Tom Sawyer is painting a fence at a rate of (200 – 4*x*) square feet per hour, where *x* is the number of hours since he started painting. If the fence is 800 square feet, how long will it take him to finish painting the fence? Round your answer to the nearest minute.

Answer

We want to find how many hours *H* it takes Tom to paint 800 square feet. In *H* hours Tom can paint

square feet, so we want to find *H* such that

Here we go:

Rearranging, we get

2*H*^{2} – 200*H* + 800 = 0

and dividing through by 2,

*H*^{2} – 100*H* + 400 = 0.

It's not obvious whether we can factor this, but we can use the quadratic formula to solve for *H*:

The two answers are

and

Since 95 hours isn't a realistic amount of time to spend painting a fence, we'll throw that answer out and keep the answer that's approximately 4 hours. The amount 0.174 hours is approximately

0.174 × 60 = 10.44 minutes,

so rounding to the nearest minute Tom spent about 4 hours and 10 minutes painting the fence.