The Question
You want to estimate the rate at which the temperature (degrees F) in a concert hall changes per hour, dT/dt, based upon how many people are in the room, p, and the total heat output of equipment in the room, measured in kilowatts, ℎ. At a given moment, people are flooding into the concert hall at a rate of 100 people per hour and electronics are consuming 500 kW each hour. It is estimated that the temperature in the room should increase by 0.1 degrees F per person and that the temperature rises by 1.2 degrees per kW of energy. Estimate dT/dt and explain precisely what it means. So far I've gathered that we have:
My Issue
So right off the bat, I am pretty sure I will need to use the chain rule to calculate dT/dt, the issue for me is isolating the necessary information from the text. So basically I'd like to know what each part of the chain rule (shown below) is in relation to this word problem.
$\frac{dT}{dt} = \frac{∂T}{∂x}\frac{dx}{dt} + \frac{∂T}{∂y}\frac{dy}{dt} $