   Chapter 9.3, Problem 51E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Heat index The highest recorded temperature in the state of Alaska was 100°F , occurring on June 27,1915, at Fort Yukon. The heat index is the apparent temperature of the air at a given temperature and humidity level. If x denotes the relative humidity (in percent), then the heat index (in degrees Fahrenheit) for an air temperature of 100°F can be approximated by the function f ( x ) =   0.009 x 2 +   0.139 x   +   91.875 (a) At what rate is the heat index changing when the humidity is 50 % ?(b) Write a sentence that explains the meaning of your answer in part (a).

(a)

To determine

To calculate: The instantaneous rate of change of heat index when humidity is 50% if the heat index for an air temperature of 100F is given by

f(x)=0.009x2+0.139x+91.875

where x is the relative humidity in percent.

Explanation

Given Information:

Humidity is 50% and the heat index for an air temperature of 100F is given by

f(x)=0.009x2+0.139x+91.875

where x is the relative humidity in percent.

Formula used:

If a function f(x) is defined in the interval [x,x+h], then its instantaneous rate of change is given by the derivative as

f(x)=limh0f(x+h)f(x)h

Calculation:

Consider the provided statement,

Humidity is 50% and the heat index for an air temperature of 100F is given by

f(x)=0.009x2+0.139x+91.875

where x is the relative humidity in percent.

Substitute x=x+h in f(x) to get,

f(x+h)=0.009(x+h)2+0.139(x+h)+91.875=0.009(x2+h2+2xh)+0.139x+0.139h+91.875=0

(b)

To determine

The meaning of instantaneous rate of change of heat index obtained when humidity is 50% if the heat index for an air temperature of 100F is given by

f(x)=0.009x2+0.139x+91.875

where x is the relative humidity in percent.

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