The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 40 mm

Question

Please help me 

Answer 1

Jessica,

This is a related rates problem.  If we know the rate of change of one variable, and we can find a relationship between the variables, then we should be able to describe how another variable changes with respect to the first one.

 

Let's look at the problem:

"The radius of a sphere is increasing at a rate of 4 mm/s."  How would we symbolize this mathematically?

Now, can we find a relationship, not between the rates, but between the radius and volume of a sphere?

 

Additionally, what SORT of rate is "4 mm/s"?  That's a time rate - so once you find the relationship between volume and radius for a sphere, if you implicitly differentiate with respect to time, you should get a nice looking relationship that you have all the pieces to solve.

 

Answer 2

Let r = radius at time t

     V = volume at time t

 

GIVEN:  dr/dt = 4

 

FIND:  dV/dt when r = 20  (when the diameter is 40, the radius is 20)

 

V = (4/3)πr³

 

So, dV/dt = (4/3)(3)πr²(dr/dt)

 

              = 4π(20mm)²(4 mm/sec) =  6400π mm³/sec