Simplify. (y^3)^5 • y^4
1. y^12
2. y^19
3. y^32
4. y^60

Answer:

Step-by-step explanation:

1/2*base*height

Anna's monthly Salary in June was $3600 as she received a 20% rate and her monthly Salary in July was $2880 as she received a 20% pay cut from June.

It is given to us that: Ana's salary in May =3000$

in June she received 20% raise

the amount she received in June after 20% raise = 3000 x 20/100

= 30 x 20 = 600

we have to first find Ana's salary at the end of June month with 20% raise. and then her salary after the end of July month with the pay cut of 20%.

Ana's salary after June month when she got 20% raise in his salary after May:

= 3000 + (20% x 3000)

= 3000 + (20/100 x 3000)

= $3600

her new salary after June will be 3600$.

therefore, she received 3600 in June.

in July she received 20% pay cut.

To find Ana's monthly salary after the two changes in June and July.

Ana's salary after June month when she got 20% raise in his salary after May = $3600

now she got a pay cut of 20% on his new salary from July month,

= 3600 - (20% x 3600)

= 3600 + (20/100 x 3600)

= 3600 - 720

= $2880

so, her salary after that would be equal to $2880.

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Answer:

Step-by-step explanation:

I have two questions; One, are you the student? Two, is this a trick question because if it is it's freaking hilarious

Answer: 55 degrees

Step-by-step explanation: 180-70-55=u

This answer is because the sum of all the angles is 180 degrees.

Answer:

u = 55°

Step-by-step explanation:

The sum of the internal angles of a triangle = 180°

Then:

70 + 55 + u = 180

125 + u = 180

u = 180 - 125

u = 55°

Answer:

False

Step-by-step explanation:

14.6/2 = 7.3

7.3 x 3 = 21.9

21.9 - 1.9 = 20

Other side:

8 - 4 = 4

4 x 3 = 12

20 does not equal 12

therefore false :)

Answer: A. Action: add 3 to both sides

Step-by-step explanation:

 The property that justifies this action is the addition property of equality, which states that if you add the same number to both sides of an equation, the equation remains true.

After tripling the sample size will reduce the standard error of the sample proportion by one-third we have:

The sample size in statistics is a measurement of the total number of samples utilised in an experiment. The sample size is 50, for instance, if we are evaluating 50 samples of city dwellers who watch TV. It is sometimes referred to as sample statistics.

One of the mathematical techniques used in statistics to calculate variability is the standard error. It is referred to as SE. The standard deviation of a sample distribution serves as the standard error of a statistic or estimate of a parameter. It may be characterised as a projection of that standard deviation.

a. true

population = 25% = 0.25

sample = n= 12

n x p

= 12 x o. 25 = 3 and 3 is less than 10

12(1 - p)

= 12 x 0.75

= 9 and is less than 10

b. True

the sample distribution of the population is normal when

sample size x population > or equal to 10

40 x 0.75

= 30 and 30 is greater than 10

c. false

50 x 0.25 = 12.5

50 x 0.20 = 10

[tex]z = 10 - \frac{12.5}{\sqrt{12.5} }[/tex]

= -2.5/3.54

= -0.70

H0: Young American family who delayed

H1: young American family who did not delay

p(z = -0.70)

0.2420>0.005

therefore we accept the null hypothesis.

d. false

150 x 0.20 = 30

150 x 0.75 = 37.5

[tex]z = 30 - \frac{37.5}{\sqrt{37.5} }[/tex]

= -7.5/6.12

= -1.22

p(z = -1.22) = 0.1112 > 0.05

therefore we do not reject the null hypothesis.

e. false

[tex]se1 = \sqrt{(p(1-p)/n}\\se2 = \sqrt{(p(1-p)/3n}\\se2 = 1/\sqrt{(3)se2}[/tex]

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Complete question:

About 25% of young Americans have delayed starting a family due to the continued economic slump. Determine if the following statements are true or false, and explain your reasoning.a. The distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump in random samples of size 12 is right skewed.b. In order for the distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump to be approximatly normal, we need random samples where the sample size is at least 40.c. A random sample of 50 young Americans where 20% have delayed starting a family due to the continued economic slump would be considered unusual.d. A random sample of 150 young Americans where 20% have delayed starting a family due to the continued economic slump would be considered unusual.e. Tripling the sample size will reduce the standard error of the sample proportion by one-third.

The measure of angle subtended by the arc JR is determined as 74 ⁰.

The measure of arc JR is calculated by applying the following principle of circle geometry for a cyclic quadrilateral as shown below.

The measure of the angle tangent to the circle opposite the arc JR is given as 37 ⁰.

The measure of angle subtended by the arc JR is calculated as

m∠ JR = 2 x 37 ⁰ ( angle at the tangent is half of the angle subtended by the arc.

m∠ JR =74 ⁰

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[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3} ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\stackrel{9\times 7}{63}\\ h=6 \end{cases}\implies V=\cfrac{(63)(6)}{3}\implies V=126~cm^3[/tex]

now, for the slanted cone, we can use Cavalieri's Principle for an slanted cylinder, thus we can say that the slanted cone will have the same volume of a non-slanted cone, now, since this one has a diameter of 6, that means is has a radius of 3.

[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3} \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=3\\ h=7 \end{cases}\implies V=\cfrac{\pi (3)^2(7)}{3}\implies V=21\pi \implies V\approx 66~cm^3[/tex]

Answer:

64

Step-by-step explanation:

You need a sample size of approximately 82 students to estimate the average time college students spend studying per week with a 99% confidence interval and a margin of error of 30 minutes (0.50 hours).

To determine the sample size, we will use the formula given below;

Where, n = sample size

Zα/2 = the z-score associated with the level of confidence

α = level of significance

σ = population standard deviation

d = margin of error of the estimation.

Substituting the given values, we have; n = [(Zα/2 * σ)/d]^2

We are given the population standard deviation, σ = 3.5 hours.

We are also given the margin of error, d = 0.5 hours.

We need to find the value of Zα/2 at 99% confidence level.

Since the sample size is large enough, we can use the z-distribution to find the z-value.

The area under the standard normal distribution curve for a 99% confidence level is as shown below;

Using the z-tables, the value of Zα/2 for a 99% confidence level is 2.576.

Substituting the values into the formula; n = [(Zα/2 * σ)/d]^2n = [(2.576 * 3.5)/0.5]^2= (9.006)^2= 81.12

Since we need a whole number for the sample size, we round up the value to get 82.

However, since the standard deviation of the population is known, we can use the z-distribution to construct a confidence interval for the population mean. This will give us an accurate interval estimate of the population mean.

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AE and CD intersect at point B then angle ABD = 152°

A straight angle in mathematics is one with a degree value of 180. Because it resembles a straight line, it is called straight. Essentially, two rays linked end to end create an angle according to the definition of angles in mathematics. As a result, where the rays are linked at endpoints, the two 180-degree angle rays are subtended in the opposite direction.

AE and CD intersect at point B,

angle DBE = 48,

and AE is a straight line,

So the sum of angles on a straight line is 180,

We have to find the angle ABD,

∠DBE+∠ABD=180

∠ABD=180-48

∠ABD=152°

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Answer:Option B is correct.

Step-by-step explanation: Given:  

x+2

1

​

+

x+3

1

​

1        =

(x+3)(x+2)

x+3+x+2

1​      =

2

+5x+6

2x+5

​

1

​

       =

2x+5

x

2

+5x+6

​

The expression 1/(1/x+2) + (1/x+3) is equivalent to (x² + 4x + 44)/x(x + 11) when x > 3.

We have,

To simplify the expression 1/(1/x+2) + (1/x+3), we can apply the concept of finding a common denominator and combining fractions.

Let's start by finding the common denominator for the two fractions.

The denominators are (1/x + 2) and (1/x + 3).

To find the common denominator, we multiply the denominators together:

Common denominator = (1/x + 2) x (1/x + 3)

Expanding the expression:

Common denominator = (1/x * 1/x) + (1/x * 3) + (2 * 1/x) + (2 * 3)

Simplifying further:

Common denominator = 1/x^2 + 3/x + 2/x + 6

Combining the fractions with the common denominator:

1/(1/x+2) + (1/x+3) = (1/x) / (1/x^2 + 3/x + 2/x + 6) + (1/x+3)

To simplify the expression further, we can multiply the first fraction by x/x to clear the fraction in the numerator:

(1/x) / (1/x² + 3/x + 2/x + 6) + (1/x+3) = (x/x) * (1/x) / (1/x² + 3/x + 2/x + 6) + (1/x+3)

Simplifying:

= 1 / (x/x * 1/x) / (1/x² + 3/x + 2/x + 6) + (1/x+3)

= 1 / (1/x²) / (1/x² + 3/x + 2/x + 6) + (1/x+3)

= 1 / (1/x²) / (1/x² + (3 + 2)/x + 6) + (1/x+3)

= 1 / (1/x²) / (1/x² + 5/x + 6) + (1/x+3)

= 1 / (1/x²) / (1/x² + 5/x + 6) + (x/x) * (1/x+3)

Simplifying further:

= 1 / 1 / (1 + 5/x + 6x/x²) + (x/x² + 3x/x)

= 1 / (1 + 5/x + 6x/x²) + (x + 3x)/x²

= 1 / (1 + 5/x + 6/x) + (4x)/x²

= 1 / (1 + (5 + 6)/x) + (4x)/x²

= 1 / (1 + 11/x) + (4x)/x²

= 1 / (x + 11)/x + (4x)/x²

= x / (x + 11) + (4x)/x²

= (x² + 4(x + 11))/x(x + 11)

= (x² + 4x + 44)/x(x + 11)

Therefore,

The expression 1/(1/x+2) + (1/x+3) is equivalent to (x² + 4x + 44)/x(x + 11) when x > 3.

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Answer:

96

Step-by-step explanation:

Because you're increasing the number, you'll add the percentage to 100%. After that, use ratio to solve.

Therefore,

100% : 64

100% + 50%(150%) : ?

= 150/100 × 64

= 15/10 × 64

= 3/2 × 64

= 3/1 × 32

= 3 × 32

= 96

The new median of the data set is 616

Multiply all the numbers by 22

looking at this number they both are in the middle. So you add both numbers and divide by two.

440 + 792 = 1232

The rectangular bird sanctuary has dimensions of  12 km by 6 km in size to reduce the amount of fencing required to enclose the other three sides.

   The given rectangular bird sanctuary has a side along a straight river. To minimize the amount of fence necessary to enclose the remaining three sides, we need to find the dimensions of the rectangle.

  Therefore, let us consider that the width of the rectangular bird sanctuary is x and the length is y.

  The area of the given bird sanctuary is given as,

                A = xy= 72 km²

   The amount of fence required for the bird sanctuary can be written as,

                 F = 2x + since we are looking to minimize the amount of fence necessary to enclose the remaining three sides, we need to differentiate the expression for a fence with respect to x and equate it to zero to obtain the value of x that will minimize the fence.

              dF/dx = 2 - 0 (since dy/dx = 0)

     Therefore, dF/dx = 2 Let us consider this to be equation (1) Differentiating the expression for a fence with respect to y and equating it to zero to obtain the value of y that will minimize the fence.

             dF/dy = 0 + 1 (since dx/dy = 0)

      Therefore, dF/dy = 1

Let us consider this to be equation (2)

      Now, we can solve equations (1) and (2) to obtain the values of x and y that will minimize the fence.

            dF/dx = 2 ⇒ 2x + y

                       = 0dF/dy

                       = 1 ⇒ 2y + x

                      = 0

       Solving the above two equations for x and y, we get x = 12 km and y = 6 km

       Therefore, the dimensions of the rectangular bird sanctuary are 12 km and 6 km to minimize the amount of fence necessary to enclose the remaining three sides.

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We can begin by defining Penny's total wealth as a mix of a set sum and a portion of Susie's wealth to develop an expression of how much money she has. We are informed that Penny has at least $4.50 and an additional 2/5 of Susie's wealth. Hence, we can write:

p = $4.50 + (2/5)s

where p is Penny's wealth, s is Susie's wealth, and s is represented by the letters.

If Susie has $11.50, we may enter s = $11.50 into the formula for p to see how much Penny has:

p = $4.50 + (2/5)($11.50)

p = $4.50 + $4.60

p = $9.10

As a result, Penny has $9.10 and Susie has $11.50.

In conclusion, the equation for Penny's wealth in terms of Susie's wealth is p = $4.50 + (2/5)s. When we enter s = $11.50 into the formula, we obtain p = $9.10, which is how much money Penny has while Susie has $11.50.

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The length of the hypotenuse is the line connecting the two points (-3,-7) and (-7,-1), which has length 2√13, as calculated above.

A triangle is a clοsed, dοuble-symmetrical shape cοmpοsed οf three line segments knοwn as sides that intersect at three places knοwn as vertices. Triangles are distinguished by their sides and angles. Triangles can be equilateral (all factiοns equal), isοsceles, οr scalene based οn their sides.

Next, we can find the length of the hypotenuse by using the distance formula, which is:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

So, in this case, we have:

d = √((-7 - (-3))² + (-1 - (-7))²)

= √((-4)² + (-6)²)

= √(16 + 36)

= √52

= 2√13

Therefore, the distance between the two points in simplest radical form is 2√13.

The length of the hypotenuse is the line connecting the two points (-3,-7) and (-7,-1), which has length 2√13, as calculated above.

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The point (2, 7) lies on the line defined by the equation Y=3/2x+4

The given equation is [tex]Y = 3/2x + 4[/tex]

To find the point that lines on the line defined by this equation, we need to solve for x. Start by subtracting 4 from both sides of the equation:

[tex]Y - 4 = 3/2x[/tex]

Next, multiplying both sides by 2:

[tex]2Y - 8 = 3x[/tex]

Finally, divide both sides by 3:

[tex]x = (2Y - 8) /3[/tex]

Now, given a  value, we can calculate the corresponding x value. For example, if Y = 7, then:

[tex]x = (2 * 7 - 8) / 3x = (14 - 8) / 3x = 6 / 3x = 2[/tex]

Therefore, the point (2, 7) lies on the line defined by this equation.

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Complete question :

What is the value of x for the point (x,7) that lies on the line defined by the equation Y = 3/2x + 4?

A. 1

B. 2

C. 3

D. 4

Answer:

A=1 B=2 C=3   Domain: [2, infinity) Range: (neg. infinity, 3]

Step-by-step explanation:

a is rate B is Horitzontal shift C is vertical shift

The linear function that represents the line is f(x) = -3x + 14

We can rewrite the point-slope equation y + 1 = -3(x - 5) in slope-intercept form, which has the general form of y = mx + b, where m is the slope and b is the y-intercept.

To do this, we first distribute the -3 on the right side of the equation:

y + 1 = -3x + 15

Then, we can isolate y by subtracting 1 from both sides:

y = -3x + 14

So the slope of the line is -3 and the y-intercept is 14. Therefore, the linear function that represents the line is:

f(x) = -3x + 14

Hence, the correct answer is f(x) = -3x + 14.

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Answer:

3x + 12

Step-by-step explanation:

3(x + 4) ​

3x + 12

So, the expand is 3x + 12

Answer:

3x + 12

Step-by-step explanation:

You multiply the outside of the perentheses by everything inside, and 3(x) + 3(4) = 3x + 12

7.68 pounds of honey remains after the bears ate 0.32 pounds.

What is a initial amount mean?

When referring to any Capital Appreciation Bond, Original Amount refers to the Bond's Accrued Value as of the date of issuance.

To find how much honey remains, we need to subtract the amount eaten by the bears from the initial amount produced:

Remaining honey = Initial honey - Honey eaten by bears

Remaining honey = 8 pounds - 0.32 pounds

Remaining honey = 7.68 pounds

Therefore, 7.68 pounds of honey remains after the bears ate 0.32 pounds.

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The points or coordinates that will fall inside the parallelogram are 0,4 and 4,7.

Based on the above, the coordinates that would fit inside the parallelogram are:

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Answer: The nth term of the sequence is given by the formula 2n + 11

Step-by-step explanation:

The given sequence is an arithmetic sequence because the difference between each pair of consecutive terms is the same. In this case, the common difference is 2, because:

15 - 13 = 2

17 - 15 = 2

and so on

The formula for the nth term of an arithmetic sequence is:

an = a1 + (n-1)d

where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

For this sequence, the first term (a1) is 13 and the common difference (d) is 2. Therefore, the formula for the nth term is:

an = 13 + (n-1)2

Simplifying this expression, we get:

an = 2n + 11

The measure of ∠R in ΔPQR is option (a) 26⁰

Since angle P is a right angle (90 degrees), triangle PQR is a right triangle.

Using the Pythagorean theorem, we can find the length of the remaining side PQ

PQ^2 = PR^2 - QR^2

PQ^2 = 12.7^2 - 14.1^2

PQ^2 = 161.29 - 198.81

PQ^2 = 37.52

PQ ≈ 6.12

Now we can use the sine function to find the measure of angle R

sin(R) = opposite/hypotenuse = PQ/PR

sin(R) = 6.12/12.7

sin(R) ≈ 0.482

Taking the inverse sine (sin^-1) of both sides, we get:

R ≈ sin^-1(0.482)

R ≈ 26 degrees

Therefore, the correct option is (a) 26⁰

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The given question is incomplete, the complete question is

What is the measure of angle R in triangle PQR?

a) 26°

b) 42°

c) 45°

d) 64°

1. (5, 12, 13) is a Pythagorean triple.

2.  (20, 21, 27) is not a Pythagorean triple.

3. Fake triples are (7, 2, 25), (280, 450, 533), and (110, 60, 651)

4.  A Pythagorean triple that I know is (3, 4, 5).

5. a.Thus, (6, 8, 10) is also a Pythagorean triple.

b.  Thus, (30, 40, 50) is also a Pythagorean triple.

c. Thus, (1.5, 2, 2.5) is still a Pythagorean triple.

d. Yes, a Pythagorean triple consisting of three even numbers is (6, 8,

10). A Pythagorean triple consisting of three odd numbers is (5, 12, 13).

The Pythagorean theorem is a fundamental theorem in mathematics that relates to the sides of a right-angled triangle.

1. To prove that (5, 12, 13) is a Pythagorean triple, we need to show that

⇒ 5² + 12² = 13²

⇒ 5² + 12² = 169       (5² = 25 and 12² = 144 and 13² = 169)

Therefore, 5² + 12² = 13², which means that (5, 12, 13) is a Pythagorean triple.

2. To prove that (20, 21, 27) is not a Pythagorean triple.

Assuming that (20, 21, 27) is a Pythagorean triple, we would have:

⇒ 20² + 21² = 27²

⇒ 400 + 441 ≠ 729

Therefore, (20, 21, 27) is not a Pythagorean triple.

3. To find the fakes among the following list of triples, we need to check if there is a whole number solution for the sides of a right-angled triangle with these measurements.

Therefore, the fakes are (7, 2, 25), (280, 450, 533), and (110, 60, 651).

4. A Pythagorean triple that I know is (3, 4, 5).

Scaling up each side by the same amount should still result in a Pythagorean triple, as the relationship between the sides will remain proportional.

 5.  a) Multiplying each side by 2, we get (6, 8, 10). Verifying Pythagoras' theorem, we have:

6^2 + 8^2 = 36 + 64 = 100

10^2 = 100

Thus, (6, 8, 10) is also a Pythagorean triple.

b) Multiplying each side by 10, we get (30, 40, 50). Verifying Pythagoras' theorem, we have:

30^2 + 40^2 = 900 + 1600 = 2500

50^2 = 2500

Thus, (30, 40, 50) is also a Pythagorean triple.

c) Halving each side of (3, 4, 5), we get (1.5, 2, 2.5). Verifying Pythagoras' theorem, we have:

1.5^2 + 2^2 = 2.25 + 4 = 6.25

2.5^2 = 6.25

Thus, (1.5, 2, 2.5) is still a Pythagorean triple.

d) The observation is that scaling up or down a Pythagorean triple by the same amount results in another Pythagorean triple. This is because the sides of a Pythagorean triple are related by the Pythagorean theorem, which is a quadratic equation. Multiplying each side by the same amount simply scales the sides up or down proportionally, while preserving the quadratic relationship.

Yes, a Pythagorean triple consisting of three even numbers is (6, 8, 10). A Pythagorean triple consisting of three odd numbers is (5, 12, 13).

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1. (5, 12, 13) is a Pythagorean triple.

2.  (20, 21, 27) is not a Pythagorean triple.

3. Fake triples are (7, 2, 25), (280, 450, 533), and (110, 60, 651)

4.  A Pythagorean triple that I know is (3, 4, 5).

5. a.Thus, (6, 8, 10) is also a Pythagorean triple.

b.  Thus, (30, 40, 50) is also a Pythagorean triple.

c. Thus, (1.5, 2, 2.5) is still a Pythagorean triple.

d. Yes, a Pythagorean triple consisting of three even numbers is (6, 8,

10. A Pythagorean triple consisting of three odd numbers is (5, 12, 13).

The Pythagorean theorem is a fundamental theorem in mathematics that relates to the sides of a right-angled triangle.

1. To prove that (5, 12, 13) is a Pythagorean triple, we need to show that

⇒ 5² + 12² = 13²

⇒ 5² + 12² = 169       (5² = 25 and 12² = 144 and 13² = 169)

Therefore, 5² + 12² = 13², which means that (5, 12, 13) is a Pythagorean triple.

2. To prove that (20, 21, 27) is not a Pythagorean triple

Assuming that (20, 21, 27) is a Pythagorean triple, we would have:

⇒ 20² + 21² = 27²

⇒ 400 + 441 ≠ 729

Therefore, (20, 21, 27) is not a Pythagorean triple.

3. To find the fakes among the following list of triples, we need to check if there is a whole number solution for the sides of a right-angled triangle with these measurements.

Therefore, the fakes are (7, 2, 25), (280, 450, 533), and (110, 60, 651).

4. A Pythagorean triple that I know is (3, 4, 5).

Scaling up each side by the same amount should still result in a Pythagorean triple, as the relationship between the sides will remain proportional.

5.  a) Multiplying each side by 2, we get (6, 8, 10). Verifying Pythagoras' theorem, we have:

6² + 8² = 36 + 64 = 100

10² = 100

Thus, (6, 8, 10) is also a Pythagorean triple.

b) Multiplying each side by 10, we get (30, 40, 50). Verifying Pythagoras' theorem, we have:

30² + 40² = 900 + 1600 = 2500

50² = 2500

Thus, (30, 40, 50) is also a Pythagorean triple.

c) Halving each side of (3, 4, 5), we get (1.5, 2, 2.5). Verifying Pythagoras' theorem, we have:

1.5² + 2² = 2.25 + 4 = 6.25

2.5² = 6.25

Thus, (1.5, 2, 2.5) is still a Pythagorean triple.

d) The observation is that scaling up or down a Pythagorean triple by the same amount results in another Pythagorean triple. This is because the sides of a Pythagorean triple are related by the Pythagorean theorem, which is a quadratic equation. Multiplying each side by the same amount simply scales the sides up or down proportionally, while preserving the quadratic relationship.

Yes, a Pythagorean triple consisting of three even numbers is (6, 8, 10). A Pythagorean triple consisting of three odd numbers is (5, 12, 13).

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The area of the parallelogram is approximately 193.73 square meters.

How to find the area?

To find the area of a parallelogram, we need to multiply the length of one side of the parallelogram by the perpendicular distance between that side and the opposite side.

In this case, we know that one side of the parallelogram is 11.8 m and the other side is 18.5 m. Let's call these sides "a" and "b" respectively.

To find the perpendicular distance between these two sides, we need to draw a perpendicular line from one of the endpoints of side "a" to side "b". Let's call this perpendicular distance "h".

We can use the formula for the area of a parallelogram:

Area = base x height

where the base is side "a" and the height is "h".

To find "h", we can use the Pythagorean theorem:

h² = b² - (a/2)²

where "a/2" is half of side "a".

Plugging in the values we know, we get:

h² = 18.5² - (11.8/2)²

h² = 302.25 - 34.81

h² = 267.44

h ≈ 16.36

Now that we know the height of the parallelogram, we can calculate its area:

Area = 11.8 x 16.36

Area ≈ 193.73 m²

Therefore, the area of the parallelogram is approximately 193.73 square meters.

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Answer:

[0,12]

Step-by-step explanation:

The unit of the domain is seconds

x is greater than or equal to zero because time can't be negative

Time stops at 12 seconds since the ball hits the ground.

Brackets are used to include both 0 and 12 as values

The height of the plane is about 3855 feet and the distance from the nearer end of the runway is about 13580 feet.

To find out the height of the plane and the distance of it from the nearer end of the runway, we have to use trigonometry.

Let's start by finding the height of the plane, h.

Using the angle of depression of 12.5 degrees, we can write this equation:

tan 12.5 = h/d

Where d is the distance from the nearer end of the runway. If we cross-multiply, we get:

d = h/tan 12.5

We can use the equation from the other angle of depression to eliminate the distance,

d.tan 17.0 = h/(d + 10,000)

By substituting the previous equation in for d, we get:

tan 17.0 = h/(h/tan 12.5 + 10,000)

Simplifying: tan 17.0 = h(tan 12.5)/(h/tan 12.5 + 10,000)

Multiplying both sides by h/tan 12.5 + 10,000:

tan 17.0 (h/tan 12.5 + 10,000) = h tan 12.5

Expanding the left side: distributed = h + 10,000 tan 17.0 tan 12.5

Simplifying and isolating h: h = 10,000 tan 17.0 tan 12.5/(tan 12.5 - tan 17.0) ≈ 3855 ft

Therefore, the height of the plane is about 3855 feet. To find the distance from the nearer end of the runway, we use the previous equation:d = h/tan 12.5 ≈ 13580 ft

Therefore, the distance from the nearer end of the runway is about 13580 feet.

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