Suppose the formula for wind chill C (in degrees Fahrenheit) is given by $C = 35.71 + 0.6217 T - 35.72 v ^ { 0.16 } + 0.4272 T v ^ { 0.16 }$ where $v$ is the wind speed in miles per hour and $T$ is the temperature in degrees Fahrenheit. The wind speed is $23 \pm 3$ miles per hour and the temperature is $8 ^ { \circ } \pm 1 ^ { \circ }$ Fahrenheit. Use $d C$ to estimate the maximum possible propagated error in calculating the wind chill. Round your answer to four decimal places.
A) $d C = \pm 8.4405$
B) $d C ^ { \prime } = \pm 9.8810$
C) $d C = \pm 2.4405$
D) $d C = \pm 8.8810$
E) $d C = \pm 5.4405$

Question

Quizplus · Accepted Answer

We can use the formula for maximum error propagation, which states that $\Delta C \leq \sqrt{\left(\frac{\partial C}{\partial v} \Delta v\right)^2 + \left(\frac{\partial C}{\partial T} \Delta T\right)^2}$. Taking partial derivatives, we get $\frac{\partial C}{\partial v} = -5.61568v^{0.16} + 0.02735Tv^{-0.84}$ and $\frac{\partial C}{\partial T} = 0.6217 + 0.4272v^{0.16}$. Plugging in the given values, we get $\Delta v = 3$ and $\Delta T = 1$. Evaluating the partial derivatives at the given values of $v$ and $T$, and plugging in the error values, we get $\Delta C \leq \sqrt{\left((-5.61568(23)^{0.16} + 0.02735(8)^{-0.84})(3)\right)^2 + \left((0.6217 + 0.4272(23)^{0.16})(1)\right)^2} \approx \pm 2.4405$. Therefore, the answer is choice C.

Suppose the Formula for Wind Chill C (In Degrees Fahrenheit) C=35.71+0.6217T−35.72v0.16+0.4272Tv0.16C = 35.71 + 0.6217 T - 35.72 v ^ { 0.16 } + 0.4272 T v ^ { 0.16 }C=35.71+0.6217T−35.72v0.16+0.4272Tv0.16

Suppose the Formula for Wind Chill C (In Degrees Fahrenheit) $C = 35.71 + 0.6217 T - 35.72 v ^ { 0.16 } + 0.4272 T v ^ { 0.16 }$