sporting goods manufacturer designs a golf ball having a volume of 2.56 cubic inches. (a) What is the radius of the golf ball? (b) The volume of the golf ball varies between 2.53 cubic inches and 2.59 cubic inches. How does the radius vary? (c) Use the ε-δ definition of limit to describe this situation. Identify ε and δ.

a) The volume is 2.56 in^3

The volume of a sphere is V = (4/3) * 3.14*r^3

where r is the radius, so here we have:

V = 2.56 = (4/3) * 3.14*r^3

r = (2.56*3 / (4*3.14)) ^ (1/3) = 0.8488 inches.

b) the minimum radius can be:

r = (2.53*3 / (4*3.14)) ^ (1/3) = 0.8454 inches

the maximum radius can be:

r = (2.59*3 / (4*3.14)) ^ (1/3) = 0.8520 inches

c) Here radius will be our x (so δ is related to the radius) and the volume is our y (so ε) is related to the volume.

Now, the ε is easy to find, the mean value of the volume is 2.56, and the range is 2.53 to 2.59, so the value of ε = 0.03

For the delta we can do the same thinking, the mean value is r = 0.8488.

The delta will be:

δ = 0.8488 - 0.8454 = 0.0034

δ = 0.8520 - 0.8488 = 0.0032

We need to take the biggest value for delta, so we have:

ε = 0.03 in^3

δ = 0.0034 in