the data below represents the number of hours a random sample of students watched netflix in a month.32 33 34 21 33 22 34 29 18 21 31 21 18 30a) create a dot plot to the right. label carefully including title.b) give the mode. c) give the median. d) give the mean and variable. e) give the range.

The frequency with the largest amplitude is `w ≈ 2.303` (rtol=0.01, atol=1e-08).

The question asks us to find the frequency w for which the particular solution to the differential equation [tex]dạy dy 2- + dt2 + 2y = eiwt dt[/tex]has the largest amplitude.

We can assume a positive frequency w > 0.

Let's find out how to solve this problem:

We need to find the particular solution in the form Aeiwt, and then minimize the modulus of the denominator of A over all frequencies w. It means the denominator of A will have a maximum amplitude if we minimize the modulus. The amplitude of the solution is given by the value of |A|.

Let us assume the particular solution to be `[tex]y = Aeiwt`.[/tex]

Substitute the above solution in the given differential equation.

Then, we get:[tex]`d^2(A e^(iwt))/dt^2 + 2(A e^(iwt)) = e^(iwt)`[/tex]Applying the differential operator on the above equation,

we get: [tex]`(iwt)^2 A e^(iwt) + 2A e^(iwt) = e^(iwt)`Therefore, `A = 1 / (1 - (w^2) + 2i)`.[/tex]

Thus, the amplitude of the particular solution is:

[tex]`|A| = 1 / sqrt((1 - w^2)^2 + 4w^2)`[/tex]

Now, we need to minimize the above expression to get the frequency w at which the amplitude of the particular solution is maximum. This can be done by minimizing the modulus of the denominator of A over all frequencies w.To minimize the above expression, we take the derivative of the expression with respect to w and equate it to zero, which gives us: [tex]`(8w^2) / ((w^2 - 1)^2 + 4w^2)^(3/2) - (2(w^2 - 1)) / ((w^2 - 1)^2 + 4w^2)^(3/2) = 0`[/tex]

Simplifying the above expression, we get: `

[tex]2w^2 = w^4 - 3w^2 + 1`[/tex]

Therefore, [tex]`w^4 - 5w^2 + 1 = 0`.[/tex]

Solving the above equation gives us:

`[tex]w^2 = (5 ± sqrt(21)) / 2`[/tex].Since we know that w > 0, we choose the positive value of w.

Thus, the value of w is: [tex]`w = sqrt((5 + sqrt(21)) / 2)` or `w ≈ 2.303[/tex]`.

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The cοmplete table with the οrdered pairs is:

(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))

(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))

A quadratic equatiοn is a secοnd-degree pοlynοmial equatiοn in οne variable οf the fοrm: [tex]ax^2 + bx + c = 0.[/tex]

Tο find the sοlutiοns tο the system οf cοnics, we can use substitutiοn tο eliminate οne variable and οbtain a quadratic equatiοn in the οther variable. Then, we can sοlve this quadratic equatiοn tο find the pοssible values οf the remaining variable.

Frοm the given system οf cοnics:

[tex]y^2/4 - x^2/2 = 1 ...(1)[/tex]

[tex]y^2 = 8(x + 1) ...(2)[/tex]

We can eliminate [tex]y^2[/tex] frοm equatiοn (1) by multiplying bοth sides by 4:

[tex]y^2 - 2x^2 = 4[/tex]

Substituting the value οf [tex]y^2[/tex]  frοm equatiοn (2), we get:

[tex]8(x + 1) - 2x^2 = 4[/tex]

Simplifying and rearranging, we get:

[tex]x^2 + 2x - 2 = 0[/tex]

We can sοlve this quadratic equatiοn using the quadratic fοrmula:

[tex]x = (-2 \± \sqrt{(2^2 - 4(1)(-2))}) / (2(1))[/tex]

x = (-2 ± √12) / 2

x = -1 ± √3

Substituting these values οf x in equatiοn (2), we can find the cοrrespοnding values οf y:

When x = -1 + √3, we get:

[tex]y^2[/tex] = 8((-1 + √3) + 1) = 8√3

y = ±2√(2√3)

Therefοre, the sοlutiοns are:

(-1 + √3, 2√(2√3)) and (-1 + √3, -2√(2√3))

When x = -1 - √3, we get:

[tex]y^2[/tex] = 8((-1 - √3) + 1) = -8√3

This equatiοn has nο real sοlutiοns fοr y.

Therefοre, the cοmplete table with the οrdered pairs is:

(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))

(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))

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Answer: 250,000 > .00000042

Step-by-step explanation:

2.5 times 10^5 means move the decimal to the right 5 times.

2.5

1. 25.

2. 250.

3. 2,500.

4. 25,000.

5. 250,000.

250000 __  4.2 times 10^-7

10^-7 would mean move the decimal to the left that many times.

4.2

1. .42

2. .042

3. .0042

4. .00042

5. .000042

6. .0000042

7. 00000042

250,000 > .00000042

The correct hypothesis statement in this case would be "The average round-trip airfare between Philadelphia and Dublin is less than $1,200".

The hypothesis statement for the test of the average round-trip airfare between Philadelphia and Dublin being less than $1,200 would be “The average round-trip airfare between Philadelphia and Dublin is less than $1,200”.What is hypothesis?A hypothesis is a suggested explanation for a phenomenon or a theory that is tested using various experiments.

It is the first step in research that aids in forming a research question and establishes a framework for the study.The hypothesis statement:In hypothesis testing, the hypothesis statement is a declarative statement or an assertion that specifies the existence or non-existence of a phenomenon. In hypothesis testing, there are two hypothesis statements: null hypothesis and alternative hypothesis.

The null hypothesis would be that the average round-trip airfare between Philadelphia and Dublin is equal to or greater than $1,200. The alternative hypothesis would be that the average round-trip airfare between Philadelphia and Dublin is less than $1,200.

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Answer: 62.5 would be the answer.

Step-by-step explanation: You have to multiply how many meters per second by the whole total meters.

50x1.25=62.5.

*********************************************************************************************

Hope this helps! ^^

Answer:

Step 1: Simplify the exponents on both sides of the equation. Recall that (a^b)^c = a^(b*c), so we can write:

(1/216)^(3v-1) = (1/6)^(2v-2)

= (6^-1)^(2v-2) (since 1/6 = 6^-1)

= 6^(-(2v-2)) (since (a^-b) = 1/(a^b))

= 6^(2-2v) (since -(2v-2) = -2v + 2)

Step 2: Rewrite 1/216 as a power of 6. We have:

1/216 = 6^(-3)

Substituting this into our equation gives:

6^(-3(3v-1)) = 6^(2-2v)

Step 3: Use the property that if a^x = a^y, then x = y. Since both sides of the equation have the same base (6), we can equate the exponents:

-3(3v-1) = 2-2v

Step 4: Solve for v. First, simplify the left-hand side of the equation:

-3(3v-1) = -9v + 3

Now we can rewrite the equation as:

-9v + 3 = 2 - 2v

Step 5: Solve for v. Add 9v to both sides and subtract 2 from both sides:

-7v = -1

Finally, divide both sides by -7 to obtain:

v = 1/7

Therefore, the solution to the equation is v = 1/7.

To verify that v = 1/7 is the solution to the equation:

(1/216)^(3v-1) = (1/6)^(2v-2)

We can substitute v = 1/7 into the equation and simplify both sides to see if they are equal:

Left-hand side:

(1/216)^(3v-1) = (1/216)^(3(1/7)-1) = (1/216)^(2/7)

Right-hand side:

(1/6)^(2v-2) = (1/6)^(2(1/7)-2) = (1/6)^(-10/7) = (6/1)^10/7

Simplifying:

(1/216)^(2/7) = (6/1)^(10/7)

Taking the seventh root of both sides:

1/6 = 1/6

Since both sides of the equation are equal, we have verified that v = 1/7 is the solution to the equation.

The exact answer on Edge is

"Use the product of powers property to simplify the numerator by removing the parentheses.  Follow the order of operations by removing the innermost parentheses first.  Cube the quantity to get the product of 2 to the third power, r to the 6th power, and t to the third power, or 2^3r^6t^3 in the numerator. "

Answer:

Step-by-step explanation:

To obtain the derivative of F(x) = 5x^10 - 2x^2 + x - 1 using the addition and subtraction rule, we can take the derivative of each term separately and then add or subtract the resulting derivatives.

The derivative of 5x^10 is:

(5x^10)' = 50x^9

The derivative of -2x^2 is:

(-2x^2)' = -4x

The derivative of x is:

(x)' = 1

The derivative of -1 is:

(-1)' = 0

Putting these derivatives together using the addition and subtraction rule, we get:

F'(x) = (50x^9) - (4x) + (1) + (0)

Simplifying this expression, we get:

F'(x) = 50x^9 - 4x + 1

Therefore, the derivative of F(x) = 5x^10 - 2x^2 + x - 1 using the addition and subtraction rule is F'(x) = 50x^9 - 4x + 1.

Answer:

x=52

Step-by-step explanation:

Set equal to 180

x-30+3x+2=180

solve for x

4x-28=180

4x=208

x=52

plug in to check

The total number of words with 5 or les sletters is  1,364

We want to see how many words of 5 or less letters can be done with 4 different characters.

There are 4 words with one letter.

Now words with two letters, suppose that the word is represented by:

_,_

Each blank represents a letter, so we have 4 options per blank, then the number of words with two letters is 4*4 = 16

And with 3 is 4*4*4 = 64

And with 4 is 4*4*4*4 = 256

And finally,with 5 is: 4*4*4*4*4 = 1,024

The total sum gives:

4 + 16 + 64 + 256 + 1,024 = 1,364

These is the number of words.

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The distance between these two streets on Broad St is 0.3 miles.

We can use proportions to solve the problem.

Let's assume x is the distance between Thomas St and Denny Way on Broad St.

We know that on Aurora Ave, the distance between Thomas St and Denny Way is 0.2 miles, and on Broad St, the distance between Mercer St and Denny Way is 0.7 miles.

So, we can set up the following proportion:

0.2/0.45 = x/0.7

Simplifying, we get:

x = (0.2/0.45) * 0.7

x = 0.3111 miles

Rounded to the nearest tenth, the distance between Thomas St and Denny Way on Broad St is 0.3 miles.

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The value of x is 24

The measure of is angle 1 is 101°

The measure of angle 2 is 79°

The measure of angle 3 is 101°

The measure of angle 4 is 101°

The measure of angle 5 is 79°

The measure of angle 6 is 79°

The measure of angle 7 is 101°

The measure of angle 8 is 79°

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

3x +19 = 5x-29 ( opposite angle)

collect like terms

5x-3x = 29+19

2x = 48

x = 24

angle 1 = 180-(3×24+7)

= 180- 79

= 101°

angle 2 = 3x+7 = 79°( opposite angles)

angle 3 = 101° ( opposite angles)

angle 4 = angle 1 = 101° ( corresponding angles)

angle 5 = 79° ( corresponding angles)

angle 6 = angle 5 = 79°( opposite angle)

angle 7 = angle 4 = 101 ( corresponding angles)

angle 8 = angle 7 = 79 ( opposite angle )

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The annual interest rate on the loan is 4.5% using simple interest formula and concepts.

To calculate the annual interest rate on the loan, we need to use the simple interest formula:

Simple Interest = (P × r × t) / 100

P stands for principal, r stands for annual interest rate, and t stands for time in years.

Given principal amount is $5,600, the simple interest earned after 4 years is $1,008. Therefore, we can plug in values into formula and solve for r:

$1,008 = (5,600 × r × 4) / 100

Multiply LHS and RHS by 100 and divide by 5,600 × 4, we will obtain:

r = $1,008 / $22400 = 0.045 or 4.5%

Hence, the annual interest rate on the loan is 4.5%.

It is significant to remember that simple interest does not compound and has a linear relationship to time. This indicates that the interest charged on the loan is the same each year and does not compound. Short-term loans and personal loans frequently employ simple interest. However, compound interest, which takes into account the cumulative interest over time, is utilised for long-term loans such as mortgages. To make wise financial decisions, it is crucial to comprehend the distinction between simple and compound interest.

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Answer:

x^2-11x+28=0

Step-by-step explanation:

(x-7)(x-4)=0

= x^2-7x-4x+28=0

= x^2-11x+28=0

The expression is equivalent to (x - 2)(x - 5)/(x - 4)(x - 1), as long as the denominators do not equal 0.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (such as addition, subtraction, multiplication, and division), as well as grouping symbols like parentheses.

We can begin by factoring the denominators of both fractions as follows:

[tex]x^{2}[/tex]- x - 12 = (x - 4)(x + 3)

[tex]x^{2}[/tex]- + 6x - 5 = (x - 1)(x + 5)

Substituting these expressions, the given expression becomes:

[(x - 1)(x-1)  [tex]x^{2}[/tex]-+ (-6)]/[(x - 4)(x + 3)] * [(x + 5)/(x - 1)(x + 5x - 5)]

Simplifying, we can cancel out the (x - 1) and (x + 5) terms in the numerator and denominator:

[(x - 1) *  [tex]x^{2}[/tex]- + (-6)]/[(x - 4)(x + 3)] * [1/(x - 5)]

Expanding the numerator:

( [tex]x^{3}[/tex]-  [tex]x^{2}[/tex]- - 6)/( [tex]x^{2}[/tex]- - x - 12) * [1/(x - 5)]

Factoring the numerator:

( [tex]x^{2}[/tex]- + x - 6)(x - 5)/( [tex]x^{2}[/tex]- - x - 12)

Factoring again:

(x + 3)(x - 2)(x - 5)/(x - 4)(x + 3)(x - 1)

Canceling out the (x + 3) terms:

(x - 2)(x - 5)/(x - 4)(x - 1)

Therefore, the expression is equivalent to (x - 2)(x - 5)/(x - 4)(x - 1), as long as the denominators do not equal 0.

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The height of the parallelogram is 3 cm.

What is area?

The region that an object's shape defines as its area. The area of a figure or any other two-dimensional geometric shape in a plane is how much space it occupies.

A parallelogram is a quadrilateral with two pairs of parallel sides. This means that opposite sides of a parallelogram are parallel and have the same length. Additionally, opposite angles of a parallelogram are congruent (i.e., have the same measure). Some common properties of parallelograms include:

The opposite sides of a parallelogram are equal in length.

The opposite angles of a parallelogram are equal in measure.

The consecutive angles of a parallelogram are supplementary (i.e., they add up to 180 degrees).

The diagonals of a parallelogram bisect each other (i.e., they intersect at their midpoint).

Here the given ,

Area of the parallelogram = 18 square centimeters.

Base b = 6 cm

Area of parallelogram = bh square unit

=> 18=6*h

=> h= 18/6 = 3 cm

Hence the height of the parallelogram is 3 cm.

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Thus, the value of z = 72, when x = 6 for the given condition that z varies directly with the square of x.

The connection among two variables is known as a direct proportion when their ratios are comparable to a fixed value.

Given data:

z varies directly as x².

z= 8 when x= 2​.

So,

z ∝ x²

z = k x²

k = z/ x² (k is the constant of proportionality)

z= 8 when x= 2​.

k = 8/(2)²

k = 2

Now, when x = 6

z = k x²

z = 2 * (6)²

z = 72

Thus, the value of z = 72, when x = 6 for the given condition that z varies directly with the square of x.

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NR = QR: This statement is true. In a rhombus, the diagonals bisect each other, so NR is equal in length to QR.

What is Rhombus ?

A rhombus is a quadrilateral with four equal-length sides. The opposite angles of a rhombus are equal to each other, and the diagonals bisect each other at right angles. This means that the diagonals of a rhombus are perpendicular to each other and divide the rhombus into four congruent right triangles.

From the given information, we know that MNPQ is a rhombus, and that NQ and PM are diagonals that intersect at point R.

NQ is perpendicular to MP: This statement is false. In a rhombus, the diagonals are perpendicular bisectors of each other, so we know that NQ is perpendicular to PM at point R, but not necessarily to the entire diagonal MP.

NP is perpendicular to QP: This statement is also false. In a rhombus, opposite sides are parallel and congruent, but not necessarily perpendicular to each other.

NR = QR: This statement is true. In a rhombus, the diagonals bisect each other, so NR is equal in length to QR.

Angle MNQ is congruent to angle PNQ: This statement is false. While we know that NQ is perpendicular to PM at point R, we cannot conclude that angle MNQ is congruent to angle PNQ. In fact, there is not enough information given to determine the measure of either angle.

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Answer:

The value of the book after 11 years is $103.11

The equation should be 258*0.92, 11 times repeating.

The answer should be 103.106443494, but if you round it the answer is 103.11

[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &258\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ t=years\dotfill &11\\ \end{cases} \\\\\\ A = 258(1 - 0.08)^{11} \implies A=258(0.92)^{11}\implies A \approx 103.11[/tex]

The null hypothesis is that the proportion of computer chips that do not fail in the first 1000 hours of use is equal to 50%. The alternative hypothesis is that the proportion is greater than 50%.

At the 0.10 level of significance, we will reject the null hypothesis if the test statistic is greater than 1.28. The test statistic can be calculated using the formula:

[tex]z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$[/tex]

where [tex]\hat{p}$[/tex] is the sample proportion, [tex]p_0$[/tex] is the hypothesized proportion under the null hypothesis, and $n$ is the sample size.

In this case, we have [tex]\hat{p} = 0.53$, $p_0 = 0.50$, and $n = 1200$.[/tex] Substituting these values into the formula, we get:

[tex]z = \frac{0.53 - 0.50}{\sqrt{\frac{0.50(1-0.50)}{1200}}} = 2.77$[/tex]

Since the test statistic is greater than 1.28, we reject the null hypothesis and conclude that there is sufficient evidence at the 0.10 level to support the company's claim that more than 50% of computer chips do not fail in the first 1000 hours of their use.

Therefore, the null hypothesis is equal to 50%, and the alternative hypothesis is 50%. The test statistic is 2.77, which is greater than the critical value of 1.28 at the 0.10 level of significance, so we reject the null hypothesis and conclude that there is sufficient evidence to support the company's claim.

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Answer:

5

Step-by-step explanation:

You have to divide 35 by 7.

35/7 is 5.

So, he puts 5 stickers on each paper.

Answer:

Step-by-step explanation:

Story Problem:

Samantha is baking cookies and her recipe calls for 3 cups of flour. She only has a bag of flour that is 7/8 full. If each cup of flour weighs the same, how much flour does Samantha have left after taking out the 3 cups needed for the recipe?

Solution:

To solve the problem, we need to multiply the amount of flour in the bag by 3/1 (which is the same as multiplying by 3).

3 x 7/8 = (3 x 7)/8 = 21/8

So, Samantha has 21/8 cups of flour in the bag.

To find out how much flour she has left after taking out the 3 cups needed for the recipe, we need to subtract 3 from 21/8:

21/8 - 3 = 21/8 - 24/8 = -3/8

Samantha has -3/8 cups of flour left in the bag, which means she doesn't have enough flour to make the recipe. She needs to get more flour before she can continue baking.

Answer:

Look below

Step-by-step explanation:

Ken drinks 7/8 of a carton of milk each day. How much milk does

he drink in 3 days?

7/8*3

21/8

2 5/8 cartons of milk a day

The equation that represents f(x) is f(x) = [tex]3x^2[/tex] + 3x + 4. To determine the equation that represents the function f(x) based on the given points, we can use the general form of a quadratic function:

f(x) =[tex]ax^2[/tex]+ bx + c

where a, b, and c are constants that we need to determine. We can use the given points to form a system of equations:

[tex]a(0)^2[/tex] + b(0) + c = 4 -- Equation 1

[tex]a(1)^2[/tex] + b(1) + c = 10 -- Equation 2

[tex]a(2)^2[/tex] + b(2) + c = 25 -- Equation 3

Simplifying each equation, we get:

c = 4 -- Equation 1a

a + b + c = 10 -- Equation 2a

4a + 2b + c = 25 -- Equation 3a

Substituting Equation 1a into Equation 2a and Equation 3a, we get:

a + b = 6 -- Equation 2b

4a + 2b = 21 -- Equation 3b

Solving for b in terms of a in Equation 2b, we get:

b = 6 - a

Substituting b = 6 - a into Equation 3b, we get:

4a + 2(6 - a) = 21

Simplifying and solving for a, we get:

a = 3

Substituting a = 3 into b = 6 - a, we get:

b = 3

Substituting a = 3 and b = 3 into Equation 1a, we get:

c = 4

Therefore, the equation that represents f(x) is:

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

This quadratic function passes through the given points {(0,4), (1,10), (2,25)}, as can be verified by plugging in the x-values of each point into the equation and checking that the y-values match.

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The probability that the mean length of the 25 items is less than 12.4 inches is approximately 0.0009

We can solve this problem by using the Central Limit Theorem (CLT), which states that the sample mean of a large enough sample size from any distribution with a finite mean and variance will follow a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

In this case, the population mean is 14.7 inches, the population standard deviation is 4.8 inches, and the sample size is 25. Thus, the mean of the sample means is also 14.7 inches, and the standard deviation of the sample means is 4.8 inches / sqrt(25) = 0.96 inches.

To find the probability that the mean length of the 25 items is less than 12.4 inches, we can standardize the sample mean using the z-score formula

z = (x - mu) / (sigma / sqrt(n))

where x is the sample mean we want to find the probability of (in this case, 12.4 inches), mu is the population mean (14.7 inches), sigma is the population standard deviation (4.8 inches), and n is the sample size (25).

Substituting these values, we get

z = (12.4 - 14.7) / (4.8 / sqrt(25)) = -3.13

We can then use a standard normal distribution table or calculator to find the probability that a standard normal variable is less than -3.13, which is approximately 0.0009.

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a. The critical value(s) is/are z = -1.645, 1.64

b. Fail to reject H0.

There is not sufficient evidence to warrant rejection of the claim that p = 1/2. The test statistic of z = -1.91 is obtained when testing the claim that p = 1/2.

We want to know whether to reject H0 or fail to reject H0 using a significance level of α = 0.10.

Using the standard normal distribution table, the critical value(s) are z = ± 1.645 (rounded to two decimal places).

Since -1.91 is less than -1.645, we fail to reject H0.

In other words, there is not sufficient evidence to warrant rejection of the claim that p = 1/2.

Therefore, the correct conclusion is "Fail to reject H0.

There is not sufficient evidence to warrant rejection of the claim that p = 1/2."

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The cost of one adult tickets would be $12.5 and the cost of one youth tickets would be $8.75 .Therefore the cost of six adult tickets and eight youth tickets would be 6(12.5) + 8(8.75) = $145.

Let the cost of adult tickets be x.

Let the cost of youth tickets be y.

So, from the given condition given in question the equations are :-

For The Jefferson family,

2x + 2y = 42.50

x + y = 21.25 —---------- equation(1)

and for The Mendoza family,

4x + 3y = 76.25 —------ equation(2)

To solve above equation 4 × equation(1) - equation(2).

y = 8.75

and x = 12.5

Hence the cost of one adult tickets would be $12.5 and the cost of one youth tickets would be $8.75 .

Therefore the cost of six adult tickets and eight youth tickets would be 6(12.5) + 8(8.75) = $145.

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—-------- Correct question format is given below —--------

(Q). Two families go to a movie on Saturday night. The Jefferson family purchases 2 adult tickets and 2 youth tickets for $42.50. The Mendoza family purchases 4 adult tickets and 3 youth tickets for $76.25. How much would it cost to purchase six adult tickets and eight youth tickets?

In this case, the bounce heights form a geometric sequence with a common ratio of 2/3. , the ball reaches a height of  [tex]8[/tex] feet on the third bounce,

The definition of a geometric sequence is a set of numbers where each term is created by multiplying the preceding term by a fixed factor.

If the bounce heights form a geometric sequence with common ratio "r", we can write:

27 = initial height

[tex]18 = 27r[/tex]

[tex]12 = 18r = 27 \times r^2[/tex]

Solving for "r", we get:

[tex]r = 2/3[/tex]

So, the fourth bounce height would be:

[tex](12) \times (2/3) = 8[/tex]

Therefore, the ball reaches a height of 8 feet on the third bounce, not the fourth bounce as stated in the original statement.

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After answering the provided question, we can conclude that triangle  2(AD² + DC²) = 2(58 + 26) = 168

A triangle is a locked, double-symmetrical target made up of 3 line segments known as sides that interlock at three points known as vertices. Triangles are distinguished by their sides and angles. Triangles can be rectangular prism (all factions equal), ellipse, or scalene based on their sides. Triangles are classified as acute (all angles are under 90 degrees), okay (one angle is equal to 90 degrees), or orbicular (all angles are greater than 90 ° c) (all angles greater than 90 degrees). The continent of a triangle can be calculated using the formula A = (1/2)bh, where an is the suburb, b is the triangle's base, and h is the triangle's height.

(a) The distance formula is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Length of AB:

AB = √[(2 - (-6))² + (5 - 3)²] = √64 = 8

Length of AC:

AC = √[(0 - (-6))² + (-5 - 3)²] = √100 = 10

Length of BC:

BC = √[(2 - 0)² + (5 - (-5))²] = √144 = 12

Midpoint of BC:

D = [(0 + 2)/2, (-5 + 5)/2] = (1, 0)

Length of AD:

AD = √[(1 - (-6))² + (0 - 3)²] = √58

Length of DC:

DC = √[(1 - 0)² + (0 - (-5))²] = √26

Now we can find the values of AC², AB², AD², and DC²:

AC² = AC² = 100

AB² = AB² = 64

AD² = AD² = 58

DC² = DC² = 26

(b) AC² + AB² = 2(AD² + DC²).

AC² + AB² = 100 + 64 = 164

2(AD² + DC²) = 2(58 + 26) = 168

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