Suppose the lowest temperature recorded in Indianapolis, Indiana, was If x is the wind speed in miles per hour and , then the wind chill (in degrees Fahrenheit) for an air temperature of can be approximated by the function . At what rate is the wind chill changing when the wind speed is 50 mph? Round your answer to two decimal places.

Question

Suppose the lowest temperature recorded in Indianapolis, Indiana, was  If x is the wind speed in miles per hour and  , then the wind chill (in degrees Fahrenheit) for an air temperature of  can be approximated by the function  . At what rate is the wind chill changing when the wind speed is 50 mph? Round your answer to two decimal places.

Quizplus · Accepted Answer

We need to use the chain rule to find the rate of change of the wind chill with respect to the wind speed. 
Let y = wind chill and f(x) = x - 0.16(31 - t)(0.1385x^0.5 + 0.4799x^(0.16)) be the function that gives the wind chill for an air temperature of t and wind speed of x. 
Then, we have y = f(50) and we want to find dy/dx when x = 50. 
Using the chain rule, we have dy/dx = df/dx * dy/df. 
df/dx = 1 - 0.022055(31 - t)(0.5x^(-0.5) + 0.079984x^(-0.84)). 
dy/df = 1. 
Substituting x = 50 and t = -11, we get: 
df/dx = 1 - 0.022055(31 - (-11))(0.5(50)^(-0.5) + 0.079984(50)^(-0.84)) 
= 0.0716. 
dy/dx = df/dx * dy/df = 0.0716 * 1 = 0.0716. 
Rounding to two decimal places, we get dy/dx ≈ 0.07, which corresponds to choice B.

Suppose the Lowest Temperature Recorded in Indianapolis, Indiana, Was