EXAMPLE 2.6.11 solution

Suppose that it takes 12 hours to decontaminate a circular chemical spill that has a radius of 6 feet. Assuming that the amount of time required to decontaminate such a spill depends upon its size (area), how many hours would it take to decontaminate a similar spill with a radius of 3 feet?

A. 4.5 hours

B. 6 hours

C. 1.5 hours

D. 3 hours

We can solve this problem in two steps. First, we find the number of hours per square foot required to decontaminate one of these spills (using the information from the larger spill). Next, we multiply the time factor from step one by the area of the smaller spill.

» .1061 hours per square foot.

Now, the area of the smaller circular spill is π(3²) »28.274 square feet.

Finally, the amount of time required to clean up the smaller spill is:

(28.274 square feet)(.1061 hours per square foot) » 2.99999 hours

The correct choice is D.

An alternative method for solving this type of problem is to use a proportion. The disadvantage to this alternative method is that it requires a bit more algebra than the other approach.

To use a proportion, we rely on the following observation:

The number of hours required to decontaminate the smaller spill, in proportion to the area of the smaller spill, should be equal to the number of hours required to decontaminate the larger spill in proportion to its area.

We solve this proportion for x by using a variation of “cross multiplication.”