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 22 ± 3 miles per hour and the temperature is 80 ± 2°. use dc to estimate the maximum possible propagated error and relative error in calculating the wind - chill.
step 1
first find the partial derivative with respect to t.
∂c/∂t=

substitute for the median value v = 22 and simplify. (round your answer to four decimal places.)
∂c/∂t=

Explanation:

Step1: Find partial derivative with respect to T

Given $C = 35.74+0.6215T - 35.75v^{0.16}+0.4275Tv^{0.16}$. When finding $\frac{\partial C}{\partial T}$, we treat $v$ as a constant. The derivative of a constant ($35.74$) with respect to $T$ is $0$. The derivative of $0.6215T$ with respect to $T$ is $0.6215$. The derivative of $- 35.75v^{0.16}$ with respect to $T$ is $0$ (since it's a function of $v$ only). The derivative of $0.4275Tv^{0.16}$ with respect to $T$ is $0.4275v^{0.16}$. So, $\frac{\partial C}{\partial T}=0.6215 + 0.4275v^{0.16}$.

Step2: Substitute $v = 22$

Substitute $v = 22$ into $\frac{\partial C}{\partial T}$. We get $\frac{\partial C}{\partial T}=0.6215+0.4275\times22^{0.16}$. First, calculate $22^{0.16}$. Let $x = 22^{0.16}$, then $\ln x=0.16\ln22$. $\ln22\approx3.09104$, so $0.16\ln22\approx0.16\times3.09104 = 0.4945664$. Then $x = e^{0.4945664}\approx1.6397$. Now, $0.4275\times1.6397\approx0.7009$. And $0.6215 + 0.7009=1.3224$.

Answer:

$1.3224$