MATH SOLVE

4 months ago

Q:
# A solid is formed by adjoining two hemispheres to the ends of a circular cylinder. the radius of the hemispheres is the same as the radius of the cylinder. the total volume of the solid is 16 cubic inches. what radius should the cylinder be to produce the minimum surface area?

Accepted Solution

A:

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

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