Total volume = Volume of Sphere + Volume of Cylinder
16 = (4/3)πr³ + πr²h
Express h in terms of r:
πr²h = 16 - (4/3)πr³
h = 16/πr² - (4/3)r
Next, let's solve for surface area:
Total Surface Area = SA of sphere + SA of cylinder
A = 4πr² + 2πrh
Substitute the expression for h:
A = 4πr² + 2πr[16/πr² - (4/3)r]
A = 4πr² + 32/r - (8/3)πr²
Find the derivative of A with respect to r and equate to zero.
dA/dr = 8πr - 32r⁻² - (16/3)πr = 0
Solve for r:
[(8/3)πr - 32/r² = 0]*r²
(8/3)πr³ - 32 = 0
r³ = 32*3/8π = 12/π
r = ∛(12/π)
r = 1.56 inches
