Q. 66 - Calculus

Question

Suppose another oil tank with a hole in it takes three

days to leak $11$ gallons of oil. Assuming that the rate of

leakage at time t is given by $r\left(t\right) = kt$ for some constant k,

set up and solve an equation involving a definite integral

to find k.

Step-by-Step Solution

Answer

The rate is $0.0042$

Step 1. Given information: An Oil tank takes thre days to leak 11 gallons of oil

An expression for  rate of leakage is given as $r\left(t\right) = kt$

Step 2. Finding rate

The integral for the given condition is ${\int }_0^{72}ktdt=11$

Now simplify the integral as,

$$\begin{gathered} {\int }_0^{72}ktdt=11 \\ k{\left[\frac{t^2}{2}\right]}_0^{72}=11 \\ k\left(\frac{(72{)}^2}{2}\right)=11 \\ k=0.0042 \end{gathered}$$