x − 2)°, what is the value of x?
22
44
46
89
The quadrilateral is inscribed inside the circle and x = 44
Given data ,
Let the quadrilateral be inscribed inside the circle as shown in the figure
Now , the measure of angles are
m∠CFE = (2x + 6)°
m∠CDE = (2x − 2)°
And , opposite sides of the angles of quadrilateral are supplementary
So , m∠CFE + m∠CDE = 180°
On simplifying , we get
2x + 6 + 2x - 2 = 180
4x + 4 = 180
4x = 176
x = 44
Hence , the angle is solved and x = 44
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The complete question is attached below :
Quadrilateral CDEF is inscribed in circle A.
If m∠CFE = (2x + 6)° and m∠CDE = (2x − 2)°, what is the value of x?
A) 22
B) 44
C) 46
D) 89
Answer: 44
Step-by-step explanation:
i took the test
(1)Consider an experiment that has N outcomes e1,e2,…,eN,where it is known that outcome ej+1 is twice as likely as the outcome ej for j=1,2,,N-1.Let Ek=E1,E2,…,Ek.Show that PEk=2k-12N-1
[tex]PEk = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex] which is the desired equation.
What is equation?An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
According to given information:Let P(ej) denote the probability of outcome ej for j=1,2,...,N. Then we know that:
P(ej+1) = 2P(ej) for j=1,2,...,N-1
Also, we know that the sum of probabilities of all outcomes is 1:
P(e1) + P(e2) + ... + P(eN) = 1
We can use the geometric sequence formula to find P(ek) in terms of P(e1) as follows:
[tex]P(ek) = P(e1) * (2^{(k-1)})[/tex]
Substituting for P(ej+1) in terms of P(ej) gives:
[tex]P(ej+1) = 2P(ej) = 2^1 * P(ej)[/tex]
[tex]P(ej+2) = 2P(ej+1) = 2^2 * P(ej)[/tex]
...
[tex]P(ek) = 2^{(k-1)} * P(e1)[/tex]
We can sum the probabilities of all outcomes to get:
[tex]1 = P(e1) + 2P(e1) + 2^2 P(e1) + ... + 2^{(N-1)}P(e1)[/tex]
Using the formula for the sum of a geometric series, we get:
[tex]1 = P(e1) * [(2^N - 1)/(2 - 1)][/tex]
Simplifying, we get:
[tex]P(e1) = 1/(2^N - 1)[/tex]
Substituting this value for P(e1) in the expression for P(ek) gives:
[tex]P(ek) = (1/(2^N - 1)) * (2^{(k-1)})[/tex]
Simplifying, we get:
[tex]P(ek) = 2^{(k-N)} / (2 - 1/2^{(N-k+1)})[/tex]
Multiplying by 2/2, we get:
[tex]P(ek) = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]
Substituting k=N, we get:
P(Ek) = 1/2
Substituting k=1, we get:
[tex]P(E1) = 2^{(N-1)} / (2^N - 1)[/tex]
Therefore, we have shown that:
[tex]P(Ek) = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]
for k=1,2,...,N.
To show that PEk=2k-1/2N-1, we can substitute k=N-j+1 for j=1,2,...,N:
[tex]PEk = P(EN-j+1) = 2^{(N-j)} / (2^j - 1)[/tex]
Letting t = j-1, we can rewrite this as:
[tex]PEk = 2^{(N-t-1)} / (2^{(t+1)} - 1[/tex]
Multiplying the numerator and denominator by 2, we get:
[tex]PEk = 2^{(N-t)} / (2^{(t+2)} - 2)[/tex]
Substituting t = N-k, we get:
[tex]PEk = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]
which is the desired result.
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adam bought a skateboard that is on sale for 35% off. the original cost is $150. what is the sale price? show your work
Answer:
$97.50
Step-by-step explanation:
Purchase Price:
$150
Discount:
(150 x 35)/100 = $52.50
Final Price:
150 - 52.50 = $97.50
LMNP is rotated 180° clockwise about the origin.
L 0.1
m 3,4
N 5.4
P 9.0
What are the coordinates of N.
In triangle ABC, a=6, b=7, and cos C=1/4 A.) Find the exact area of the triangle. B.) Find C C.) Find sin A in simplest radical form.
A.) Area = (1/2) × 7 × 6 × √(15)/4 = 21√(15)/4
B.) Angle C is 75 degrees.
C.) Sin(A) is 3√(255)/85 in simplest radical form.
What is trigonometry?Trigonometry is a branch of mathematics that examines triangle-related relationships and calculations, particularly the measurement of triangle sides and angles.
A.) We can apply the following formula to determine the triangle's area:
Area = (1/2) × b × a × sin(C)
Since we know b=7 and a=6, we have:
Area = (1/2) × 7 × 6 × sin(C)
To find sin(C), we can use the fact that cos(C) = 1/4 and the Pythagorean identity:
sin²(C) + cos²(C) = 1
sin(C) = √(1 - cos²(C)) = √(1 - 1/16) = √(15/16) = √(15)/4
B.) To find C, we can use the Law of Cosines:
c² = a² + b² - 2ab×cos(C)
c² = 6² + 7² - 2(6)(7)(1/4) = 85/2
c = √(85)/2
Now we can use the Law of Cosines again to find angle C:
cos(C) = (a² + b² - c²)/(2ab)
cos(C) = (6² + 7² - (√(85)/2)²)/(267) = 1/4
C.) To find sin(A), we can use the Law of Sines:
sin(A)/a = sin(C)/c
sin(A) = asin(C)/c = 6(√(15)/4)/(√(85)/2) = 3×√(15)/√(85) = √(255)/85
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Indirect measurement, pls help, I am stuck on this question.
The building is, 246 feet tall.
Since, We know that;
Two or more triangles will always be referred to as similar if on comparing their corresponding properties, some common relations holds. Some of the relations are relations between their length of sides, relations among their internal angles etc.
To determine the height of the building, we have;
height of pole = 7 ft
total length of the building's shadow = 167 ft
the length of the shadow of the pole = 4.75 ft
Let the height of the building be represented by h. On comparison, we have;
4.75/ 167 = 7/h
4.75h = 167 x 7
= 1169
h = 1169/ 4.75
h = 246.11
Hence, The building is 246 ft. tall.
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Determine the volume of the "leaning" regular hexagonal prism.
It has a base perimeter of 36 inches, a slanted height of 11 inches, and is leaning at
70°. The base is a regular hexagon with a perimeter of 36 inches.
6.70%
11"
The volume of the leaning regular hexagonal prism, can be found to be 351. 89 inch ³.
How to find the volume ?A right triangle can be constructed in a regular hexagon by linking the midpoint of one side and a vertex to the center. This specially-crafted triangle comprises two legs, one of which is half the size of the primary side (acting as the hypotenuse) while the remaining leg lies parallel to the hexagon's apothem (height).
The height of the prism can be found with cosine to be:
h = slanted height x Cos (angle )
h = 11 x Cos ( 70 degrees )
h = 3. 762 inches
We can then find the volume of the leaning hexagonal prism to be:
= Area x Height
= 93. 528 x 3. 762
= 351. 89 inch ³
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Solve the equation.
Sin 0= -0.5; The angle must be 270< 0 < 360°
The value of tetha is 330°
What are positive Angles?If a ray rotates in an anticlockwise direction, then the angle formed as a product of the rotation is called a positive angle.
In other words if it rotates in clockwise direction, it is a negative angles.
If sin(tetha) = -0.5
to find the value of tetha we multiply both sides with the inverse of sin
therefore;
tetha = sin^-1(-0.5)
tetha = -30
therefore -30 is a negative angle, to find the positive angle of tetha, we add 360
therefore tetha = -30+360
= 330°
therefore the value if tetha that will be greater than 279 but less than 360 is 330°
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|y-x| if y>x Write each of the following expressions without using the absolute value
The solution of the expression is; |y-x| = y - x without using absolute value.
What is mean by expression?Numerical articulation is characterized as the assortment of the numbers factors and works by utilizing tasks like expansion, deduction, increase, and division.
Given that;
from the question the expression is;
⇒ |y−x| , if y > x
⇒ If y > x,
⇒ y - x > 0
then |y-x| is equal to (y-x),
since (y-x) is always positive in this case.
Therefore, The solution of the expression is; |y-x| = y - x
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A data set of the least purchased menu items at a bakery is best shown as a _____
Answer:
A data set of the least purchased menu items at a bakery is best shown as a (Report)
Step-by-step explanation:
Have A Good Day :)
The table shown represents a proportional realshipbship between a and b. Form an equation that represents the relationship between a and b and identify the constant of proportionality. Write your answer in the space provided
Answer:
y=3x
Step-by-step explanation:
y=kx
21/7=3
15/5=3
6/2=3
Each relationship in the table has a constant of proportionality of 3.
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Answer:
This is the equation we have to use in order to find the eigenvalues and eigenvectors of matrix A:
det(A - λI) = 0
where λ is the eigenvalue and I is the identity matrix.
So we have:
A - λI = [ -5-λ 1 2 ]
[ 0 -2-λ 0 ]
[ 4 2 -3-λ ]
And the determinant is:
det(A - λI) = (-5-λ)(-2-λ)(-3-λ) - 8(2+λ) = -λ^3 + 4λ^2 + 23λ - 40
We can factor this to get:
det(A - λI) = -(λ - 5)(λ - 2)(λ + 4)
So the eigenvalues are λ1 = 5, λ2 = 2, and λ3 = -4.
To find the eigenvectors for each eigenvalue, we need to solve the system of linear equations:
(A - λI)x = 0
For λ1 = 5, we have:
[ -5-5 1 2 ] [ x1 ] [ 0 ]
[ 0 -2-5 0 ] [ x2 ] = [ 0 ]
[ 4 2 -3-5] [ x3 ] [ 0 ]
which simplifies to:
[ -10 1 2 ] [ x1 ] [ 0 ]
[ 0 -7 0 ] [ x2 ] = [ 0 ]
[ 4 2 -8 ] [ x3 ] [ 0 ]
We can solve this system to get:
x1 = -1/2 x2
x3 = -1/2 x2
So the eigenvector for λ1 = 5 is:
v1 = [ x1, x2, x3 ] = [ -1/2, 1, -1/2 ]
For λ2 = 2, we have:
[ -5-2 1 2 ] [ x1 ] [ 0 ]
[ 0 -2-2 0 ] [ x2 ] = [ 0 ]
[ 4 2 -3-2] [ x3 ] [ 0 ]
which simplifies to:
[ -7 1 2 ] [ x1 ] [ 0 ]
[ 0 -4 0 ] [ x2 ] = [ 0 ]
[ 4 2 -5 ] [ x3 ] [ 0 ]
We can solve this system to get:
x1 = -2/3 x2
x3 = -4/3 x2
So the eigenvector for λ2 = 2 is:
v2 = [ x1, x2, x3 ] = [ -2/3, 1, -4/3 ]
For λ3 = -4, we have:
[ -5+4 1 2 ] [ x1 ] [ 0 ]
[ 0 -2+4 0 ] [ x2 ] = [ 0 ]
[ 4 2 -3+4] [ x3 ] [ 0 ]
which simplifies to:
[ -1 1 2 ] [ x1 ] [ 0 ]
[ 0 2 0 ] [ x2 ] = [ 0 ]
[ 4 2 1 ] [ x3 ] [ 0 ]
We can solve this system to get:
x1 = -2/5 x2
x3 = 1/2 x2
So the eigenvector for λ3 = -4 is:
v3 = [ x1, x2, x3 ] = [ -2/5, 0, 1/2 ]
To determine matrix A's sign definiteness, we must compute the determinant of all of A's primary submatrices. A square submatrix generated by eliminating the equal amount of rows and columns from the top left corner of A is known as a primary submatrix.
The principal submatrices of A are:
A1 = [-5]
A2 = [ -5 1 ]
[ 0 -2 ]
A3 = [ -5 1 2 ]
[ 0 -2 0 ]
[ 4 2 -3 ]
The determinants of these matrices are:
det(A1) = -5
det(A2) = (-5)(-2) - (0)(1) = 10
det(A3) = (-5)(-2)(-3) + 2(0)(4) + (1)(0)(2) - (-3)(1)(4) - 2(0)(-5) - (-2)(2)(-5) = -92
Because A1's determinant is negative, A is not positive definite. A is not negative definite since the determinants of A1 and A2 have different signs. As a result, A is infinite.
Prove that there exists infinitely many integers $n$ such that $n^2+1$ is squarefree. (a square-free integer is an integer which is divisible by no square number other than 1.)
Answer:
To prove that there exist infinitely many integers n such that n^2 + 1 is squarefree, we can use a proof by contradiction.
Assume the contrary, that there are only finitely many integers n such that n^2 + 1 is squarefree. Let's denote these integers as n_1, n_2, ..., n_k, where k is a finite positive integer.
Consider the number N = (n_1^2 + 1) * (n_2^2 + 1) * ... * (n_k^2 + 1) + 1.
Note that N is an integer and is greater than each of the numbers n_i^2 + 1 for i = 1 to k. Since n_i^2 + 1 is squarefree for each i, N is also squarefree, as it is not divisible by any square number other than 1.
However, N cannot be equal to any of the n_i^2 + 1, as N is strictly greater than each of them. This contradicts our assumption that n_1, n_2, ..., n_k are all the integers such that n^2 + 1 is squarefree.
Therefore, our assumption must be false, and there exist infinitely many integers n such that n^2 + 1 is squarefree. This completes the proof by contradiction.
Step-by-step explanation: