Question

Erik got caught in a bad snow storm in downtown Chicago. He nearly froze during his 12 minute walk from his work at the museum to his car in the parking garage. The temperature was dropping by 4 degrees every minute. When he left the museum, the temperature was 36º F. What was the temperature in Chicago when Erik got to his car?

Answer 1

Given:

The initial temperature (when Erik left the museum) is 36º F.

Time taken for Erik to reach his car from the museum is 12 minutes.

At this time, the temperature was falling at the rate of 4 degrees per minute. In other words, the temperature T was changing at the rate of -4° per minute.

Concept:

The concept of differentiation will be used for this problem.

Now, differentiation of a function refers to the rate of change of that function with respect to one of the variables the function depends on.

So, in this case, temperature function 'T' falls with the passage of time 't'. And so, the rate of change of this temperature function with respect to time may be denoted as the differential.

In other words, we may say $\frac{dT}{dt}$ equals the rate of change of the temperature with time.

Calculation:

Let us denote the temperature at the time Erik left his workplace as T₀ and the temperature at the time Erik reached his car as T₁₂.

Therefore, T₀ = 36 and T₁₂ is what we have to find.

Now, we have been given that the temperature T was changing at the rate of -4° per minute. We may express this as

$\frac{dT}{dt}=-4$

where t denotes time.

Integrating the above equation,

$\int\ {\frac{dT}{dt} } \, dt=\int\ {-4} \, dt\\  $

Therefore, $T=-4t$

This means that at time 't', the temperature change can be calculated as -4 x t

So, when the time 't' is 12 minutes (that is, the time it took Erik to reach his car from the museum), temperature change will be T = (-4)(12) = - 48

This means that the temperature has fallen by a total of 48 degrees from the initial temperature of 36 degress.

So, temperature when Erik reached his car T₁₂ = 36 - 48 = -12 degree F