Total volume = Volume of Sphere + Volume of Cylinder
16 = (4/3)ĻrĀ³ +Ā ĻrĀ²h
Express h in terms of r:
ĻrĀ²h = 16 - (4/3)ĻrĀ³
h = 16/ĻrĀ² - (4/3)r
Next, let's solve for surface area:
Total Surface Area = SA of sphere + SA of cylinder
A = 4ĻrĀ² + 2Ļrh
Substitute the expression for h:
A = 4ĻrĀ² + 2Ļr[16/ĻrĀ² - (4/3)r]
A = 4ĻrĀ² + 32/r - (8/3)ĻrĀ²
Find the derivative of A with respect to r and equate to zero.
dA/dr = 8Ļr - 32rā»Ā² - (16/3)Ļr = 0
Solve for r:
[(8/3)Ļr - 32/rĀ² = 0]*rĀ²
(8/3)ĻrĀ³ - 32 = 0
rĀ³ = 32*3/8Ļ = 12/Ļ
r =Ā ā(12/Ļ)
r = 1.56 inches
