Answer:
- radius: 3.671 cm
- height: 11.810 cm
Explanation:
The total cost of producing a cylindrical can with radius r and height h will be ...
cost = (lateral area)×(side cost) +(end area)×(end cost) +(seam length)×(seam cost)
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The lateral area (LA) is ...
LA = 2πrh
Since the volume of the can is fixed, we can write the height in terms of the radius using the volume formula.
V = πr²h
h = V/(πr²)
Then the lateral area is ...
LA = 2πr(V/(πr²)) = 2V/r = 2·500/r = 1000/r
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The end area (EA) is twice the area of a circle of radius r:
EA = 2×(πr²) = 2πr²
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The seam length (SL) is twice the circumference of the end:
SL = 2×(2πr) = 4πr
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So, the total cost in cents of producing the can, in terms of its radius, is ...
cost = (1000/r)(0.01) +(2πr²)(.012) +(4πr)(0.015)
We can find the minimum by setting the derivative to zero.
d(cost)/dr = -10/r² +0.048πr +.06π = 0
Multiplying by r² gives the cubic ...
0.048πr³ +0.06πr² -10 = 0
r³ +1.25r² -(625/(3π)) = 0 . . . . . . divide by .048π
This can be solved graphically, or using a spreadsheet to find the value of r to be about 3.671 cm. The corresponding value of h is ...
h = 500/(π·3.671²) ≈ 11.810 . . . cm
The minimum-cost can will have a radius of about 3.671 cm and a height of about 11.810 cm.
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A graphing calculator can find the minimum of the cost function without having to take derivatives and solve a cubic.