Water flows from the bottom of a storage tank at a rate of r(t) = 200-4t liters per minute, where...

Question:

Water flows from the bottom of a storage tank at a rate of {eq}r(t) = 200-4t {/eq} liters per minute, where {eq}0 \leq t \leq 50. {/eq} Find the amount of water that flows from the tank during the first {eq}40 {/eq} minutes.

Definite Integrals

The definite integral of a function {eq}\displaystyle f(x), {/eq} over an interval {eq}\displaystyle [a,b] {/eq} is given as {eq}\displaystyle \int_a^b f(x)\ dx. {/eq}

If {eq}\displaystyle f(x) {/eq} is a rate of change of a quantity, {eq}\displaystyle F(x): f(x)=F'(x), {/eq}

then the definite integral above represent the net amount of {eq}\displaystyle F(x) {/eq} between a and b.

If {eq}\displaystyle r(t)=200-4t {/eq} is the rate of water amount that flows out of a tank (liters/minute), then the amount of water drained in the first 40 minutes

is given by the following definite integral

{eq}\displaystyle \text{Amount of Water}=\int_0^{40} (200-4t)\ dt=\left(200t-2t^2\right)\bigg\vert_0^{40}=200\cdot 40-2\cdot 40^2= \boxed{4800\ \rm liters}. {/eq}