.
A cylinder shaped can needs to be constructed to hold 500 cubic centimeters of soup.
The material for the sides of the can costs 0.04 cents per square centimeter.
The material for the top and bottom of the can need to be thicker, and costs 0.07 cents per square centimeter.
Find the dimensions for the can that will minimize production cost (round to 4 digits).
~~~~~~~~~~~~~~~
r and h are in centimeters.
Cost for the top and bottom materials, total is + = cents.
Cost for the lateral surface is cents.
The function to minimize is F(r,h) = . (1)
The restriction is = 500 (2) (the volume).
From the restriction, h = . (3).
Substitute (3) into (2). Then the function to minimize is
f(r) = = .
To find the minimum of f(r), take the derivative and equate it to zero. You will get
f'(r) = - = 0,
which implies
= 20, = = = 22.74795,
r = = 2.83 cm.
Then h = = = 19.88 cm.
ANSWER. r = 2.83 cm; h = 19.88 cm.
Solved.