Question

A measure of what hot weather feels like to two average persons is the Apparent Temperature Index. A model for this index is \(A=0.885 t-22.4 h+1.20 t h-0.544\) where \(A\) is the apparent temperature in degrees Celsius, \(t\) is the air temperature, and \(h\) is the relative humidity in decimal form. (Source: The UMAP Journal, Fall 1984) (a) Find \(\partial A / \partial t\) and \(\partial A / \partial h\) when \(t=30^{\circ}\) and \(h=0.80\). (b) Which has a greater effect on \(A,\) air temperature or humidity? Explain.

Short answer

At an air temperature of 30 degrees Celsius and relative humidity of 0.80, the rate of change of the apparent temperature with respect to air temperature is 1.845 and with respect to humidity is -14.4. Therefore, humidity has a greater effect on the apparent temperature compared to air temperature.

Step-by-step solution

Calculate the partial derivative of A with respect to t

This involves treating h as a constant and differentiating A with respect to t. So, the partial derivative is obtained as follows: \(\partial A / \partial t = 0.885 + 1.20h\). This shows how much A changes for a unit change in t while keeping h constant.

Calculate the partial derivative of A with respect to h

This involves treating t as a constant and differentiating A with respect to h. So, the partial derivative is obtained as follows: \(\partial A / \partial h = -22.4 + 1.20t\). This shows how much A changes for a unit change in h while keeping t constant.

Evaluate the partial derivatives at given values

Evaluate \(\partial A / \partial t\) and \(\partial A / \partial h\) at \(t = 30\) and \(h = 0.80\). Upon substitution, \(\partial A / \partial t = 0.885 + 1.20(0.80) = 1.845\) and \(\partial A / \partial h = -22.4 + 1.20(30) = -14.4\).

Comparing the effects of air temperature and humidity on the apparent temperature

Upon observing the absolute values of the partial derivatives, it can be seen that the rate of change of A is higher with respect to humidity than air temperature at the given points. Therefore, humidity has a greater effect on the apparent temperature compared to air temperature.