The diameter of a circle is 4ft , Find it’s circumference in terms of pi.

Answer:

The length should be 11 feet and the width should be 6 feet. 11 feet is 5 feet more than 6 feet. 11 feet times 6 feet is 66 square feet.

Step-by-step explanation:

Answer:

a)  ∩

Step-by-step explanation:

set A : {6,8,10,12}

set B : {5,6,7,8,9}

set C : {6,8}

A ∪ B = {6,8,10,12} ∪ {5,6,7,8,9} = {5,6,7,8,9,10,12} ≠ C

A ∩ B = {6,8,10,12} ∩ {5,6,7,8,9} = {6, 8} = C

Therfore{6,8,10,12} ∩ {5,6,7,8,9} = {6, 8}

Answer:

a. 0.9375, 0.46875

b. geometric sequence

c. equation: [tex] 7.5 * (\frac{1}{2})^(n-1)[/tex]

Step-by-step explanation:

a.

The table can be finished as follows:

n t(n)

1 7.5

2 3.75

3. 1.875

4. 0.9375

5 0.46875

b.

The type of sequence is a geometric sequence.

A geometric sequence is a sequence of numbers where the ratio between any two consecutive terms is constant.

In this case, the ratio between any two consecutive terms is 3.75/7.5=½ ,

so the sequence is geometric.

c.

The equation for the sequence is t(n) = 7.5 * (1/2)^n.

This equation can be found by looking at the first term of the sequence (7.5) and the common ratio (1/2).

t(1) = 7.5

t(2) = 7.5 * (1/2) = 3.75

t(3) = 7.5 * (1/2)^2 = 1.875

The equation can also be found by looking at the general formula for a geometric sequence,

which is [tex]t(n) = a*r^{n-1}[/tex]

In this case,

t(n) =[tex] 7.5 * (\frac{1}{2})^{n-1}[/tex]

This is the required equation.

Answer:

[tex]\textsf{a.}\quad \begin{array}{|c|c|c|c|c|c|}\cline{1-6}\vphantom{\dfrac12} n&1&2&3&4&5\\\cline{1-6}\vphantom{\dfrac12}t(n)&7.5&3.75&1.875&0.9375&0.4687\\\cline{1-6}\end{array}[/tex]

[tex]\textsf{b.} \quad \textsf{Geometric sequence.}[/tex]

[tex]\textsf{c.} \quad t(n)=7.5(0.5)^{n-1}[/tex]

Step-by-step explanation:

Before we can complete the table, we need to determine if the sequence is arithmetic or geometric.

To determine if a sequence is arithmetic or geometric, examine the pattern of the terms in the sequence.

Calculate the difference between consecutive terms by subtracting one term from the next:

[tex]t(2)-t(1)=3.75-7.5=-3.75[/tex]

[tex]t(3)-t(2)=1.875-3.75 = -1,875[/tex]

As the difference is not common, the sequence is not arithmetic.

Calculate the ratio between consecutive terms by dividing one term by the previous term.

[tex]\dfrac{t(2)}{t(1)}=\dfrac{3.75}{7.5}=0.5[/tex]

[tex]\dfrac{t(3)}{t(2)}=\dfrac{1.875}{3.75}=0.5[/tex]

As the ratio is common, the sequence is geometric.

To complete the table, multiply the preceding term by the common ratio 0.5 to calculate the next term:

[tex]t(4)=t(3) \times 0.5=1.875 \times 0.5=0.9375[/tex]

[tex]t(5)=t(4) \times 0.5=0.9375 \times 0.5=0.46875[/tex]

Therefore, the completed table is:

[tex]\begin{array}{|c|c|c|c|c|c|}\cline{1-6}\vphantom{\dfrac12} n&1&2&3&4&5\\\cline{1-6}\vphantom{\dfrac12}t(n)&7.5&3.75&1.875&0.9375&0.4687\\\cline{1-6}\end{array}[/tex]

To find an equation for the sequence, use the general form of a geometric sequence:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

In this case, the first term is the value of t(n) when n = 1, so a = 7.5

We have already calculated the common ratio as being 0.5, so r = 0.5.

Substitute these values into the formula to create an equation for the sequence:

[tex]t(n)=7.5(0.5)^{n-1}[/tex]

Answer: 33.7

Step-by-step explanation:

Let the height of the house be h, and the distance from the foot of the rock to the house be x.

From the given information, we have the following diagram:

```

*

/ \

/ \

/ 63° \

/ \

/θ \

A ----------------------- B

70m x

Angle A = 60° (complementary to the angle of elevation of the foot of the house)

Angle B = 63° (angle of elevation of the top of the house)

Using trigonometry, we have:

tan(63°) = h/x ----(1) (for triangle AOB)

tan(60°) = h/(x + 70) ----(2) (for triangle ABD)

Solving equations (1) and (2) simultaneously, we get:

h = (70 tan(63°) - 70 tan(60°)) meters

h ≈ 33.7 meters (rounded to one decimal place)

Therefore, the height of the house is approximately 33.7 meters.

Answer:

(x+3)^2 + (y+1)^2 = 16

Step-by-step explanation:

The equation of a circle is (x – h)^2 + (y – k)^2 = r^2, where h is the x value of the center, k is the y value of the center, and r is the radius.

We can see from the picture that the radius is at about (-3, -1) and the radius is about 4, so we can plug those in:
(x – (-3))^2 + (y – (-1))^2 = 4^2

Simplify:
(x+3)^2 + (y+1)^2 = 16

Answer:

Equation of circle:[tex](x + 3)^2 + (y + 1)^2 = 16[/tex]

Step-by-step explanation:

Given:

Center of the circle = (-3, -1)

Point on the circle = (1, -1)

In order to find the radius of the circle, we can use the distance formula.

distance =[tex] \boxed{\bold{\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}}}[/tex]

where:

In this case, the distance formula becomes:

radius = [tex]\sqrt{(-3 - 1)^2 + ((-1) - (-1))^2}= \sqrt{16}=4[/tex]

Therefore, the radius of the circle is 4 units.

Now that we know the radius of the circle, we can find the equation of the circle using the following formula:

[tex]\boxed{\bold{(x - h)^2 + (y - k)^2 = r^2}}[/tex]

where:

In this case, the equation of the circle becomes:

=[tex](x + 3)^2 + (y + 1)^2 = 4^2[/tex]

=[tex](x + 3)^2 + (y + 1)^2 = 16[/tex]

This is the equation of the circle.

The values of p and q are 1 and 3, respectively.

Given the equation x^2-4x + 1 = (x-p)- q, we can compare the coefficients of the corresponding terms on both sides of the equation.

We compared the coefficients of the x^2 terms on both sides of the equation. The coefficient of the x^2 term on the left-hand side is 1, and the coefficient of the x^2 term on the right-hand side is 1. This means that the two terms are equal, and therefore p = 1.

We compared the coefficients of the x terms on both sides of the equation. The coefficient of the x term on the left-hand side is -4, and the coefficient of the x term on the right-hand side is -1. This means that the two terms are equal, and therefore q = 3.

For such more question on values:

https://brainly.com/question/26352252

#SPJ8

Avoidable costs are costs that can be eliminated if the activity that caused them is discontinued. The examples are listed.

Direct materials: Direct materials are the materials that go into a product and can be easily traced to it. For example, the wood used to make a table is a direct material. If a company decides to stop making tables, it can avoid the cost of buying wood.

Direct labor: Direct labor is the labor that is directly involved in making a product. For example, the wages paid to the workers who assemble a car are direct labor. If a company decides to stop making cars, it can avoid the cost of paying direct labor.

Variable overhead: Variable overhead is the overhead costs that vary with the number of units produced. For example, the cost of electricity used to power a factory is a variable overhead cost. If a company decides to stop producing a product, it can avoid the variable overhead costs associated with that product.

Sunk costs: Sunk costs are costs that have already been incurred and cannot be recovered. For example, the cost of research and development for a new product is a sunk cost. If the company decides not to launch the product, it cannot recover the cost of research and development.

Find out more on Avoidable Costs here: https://brainly.com/question/29807108

#SPJ1

The radius of the semicircle protractor is approximately 4.693 cm.

Given,Perimeter of a semicircle protractor = 14.8 cm.

To find:The radius of a semicircle protractor.Solution:We know that the perimeter of a semicircle protractor is the sum of the straight edge of a protractor and half of the circumference of the circle whose radius is the radius of the protractor.

Circumference of a circle = 2πrWhere, r is the radius of the circle.If the radius of the semicircle protractor is r, then Perimeter of a semicircle protractor = r + πr [∵ half of the circumference of a circle =[tex](1/2) × 2πr = πr]14.8 = r + πr14.8 = r(1 + π) r = 14.8 / (1 + π)r ≈ 4.693[/tex] cm.

for more questions on semicircle protractor

https://brainly.com/question/30188078

#SPJ8

The equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

To find the equation of a line passing through two points, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope of the line and b is the y-intercept.

Given points:

Point 1: (1, 0)

Point 2: (3, 6)

Step 1: Calculate the slope (m) using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates:

m = (6 - 0) / (3 - 1)

m = 6 / 2

m = 3

Step 2: Substitute one of the given points and the slope into the equation y = mx + b to find the y-intercept (b).

Using Point 1 (1, 0):

0 = 3(1) + b

0 = 3 + b

b = -3

Step 3: Write the equation of the line using the slope (m) and the y-intercept (b):

y = 3x - 3

Therefore, the equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

This equation represents a line with a slope of 3, indicating that for every increase of 1 unit in the x-coordinate, the y-coordinate increases by 3 units. The y-intercept of -3 means that the line crosses the y-axis at the point (0, -3). By substituting any x-value into the equation, we can determine the corresponding y-value on the line.

Hence, the equation of the line passing through (1, 0) and (3, 6) is y = 3x - 3.

For more such questions on equation of a line, click on:

https://brainly.com/question/18831322

#SPJ8

If Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

To show that x is also an eigenvector of QT, we need to demonstrate that QT * x is a scalar multiple of x.

Given that Q is an orthogonal matrix, we know that QT * Q = I, where I is the identity matrix. This implies that Q * QT = I as well.

Let's denote x as the eigenvector corresponding to the eigenvalue λ1 This means that Q * x = λ1 * x.

Now, let's consider QT * x. We can multiply both sides of the equation Q * x = λ1 * x by QT:

QT * (Q * x) = QT * (λ1 * x)

Applying the associative property of matrix multiplication, we have:

(QT * Q) * x = λ1 * (QT * x)

Using the fact that Q * QT = I, we can simplify further:

I * x = λ1 * (QT * x)

Since I * x equals x, we have:

x = λ1 * (QT * x)

Now, notice that λ1 * (QT * x) is a scalar multiple of x, where the scalar is λ1. Therefore, we can rewrite the equation as:

x = λ2 * x

where λ2 = λ1 * (QT * x).

This shows that x is indeed an eigenvector of QT, with the eigenvalue λ2 = λ1 * (QT * x).

In conclusion, if Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

for more such question on matrix visit

https://brainly.com/question/2456804

#SPJ8

Answer:

7: 63

8: 73

arithmetic sequence

y = 10x - 7

or f(n) = 10x -7

or

[tex]a_{n}[/tex] = 3 + (n-1)10

Step-by-step explanation:

the output increases by 10 every time that the input increases by 1.  That gives us our common difference or slope.  The y intercept is -7. That is the value is you worked backwards until you get to n = 0.  The initial value  is 3.  That is when n is 1.

When n is 3, f(n) is 23

When n is 2, f(n) is 13

When n is 1, f(n) is 3

When n is 0, f(n) is -7

I am not sure if this is clear.  I am assuming that you have a lot of knowledge of linear equations and how to write arithmetic sequence.  If my explanation is confusing it is me and not you.

Answer:

a. 63,73

b. Arithmetic sequence

c.t(n)=10n-7

Explanation:

a. Here is the completed table:

n | t(n)

4 | 33

5 | 43

6 | 53

7 | 63

8 | 73

b.

The type of sequence is arithmetic.

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant.

In this case, the difference between any two consecutive terms is 10.

c.

The equation for the arithmetic sequence is:

t(n)=a+(n-1)d

where:

For Question:

Now

equation becomes:

t(4) = a+(4-1)10

33=a+30

a=33-30

a=3

Now, the Equation becomes

t(n) = 3+(n-1)10

t(n) = 3+10n-10

t(n)=10n-7

The calculated value of x in the figure is 12

From the question, we have the following parameters that can be used in our computation:

The figure

The scale is given as

A : B = 8 : 9

So, we have

48 : 2x + 30 = 8 : 9

Next, we have

(2x + 30)/48 = 9/8

This gives

2x + 30 = 48 * 9/8

2x + 30 = 54

This gives

2x = 24

So, we have

x = 12

Hence, the value of x in the figure is 12

Read more about similar shapes at

https://brainly.com/question/32122930

#SPJ1

Answer:

The product of each pair of complex conjugates is:

(3 + 8i)(3 – 8i) = (9 - 8^2) + i(9 + 8^2) = (-81) + i(99) = (-81) + i(11) = (-81) + 11i

And

(4 + 5i)(4 – 5i) = (16 - 25) + i(16 + 25) = (-9) + i(41) = (-9) + 41i

So, the products are:

(-81) + 11i

and

(-9) + 41i

Answer:

73

41

Step-by-step explanation:

The complete factorization of the polynomial x³+2x²+x+2 is(x-1)(x+1)(x+2).

To factorize the polynomial x³+2x²+x+2 completely, one may use the rational root theorem, and synthetic division method.

Therefore, first we list the factors of 2 and divide it by the factors of 1 as the leading coefficient:Factors of 2 = ±1, ±2Factors of 1 = ±1Therefore, the possible rational roots for the given polynomial equation are± 1, ±2From the given polynomial, x³+2x²+x+2,

we can easily see that there is no remainder when divided by (x-1). Hence, (x-1) is a factor of the polynomial and we can use synthetic division to factorize the polynomial.x³+2x²+x+2 = (x-1)(x²+3x+2)Now we can use the quadratic formula or factoring to find the remaining factors. We get,x²+3x+2 = (x+1)(x+2).

for more question on polynomial

https://brainly.com/question/4142886

#SPJ8

Answer: x = 3 and x = -6

Step-by-step explanation:

the first step is to square root both sides of the equation to get rid of the exponent (2) on the left side of the equation, however lets break it down:

[tex](2x + 3)^2 = 81\\\\[/tex]

81 can be rewritten as 9^2

[tex](2x + 3)^2= 9^2[/tex]

and now lets square root both sides:

[tex]\sqrt{(2x+3)^2} =\sqrt{9^2}[/tex]

The squares (the exponent 2) cancels out with the square root:

2x + 3 = +/- 9

now lets isolate x by subtracting 3 from both sides:

2x + 3 = +/- 9

       -3  -3

2x = -3 +/- 9

2x = -3 + 9

2x = 6

2x = -3 - 9

2x = 12

And after simplifying, you can divide two on both sides:

2x = 6

/2    /2

x = 3

2x = -12

/2    /2

x = -6

x = 3 and x = -6

The notation of the transformation of the polygons is (x, y) = (x - 9, y + 2)

From the question, we have the following parameters that can be used in our computation:

The polygons 1 and 2

In the graph, we can see that

using the above as a guide, we have the following:

(x, y) = (x - 9, y + 2)

This means that the transformation of the polygons is (x, y) = (x - 9, y + 2)

Read more about transformation at

brainly.com/question/27224272

#SPJ1

Answer: 1000

Step-by-step explanation:

If one person can pick up a plate every 10 seconds, then the rate of picking up plates can be represented as 1 plate/10 seconds.

When two people are picking up plates, the rate doubles, so it becomes 2 plates/10 seconds or 1 plate/5 seconds.

To find how long it will take for 200 people to pick up plates, we need to use the formula:

time = amount ÷ rate

Plugging in the values, we get:

time = 200 plates ÷ (1 plate/5 seconds)

time = 200 plates × 5 seconds/plate

time = 1000 seconds

Therefore, it will take 1000 seconds for 200 people to pick up plates at a faster rate.

I hope that this answer has helped you!

State the null and alternative hypotheses.

c. H0: µ = 0, Ha: µ > 3

The null hypothesis (H0) states that the population mean rating is equal to 3, while the alternative hypothesis (Ha) suggests that the population mean rating is greater than 3.

Specify the rejection region for α = 0.01. Reject H0 if

a. z > 2.33

The rejection region for a one-tailed test with a significance level of 0.01 (α) is in the upper tail of the distribution. In this case, we reject the null hypothesis if the test statistic (z-score) is greater than 2.33.

Calculate the test statistic

c. 0.44

To calculate the test statistic, we use the formula:

z = (sample mean - population mean) / (sample standard deviation / √n)

Plugging in the given values, we get:

z = (3.31 - 3) / (0.70 / √172) ≈ 0.44

What is your conclusion?

b. Fail to Reject H0

Since the calculated test statistic (0.44) does not exceed the critical value (2.33) in the rejection region, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the average rating is significantly greater than 3 in the population.

Obtain the lower bound of a 99% confidence interval for the mean rating.

d. 3.17

To obtain the lower bound of a 99% confidence interval, we subtract the margin of error from the sample mean. The margin of error can be calculated by multiplying the critical value (obtained from the z-table for a 99% confidence level) with the standard error (sample standard deviation divided by the square root of the sample size).

The lower bound is given by:

3.31 - (2.33 * (0.70 / √172)) ≈ 3.17

Obtain the upper bound of a 99% confidence interval for the mean rating.

c. 3.44

To obtain the upper bound of a 99% confidence interval, we add the margin of error to the sample mean. Using the same calculation as above, the upper bound is given by:

3.31 + (2.33 * (0.70 / √172)) ≈ 3.44

What assumption(s) do you need to make in order to answer the above questions?

b. The population distribution is assumed to be normal.

To perform hypothesis testing and construct confidence intervals, it is typically assumed that the population distribution is approximately normal. Additionally, assumptions such as random sampling and independence of observations are generally made.

For such more question on hypothesis

https://brainly.com/question/606806

#SPJ8

Answer: true

Step-by-step explanation:

Answer: y = (-28)x + 14

Step-by-step explanation:

The equation for the line tangent at point (a, b) can be found using the formula y = mx + c, where m is the slope of the line and c is the y-intercept. At point (6, 14), we need to find the value of m.

We start by finding the derivative of the original function g(x). Its derivative is given by:

dg(x)/dx=7/(x^2-3x+1)

Then, substitute x = 6 into the derivative expression to obtain:

dg(6)/dx = d/dx [7 * ln|x-3| ] evaluated at x = 6 = 7/3

Next, evaluate the original function g(x) at x = 6 to get g(6) = 7 * ln |6 - 3| / (6 - 3) = 7 * ln 3.

Since we know the coordinates of the point of tangency (6, 14), we can substitute them into the general form of the linear equation y = mx + c:

14 = 7 * 6 + c

14 = 42 + c

c = -28

The final equation of the line tangent at point (6, 14) is therefore:

y = (-28)x + 14

The scaled triangle will be larger than the initial size by a factor 2.

The scaled square will be smaller than the initial side by a factor 4

Dilation refers to a transformation that changes the size of a geometric figure without altering its shape.

Dilation involves scaling an object by a certain factor, that might result in  enlarging or reducing its dimensions uniformly in all directions.

Based on the given diagram, the new length and size of the object is calculated as follows;

For the triangle, (measure the length with ruler)

For the square; (measure the length with ruler)

Learn more about dilation here: https://brainly.com/question/20482938

#SPJ1

The equation of the line that passes through the point (-3, 7) and has a slope of -5/3 is y - 7 = (-5/3)(x + 3).

We are given the point (-3, 7) and the slope of the line as -5/3.The slope-intercept form of a line is y = mx + b where m is the slope and b is the y-intercept.

To obtain the equation of the line, we need to substitute the values of slope and point in the slope-intercept form and solve for b.(7) = (-5/3)(-3) + b 21/3 = b.

Now we have the value of b, and we can substitute the values of m and b in the slope-intercept form.y = (-5/3)x + 21/3 is the equation of the line in slope-intercept form.

To obtain the equation in the standard form Ax + By = C, we multiply each term by 3.3y = -5x + 7Add 5x to both sides5x + 3y = 7.

This is the equation of the line in standard form.

For more question on equation

https://brainly.com/question/17145398

#SPJ8

Answer: The two solutions are (2, 4) and (-1, 1).

Step-by-step explanation:

The given is a system of equations, y = x + 2, and, y = x^2

A system of equations comprises two or more equations and seeks common solutions to the equations.

To solve you can use substitution. And by doing so you can replace y of one equation with what the other equation equals:

y = x + 2

y = x^2

->   x + 2 = x^2

Now lets get all variables and constant to one side of the equation, set it equal to zero.

x + 2 = x^2

-x -2           -x -2

x^2 -x -2 = 0

Lets factor to find the solution

Factoring is used to simplify an algebraic expression by finding the greatest common factors that are shared by the terms in the expression.

We can factor, x^2 -x -2 = 0, into:

(x - 2)(x + 1) = 0

-------

SIDE NOTE:

If confuse on factoring please see attached image.

------

with (x - 2)(x + 1) = 0 we can find the solution by setting each component in the parentheses to 0:

x - 2 = 0

x + 1 = 0

Now you must solve for x.

x - 2 = 0

+2      +2

x = 2

x + 1 = 0

  -1      -1

x = -1

x = 2 and x = -1

However, we are not done yet, we need to find what y is, and by doing so we can plug in the x values we got to find the corresponding y value to create points (coordinates).

-

When x = 2

y = x + 2

y = (2) + 2

y = 4

(2, 4)

-

When x = -1

y = x + 2

y = -1 + 2

y = 1

(-1, 1)

-

The two solutions are (2, 4) and (-1, 1).

The valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Let's solve each inequality step by step:

1. Two less than a number (x) is no more than 5:

The inequality is given as x - 2 ≤ 5. Adding 2 to both sides, we get x ≤ 7. This means that any number less than or equal to 7 satisfies the inequality.

2. Seven subtracted from 4 times a number (x) is more than 13:

The inequality is given as 4x - 7 > 13. Adding 7 to both sides, we get 4x > 20. Dividing both sides by 4, we obtain x > 5. So any number greater than 5 satisfies the inequality.

3. The sum of two times a number (x) and -2 is at least 8:

The inequality is given as 2x - 2 ≥ 8. Adding 2 to both sides, we get 2x ≥ 10. Dividing both sides by 2, we have x ≥ 5. So any number greater than or equal to 5 satisfies the inequality.

4. Four added to 3 times a number (x) is less than 19:

The inequality is given as 3x + 4 < 19. Subtracting 4 from both sides, we get 3x < 15. Dividing both sides by 3, we obtain x < 5. So any number less than 5 satisfies the inequality.

Based on the above solutions, the valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. These numbers can be represented on a number line graph by marking the appropriate points and shading the corresponding regions.

For more such questions inequalities,click on

https://brainly.com/question/30238989

#SPJ8

The members of the given set in this problem are given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

The union and intersection sets of multiple sets are defined as follows:

The intersection of the sets Y and Z for this problem is given as follows:

Y ∩ Z = {0, 6, 23, 26}

(which are the elements that belong to both of the sets).

The union of the above set with the set X is given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

(elements that belong to at least one of the sets).

More can be learned about union and intersection at brainly.com/question/4699996

#SPJ1

Matching of the equations of the Ellipses to their equivalent equations in standard form are:

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² - 36y + 9y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

The general form and the standard form of the ellipse are:

- The general form is: Ax² + Bxy + Cy² + Dx + Ey + F = 0

- The standard form is: (x - h)²/a² + (y - k)²/b² = 1

1) x²/3² + (y - 2)²/2² = 1

Expanding gives:

x²/9 + (y - 2)²/4 = 1

Multiply through by 36 to get:

4x² + 9(y - 2)² = 36

Expand bracket to get:

4x² + 9y² - 36y + 36 = 36

Subtract 36 from both sides to get:

4x² - 36y + 9y² = 0

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² + 9y² - 36y = 0

2) (x + 7)²/7² + y²/6² = 1

(x + 7)²/49 + y²/36 = 1

Multiply through by 1764 to get:

36(x + 7)² + 49y² = 1764

Expand bracket to get:

36x² + 504x + 1764 + 49y² = 1764

Subtract 1764 from both sides to get:

36x² + 504x + 49y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

3) x²/6² + (y + 4)²/4² = 1

x²/36 + (y + 4)²/16 = 1

Multiply through by 576 to get:

16x² + 36(y + 4)² = 576

Expand bracket to get:

16x² + 36y² + 288y + 576 = 576

Subtract 576 from both sides to get:

16x² + 288y + 36y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

4)  (x - 3)²/3² + y²/5² = 1

(x - 3)²/9 + y²/25 = 1

Multiply through by 225 to get:

25(x - 3)² + 9y² = 225

Expand the bracket power to get:

25x² - 150x + 225 + 9y² = 225

Subtract 225 from both sides

25x² - 150x + 9y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

Read more about Equation of Ellipse at: https://brainly.com/question/16904744

#SPJ1

Match the equations of ellipses to their equivalent equations in standard form.

25x² - 150x + 9y² = 0

16x² + 288y + 36y² = 0

49x² + 686 + 36y² = 0

4x² - 36y + 9y² = 0

36x² + 504x + 49y² = 0

9x² - 54x + 25y² = 0

x²/3² + (y - 2)²/2² = 1  ⇒

(x + 7)²/7² + y²/6² = 1 ⇒

x²/6² + (y + 4)²/4² = 1 ⇒

(x - 3)²/3² + y²/5² = 1 ⇒  

Answer:

Step-by-step explanation:

According to the triangle proportionality theorem (or thales theorem), the lines are parallel if AD/BD = AE/CE

15/12= 1.25

10/8 = 1.25

Because the ratios are equal, the lines are parallel to each other.  

In 2025, both colleges will have an enrollment of 21,200 students.

To determine when the colleges will have the same enrollment, we need to set up an equation based on the projected enrollment increases and declines.

Let's assume the number of years from 2020 is represented by 'x'.

For College A, the projected enrollment can be represented by: 18,700 + 500x.

For College B, the projected enrollment can be represented by: 26,200 - 1000x.

To find when the colleges will have the same enrollment, we need to solve the equation:

18,700 + 500x = 26,200 - 1000x

Combining like terms, we get:

1500x = 7,500

Dividing both sides by 1500, we find:

x = 5

Therefore, the colleges will have the same enrollment in 5 years, or in the year 2025.

To determine the enrollment at that time, we substitute x = 5 into either equation. Let's use College A's equation:

Enrollment in College A in 2025:

18,700 + 500(5) = 18,700 + 2500 = 21,200 students

Enrollment in College B in 2025:

26,200 - 1000(5) = 26,200 - 5000 = 21,200 students

Both colleges will have an enrollment of 21,200 students in the year 2025 when they have the same enrollment.

Know more about equation here:

https://brainly.com/question/29174899

#SPJ8