   Chapter 14.3, Problem 90E

Chapter
Section
Textbook Problem

The wind-chill index is modeled by the functionW = 13.12 + 0.6215T − 11.37v0.16 + 0.3965Tv0.16where T is the temperature (°C) and v is the wind speed (km/h). When T = − 15°C and v = 30 km/h, by how much would you expect the apparent temperature W to drop if the actual temperature decreases by 1°C? What if the wind speed increases by 1 km/h?

To determine

To find: The rate of temperature when the actual temperature decreases by 1°C and if the wind speed increases by 1km/h when T=15°C and v=30km/h .

Explanation

Given:

The wind-chill index function is, W=13.12+0.6215T11.37v0.16+0.3965Tv0.16 , where T is temperature (°C) and v is the wind speed (km/h) .

The value of T=15°C and v=30km/h .

Calculation:

The wind-chill index function is, W=13.12+0.6215T11.37v0.16+0.3965Tv0.16 .

Take the partial derivative with respect to T and obtain WT .

WT=T(13.12+0.6215T11.37v0.16+0.3965Tv0.16)=T(13.12)+T(0.6215T)T(11.37v0.16)+T(0.3965Tv0.16)=0+0.6215T(T)0+0.3965v0.16T(T)=0.6215+0.3965v0.16

Substitute T=15°C and v=30km/h in the above equation,

WT=0.6215+0.3965(30)0.16=0.6215+0.3965(1.7232)=0.6215+0.6833=1.3047

Thus, it is clear that the rate of temperature decreases approximately by 1.3°C when the actual temperature decreases by 1°C

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