Question

An upscale resort has built its circular swimming pool around a central area that contains a restaurant. The central area is a right triangle with legs of 60 feet, 120 feet, and approximately 103.92 feet. The vertices of the triangle are points on the circle. The hypotenuse of the triangle is the diameter of the circle. The center of the circle is a point on the hypotenuse (longest side) of the triangle. The building permit the resort obtained requires that the resort state how much water the pool will hold so the city can manage the resort’s water rights effectively.
The resort owner wants to line the largest circular edge of the pool with 2 inches of 24K gold leaf. Explain how to determine how many linear feet of gold leaf you should plan to cover.

What is the area of the largest section of the pool? Explain if you feel this area would be large enough to add a waterslide.

Inset in the floor of the restaurant is a circular fish tank with a diameter of 10 feet. The fish tank is concentric with the pool. Describe how to find the center point for that circle.

If the pool’s depth averages 4 feet, how much water will it take to fill the pool? (Hint: the volume of a cylinder is the area of the base times the depth.)

City ordinances only allow the resort to use 40,000 cubic feet of water in the pool without the fish tank. Show how to calculate the maximum depth the pool can be to stay under the 40,000-cubic-foot threshold.

How much water do you need to fill just the pool without the fish tank, if the average depth is 4 feet? Explain the difference in your calculations for questions 5 and 6. Does it change the maximum average depth of your water?

Answer 1

The triangle is inside the circle with the vertices of the triangle on the circle - since the center of the circle is on the hypotenuse, the hypotenuse is the diameter(D) of the circle

:

D = 120 feet

:

circumference of the circle = 22/7 * 120 = 377.14 feet

perimeter of triangle = 120 + 60 + 103.92 = 283.92

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1) total distance around circle + triangle = 661.06 feet

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2) the vertices of the triangular central area must lie on the

circular pool - construct perpendicular bisectors of two sides

of the triangular area, where they intersect is the center of

the circular pool, the distance from the center to any one of

the vertices of the triangular area is the radius of the

circular pool

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3) the intersection of the angle bisectors of the triangular area

is the center of the shark tank

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4) volume of pool = ((pi * 60^2) - area of triangle) * 4 =

(11314.285) - (1/2 * 103.92 * 60) * 4 = 32786.74 cubic feet

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5) 8196.685 * x < 30000

x < 3.66 feet

average depth must be less than 3.66 feet

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