# Definite Integrals and Average Temperature

Here is a problem:

Suppose that at the beginning of day 0, some time last summer, the temperature in Boston was y(0) = 15◦ Celsius and that over a 50-day period, the temperature increased according to the rule y'(t) = y(t)/50, with time t measured in days. Find the formula for y. By definition, day 0 is 0 < t < 1. Find the average temperature on day 3. Find the average on day 4.

Is this correct for the formula for y: y(t) = 15 + t

And then is the average for day 3 the definite integral of that function with limits from 0 to 3?

• You should always test to see if your suggested solution fits the differential equation. If $y_p(t) = 15 + t$, then $y'_p(t) = 1 \neq y_p(t)/50$ – jameselmore Apr 23 '16 at 16:42

To obtain an expression for $y$, you have to solve the differential equation : $$y' = \frac{y}{50}$$ It is a first order linear equation so the solution is of the form : $$y(t)=Ke^{\frac{1}{50}t}, K \in \mathbb{R}$$ And with condition $y(0)=15$ you should be able to find the correct value of $K$. Can you continue from there ?
And the average for day $3$, is the integral on $[2,3]$, the integral from $0$ to $3$ would be the average on the days $1,2$ and $3$.