SOLUTION: This is a calculus problem. A cylindrical container is to hold 20π cm3. The bottom is made of a material that costs $0.80 per cm2, and the top is left open (no material needed

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Question 858725: This is a calculus problem. A cylindrical container is to hold 20π cm3. The bottom is made of a material that costs $0.80 per cm2, and the top is left open (no material needed). The material for the curved side costs $0.32 per cm2.
Find the height in centimeters of the most economical container.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the volume of a cylinder, V=pi%2AR%5E2%2AH=20%2Api
R%5E2%2AH=20
Find the total cost by calculating the area of the bottom and area of the side wall and multiplying by cost per unit area.
T=pi%2AR%5E2%280.80%29%2B2%2Api%2AR%2AH%2A%280.32%29
T=0.80%2Api%2AR%5E2%2B0.64%2Api%2AR%2AH
From the volume equation, you can find a relationship between H and R.
H=20%2FR%5E2
Substitute,
T=0.80%2Api%2AR%5E2%2B0.64%2Api%2AR%2A%2820%2FR%5E2%29
T=0.80%2Api%2AR%5E2%2B%2812.8%2Api%29%2AR%5E%28-1%29
Now you have total cost T as a function of one variable.
Take the derivative and set it to zero.
dT%2FdR=1.60%2Api%2AR-12.8%2Api%2AR%5E%28-2%29
1.60%2Api%2AR-12.8%2Api%2AR%5E%28-2%29=0
1.6%2AR=12.8.R%5E2
R%5E3=12.8%2F1.6
R%5E3=8
R=2
Then,
H=20%2F%282%5E2%29=5
.
.
.
T=0.80%2Api%2A4%2B0.64%2Api%2A2%2A5
T=10.053%2B20.106
T=30.16