Answer:
Step-by-step explanation:
The formula for the volume of a cone of radius r and height h is ...
V = (1/3)πr²h
Then r² can be found in terms of h and V as ...
r² = 3V/(πh)
The lateral surface area of the cone is ...
A = (1/2)(2πr)√(r² +h²) = πr√(r² +h²)
The square of the area is ...
T = A² = π²r²(r² +h²)
Substituting for r² using the expression above, we have ...
T = π²(3V/(πh))((3V/(πh) +h²) = 9V²/h² +3πVh
We want to find the minimum, which we can do by setting the derivative to zero.
dT/dh = -18V²/h³ +3πV
This will be zero when ...
3πV = 18V²/h³
h³ = 6V/π . . . . . multiply by h³/(3πV)
For V = 27 cm³, the value of h that minimizes paper area is ...
h = 3∛(6/π) ≈ 3.7221029
The corresponding value of r is ...
r = √(3V/(πh)) = 9/√(π·h) ≈ 2.6319242
The optimal radius is 2.632 cm; the optimal height is 3.722 cm.
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The second derivative test applied to T finds that T is always concave upward, so the value we found is a minimum.
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Interestingly, the ratio of h to r is √2.
