a) Our assumption that there exist integers x and y such that x² - 4y = 2 is false, and we can conclude that for all integers x and y, x² - 4y ≠ 2.
b) 46¹⁰ ≡ 1 (mod 7).
345 mod 9 ≡ 1 (mod 9).
How evaluate each part of the question?(a) Proof by contradiction:
Assume that there exist integers x and y such that x² - 4y = 2.
Then x² = 2 + 4y.
Since 2 is an even integer, 4y must also be an even integer, which means that y is an even integer.
Let y = 2k, where k is an integer.
Then x² = 2 + 8k.
If x² is an even integer, then x must also be an even integer (by the given fact).
Let x = 2m, where m is an integer.
Then (2m)² = 2 + 8k.
Simplifying this equation, we get:
4m² = 1 + 4k.
This equation implies that 4m² is an odd integer, which is a contradiction.
Therefore, our assumption that there exist integers x and y such that x² - 4y = 2 is false, and we can conclude that for all integers x and y, x² - 4y ≠ 2.
(b)
(i) 46¹⁰ mod 7:
We can use the property that [tex]a^{b+c} = (a^b)*(a^c)[/tex] to simplify the exponent:
46¹⁰ = (46⁵)²
To find 46⁵ mod 7, we can reduce the base modulo 7:
46 ≡ 4 (mod 7)
Then, we can use the property that (a*b) mod m = ((a mod m) * (b mod m)) mod m:
46⁵ ≡ 4⁵ (mod 7)
≡ (44444) mod 7
≡ (-1)(-1)(-1)(-1)(-1) mod 7
≡ -1 mod 7
≡ 6 (mod 7)
Substituting this value back into the original expression:
46¹⁰ ≡ (46⁵)²
≡ 6² (mod 7)
≡ 36 (mod 7)
≡ 1 (mod 7)
Therefore, 46¹⁰ ≡ 1 (mod 7).
(ii) 345 mod 9:
We can use the property that [tex]a^{b+c} = (a^b)*(a^c)[/tex] to simplify the exponent:
345 = (3100 + 410 + 5)
Therefore, we can break down 345 into its digits and calculate each digit modulo 9:
3100 mod 9 ≡ 0 (mod 9)
410 mod 9 ≡ 5 (mod 9)
5 mod 9 ≡ 5 (mod 9)
Then, we can use the property that (a+b) mod m = ((a mod m) + (b mod m)) mod m:
345 mod 9 ≡ (0 + 5 + 5) mod 9
≡ 10 mod 9
≡ 1 (mod 9)
Therefore, 345 mod 9 ≡ 1 (mod 9).
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Kathy can run 4 mi to the beach in the same amount of time Dennis can ride his bike 14 mi to work. Kathy runs 5 mph slower than Dennis rides his bike. Find
their speeds.
Kathy runs at a speed of 2 mph, and Dennis rides his bike at a speed of 7 mph.
How to find the speeds ?To find the speed that Kathy is running and that Dennis is riding, the first relationship is:
K = D - 5
Then use the formula for time:
Time for Kathy = Time for Dennis
4 mi / K = 14 mi / D
4 mi / (D - 5) = 14 mi / D
4D = 14(D - 5)
4D = 14D - 70
-10D = -70
D = 7 mph
Then we can find Kathy's speed :
K = 7 - 5
K = 2 mph
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21.5 ÷ 5 + (80.6 - 12.5 ÷ 2)
PEMDAS
Answer:
78.65
Step-by-step explanation:
Question 4 Graph and label each figure and its image under a reflection in the given line. Give the coordinates of the image. Rhombus WXYZ with verlices 1.5), X[6,3). 1. 1), and Z 4,3): x-axis WC X' YT Z'
First, let's identify the coordinates of the vertices of rhombus WXYZ:
W(1,5), X(6,3), Y(1,1), and Z(4,3)
Now, we will perform a reflection over the x-axis. To do this, we simply need to negate the y-coordinate of each vertex while keeping the x-coordinate the same.
Step-by-step:
1. Reflect point W(1,5):
The x-coordinate stays the same: 1
Negate the y-coordinate: -5
New coordinates for W': W'(1,-5)
2. Reflect point X(6,3):
The x-coordinate stays the same: 6
Negate the y-coordinate: -3
New coordinates for X': X'(6,-3)
3. Reflect point Y(1,1):
The x-coordinate stays the same: 1
Negate the y-coordinate: -1
New coordinates for Y': Y'(1,-1)
4. Reflect point Z(4,3):
The x-coordinate stays the same: 4
Negate the y-coordinate: -3
New coordinates for Z': Z'(4,-3)
The coordinates of the image of rhombus WXYZ under the reflection in the x-axis are W'(1,-5), X'(6,-3), Y'(1,-1), and Z'(4,-3).
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Suppose you have a regression model with an interaction term and a dummy variable. In this case, we can have a only one slope and only one intercept b.only one slope, but more than one intercept. c. more than one slope, but only one intercept d. more than one slope and more than one intercept.
When a regression model has an interaction term and a dummy variable in statistics and probability, there will be more than one slope and more than one intercept (D)
When there is an interaction term and a dummy variable in a regression model, we can have more than one slope and more than one intercept. The interaction term allows for different slopes for different levels of the dummy variable, while the intercepts represent the expected value of the dependent variable when the dummy variable is equal to zero for each level of the interaction term.
When a regression model has an interaction term and a dummy variable, it means that the effect of one independent variable on the dependent variable varies depending on the value of the other independent variable. In other words, the slope and intercept of the regression line will change depending on the value of the dummy variable.
More specifically, the model will have one intercept and two slopes: one for the dummy variable and one for the interaction term. As a result, the relationship between the dependent variable and the independent variables will vary depending on the value of the dummy variable, which will result in different slopes and intercepts.
Therefore, the correct answer is (d): more than one slope and more than one intercept.
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Colin, Dave and Emma share some money.
Colin gets 3⁄10 of the money.
Emma and Dave share the rest of the money in the ratio 3 : 2 What is Dave's share of the money
Make the amount of money they have £100 because this makes the question easier.
Colin gets 3/10 of the money, so Colin will get £30.
After Colin has taken his share £70 will be left over.
The ratio give is 3 : 2. So 3 + 2 is equal to 5.
The amount of money left over is then divided by the ratio added in this case its 70/5.
70/5 gives us an answer of 14 .
This means that each share is equal to £14.
Emma gets the ratio of 3 so we do 3 x 14 which gives us he answer of £42.
And if we do 3 x 2 we get the answer of £28.
We then know Dave gets £26 pounds from the £100 at the start.
26/100 converted to a percentage is 26%.
Help please, i don't get it i need it done asap
The number of boxes that can fit into the crate is 7 boxes.
What is the shape of a cuboid?A cuboid has a hexahedron six-faced solid shape and the volume is determined by multiplying the length by width by height. Here; the volume of the crate is determined by finding the volume of the cuboid.
Volume of the cuboid is: 2.4 m × 1.8 m × 1.1 m
Volume of the cuboid = 4.752 m³
To cm, volume of the cuboid = 475.2 cm³
Now, since the cube has a length of 60 cm, then the number of boxes that will fit into the crate can be estimated by dividing the volume of the cuboid shape by the length of the cube.
Thus, the number of boxes that can fit into the crate is:
= 475.2 cm/ 60 cm
= 7. 92
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find a particular solution to ″ 6′ 9=^−3/^3
The particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex] based on integration.
To find a particular solution to given equatio we need to integrate twice. First, we integrate with respect to x to get [tex]-3x^(-2)[/tex].
Then, we integrate again with respect to x to get 3x^(-1) + C1, where C1 is a constant of integration.
Next, we use the initial condition 6′ 9 to solve for C1. Taking the derivative of [tex]3x^(-1) + C1[/tex], we get [tex]-3x^(-2)[/tex]. Plugging in x = 9, we get [tex]-3(9)^(-2) = -1/27[/tex].
Therefore, [tex]-1/27 = -3(9)^(-2) + C1[/tex], and solving for C1, we get[tex]C1 = -1/27 + 3(9)^(-2)[/tex].
Thus, the particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex].
Hi! It seems there might be a typo in your question, making it difficult to understand the exact problem you need help with. However, I will try to address the terms "solution" and "particular."
A "solution" refers to the result or answer obtained when solving an equation, problem, or system of equations. It is the value or values that satisfy the given conditions or equations.
A "particular solution" is a specific instance of a solution, usually when there are multiple solutions or when dealing with differential equations. It is a single example of a valid answer that meets the given criteria.
If you can provide more clarification on your question, I would be happy to help you find the particular solution you're looking for!
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The graph shows the height y in feet of a gymnast jumping off of a vault after x seconds.
a) How long does the gymnast stay in the air?
b) What is the maximum height that the gymnast reaches?
c) In how many seconds does it take for the gymnast to start descending?
d) What is the quadratic function that models this situation?
Using the graph, we can find the following:
a) The gymnast stays 4 seconds in the air.
b) The maximum height that the gymnast reaches is 10 ft.
c) After 2 seconds the gymnast starts to descend.
d) The quadratic function that models this situation is:
y = mx + c
Define graphs?Quantitative data can be represented and analysed graphically. In a graph, variables representing data are drawn over a coordinate plane. Analysing the magnitude of one variable's change in light of other variables' changes became simple.
Here in the question,
a. We can see from the graph that the curve above x-axis starts from the origin (0,0) and ends at (4,0) on the x-axis.
So, the gymnast stays 4 seconds in the air.
b. As we can see from the graph that it rises and then at point (2,10) it starts to descend.
So, the maximum height that the gymnast reaches is 10 ft.
c. As we can see from the graph that it rises and then at point (2,10) it starts to descend.
So, after 2 seconds the gymnast starts to descend.
d. The quadratic function that models this situation is:
y = mx + c
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if x(t) = 2·tri(t/4)*δ(t – 2), find the values of a. x(1) b. x(–1)
The values for x(1) and x(-1) are both 0.
To find the values of x(1) and x(-1) given that x(t) = 2·tri(t/4)*δ(t – 2), we will evaluate the function at these points.
a. x(1):
To find the value of x(1), we need to substitute t = 1 into the function:
x(1) = 2·tri(1/4)*δ(1 - 2)
Since δ(1 - 2) is the Dirac delta function at a point different from zero (specifically, -1), its value is 0.
Therefore,
x(1) = 2·tri(1/4) * 0 = 0
b. x(-1):
To find the value of x(-1), we need to substitute t = -1 into the function:
x(-1) = 2·tri(-1/4)*δ(-1 - 2)
Again, since δ(-1 - 2) is the Dirac delta function at a point different from zero (specifically, -3), its value is 0.
Therefore,
x(-1) = 2·tri(-1/4) * 0 = 0
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given that income is 500 and px=20 and py=5 what is the market rate of subsitutino between good x and y? a. 100
b. -4
c. -20
d. 25
The market rate of substitution between good x and y is represented by the ratio of their prices, which is px/py. Therefore, in this case, the market rate of substitution is 20/5 = 4. However, this answer choice is not listed. The closest answer choice is b. -4, which is the negative inverse of the market rate of substitution (-1/4).
The market rate of substitution between good X and Y is represented by the marginal rate of substitution (MRS), which is the ratio of the marginal utilities of both goods. In this case, we are given income (I) = 500, the price of good X (Px) = 20, and the price of good Y (Py) = 5.
To find the MRS, we can use the formula:
MRS = - (Px / Py)
Plugging in the values, we get:
MRS = - (20 / 5)
MRS = -4
So, the market rate of substitution between good X and Y is -4 (option b).
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Find the GLB of the set: {x:|x - 4|<1}. a. -1 b. 1 c. -3 d. 3 e. 0
The GLB of the set {x : |x - 4| < 1} is option (d) 3
The set {x : |x - 4| < 1} can be written as the open interval (3, 5), which contains all values of x that satisfy the inequality |x - 4| < 1.
The greatest lower bound (GLB), also known as the infimum, is a concept in mathematics that applies to sets of numbers or other mathematical objects that are partially ordered.
To find the GLB (greatest lower bound) of this interval, we need to look for the greatest value that is less than or equal to every element in the interval.
Since the interval contains all real numbers greater than 3 and less than 5, its GLB is 3. Therefore, the answer is option (d) 3.
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Find the value of x.
If necessary, you may learn what the markings on a figure indicate.
to
73°
X =
The value of the angle is 34 degrees
How to determine the valueTo determine the value of the variable, we need to the following;
The sum of triangle theorem states that the sum of the angles in a triangle is 180 degreesAlternate angles are know to be equalAn isosceles triangle has two of its sides equalTwo of its angles are equalFrom the information given, we have that the angles are;
73 degrees
73 degrees
x degrees
Equate the angles
73 + 73 +x = 180
collect the like terms
x = 180 - 146
subtract the values
x = 34 degrees
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Find the area of this sector.
Give your answer in terms of
π
.
Answer:245/36 π
Step-by-step explanation: you do 50/360 times π(7)^2
solve each system of inequalities and indicate all the integers that are in the solution set. 2-6y<14 and 1<21-5y
Answer:
{-1, 0, 1, 2, 3}
Step-by-step explanation:
. 2-6y<14
-6y < 12
y > 12/-6
y > -2.
1<21-5y
-5y > -20
y < 4.
So -2 < y < 4
and the solution set of integers is
{-1, 0, 1, 2, 3}
Ashlee purchased a house for $875 000. She made a down payment of 15% of the purchase price and took out a mortgage for the rest. The mortgage has an interest rate of 6.95% compounded monthly, and amortization period of 20 years, and a 5 year term. Calculate Ashley’s monthly payment.
$5744 is Ashley’s monthly payment.
The amount of the down payment made by Ashlee is 15% of $875,000, which is:
Down payment = 0.15 x $875,000 = $131,250
The amount that Ashlee took out on a mortgage is:
Mortgage amount = Purchase price - Down payment
= $875,000 - $131,250
= $743,750
The monthly payment on a mortgage:
[tex]M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ][/tex]
where:
M = monthly payment
P = principal amount (mortgage amount)
i = monthly interest rate (annual interest rate / 12)
n = total number of monthly payments (amortization period x 12)
In this case, the annual interest rate is 6.95% and the term is for 5 years, so we need to first calculate the monthly interest rate and the total number of monthly payments.
Monthly interest rate = 6.95% / 12 = 0.57917%
Total number of monthly payments = 20 years x 12 = 240
Substituting these values into the formula, we get:
M = $743,750 [ 0.0057917 (1 + 0.0057917)^240 ] / [ (1 + 0.0057917)^240 - 1 ]
= $5744.002
Therefore, Ashley's monthly payment on the mortgage is $5744.
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Special right triangle
Answer:
s = 5[tex]\sqrt{6}[/tex]
Step-by-step explanation:
using the cosine ratio in the right triangle and the exact value
cos30° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
cos30° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{s}{10\sqrt{2} }[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2s = 10[tex]\sqrt{2}[/tex] × [tex]\sqrt{3}[/tex] = 10[tex]\sqrt{6}[/tex] ( divide both sides by 2 )
s = 5[tex]\sqrt{6}[/tex]
exercise 1.1.10. solve ,dxdt=sin(t2) t, .x(0)=20. it is ok to leave your answer as a definite integral.
The solution of the differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
To solve the given differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 and leaving the answer as a definite integral, follow these steps:
1. Identify the given differential equation:
dx/dt = sin(t²)×t.
2. Recognize the initial condition:
x(0) = 20.
3. Integrate both sides of the equation with respect to t:
∫dx = ∫sin(t²)×t dt.
4. Apply the initial condition to determine the constant of integration:
x(0) = 20.
5. Write the final solution:
[tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
So, the solution is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
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S
1
.
2
3
Which translation maps the graph of the function f(x) = x² onto the function g(x) = x² − 6x + 6?
Oleft 3 units, down 3 units
Oright 3 units, down 3 units
Oleft 6 units, down 1 unit
Oright 6 units, down 1 unit
Answer:
right 3 units, down 3 unitsStep-by-step explanation:
You want the translation that maps f(x) = x² to g(x) = x² -6x +6.
GraphA graph of the two functions shows g(x) is right 3 units and down 3 units from f(x).
Vertex formWe know the vertex of f(x) = x² is the origin (0, 0). The vertex of g(x) will tell us the translation. Putting that function in vertex form, we have ...
g(x) = x² -6x +6
g(x) = (x² -6x) +6
g(x) = (x² -6x +9) +6 -9 . . . . . add and subtract 9 to complete the square
g(x) = (x -3)² -3
Compare this to ...
y = (x -h)² +k . . . . . . has vertex (h, k)
We see that (h, k) = (3, -3).
g(x) is translated right 3 units and down 3 units.
Complete the inductive step, identifying where you use the inductive hypothesis. (You must provide an answer before moving to the next part.) Multiple Choice O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the Inductive hypothesis, we have (kok+") + (x + 1)2 = (x + 1)2 +68+4)* (+1)(x+2) as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what It equals by virtue of the inductive hypothesis, we have (k++) + (x + 1)2 = (x + 1)2 *****!) -(+1Xk+2) * as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the inductive hypothesis, we have ( kk+) + (k+ 1)2 = (x + 1)2( 344x+2) = (x+1}+2) as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the inductive hypothesis, we have (64 + (k+ 1)2 = (k+ 1)2 (+4x+1) = (+1}x+2) as desired.
Completing the inductive step and identifying where the inductive hypothesis is used, the correct multiple choice answer is: Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the Inductive hypothesis, we have (kok+") + (x + 1)² = (x + 1)² +68+4)* (+1)(x+2) as desired.
In an induction proof, the inductive step involves assuming a statement is true for some arbitrary value, say k, and then proving it's true for the next value, k+1.
Here, the inductive hypothesis corresponds to the term (kok+"). By replacing this term on the left-hand side of part (c) with its equivalent based on the inductive hypothesis, we can show that the equation holds for the (k+1) case as well. This is crucial for proving the statement using induction, as it establishes the necessary pattern for all cases.
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If fy (a,b) = f, (a,b) = 0, does it follow that f has a local maximum or local minimum at (a,b)? Explain. Choose the correct answer below. A. No. It follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum. B. Yes. The point (a,b) is a critical point and must be a local maximum or local minimum. C. Yes. The tangent plane to f at (a,b) is horizontal. This indicates the presence of a local maximum or a local minimum at (a,b). D. No. One (or both) of fy and f, must also not exist at (a,b) to be sure that f has a local maximum or local minimum at (a,b).
If fy (a,b) = f, (a,b) = 0, does it follow that f has a local maximum or local minimum at (a,b) then it does not follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum. Therefore, the correct option is option A. No. It follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum.
If fy (a,b) = f, (a,b) = 0, does it follow that f has a local maximum or local minimum at (a,b) then it does not follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum.
This is because fy (a,b) = f, (a,b) = 0 indicates that the partial derivatives of f with respect to both a and b are zero at (a,b), which makes (a,b) a critical point of f. However, this does not guarantee that f has a local maximum or local minimum at (a,b), as further analysis is required to determine the nature of the critical point.
Just because both partial derivatives are zero does not guarantee that (a,b) is a local maximum or minimum. It could also be a saddle point or an inflection point. To determine whether it is a local maximum, local minimum, or neither, you would need to use the second partial derivative test or examine the nature of the function around the point (a,b).
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Help ASAP! I need this badly, my last question!
Answer:
Step-by-step explanation:
...... The problem is glitched re send the image/
Test the series for convergence or divergence. [infinity]
Σ (-1)^n+1/3n^4 . n=1 - converges
- diverges
The answer is: Test the series for convergence or divergence. [infinity] Σ (-1)n+1/3n² - converges.
To test the series Σ (-1)n+1/3n² for convergence or divergence, we can use the alternating series test. This test states that if a series alternates in sign and the absolute value of its terms decreases monotonically to zero, then the series converges.
In this case, the series Σ (-1)n+1/3n² alternates in sign and the absolute value of its terms is given by 1/3n², which decreases monotonically to zero as n increases. Therefore, we can apply the alternating series test and conclude that the series converges.
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What kind of geometric transformation is shown in the line of music ?
The kind of geometric transformation shown by the line of music is: Reflection
How to find the geometric transformation?A transformation is a mathematical manipulation that moves a geometric shape or function from one space to another.
Now, a set of image transformations where the geometry of image is changed without altering its actual pixel values are commonly referred to as “Geometric transformations"
Looking at the music note, we can see that the note didn't change in size but looks to be a mirror image of its' previous notes.
Since it is a mirror image, the obvious transformation will be a reflection
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how many different ways can be people be chosen as president, vice president, and secretary from a class of 40 students?
By using the Concept of Permutations,There are 59,280 different ways to choose the president, vice president, and secretary from a class of 40 students.
To determine the number of different ways people can be chosen as president, vice president, and secretary from a class of 40 students:
You can use the concept of permutations.
Step 1: Choose the president. There are 40 students in the class, so there are 40 choices for the president position.
Step 2: Choose the vice president. Since the president has been chosen, there are now 39 remaining students to choose from for the vice president position.
Step 3: Choose the secretary. After selecting the president and vice president, there are 38 remaining students to choose from for the secretary position.
Now, multiply the number of choices for each position:
40 (president) x 39 (vice president) x 38 (secretary) = 59,280 different ways to choose the president, vice president, and secretary from a class of 40 students.
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Divide £200 in the ratio 3:5
Answer:
£75 and £125
Step-by-step explanation:
To divide £200 in the ratio 3:5, you need to first find the total parts of the ratio, which is 3+5=8.
Then you can divide £200by 8 to find the value of each part of the ratio:
£200/8 =£25
So, each part of the ratio 3:5 is worth £25.
To find the share of each part of the ratio, you can multiply the value of each part by the corresponding ratio number:
The share of the first part (3) is £25*3=£75
The share pf the second part (5) is £25*5=£125
Therefore, the £200 is divided into the ratio of 3:5 as £75 for the first part and £125 for the second part.
Please Help!!!!! Find the Value of X!!!
The value of x from the Intersecting chords that extend outside circle is 13
From the question, we have the following parameters that can be used in our computation:
Intersecting chords that extend outside circle
Using the theorem of intersecting chords, we have
8 * (3x - 2 + 8) = 12 * (x + 5 + 12)
This gives
8 * (3x + 6) = 12 * (x + 17)
Using a graphing tool, we have
x = 13
Hence, the value of x is 13
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find the x-coordinates of the inflection points for the polynomial p(x)= x^5/20
The inflection point of the polynomial p(x) = [tex]x^5/20[/tex] is at x = 0. This is the only one inflection point.
To find the x-coordinates of the inflection points for the polynomial p(x) = [tex]x^5/20[/tex], we'll need to follow these steps:
1. Find the first derivative, p'(x), to determine the slope of the function.
2. Find the second derivative, p''(x), to determine the concavity of the function.
3. Set p''(x) equal to zero and solve for x to find the inflection points.
Step 1: Find the first derivative, p'(x):
p'(x) = [tex]d(x^5/20)/dx = (5x^4)/20 = x^4/4[/tex]
Step 2: Find the second derivative, p''(x):
p''(x) = [tex]d(x^4/4)/dx = (4x^3)/4 = x^3[/tex]
Step 3: Set p''(x) equal to zero and solve for x:
[tex]x^3[/tex] = 0
x = 0
There is only one inflection point, and its x-coordinate is 0.
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Solids A and cap B are similar.
The volume of the cone B using scale factor k = 3/2 is equal to 54π cubic centimeters.
Volume of cone A = 16π cubic centimeters
Scale factor 'k' = 3/2
Two solids A and B are similar.
This implies ,all corresponding lengths in solid B are 3/2 times the lengths in solid A.
The volume of a cone is given by the formula
V = (1/3)πr²h,
where r is the radius and h is the height.
Volume of cone A is 16π cubic centimeters.
Let r₁ and h₁ be the radius and height of cone A and r₂ and h₂ of cone B.
⇒16π = (1/3)π(r₁²)(h₁)
Multiplying both sides by 3 and dividing by π, we get,
⇒48 =(r₁²)(h₁)
Since solid A and B are similar with a scale factor of 3/2, we have,
h₂ = (3/2)h₁ and r₂= (3/2)r₁
Using these relationships, the volume of cone B is,
Volume of cone B = (1/3)π(r₂²)(h₂)
Volume of cone B = (1/3)π[(3/2)r₁]²[(3/2)h₁]
Volume of cone B = (1/3)π(9/4)(r₁²)(3/2)(h₁)
Volume of cone B = (27/8)(1/3)π(r₁²)(h₁)
Substituting Volume of cone A = 16π, we get,
Volume of cone B = (27/8)(16π)
Volume of cone B = 54π
Therefore, the volume of cone B is 54π cubic centimeters.
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Evaluate the iterated integral by changing to cylindrical coordinates.∫ ^2_0 ∫ ^√(4 − y^2)_0 ∫ ^(16 − x^2 − y^2)_0 1 dz dx dy
To convert the integral to cylindrical coordinates, we use the following conversions:
x = r cos(theta)
y = r sin(theta)
z = z
And we also replace dV with r dz dr d(theta).
The limits of integration are:
0 ≤ r ≤ 2 (since the bounds on x and y are from 0 to 2)
0 ≤ theta ≤ 2pi (since we integrate over the entire circle)
0 ≤ z ≤ 16 - r^2 (since the bounds on z are from 0 to 16 - x^2 - y^2, which in cylindrical coordinates is 16 - r^2)
Thus, the integral becomes:
∫^(2pi)_0 ∫^2_0 ∫^(16-r^2)_0 r dz dr d(theta)
Integrating with respect to z, we get:
∫^(2pi)_0 ∫^2_0 (16 - r^2)r dr d(theta)
Integrating with respect to r, we get:
∫^(2pi)_0 [8r^2 - (1/3)r^4]∣_0^2 d(theta)
= ∫^(2pi)_0 (32/3) d(theta)
= (32/3) ∫^(2pi)_0 d(theta)
= (32/3)(2pi)
= (64/3)pi
Therefore, the value of the iterated integral in cylindrical coordinates is (64/3)pi.
Three factories produce the same tool and supply it to the market. Factory A produces 30% of the tools for the market and the remaining 70% of the tools are produced in factories B and C. 98% of the tools produced in factory A, 95% of the tools produced in factory B and 97% of the tools produced in factory C are not defective. What percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96%?
The percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96% = 0.96 are 5% and 95% respectively.
We have three factories produce the same tool and supply it to the market. Let's consider three events defined as, A = event for tools produced by factory A
B = event for tools produced by factory B
C = event for tools produced by factory C and N be the count that tools produced by all factories is not defective.
The probability that the tools produced by factory A for the market, P( A) = 30%
= 0.30
The probability that the tools produced by factories B and C for the market, P( B and C) = 70% = 0.70
The Probability that tools are non- defective and that are produced in factory A, P( N/A) = 98%
= 0.98
The Probability that tools are non- defective and that are produced in factory B, P( N/B) = 95%
= 0.95
The Probability that tools are non- defective and that are produced in factory C, P( N/C) = 97% = 0.97
Now, since only three factories supply to the whole market, then by probability law, P(A) + P(B) + P( C) = 1
=> 0.3 + P(B) + P( C) = 1
=> P(B) = 0.7 - P(C) --(1)
We have to determine percent of tools should be produced by factories B and C that is P(C) and P(B) when probability of non defective, P(N) is 96% = 0.96. From the law of total probability law, P(N) is written by, P( N) = P( N/A) P(A) + P( N/B) P(B) + P( N/C) P( C)
=> 0.96 = 0.98 × 0.3 + 0.95 × ( 0.7 - P(C) ) + 0.97 × P(C)
=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)
=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)
=> 0.96 = 0.294 + 0.665 + 0.02 × P(C)
=> 0.96 = 0.959 + 0.02 × P(C)
=> 0.02 × P(C) = 0.96 - 0.959
=> 0.02 × P(C) = 0.001
=> P(C) = 0.05 = 5%
from equation (1), P(B) = 1 - P(C)
=> P( B) = 1 - 0.05 = 0.95
Hence, required percentage is 95%.
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