# Thread: Word Problem involving Integration

1. ## Word Problem involving Integration

Any help as to how to get started or get through the problem will be much appreciated:

An oil spill is spreading in a circular pattern from a damaged tanker on a calm sea. The radius of the circle is growing at a rate of r'(t)=40 times the radical of t feet per minute.

A)How much does the radius grow from time t= 100 to t= 225?
B) When will the radius be 1,000 feet long?

2. Originally Posted by blurain
Any help as to how to get started or get through the problem will be much appreciated:

An oil spill is spreading in a circular pattern from a damaged tanker on a calm sea. The radius of the circle is growing at a rate of r'(t)=40 times the radical of t feet per minute.

A)How much does the radius grow from time t= 100 to t= 225?
B) When will the radius be 1,000 feet long?
$\displaystyle r(t) = \int r'(t)~dt = \int 40 \sqrt{t}~dt = 40 \int t^{\frac 12}~dt$ ----> use the power rule

(A) we want $\displaystyle \int_{100}^{225} r'(t)~dt = r(225) - r(100)$

or you can use the second fundamental theorem of calculus if you which, but direct integration is more fun

(B) we want $\displaystyle r(t) = 1000$ and solve for t

3. Is r(t)=80/3 t^3/2 the correct function for the derivative r'(t)= 40 times the radical of t?

4. Originally Posted by blurain
Is r(t)=80/3 t^3/2 the correct function for the derivative r'(t)= 40 times the radical of t?
yes, but it should really be $\displaystyle \frac {80}3t^{3/2} + C$ but we know that C = 0, so it's fine