Free response: 3 questions, 52 points, 30 minutes
1. blinko is a game in which three dice are rolled. to win the game, a player must
roll triples – that is, three of the same number. you roll the dice 20 times.
a. verify that the situation is binomial. (8points)
b. out of 20 rolls, what is the probability you win exactly 4 times? show your
work using the formula for binomial probability. (6 points)
c. what is the probability that you win at least 3 times? (6 points)
d. what is the mean number of wins in 20 rolls of the dice?

The platelet count is 652 x 10^3.

Based on the information provided, it seems that the platelet count from a test result is 652, and the normal range for platelet count is 150,000 to 450,000 platelets per microliter of blood. The value "10(3)" likely refers to the measurement being in thousands.

To convert the measurement to the standard unit of platelet count, we need to multiply the given value by 1,000. Thus, 652 * 1,000 = 652,000 platelets per microliter.

It's important to note that platelet counts can vary depending on various factors, and medical professionals typically interpret the results in conjunction with other clinical information. If the obtained value of 652,000 platelets per microliter is accurate, it would be significantly higher than the upper limit of the normal range.

It's crucial to consult a healthcare professional or a qualified medical practitioner for an accurate interpretation of test results and any necessary medical advice or treatment.

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The probability that at least one of Tom or Jerry are late for school is 0.4.

To find the probability that at least one of Tom or Jerry are late for school, we can use the formula:
P(T or J) = P(T) + P(J) - P(T and J)

Since we don't know the probability of Jerry being late (P(J)), we can use the complement rule:
P(J) = 1 - P(JC)
where JC represents the event that Jerry is not late for school.

Substituting the given probabilities:
P(T or J) = P(T) + [1 - P(JC)] - P(T and J)
P(T or J) = 0.25 + 0.3 - 0.15
P(T or J) = 0.4

Therefore, the probability that at least one of Tom or Jerry are late for school is 0.4.

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the partial derivative ∂z/∂y using implicit differentiation of e^z + sin(5x^2) + 2y = 0 is:
To calculate the partial derivative ∂z/∂y using implicit differentiation of e* + sin (5x^2) + 2y = 0, we first need to differentiate both sides of the equation with respect to y.

We get:

d/dy(e^z + sin(5x^2) + 2y) = d/dy(0)

Using the chain rule, the left-hand side becomes:

∂(e^z)/∂z * ∂z/∂y + ∂(sin(5x^2))/∂y + 2

We can simplify this by recognizing that ∂(sin(5x^2))/∂y = 0, since sin(5x^2) does not depend on y. Thus, we are left with:

∂(e^z)/∂z * ∂z/∂y + 2 = 0

Now, we need to solve for ∂z/∂y:

∂z/∂y = -2 / ∂(e^z)/∂z

To find ∂(e^z)/∂z, we differentiate e^z with respect to z, giving:

∂(e^z)/∂z = e^z

Substituting this into the expression for ∂z/∂y, we get:

∂z/∂y = -2 / e^z

Therefore, the partial derivative ∂z/∂y using implicit differentiation of e^z + sin(5x^2) + 2y = 0 is:

∂z/∂y = -2 / e^z

Note that we cannot simplify this any further without knowing the value of z.
To find the partial derivative ∂z/∂y using implicit differentiation for the equation e^z + sin(5x^2) + 2y = 0, we will first differentiate the equation with respect to y, treating z as a function of x and y.

Differentiating both sides with respect to y:

∂/∂y (e^z) + ∂/∂y (sin(5x^2)) + ∂/∂y (2y) = ∂/∂y (0)

Using the chain rule for the first term, we get:

(e^z) * (∂z/∂y) + 0 + 2 = 0

Now, solve for ∂z/∂y:

∂z/∂y = -2 / e^z

So, the partial derivative ∂z/∂y for the given equation is:

∂z/∂y = -2 / e^z

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The expression of the width of the rectangle is 2x¹²y⁸z¹¹/3.

Given that the area of a rectangle is 72x¹³y⁹z¹⁶ sq. yds and the length of the rectangle is 3x9y4z⁵, we need to find the width,

Using these expressions, we have,

Area = length × width

72x¹³y⁹z¹⁶ / 3x9y4z⁵ = width

Width = 72x¹³/3x × y⁹/9y × z¹⁶/4z⁵

Width = 24x¹²y⁸z¹¹/36 = 2x¹²y⁸z¹¹/3

Hence the expression of the width of the rectangle is 2x¹²y⁸z¹¹/3.

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The relative frequency of people who strongly agree with the statement "I feel respected by those I work for" is 0.312, or 31.2%.

This means that out of the 500 employees surveyed, 156 strongly agreed with the statement. To find the relative frequency, you simply divide the number of people who strongly agree by the total number of people surveyed (156/500).

This result suggests that the majority of employees feel respected by their employers, which is a positive sign for the workplace culture.

However, it's important to note that there are still a significant number of employees who either disagree or feel neutral about this statement, indicating that there may be room for improvement in terms of fostering a more respectful and supportive work environment.

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The decimal form of 4/5 is 0.80, and it is equivalent to 80% as a percent.

To convert the fraction 4/5 into a decimal and a percent, we start by dividing the numerator (4) by the denominator (5). The result is 0.80 as a decimal.

In decimal form, 4/5 is written as 0.80. The "0" before the decimal point represents whole units, and the "80" after the decimal point represents hundredths.

To express this decimal as a percent, we multiply it by 100, as a percent is a representation of parts per hundred. So, 0.80 multiplied by 100 equals 80. Therefore, 4/5 is equivalent to 80% when expressed as a percentage.

In summary, 4/5 as a decimal is 0.80, which means it represents 80 hundredths, and as a percent, it is 80%, which signifies 80 parts per hundred. This conversion is particularly useful in various mathematical and real-world applications, such as calculating discounts, grades, proportions, and percentages in everyday life and business contexts.

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Quadrilaterals have four sides, two diagonals, and the sum of their interior angles is 360 degrees. They can be related to triangles through the fact that each triangle has a total of 180 degrees and a quadrilateral has 360 degrees, so there are two triangles in a quadrilateral with their angles adding up to 360 degrees.

However, triangles and quadrilaterals differ in terms of their area formulas, where the area of a quadrilateral is calculated as the product of its base and height (A = BxH), while the area of a triangle is half the product of its base and height (A = (BxH)/2). Quadrilaterals have different types and properties, although they share the common characteristics mentioned above.

- Quadrilaterals have four sides and two diagonals. The sum of the interior angles in a quadrilateral is always 360 degrees.

- Triangles have three sides and the sum of their interior angles is always 180 degrees.

- The relationship between triangles and quadrilaterals is based on the fact that a quadrilateral can be divided into two triangles. Each triangle within the quadrilateral contributes 180 degrees to the total sum of 360 degrees.

- The formula for calculating the area of a quadrilateral is A = BxH, where A represents the area, B represents the base, and H represents the height.

- In contrast, the formula for calculating the area of a triangle is A = (BxH)/2, where A represents the area, B represents the base, and H represents the height. This formula demonstrates that the area of a triangle is half the area of a quadrilateral with the same base and height.

- While all quadrilaterals share the characteristics of having four sides, two diagonals, and interior angles summing up to 360 degrees, they have different types and properties.

Examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and rhombuses. Each type has its own unique properties and relationships to other quadrilaterals.

In conclusion, quadrilaterals and triangles are related through the concept of dividing a quadrilateral into two triangles. They differ in their area formulas, and although all quadrilaterals have four sides, two diagonals, and interior angles summing up to 360 degrees, they have different types and properties.

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The new coordinate of the vertex after dilation is (-5, -3).

To find the new coordinate, you'll need to use the given scale factor of 1/3 and apply it to the original vertex coordinates (-15, -9). Here's a step-by-step explanation:

1. Identify the original vertex coordinates: (-15, -9).
2. Identify the scale factor for dilation: 1/3.
3. Apply the scale factor to the x-coordinate: (-15) * (1/3) = -5.
4. Apply the scale factor to the y-coordinate: (-9) * (1/3) = -3.
5. The new coordinates after dilation are (-5, -3).

By following these steps, you can find the new coordinates of the vertex after dilation at the origin using the given scale factor.

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57 students most likely got heads between 45 and 60 times.

To determine the number of students who got heads between 45 and 60 times, we'll use the normal distribution properties. First, we need to calculate the z-scores for 45 and 60:

Z = (X - μ) / σ

For 45 heads:
Z1 = (45 - 50) / 5 = -1

For 60 heads:
Z2 = (60 - 50) / 5 = 2

Next, we need to find the probability that a student falls between these z-scores. We can do this by looking up the z-scores in a standard normal distribution table or using a calculator. The probabilities corresponding to these z-scores are:

P(Z1) = 0.1587
P(Z2) = 0.9772

Now, subtract P(Z1) from P(Z2) to get the probability of a student's result falling between 45 and 60 heads:

P(45 ≤ X ≤ 60) = P(Z2) - P(Z1) = 0.9772 - 0.1587 = 0.8185

Finally, multiply this probability by the total number of students (70) and round to the nearest whole number:

Number of students = 0.8185 * 70 ≈ 57

So, approximately 57 students most likely got heads between 45 and 60 times.

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The correct answer is (C) as the height of the pole is 8.7 meters.

What are Trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles and sides of a right triangle.

Tangent (tan): the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.

We can use the tangent function to solve this problem.

Let's call the height of the pole "h" and the length of the rope "r".

We know that the length of the shadow cast by the pole is 11.6 meters, and the angle between the sun's rays and the ground is 36.8°. This means that:

tan(36.8°) = h / 11.6

To solve for "h", we can rearrange this equation to get:

h = 11.6 * tan(36.8°)

Using a calculator, we find that h ≈ 8.7 meters.

So the height of the pole is approximately 8.7 meters. Therefore, the correct answer is (C) 8.7.

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The Laplace transform of f(x)= (-2x-3)/4 is (-2L{x}-3L{1})/4, where L{x} is the Laplace transform of x and L{1} is the Laplace transform of 1.
Hi! The Laplace transform of a given function f(t) is denoted by L{f(t)} and is defined as the integral of f(t) multiplied by e^(-st), where s is a complex variable. For the function f(x) = (-2x - 3)/4, the Laplace transform can be calculated as follows:

L{f(t)} = L{(-2t - 3)/4}

To find the Laplace transform, we will treat the function as two separate parts:

L{(-2t - 3)/4} = (-2/4) * L{t} + (-3/4) * L{1}

The Laplace transforms of t and 1 are well-known:

L{t} = 1/s^2
L{1} = 1/s

Now, substitute these transforms back into our expression:

L{f(t)} = (-1/2) * (1/s^2) + (-3/4) * (1/s)

L{f(t)} = -1/(2s^2) - 3/(4s)

And that's the Laplace transform of f(x) = (-2x - 3)/4.

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Therefore , the solution of the given problem of unitary method  comes out to be  $52 10 units of coverage will be purchased.

To complete the work, the well-known straightforward strategy, actual variables, and any essential components from the very first and specialised inquiries can all be utilised. In response, customers might be given another opportunity to sample the product. Otherwise, important advancements in our comprehension of algorithms will be lost.

Here,

We are informed that the first unit of coverage will cost $16 and each additional unit will cost $4. We may calculate the price of x units of coverage using the following formula:

=> C(x) = 16 + 4(x-1)

The number of subsequent units of coverage following the initial unit is indicated by the (x-1) term in the calculation.

We may enter x=10 into the algorithm to get the price for 10 units of coverage:

=> C(10) = 16 + 4(10-1)

=> C(10) = 16 + 4(9)

=> C(10) = 16 + 36

=> C(10) = 52

For $52, 10 units of coverage will be purchased.

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In the last 215 days, builders have completed 700 m2 of the alligator habitat that will eventually be 1,200 m2.
It will take approximately 153 days to complete the remaining part of the alligator habitat.

Determine how much longer it will take to complete the alligator habitat, first, we need to find the rate at which the builders are working.
Calculate the work rate
The builders have completed 700 m2 of the 1,200 m2 alligator habitat in 215 days.
Work rate = (completed work) / (number of days)
Work rate = 700 m2 / 215 days = 3.26 m2/day (approximately)
Calculate the remaining work
The total area of the alligator habitat is 1,200 m2, and 700 m2 has been completed.
Remaining work = Total area - Completed work
Remaining work = 1,200 m2 - 700 m2 = 500 m2
Calculate the time to complete the remaining work
Time to complete = (remaining work) / (work rate)
Time to complete = 500 m2 / 3.26 m2/day ≈ 153.37 days
It will take approximately 153 days to complete the remaining part of the alligator habitat.

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It will take approximately 394 more days to complete the alligator habitat.

We can start by finding the proportion of the habitat that has already been completed:

proportion completed = 700 m^2 / 1200 m^2 = 0.5833

This means that there is still 1 - 0.5833 = 0.4167 (or 41.67%) of the habitat left to complete.

Next, we can use a proportion to find out how long it will take to complete the remaining 41.67% of the habitat:

215 days / 0.5833 = x days / 0.4167

Solving for x, we get:

x = 215 days * 0.4167 / 0.5833 ≈ 153 days

Therefore, the total time it will take to complete the alligator habitat is approximately 215 + 153 = 368 days, or about 394 more days from the start.

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The length of the base is 16 inches.

To find the length of the base of the triangular pane of glass, we can use the formula for the area of a triangle which is:

Area = (1/2) x base x height

We are given that the height of the pane is 32 inches and the area is 256 square inches. Substituting these values into the formula, we get:

256 = (1/2) x base x 32

To isolate the base, we can divide both sides by (1/2) x 32, which simplifies to 16. This gives us:

256 ÷ 16 = base

Simplifying the left side of the equation, we get:

16 = base

Therefore, the length of the base of the pane is 16 inches.

In summary, the triangular pane of glass has a height of 32 inches and an area of 256 square inches. To find the length of the base, we use the formula for the area of a triangle and solve for the base.

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To find the value of a² + bc√(-d) when a = -3, b = 2, c = 100, and d = -2, follow these steps:

Step 1: Substitute the values into the expression.
a² + bc√(-d) = (-3)² + (2)(100)√(-(-2))

Step 2: Simplify the expression.
(-3)² + (2)(100)√(2) = 9 + 200√2

So, the value of a² + bc√(-d) when

a = -3,

b = 2,

c = 100,

d = -2 is 9 + 200√2.

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Answer:

True

Step-by-step explanation:

Parallelogram: A parallelogram is 4-sided shape that has parallel opposite sides

Square: A square is 4-sided shape where all sides are congruent and has 4 right angles.  A square also has 2 pairs of opposite parallel sides.

Based on these definitions, we can see that not ALL parallelograms can be squares because a parallelogram can have no right angles, or can have all 4 angles be right angles.

All squares have to be parallelograms though because a square and a parallelogram both have 2 pairs of opposite parallel sides.

Hope this helps :)

Answer:

Step-by-step explanation:

The lateral surface area of a cylinder is the area of the sides of the cylinder, not including the top or bottom. In the case of a can of beans, the label that wraps around the can represents this lateral surface area.

To find the lateral surface area of the can, we need to use the formula for the lateral surface area of a cylinder: LSA = 2πr*h, where r is the radius of the base of the cylinder and h is the height of the cylinder.

Since we are given the diameter of the can (7 cm), we need to divide it by 2 to get the radius: r = 7/2 = 3.5 cm. The height of the can is given as 11 cm.

Now we can plug these values into the formula to find the lateral surface area of the can: LSA = 2π(3.5)(11) ≈ 242.95 cm².

So the lateral surface area of the can of beans is approximately 242.95 cm². This is the area of the sides of the can that the label would wrap around.

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Answer:

a. x = 68

b. x = 55

c. x = 18

Step-by-step explanation:

Formula

Inscribed angle = Central angle/2

a.

x = 136/2

x = 68

b.

x = ( 360 - 150 - 100 )/2

= 110/2

x = 55

c.

x = 18

Using trigonometry, we can solve for the missing angles to find them as 18.43°, 45°, 45°, and 18.43°.

The sign is mounted on a sloped surface, which means that we'll need to use some trigonometry to find the missing angles.

Let's concentrate on the sign's upper right corner, where the letters x and y are absent from two perspectives. The magnitude of angle x may be determined using trigonometry.

Let's begin by sketching a right triangle that has an angle x. The triangle's two sides may be represented by the sign's vertical and horizontal lines, with the addition of a third side to join the top right corner of the sign to the sloping area below.

Since the sign is an octagon, we know that each interior angle is 135°. Therefore, the measure of angle y must be:

y = 180 - 135 = 45°

Now, let's look at the right triangle that includes angle x. We know that the hypotenuse of the triangle is the sloped surface of the sign, which has a length of 4.5 meters. We also know that the opposite side of the triangle is the height of the sign above the ground, which has a length of 1.5 meters.

Using trigonometry, we can find the measure of angle x by taking the inverse tangent of the opposite side over the adjacent side:

tan(x) = opposite/adjacent = 1.5/4.5 = 1/3

x = tan⁻¹(1/3) ≈ 18.43°

Therefore, the measure of angle x is approximately 18.43 degrees.

Hence, using trigonometry, we can solve for the missing angles to find them as 18.43°, 45°, 45°, and 18.43°.

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The value of x which is not a solution to the inequality 5 - 2x ≥ -3, is x = 6.

To find out which value is not a solution to the inequality 5 - 2x ≥ -3, we can substitute each value into the inequality and see if it is true or false.

Let's start with the first value, [tex]x=4[/tex]:

5 - 2(4) ≥ -3
5 - 8 ≥ -3
-3 ≥ -3

Since -3 is greater than or equal to -3, x = 4 is a solution of the inequality.

Now let's try x = 6:

5 - 2(6) ≥ -3
5 - 12 ≥ -3
-7 ≥ -3

Since -7 is less than -3, x = 6 is not a solution of the inequality.

Therefore, the answer is x = 6.

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The volume of the obtained solid is 36 units³.

Given that a rectangle of dimension 9 units x 2 units, has been rotated to form a solid we need to find its volume,

So we know that a rectangle rotated to form a rectangular prism.

Volume of a rectangular prism = product of the dimensions.

The dimensions of the obtained solid will be 9 units x 2 units x 2 units,

So the volume = 9 x 2 x 2 = 36 units³

Hence the volume of the obtained solid is 36 units³.

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When the value of x is 7 the value of p will be equal to 8.

It is given that P varies inversely with x, so we can write that

p=k/x

where k is the proportionality constant.

here we can find the value of k by substituting the value of p and x with 7 and 8 in the relation that is given above, we get:

7=k/8

k=7*8

k=56

we the value of k to be 56 after putting the values in the relation.

Now if x is changed to 7, and k is equal to 56 we can get the value of p by putting the known values in the same relation.

p=k/x

p=56/7

p=8.

Therefore, when the value of x is 7 the value for p will be equal to 8.

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The volume of raspberries in Mateo's box is 384 cubic centimeters.

To calculate the volume of raspberries in Mateo's box, we need to use the formula for the volume of a rectangular prism, which is length x width x height. In this case, the length and width are both 8 centimeters, as the box has a square bottom. The height is 6 centimeters, as Mateo fills the box to that height with raspberries.

So, the volume of raspberries in Mateo's box is:

Volume = length x width x height
Volume = 8 cm x 8 cm x 6 cm
Volume = 384 cubic centimeters

Therefore, the volume of raspberries in Mateo's box is 384 cubic centimeters. This calculation assumes that the raspberries are tightly packed in the box, without any gaps or air pockets. In reality, the actual volume of raspberries may be slightly less than this, depending on how they are arranged in the box. Nonetheless, this calculation provides a reasonable estimate of the amount of raspberries that Mateo is able to pick and fit in the box.

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Answer:

we know that given triangle is right angled triangle.

➢ By using Phythagoras Theorem:-

[tex] \sf \longrightarrow \: (AC)^2 = (AB)^2 + (BC)^2[/tex]

[tex] \sf \longrightarrow \: ( \sqrt{80} )^2 = (x)^2 + (8)^2[/tex]

[tex] \sf \longrightarrow \: 80 = (x)^2 + (8)^2[/tex]

[tex] \sf \longrightarrow \: 80 \: = x^2 \: + \: 8^2[/tex]

[tex] \sf \longrightarrow \: 80 \: = x^2 \: + \: 64[/tex]

[tex] \sf \longrightarrow \: 80 \: - 64 = x^2 \:[/tex]

[tex] \sf \longrightarrow \: 16 = x^2 \:[/tex]

[tex] \sf \longrightarrow \: x^2 \: = 16[/tex]

[tex] \sf \longrightarrow \: x \: = \sqrt{16} [/tex]

[tex] \sf \longrightarrow \: x \: = 4 \: units [/tex]

Answer is The smaller angle is 35°.

To solve for the acute angles in a right triangle when sin(5x)° = cos(3x + 2)°, we need to use trigonometric identities.

Recall that sin(x) = cos(90 - x) for any angle x. Using this identity, we can rewrite sin(5x)° as cos(90 - 5x)°.

Similarly, cos(x) = sin(90 - x) for any angle x. Using this identity, we can rewrite cos(3x + 2)° as sin(90 - (3x + 2))°, which simplifies to sin(88 - 3x)°.

Now we have cos(90 - 5x)° = sin(88 - 3x)°. Since these two expressions are equal, we can set them equal to each other and solve for x:

cos(90 - 5x)° = sin(88 - 3x)°

sin(5x)° = sin(88 - 3x)°

5x = 88 - 3x

8x = 88

x = 11

Therefore, the acute angles in the right triangle are 5x° and 90 - 5x°, or 55° and 35°. The smaller angle is 35°.

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The expression (x + 1)(x - 3) represents the Area of base of the prism.

a crystal is a polyhedron containing a n-sided polygon base, a respectable halfway point which is a deciphered duplicate of the first, and n different countenances, fundamentally all parallelograms, joining relating sides of the two bases. Translations of the bases exist in every cross-section that runs parallel to the bases.

According to question:

The volume of a rectangular prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a square with edge length (x + 1), so its area is (x + 1)^2. The volume of the prism is given as (x + 1)^2(x - 3).

We can find the height of the prism by dividing the volume by the area of the base:

B = V/h = (x + 1)^2(x - 3)/(x + 1) = (x + 1)(x - 3)

Therefore, the expression (x + 1)(x - 3) represents the Area of base of the prism.

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The ladder reaches up 20ft the side of the house.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem can be used to solve problems involving right triangles, such as finding the length of the sides or the height of an object.

In this problem, we are given the length of the ladder and the height up the side of the house that it reaches.

We can use the Pythagorean theorem to find the distance from the base of the ladder to the side of the house.

We can then use this distance and the height up the side of the house that the ladder reaches to find the length of the ladder using the Pythagorean theorem again.

Let's call the distance from the base of the ladder to the side of the house "x". We can then use the Pythagorean theorem to find the height that the ladder reaches up the side of the house.

According to the Pythagorean theorem, the length of the ladder (which is the hypotenuse of the right triangle formed by the ladder, the ground, and the side of the house) is equal to the square root of the sum of the squares of the other two sides.

So, if we let "h" be the height up the side of the house that the ladder reaches, we have:

ladder length = √(x^2 + h^2)

We know that the ladder is 20ft long and reaches up 16ft, so we can set up the equation:

20 = √(x^2 + 16^2)

Squaring both sides of the equation, we get:

400 = x^2 + 256

Subtracting 256 from both sides, we get:

144 = x^2

Taking the square root of both sides, we get:

x = 12

So the ladder is leaning against the house 12ft away from the base, and we can use the Pythagorean theorem to find the height up the side of the house that the ladder reaches:

ladder length = √(12^2 + 16^2) = √(144 + 256) = √400 = 20

Therefore, the ladder reaches up 20ft the side of the house.

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The simplified expression for p(z), the profit the company earns, in dollars is: p(z) = -3z² + 18z

The set X is known as the domain of the function, and the set Y is known as the codomain of the function. A function from a set X to a set Y assigns each element of X to exactly one element of Y. Originally, functions were an idealization of how one variable depends on another.

First, let's substitute the expressions for r(z) and c(z) in the expression for p(z):

p(z) = r(z) - c(z)

p(z) = 60z - (3z² + 42z)

p(z) = 60z - 3z² - 42z

p(z) = -3z² + 18z

Therefore, the simplified expression for p(z), the profit the company earns, in dollars is:

p(z) = -3z² + 18z

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