Rachel, Phoebe and Monica are sunflower farmers in the village of Girasol. They each have zero wealth, so their consumption is equal to the income they earn from their economic activity. Each of them must choose one (and only one) of the following three activities:
Activity 1: Full time farming. Sunflower farming is risky because of a combination of weather and pests. Under full time farming, the farmer works 7 days per week on their farm. There is a 50% probability of having a GOOD harvest and a 50% chance of having a BAD harvest. If the harvest is GOOD, the farmer earns an income of $100. If the harvest is BAD, the farmer earns an income of only $20.
Activity 2: Full time construction work. This activity has no risk. An individual who decides to work full time in construction earns $40 with certainty.
Activity 3: Part-time farming. In this third activity, the farmer works during the week as a sunflower farmer and works in construction during the weekend. Since she is not able to work full time on the farm, the probability of having a GOOD harvest and earning $100 drops to 25%, and the probability of having a BAD harvest and earning only $20 increases to 75%. The individual also earns $10 with certainty as a construction worker (the person earns this $10 from construction in addition to her farm income under both a GOOD and BAD harvest).
Q: What is the expected value of consumption for each activity?

Answer: D

Step-by-step explanation:

7x + 4 = 88

x = 12

angle 1 = 180 - 7(12) + 2

angle 1 = 98

So it's D

Answer: The angle measure for angle 1 would be 98 degrees.

Step-by-step explanation: First, you set 88 equal to (7x+4) since they are vertical angles. (Vertical angles are congruent.) Then you solve and find that x=12. You then plug 12 in to the second equation, (7x-2) and get 82. Since angle 1 is supplementary to the angle that measures 82 degrees, you would take 180-82= Angle 1.

Answer:

:-)

Step-by-step explanation:

Straight line=180 so 180-75-50=55

Since angles in a triangle=180 and since a=55 and c=50 b must =75

According to alternate angle theorem since angle next to a =50 angle c equals 50.  

d=50 because the opposite angle theorem states that d=c and c=50

Answer: Your welcome!

Step-by-step explanation:

14

13

12

11

10

9

8

7

6

O

6

7 8 9 10 11 12 13

10

1415

The answer is 14, 13, 12, 11, 10, 9, 8, 7, 6, O, 6, 7, 8, 9, 10, 11, 12, 13, 10, 14, 15. This is the result of translating the shape by the vector (5, 4). The original shape was shifted 5 units to the right and 4 units down, resulting in the new shape.

Answer: 1

Step-by-step explanation:

3(x-2) = 4x+2

3x-6 = 4x+2

To move 3x, deduct it on both sides of the equation.

3x - 6 - (3x) = 4x + 2 - (3x)

0 - 6 = [1]x +2

Given:-

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

Answer:

  76 ft²

Step-by-step explanation:

You want the area of a trapezoid with bases 6 ft and 13 ft, and height 8 ft.

The area formula for a trapezoid is ...

  A = 1/2(b1 +b2)h

  A = 1/2(6 ft +13 ft)(8 ft) = 76 ft²

The area of the figure is 76 square feet.

__

Additional comment

The length of the longer base on the right is the sum of 6 ft and 7 ft. It is 13 ft.

The area of the figure is 76 square feet.

the area of a trapezoid is: 76 ft².

Here, we have,

from the given figure, we get,

The length of the longer base on the right is the sum of 6 ft and 7 ft. It is 13 ft.

We have to find the area of a trapezoid with bases 6 ft and 13 ft, and height 8 ft.

Trapezoid

The area formula for a trapezoid is

 A = 1/2(b₁ +b₂)h

 A = 1/2(6 ft +13 ft)(8 ft)

     = 76 ft²

so, we get,

The area of the figure is 76 square feet.

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Answer:

To show that T<0,2> (0,1), we need to evaluate the transformation T at the point (0,1) and check if the result is in the range of T.

T<0,2> (0,1) means that the transformation T takes the point (0,1) in the input space to a point in the output space that has coordinates (0,2).

Let's evaluate T at (0,1):

T<0,2> (0,1) = D3(0,1) + (0,1)

= (0+0, 3+1)

= (0, 4)

The output point (0,4) is not equal to (0,2), which means that T<0,2> (0,1) is false. Therefore, T<0,2> does not map the point (0,1) to (0,2).

To determine if T<0,2> (x,y) = D3 (x,y) is a true statement for any point (x,y), we need to check if the transformation T always equals the function D3.

T<0,2> (x,y) = (x, 3+y)

D3(x,y) = (x,y,3)

Since T and D3 have different ranges (T has a range of R^2 and D3 has a range of R^3), the statement T<0,2> (x,y) = D3(x,y) is not true for any point (x,y).

Therefore, we cannot equate T<0,2> and D3, and the statement T<0,2> (x,y) = D3 (x,y) is false for any point (x,y).

Answer:

cos G = 3/5

sin G = 4/5

tan G = 4/3

Step-by-step explanation:

I(9, 12)

GH = 9

HI = 12

(GI)² = 9² + 12²

GI = 15

For <G:

opp = HI = 12

adj = GH = 9

hyp = GI = 15

cos G = adj/hyp = 9/15 = 3/5

sin G = opp/hyp = 12/15 = 4/5

tan G = opp/adj = 12/9 = 4/3

Step-by-step explanation:

remember the triangle in a circle, when you learned about sine, cosine and the other trigonometric functions ?

this looked the same way, just that the circle was the norm-circle with radius 1.

here, now, the radius is larger, so every function line is also multiplied by the actual radius.

but the principles are the same.

sine is the up/down leg of the right-angled triangle.

cosine is the left/right leg.

tangent is sine/cosine.

I = (9, 12)

so, Pythagoras gives us the radius (line GI) :

GI² = GH² + HI² = 9² + 12² = 81 + 144 = 225

GI = sqrt(225) = 15

sin(G) × GI = HI

sin(G) × 15 = 12

sin(G) = 12/15 = 4/5 = 0.8

cos(G) × GI = GH

cos(G) × 15 = 9

cos(G) = 9/15 = 3/5 = 0.6

tan(G) = sin(G)/cos(G) = 4/5 / 3/5 = (4×5)/(5×3) =

= 4/3 = 1.333333333...

This confirms that our polynomial function has the correct y-intercept.

To find a polynomial function with the given properties, we can use the fact that the zeros of a polynomial function are the values of x that make the function equal to zero. The multiplicity of a zero is the number of times that zero appears as a factor in the polynomial.

Since we are given a triple zero at x = 1, this means that (x - 1)^3 is a factor of the polynomial. Similarly, since we are given a double zero at x = 3, this means that (x - 3)^2 is a factor of the polynomial.

To find the y-intercept, we can set x = 0 and solve for y. Since the y-intercept is given as (0, 27), this means that the constant term of the polynomial is 27.

Putting all of this information together, we can write the polynomial function as:

f(x) = 27(x - 1)^3(x - 3)^2

This is a polynomial function that satisfies the given properties. It has a triple zero at x = 1, a double zero at x = 3, and a y-intercept of (0, 27).

We can check our answer by plugging in the values of x and seeing if the function equals zero:

f(1) = 27(1 - 1)^3(1 - 3)^2 = 0
f(3) = 27(3 - 1)^3(3 - 3)^2 = 0

Both of these values equal zero, which confirms that our polynomial function has the correct zeros. Additionally, when we plug in x = 0, we get:

f(0) = 27(0 - 1)^3(0 - 3)^2 = 27

This confirms that our polynomial function has the correct y-intercept.

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The total surface area of the cylinder is 548.13 square meters.

The curved surface area of a cylinder can be calculated using the formula:

CSA = 2πrh

where CSA is the curved surface area, r is the radius, and h is the height or length of the cylinder.

We are given that the curved surface area of the cylinder is 408 m² and the length is 17 m.

We can use this information to solve for the radius:

CSA = 2πrh

408 = 2πr(17)

r = 408 / (2π × 17)

r ≈ 3.00 m (rounded to 2 decimal places)

Now that we know the radius, we can calculate the total surface area of the cylinder by adding the areas of the two bases and the curved surface area:

Total surface area = 2πr² + 2πrh

Total surface area = 2π(3.00)² + 2π(3.00)(17)

Total surface area ≈ 226.19 + 321.94

Total surface area ≈ 548.13

Therefore, the total surface area of the cylinder is approximately 548.13 square meters.

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A. If a friend's kid has no curfew, what is the probability that they don't have chores? B. the probability that a friend's kid doesn't have chores given that they have no curfew is 1/3. C. I chose this question because it involves calculating a conditional probability based on a given set of data.

Probability can also be used to describe more complex events, such as the likelihood of a stock price increasing by a certain amount in a given period of time, or the probability of a medical treatment being effective for a certain disease.

My Conditional Frequency Question is:

If a friend's kid has no curfew, what is the probability that they don't have chores?

Solution:

The conditional probability of a kid not having chores given that they have no curfew can be found using the formula:

P(No Chores | No Curfew) = P(No Chores and No Curfew) / P(No Curfew)

From the table, we can see that the number of kids who have no curfew and no chores is 2, and the total number of kids who have no curfew is 6. Therefore:

P(No Chores | No Curfew) = 2/6 = 1/3

So the probability that a friend's kid doesn't have chores given that they have no curfew is 1/3.

I chose this question because it involves calculating a conditional probability based on a given set of data. This is a common type of question in statistics and probability, and it requires an understanding of the basic concepts of probability such as conditional probability and independence. The table provided gives us the data we need to calculate the conditional probability, and the solution involves applying the formula for conditional probability and simplifying the fraction.

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The pretax price of the item is $18. The tip is $3.60 (0.20 * 18) and the tax is $1.44 (0.08 * 18), which when added to $18 gives us a total cost of $23.04.

For a 20% tip, 8% tax sales. Total $23. 04, we can say that The pre-tax cost is $19.20.

Let x be the pre-tax cost of the sales. Then we can set up the equation:

x + 0.20x + 0.08x = 23.04

Simplifying and solving for x, we get:

1.28x = 23.04

x = 23.04 / 1.28

x = 18

The pretax price is the amount charged for a product or service before any taxes are added. It is the price that a customer would pay for a product or service without any additional charges imposed by the government. In most countries, taxes such as sales tax, value-added tax (VAT), or goods and services tax (GST) are added to the pretax price to determine the final price that a customer pays.

Pretax prices are important for businesses to determine their profit margins and pricing strategies. It is also useful for customers to compare prices between different products or services before taxes are added. However, it is important to note that the final price paid by the customer may vary based on the applicable tax rate.

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Complete Question: -

Need help on math You have a meal at a restaurant. The sale tax is 8% . You leave a tip for the waitress that is 20% of the pretax price . You spend $23.04 . What is the pretax price of the meal?

To solve the compound inequality, we need to solve each inequality separately and then combine the solutions using the word "OR."

For the first inequality, 8x+7>55, we can isolate the variable on one side of the inequality by subtracting 7 from both sides:

8x > 48

Next, we can divide both sides by 8 to solve for x:

x > 6

For the second inequality, -5x-1>=-26, we can isolate the variable on one side of the inequality by adding 1 to both sides:

-5x >= -25

Next, we can divide both sides by -5 to solve for x. Remember that when we divide or multiply both sides of an inequality by a negative number, we need to reverse the inequality sign:

x <= 5

Now we can combine the solutions using the word "OR." In interval notation, this would be (-∞, 5] U (6, ∞). This means that the solution includes all values of x less than or equal to 5 or greater than 6.

So the final answer is:

x ∈ (-∞, 5] U (6, ∞)

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It is feasible to close Hargett St. between Blount Street and Person Street for a day.

Given the traffic flow model described, it is feasible to close Hargett St. between Blount Street and Person Street for a day. The model states that the average daily traffic flow on Hargett St. is 8,000 vehicles per day, with 3,000 vehicles per hour during peak hours. By closing Hargett St. between Blount Street and Person Street for a day, the average daily traffic flow would be reduced by 8,000 vehicles per day. Thus, the total number of vehicles that would be diverted from Hargett St. on the day of the event would be 8,000 vehicles. This amount of traffic would not be too overwhelming for the surrounding streets to handle, since the model states that the average daily traffic flow for those streets is approximately 4,000 vehicles per day.

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Answer:

x = 8

Step-by-step explanation:

4(x + 3) = 44 ( divide both sides by 4 )

x + 3 = 11 ( subtract 3 from both sides )

x = 8

To find the scale factors with fractions, you first need to understand what scale factors are. Scale factors are the ratios of corresponding lengths in two similar figures. They tell you how much larger or smaller one figure is compared to another.

To simplify fractions, you need to find the greatest common factor (GCF) of the numerator and denominator and divide them by it.

For example, if we have the fraction 6/12, we can simplify it by finding the GCF of 6 and 12, which is 6. Then, we divide both the numerator and denominator by 6:

6/12 = (6 ÷ 6) / (12 ÷ 6) = 1/2

So, 6/12 simplifies to 1/2.

If the fraction is already in its simplest form, then there is no need to simplify it further.

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The complete question i:

How do you find the scale factors with fractions? Give example.

The two functions with the end behavior x→+∞ , f(x) →+∞ are:

f(x) = (1/4)*x + 3

h(x) = 2x - 1

The end behavior of a function studies how the function behaves as x tends to infinity or negative infinity.

The function will tend to positive infinity as x tends to positive infinity if the function increases for x > 0.

Then we need to identifty which of these two functions are increasing, the two increasing ones are:

f(x) = (1/4)*x + 3

h(x) = 2x - 1

These two are linear equations with positive solpes, so these are increasing functions, and as x→+∞ , f(x) →+∞

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To model the path of the rock, we can use a quadratic equation of the form y = ax^2 + bx + c, where y is the height of the rock, x is the distance from the catapult, and a, b, and c are constants.
We can use the given information to find the values of a, b, and c.

First, we know that the rock launches out of the catapult at a height of 0 feet, so when x = 0, y = 0. This means that c = 0.
Next, we know that the rock reaches its maximum height of 50 feet when it is 75 feet away from the catapult, so when x = 75, y = 50. We can plug these values into the equation and simplify:
50 = a(75)^2 + b(75) + 0
50 = 5625a + 75b

Finally, we know that the rock hits the wall 8 feet from the top when it is 120 feet away from the catapult, so when
x = 120, y = 40 - 8 = 32. We can plug these values into the equation and simplify:
32 = a(120)^2 + b(120) + 0
32 = 14400a + 120b
We now have a system of two equations with two unknowns:
5625a + 75b = 50
14400a + 120b = 32

We can use substitution or elimination to solve for a and b. Using elimination, we can multiply the first equation by -1.6 to eliminate the b term:
-9000a - 120b = -80
14400a + 120b = 32
Adding the two equations together gives:
5400a = -48
Solving for a gives:
a = -48/5400 = -0.00888888889

We can then plug this value of a back into one of the original equations to solve for b:
5625(-0.00888888889) + 75b = 50
-50 + 75b = 50
75b = 100
b = 100/75 = 1.33333333333

So the equation that models the path of the rock is:
y = -0.00888888889x^2 + 1.33333333333x + 0
Or, rounding to three decimal places:
y = -0.009x^2 + 1.333x

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The correct answer is d. 9.9498.

To find the standard deviation of a distribution, we use the formula:

Standard deviation = √[(∑(x - mean)2)/n]

First, we need to find the mean of the distribution. The mean is the sum of all the values divided by the number of values. In this case, the mean is:

mean = (35 + 250)/5 = 57

Next, we need to find the sum of the squared differences between each value and the mean. This is:

∑(x - mean)2 = (35 - 57)2 + (250 - 57)2 = 485 + 37249 = 37734

Finally, we divide this sum by the number of values and take the square root to get the standard deviation:

Standard deviation = √(37734/5) = √7546.8 = 9.9498

So the standard deviation of the distribution is 9.9498.

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Answer:

Let's use the variable "r" to represent the number of red flowers in each red flower bed.

Then, the total number of red flowers in the garden would be 3r, since there are 3 beds of red flowers.

Similarly, the total number of yellow flowers in the garden would be 3 * 30 = 90, since there are 3 beds of yellow flowers, and 30 flowers in each bed.

So the total number of flowers in the garden is the sum of the red and yellow flowers:

Total number of flowers = 3r + 90

Since there are 135 flowers in the garden, we can set the equation equal to 135 and solve for "r":

3r + 90 = 135

3r = 45

r = 15

Therefore, there are 15 red flowers in each red flower bed, and the total number of red flowers in the garden is 3r = 45. The equation representing the number of red and yellow flowers is:

Total number of flowers = 3r + 90 = 3(15) + 90 = 45 + 90 = 135.

So this equation checks out, as we expect the total number of flowers to be 135.

Answer:

x = 10

Step-by-step explanation:

Since KL is parallel to HI

m∠JKL = m∠JHI

m∠JLK = m∠LIH

For the two triangles JKL and JHI, we have the common angle J and two angles equal to the corresponding two angles

Therefore the triangles are similar

Ratio of the sides to the corresponding sides must be equal

Hence
[tex]\dfrac{JH}{JK} = \dfrac{JI}{JL}\\[/tex]

JH = JK + KH = 28 + 20 = 48

JI = JL + JL = 14 + x

Therefore
[tex]\dfrac{JH}{JK} = \dfrac{JI}{JL}\\\\= > \dfrac{48}{28} = \dfrac{14 + x}{14}[/tex]

[tex]\dfrac{48}{28} = \dfrac{12}{7}[/tex]  by dividing numerator and denominator by 4

Hence

[tex]\dfrac{48}{28} = \dfrac{14 + x}{14}\\\\\rightarrow \quad \dfrac{12}{7} = \dfrac{14 + x}{14}\\\\[/tex]

Multiply both sides by 14:
[tex]14 \cdot \dfrac{12}{7} = 14 \cdot \dfrac{14 + x}{14}\\\\2 \cdot 12 = 14 + x\\\\24 = 14 + x\\\\\text{Subtract 14 both sides:}\\\\24 - 14 = x\\\\or\\\\x = 10\\\\[/tex]

No, the result is not significant at 1 percent level for the p-value of 0.802.

Results are significant at a certain level,

Compare the p-value to the significance level (known as alpha).

The significance level is typically set at 0.05 or 0.01.

Here, the p-value is 0.802, which is much larger than 0.01.

This implies here we cannot reject the null hypothesis at a 1% significance level.

Or , the result is not statistically significant at a 1% level.

If the significance level was 5%  that is equal to 0.05.

The result would not be significant at 5% level either.

If the significance level was 10% which is equal to 0.10.

Then the result would be significant as the p-value is less than 0.10.

Therefore, for the given p-value 0.802 result is not significant at 1%.

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The final answer is degree of P(x) is 10.

The degree of a polynomial is the highest power of x that appears in the polynomial.

In the case of P(x), we can find the degree by multiplying the degrees of each factor together.

The degree of (2x+5) is 1,

the degree of (5x-2)^(2) is 2,

the degree of (x+7) is 1,

and the degree of Q(x) is 6.

Therefore, the degree of P(x) is 1*2*1*6 = 12.

However, we need to subtract 2 from the degree because (5x-2)^(2) is raised to the power of 2.

So the degree of P(x) is 12-2 = 10.

Therefore, the degree of P(x) is 10.

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The distance of the hot-air balloon from Jenny is approximately 12.500 miles

To find the distance of the hot-air balloon from Jenny, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle. In this case, the triangle is formed by Susie, Jenny, and the hot-air balloon.

Let the distance from Susie to the hot-air balloon be x, and the distance from Jenny to the hot-air balloon be y. Using the Law of Sines, we can write the following equation:

(x/sin(40)) = (y/sin(30)) = (25/sin(110))

Now, we can cross multiply and solve for y:

(y)(sin(110)) = (25)(sin(30))

y = (25)(sin(30))/(sin(110))

y = 12.500

Therefore, the distance of the hot-air balloon from Jenny is approximately 12.500 miles.

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Answer:

4.5 teaspoons of vanilla

Step-by-step explanation:

We can do this by multiplying:

3/4*(6)

18/4

9/2

4.5 teaspoons of vanilla

A system of inequalities to represent the constraints of this problem are x ≥ 0 and y ≥ 0.

A graph of the system of inequalities is shown on the coordinate plane below.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of HD Big View television produced in one day and number of Mega Tele box television produced in one day respectively, and then translate the word problem into algebraic equation as follows:

Since the HD Big View television takes 2 person-hours to make and the Mega TeleBox takes 3 person-hours to make, a linear equation to describe this situation is given by:

2x + 3y = 192.

Additionally, TVs4U’s total manufacturing capacity is 72 televisions per day;

x + y = 72

For the constraints, we have the following system of linear inequalities:

x ≥ 0.

y ≥ 0.

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The lengths of the sides of the of the right triangles are found using the Pythagorean Theorem as follows;

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the lengths of the legs of the right triangle.

1) The location of the angles and the congruent 90° angle and a second congruent angle indicates;

ΔABN ~ ΔTBA ~ ΔTAN

The missing values of x can be obtained using Pythagorean Theorem as follows;

2) AB = √(10² + 5²) = √(125) = 5·√5

[tex]\overline{AB}[/tex]² = (5 + x)² - [tex]\overline{AN}[/tex]²

[tex]\overline{AN}[/tex]² = (5 + x)² - [tex]\overline{AB}[/tex]²

[tex]\overline{AN}[/tex]² = 10² - x²

10² + x² = (5 + x)² - [tex]\overline{AB}[/tex]²

10² + x² = (5 + x)² - 125

[tex]\overline{AB}[/tex]² = (5 + x)² - (10² + x²) = 10·x - 75

10·x - 75 = (5·√5)² = 125

10·x - 75 = 125

10·x = 125 + 75 = 200

x = 200/10 = 20

x = 20

3) The hypotenuse side of the right triangle with sides 8 and 4 can be found as follows;

Length of the hypotenuse = √(8² + 4²) = 4·√5

Length of the leg of the larger right triangle is, length = √(8² + x²)

Therefore;

(x + 4)² = (8² + x²) + (4·√5)²

(x + 4)² - (8² + x²) = (4·√5)²

8·x - 48 = 80

8·x = 80 + 48 = 128

x = 128/8 = 16

x = 16

4) The leg of the larger right triangle = (12 + 2)² - x² = 14² - x²

14² - x² - 12² = x² - 2²

2·x² = 14² - 12² + 2² = 56

x² = 56/2 = 28

x = √(28) = 2·√7

x = 2·√7

5) The length of the shorter leg of the larger right triangle can be found as follows;

Length of the shorter leg = (20 + 25)²- x²

x² - 25² = (20 + 25)²- x² - 20²

2·x² = (20 + 25)² + 25² - 20² = 2250

x² = 2250/2 = 1125

x = √(1125) = 15·√5

x = 15·√5

6) x² - 4² = (32 + 4)² - x² - 32²

2·x² = (32 + 4)² + 4² - 32² = 288

x² = 288/2 = 144

x = √(144) = 12

x = 12

7) Let x represent the length of the right tringle and let h represent the altitude of the right triangle

(PR)² = 16² - x²

16² - x² - 12² = x² - 4²

2·x² = 16² - 12² + 4² = 128

x² = 128/2 = 64

x = √(64) = 8

The length of the short leg is; x = 8

Length of the longer leg, PR = √(16² - x²)

PR = √(16² - 8²) = 8·√3

Length of the longer leg = 8·√3

The square of the altitude = 16² - x² - 12²

Length of the altitude = √(16² - 64 - 12²) = 4·√3

8) Let x represent the length of the shorter leg, we get;

(PR)² = 18² - x²

The square of the altitude, (PS)² = 18² - x² - 15² = x² - 3²

2·x² = 18² - 15²+ 3² = 108

x² = 108/2 = 54

x = √(54) = 3·√6

Length of the shorter leg, x = 3·√6

PR = √(18² - 54) = 3·√(30)

Length of the longer leg, PR = 3·√(30)

Length of the altitude, PS = √(54 - 3²) = 3·√(15)

9) Let PQ = x, we get;

(QR)² = 30² - x²

30² - x² - (30 - 6)² = x² - 6²

2·x² = 30² - (30 - 6)² + 6² = 360

x² = 360/2 = 180

x = √(180) = 6·√5

PQ = x = 6·√5

(QR)² = 30² - 180 = 720

QR = √(720) = 12·√5

QS = √(720 - (30 - 6)²) = 12

The altitude, QS = 12

The geometric mean of 2 numbers is the square root of the product of the numbers;

10) The geometric mean of 5 and 8 = √(5 × 8) = 4·√10

11) 7 and 11

The geometric mean = √(7 × 11) = √(77)

12) 4 and 5

The geometric mean is; √(4 × 5) = 2·√5

13) 2 and 25

The geometric mean is; √(2 × 25) = √(50) = 5·√2

14) 6 and 8

The geometric mean is; √(6 × 8) = √(48) = 16·√3

15) 8 and 32

The geometric mean is; √(8 × 32) = 16

16) (15 + 5)² - y² = z²

15² + x² = y²

x² + 5² = z²

Therefore;

(15 + 5)² - 15² - x² = x² + 5²

2·x² = (15 + 5)² - 15² - 5² = 150

x² = 150/2 = 75

x = √(75) = 5·√3

x = 5·√3

z² = x² + 5²

z² = 75 + 25 = 100

z = √(100) = 10

z = 10

15² + x² = y²

15² + 75 = 300 = y²

y = √(300) = 10·√3

y = 10·√3

17) 12² - y² = z²

z² - 9² = x²

z² = 9² + x²

x² + 3² = y²

12² - x² - 3²  = z²

12² - x² - 3²  = 9² + x²

2·x² = 12² - 3² - 9² = 54

x² = 54/2 = 27

x = √(27) = 3·√3

x = 3·√3

z² = 9² + x²

z² = 9² + 27 = 108

z = 6·√3

x² + 3² = y²

y² = x² + 3²

y² = 27 + 3² = 36

y = √(36) = 6

y = 6

18) x² - 6² = y²

z² - 24² = y²

(6 + 24)² - x² = z²

Therefore; x² - 6² = z² - 24² = (6 + 24)² - x² - 24²

x² - 6² = (6 + 24)² - x² - 24² = 30² - x² - 24²

x² - 6² = 30² - x² - 24²

2·x² = 30² + 6² - 24² = 360

x² = 360/2 = 180

x = √(180) = 6·√(5)

x = 6·√(5)

z² = (6 + 24)² - x²

z² = (6 + 24)² - 180 = 720

z² = 720

z = √(720) = 12·√5

z = 12·√5

z² - 24² = y²

y² = 720 - 24² = 144

y² = 144

y = 12

19) Let x represent GH, we get;

32² - x² = (HK)²

32² - x² - (32 - 8)² = x² + 8²

2·x² = 32² - (32 - 8)² - 8² = 384

x² = 184

x = √(184) = 2·√(46)

GH = x = 2·√(46)

GH = 2·√(46)

(HK)² = (32 - 8)² + (x² - 8²)²

(HK)² = (32 - 8)² + (184 - 8²) = 696

HK = √(696) = 2·√(174)

HK = 2·√(174)

20) Let x represent the length of the lake, we get;

x² + 6² = (x + 4)² - (4² + 6²) = x² + 8·x - 36

x² + 6² = x² + 8·x - 36

8·x = 6² + 36 = 72

x = 72/8 = 9

The length of the lake, x = 9 km

Learn more on Pythagorean Theorem here: https://brainly.com/question/26830232

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