Question

A candy company needs a custom box for their truffles. The box they've chosen 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) (4/3) (pi r^3) + pi r^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 inches, 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 inches but the volume remains constant.

B. Use the value dr/dy = -0.2 to approximate the new radius for these boxes.
(I found this derivative in part A, and it is correct).

Answer 1

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

Step-by-step explanation: