Answer:
r = 1,248 in
Explanation:
v(c) = 12 in³
The surface area of a right cylinder is:
Area of the base and top + lateral area
S(a) = 2*π*r² + 2*π*r*h (1)
v(c) = 12 in³ = π*r²*h h is the height of the cylinder, then
h = 12 / π*r²
By substitution, in equation (1) we get the Surface area as a function of r
S(r) = 2*π*r² + 2*π*r* ( 12 / π*r²)
S(r) = 2*π*r² + 24 /r
Tacking derivatives on both sides of the equation we get:
S´(r) = 4*π*r - 24 /r²
S´(r) = 0 4*π*r - 24 /r² = 0 π*r - 6/r² = 0
π*r³ - 6 = 0
r³ = 1,91
r = 1,248 in
How do we know that the value r = 1,248 makes Surface area minimum??
We get the second derivative
S´´(r) = 4*π + 48/r³ S´´(r) will be always positive therefore we have a minumum of S at the value of r = 1,248 in