Wednesday, February 16, 2005

Math Problem III

An indoor swimming pool is 12 feet in width by 30 feet in length. The shallow end is 3.5 feet deep and the deep end is 8 feet deep, and the depth of the pool changes linearly from the deep end to the shallow end.

One standard box of Jell-O brand gelatin (6 ounces) makes four cups of Jell-O when set up.

Calculate the number of boxes of Jell-O which would be required to completely fill the swimming pool. Assume that the volume of the filtration equipment is negligible and that the pool is to be filled with Jell-O completely.

Conversion factors you may need:

1 cubic foot = 7.481 gal
1 gal = 16.0 cups

Advanced students:
A) One package of Jell-O, prepared according to the directions, makes 8 servings of Jell-O which each have 80 calories. What would be the total caloric content of the swimming pool so filled?
B) If a standard caloric intake for an adult is 2000 calories per day, for how many days could a single adult live off of the swimming pool filled with Jell-O? Assume that there is no spoilage and that the Jell-O in the swimming pool is nutritionally adequate to support life.

Answers are located in the comments.

1 comment:

Jessi Guilford said...

Figuring the volume of the pool first:

The pool is in the shape of a trapezoidal prism. The area of a trapezoid is given by the formula (1/2)(H)(B1+B2), where H is the height of the trapezoid and B1 and B2 are the lengths of the two parallel sides.

In this instance, the trapezoid is oriented vertically with respect to the surface of the pool, but the calculations are otherwise straightforward: H = 30 feet, B1 = 3.5 feet, and B2 = 8 feet. Entering these numbers into the equation, we get A=(0.5)(30)(11.5) = 172.5 ft^2.

This figure then needs to be multiplied by the width of the pool to give the volume:

(172.5 ft^2)(12 ft) = 2070 ft^3.

The problem as presented is a simple division problem: the volume of the pool divided by the volume of a Jell-O unit. However, to calculate this, we need to have common units for both volumes. This can be performed all in a single calculation:

(2070 ft^3)/(4 cups / 1 Jell-O unit)*(16 cups / 1 gal)*(7.481 gal / ft^3) = 61942.7 Jell-O units.

Since one cannot buy .7 of a package of Jell-O, we need to round up. The answer is therefore 61,943 packages of Jell-O to fill the pool.

Advanced questions:

A) Each package contains 640 calories (8*80=640). Multiplying this out by the number of packages, we get 39,643,520 calories for the whole pool. (640*61943=39,643,520)

B) Dividing the answer from A) by 2000, we find that the swimming pool could provide 2000 calories per day to an adult for (39,643,520/2000 = 19,822) days, or about 54 years and 3 months. (In actuality, the person would die of technical malnutrition long before this, as Jell-O contains no significant amounts of fat, vitamins, or minerals, and is an incomplete protein.)