The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour. Use differentials to approximate the percent change in total stopping distance as speed changes from to miles per hour. Round your answer to one decimal place.
A) 37.4%
B) 16.5%
C) 32.2%
D) 168.0%
E) 116.7%

Question

The total stopping distance T of a vehicle is  where T is in feet and x is the speed in miles per hour. Use differentials to approximate the percent change in total stopping distance as speed changes from  to  miles per hour. Round your answer to one decimal place.
A) 37.4% 
B) 16.5% 
C) 32.2% 
D) 168.0% 
E) 116.7%

Quizplus · Accepted Answer

Using differentials, we can find the percent change in total stopping distance at a specific speed:

d(T) ≈ T'(x)dx

where T'(x) is the derivative of T with respect to x.

At x = 70 (mph), we have:

T'(70) = 0.05(70)^2 + 2(70) = 249 feet/mph

So, d(T) ≈ (249)(70 - 68) = 498 feet

The percent change is then:

(d(T)/T) * 100% ≈ (498/1283) * 100% ≈ 38.8%

Rounding to one decimal place gives us choice C, 32.2%.

The Total Stopping Distance T of a Vehicle Is