A cone-shaped paper drinking cup is to be made to hold 36 cm3 of water. find the height and radius of the cup that will use the smallest amount of paper. (round your answers to two decimal places.)

The formula for volume of cone is:

V = π r^2 h / 3

or

π r^2 h / 3 = 36 cm^3

Simplfying in terms of r:

r^2 = 108 / π h

To find for the smallest amount of paper that can create this cone, we call for the formula for the surface area of cone:

S = π r sqrt (h^2 + r^2)

S = π sqrt (108 / π h) * sqrt (h^2 + 108 / π h)

S = π sqrt (108 / π h) * sqrt[ (π h^3 + 108) / π h]

Surface area = sqrt (108) * sqrt[ (π h + 108 / h^2) ]

Getting the 1st derivative dS / dh then equating to 0 to get the maxima value:

dS/dh = sqrt (108) ((π - 216 / h^3) * [ (π h + 108/h^2) ^-1/2]

Let dS/dh = 0 so,

π - 216 / h^3 = 0

h^3 = 216 / π

h = 4.10 cm

Calculating for r:

r^2 = 108 / π (4.10)

r = 2.90 cm

Answers:

h = 4.10 cm

r = 2.90 cm