Question

the formula for wind - chill c (in degrees fahrenheit) is given by c = 35.74+0.6215t - 35.75v^0.16+0.4275tv^0.16 where v is the wind speed in miles per hour and t is the temperature in degrees fahrenheit. the wind speed is 27 ± 3 miles per hour and the temperature is 8° ± 3°. use dc to estimate the maximum possible propagated error (round your answer to four decimal places) and relative error in calculating the wind - chill (round your answer to two decimal places). + dc =
\frac{dc}{c}=
%

Explanation:

Step1: Identify the function

The wind - chill function is $C = 35.74+0.6215T - 35.75V^{0.16}+0.4275TV^{0.16}$, where $V$ is the wind speed and $T$ is the temperature.

Step2: Calculate the partial derivatives

First, find $\frac{\partial C}{\partial T}=0.6215 + 0.4275V^{0.16}$ and $\frac{\partial C}{\partial V}=-35.75\times0.16V^{- 0.84}+0.4275T\times0.16V^{-0.84}=( - 5.72 + 0.0684T)V^{-0.84}$.
Given $V = 27$ mph and $T = 8^{\circ}F$.
$\frac{\partial C}{\partial T}=0.6215+0.4275\times27^{0.16}\approx0.6215 + 0.4275\times1.9777=0.6215+0.8451 = 1.4666$.
$\frac{\partial C}{\partial V}=(-5.72 + 0.0684\times8)\times27^{-0.84}=(-5.72+0.5472)\times0.0814=(-5.1728)\times0.0814=-0.4211$.

Step3: Use the total - differential formula

The total - differential formula is $dC=\frac{\partial C}{\partial T}dT+\frac{\partial C}{\partial V}dV$.
Given $dT = 3$ and $dV = 3$.
$dC=1.4666\times3+( - 0.4211)\times3=(1.4666 - 0.4211)\times3=1.0455\times3 = 3.1365$.
The actual value of $C$ at $V = 27$ and $T = 8$ is $C_1=35.74+0.6215\times8-35.75\times27^{0.16}+0.4275\times8\times27^{0.16}$
$C_1=35.74 + 4.972-35.75\times1.9777+0.4275\times8\times1.9777$
$C_1=35.74 + 4.972-70.703+6.841=-29.028$.
The relative error is $\frac{dC}{C}=\frac{3.1365}{\vert - 29.028\vert}\approx0.11$.
The propagated error $dC\approx3.14$.
The relative error $\frac{dC}{C}\times100\%\approx10.80\%$.

Answer:

The propagated error $dC\approx3.14$ and the relative error is approximately $10.80\%$.