A tank is being filled with water at the rate of 300t^(1/2) gallons per hour with t 0 measured in hours. If the tank is originally empty, how many gallons of water are in the tank after 4 hours? The answer is 1600, but I need to understand why.. Calculus help s;

Question

A tank is being filled with water at the rate of 300t^(1/2) gallons per hour with t > 0 measured in hours. If the tank is originally empty, how many gallons of water are in the tank after 4 hours? The answer is 1600, but I need to understand why.. Calculus help s;

anonymous · Answer

It looks like the function would be a graph of a curve so to find the amount of water in the tank you must take the integral of the function to find the area under the curve which would represent the amount of water in the tank. THe integral must have the limits from 0 to 4 which is the time frame you are working with. That is how much water went into the tank from time 0 to time 4 hours.

anonymous · Answer

1600

anonymous · Answer

Let $f(t)$ be the amount of water in the tank.
We want to find $f(4)$ and we know $f(0)=0$ because the tank starts.

We know $f'(t) = 300t^{1/2}$

anonymous · Answer

Fundamental theorem of calculus says: $$
\int_0^4 f'(t) = f(4)-f(0)
$$

anonymous · Answer

So we know $$
f(4) = \int_0^4300t^{1/2}dt + f(0) = \int_0^4300t^{1/2}dt
$$

anonymous · Answer

i got it, thanks!