1. Calculus Word Problems

1. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of 4 cubic inches per second. At what rate is the radius increasing when the radius is 1 inch?

2. Two cars leave at an intersection at 1 pm. The first car is traveling east at 45 miles/hour while the second is traveling north at 60 miles/hour. How fast is the distance between the cars increasing at 3 pm?

3. A truck is dumping sand into a conical pile at the rate of 30 cubic foot/second, and in such a way that the height of the piles is always equal to twice the radius. At what rate is the height increasing when the sand pile has a volume of 300 cubic ft?

4. A conical paper cup has a height of 6 inches, while the radius of the circular opening at the top is 2 inches. The cup was initially filled with water but there is a leak at the bottom. At what rate is water out of the cup when the water level in the cup is 3 inches, and is dropping at the rate of 1 inch per minute?

Thanks for any help.

2. Any help?

3. Originally Posted by teddybear
1. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of 4 cubic inches per second. At what rate is the radius increasing when the radius is 1 inch?
With related rate questions, always start with some kind of expression that you can take the derivative of and hopefully a lot things will make sense from it. Easier said than done sometimes I know.

For this one, try starting with the formula for the volume of a sphere.

$\displaystyle V= \frac{4}{3} \pi r^3$

Now in the problem you are given information on how the volume is changing with respect to time, so "t" for time is the variable which with respect to you need to differentiate the volume.

$\displaystyle \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$

Notice you have to use the chain rule here when taking the derivative of the radius with respect to another variable.

Now we can plug in the info you were given $\displaystyle \frac{dV}{dt}=4$ always, and we are interested when r=1. That leaves us with one variable, $\displaystyle \frac{dr}{dt}$ which is what you are asked to find.