A candy company needs a custom box for their truffles. The box they've chose is in the shape of a cylinder with a hemisphere of the same radius on top. The total volume of the box is V= (1/2) ((4pir^2)/(3))+ pir^2 (y-r), where y is the height of the box and r is the radius of the box. Originally, the candy box was designed to have a height of 6 inches and a radius of 2 in, but the shipper suggests that the boxes be made slightly shorter. You now need to adjust the radius so that the height is reduced to 5.75 in. but the volume remains constant.

Question

A candy company needs a custom box for their truffles. The box they've chose is in the shape of a cylinder with a hemisphere of the same radius on top. The total volume of the box is V= (1/2) ((4pir^2)/(3))+ pir^2 (y-r), where y is the height of the box and r is the radius of the box. Originally, the candy box was designed to have a height of 6 inches and a radius of 2 in, but the shipper suggests that the boxes be made slightly shorter. You now need to adjust the radius so that the height is reduced to 5.75 in. but the volume remains constant.
A.Find the value of dr/dy at the point r=2, y=6.
B.Use your value of dr/dy to approximate the new radius for these boxes.

Steve · Answer

The formula has a typo. Should be
V= (1/2) ((4pir^3)/(3))+ pi r^2 (y-r)
or
v = 2π/3 r^3 + πr^2 (y-r)
= πr^2 y - π/3 r^3

2πryr' + πr^2 - πr^2 r' = 0
r' = r/(r-2y)
r'(6) = 2/(2-12) = -1/5

dr = dr/dy * dy
= -1/5 (-.25) = 0.05
so, r = 2.05

Austin gold · Answer

the answer is 2.05 inches