EXAMPLE 2.6.3

The area enclosed by the racetrack below will be landscaped. Find the total cost if landscaping costs $2.00 per square yard. Chose the closest answer.

A. $41,258

B. $60,944

C. $20,315

D. $123,774

SOLUTION

In order to solve this problem we need to find the area (in square yards) of the region contained within the race track, and then multiply by the cost factor of $2 per square yard. However, the region does not correspond to any familiar object from elementary geometry, such as a rectangle, circle, triangle, or parallelogram, for which there is a readily-available area formula. In order to be able to compute the area of this object, we are going to have to make a reasonble assumption: we will assume that the race track figure consists of a rectangle with a half-circle joined to each end:

Thus we compute the area of the figure based on this observation:

Area of figure = area of rectangle + area of two half-circles

Since two half-circles contain the same area as one full circle, the observation above can be restated in this more direct form:

Area of figure = area of rectangle + area of circle

Recall the formulas for the areas of a rectangle and circle, respectively:

Before we use either formula, however, need to to convert units. We want to compute area in square yards, but the measurements in the figures are given in terms of feet. This means that we should convert measurements from feet to yards before using the formulas for area. To convert measurements from feet to yards we divide by 3.

Now we find the area of the rectangle:

Area of rectangle = (100 yd.)(66.66 yd.) = 6,666 sq. yd.

To find the area of the circle, we need to use its radius. The diagram, shows that the diameter of the circle is 66.66 yd. Since radius is half of diameter, the radius of the circle is

(1/2)(66.66) = 33.33 yd.

Now we use the formula for the area of a circle:

Thus the area of the figure is

6,666 + 3490 = 10,156 sq. yd.

Finally, we multiply in order to find the total cost:

Cost = (10,156 sq. yd.)($2 per sq. yd.) =$20,312

This answer doesn't agree exactly with any of the multiple-choice options, due to differences in degree of accuracy in rounding. Clearly, the best choice is C.