A cylindrical package with half a hemisphere to be sent by a postal service can have a maximum combined length (height of the cylinder) and girth (perimeter of a cross section of the cylinder) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. Assume the cross section is a circle. (Make sure you verify that the value you found is the maximum).

Radius of package is 36/π inches

Height of package is 36 inches

Step-by-step explanation:

Here we have

Sum of height plus perimeter of package = 108 inches that is

2πr + h = 108 inches

Where:

Height of package = h

Radius of package = r

Volume of package = πr²h

However, h = 108 - 2πr

Therefore, the equation for the volume of the package is;

Volume of package, V = πr²(108 - 2πr) = πr²×108 - 2π²r³

Differentiating the above equation and equating to zero to find a maximum value, we have;

2πr×108 - 6r²π² = 0

2πr×108 = 6r²π²

36 = r·π

r = 36/π inches

h = 108 - 2πr = 108 - 72 = 36 inches

Radius of package, r = 36/π inches

Height of package, h = 36 inches.

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