An opentop cylindrical container is to have a volume 1331 cm

An open-top cylindrical container is to have a volume 1331 cm^3. What dimensions (radius and height) will minimize the surface area? The radius of the can is about cm and its height is about cm. (Do not round until the final answer. Then round to two decimal places as needed.)

Solution

Volume of the cylinder = 1331 cm³

pi * r² *h = 1331

22/7 * r² *h = 1331

r² h = 847 / 2

h = 847 / 2r²            ... (1)

Surface area with open top of a cyliner (A) = A_base + A_side

                                                               = pi*r² + 2 * pi * r * h

                                                                 = pi*r(r+ 2h)

                                                                 = pi *r (r + 2*847/2r²)           from (1)

                    = pi*r (r + 847/r²)

                   = pi*r² + 847 / r

A = pi * r² + 847 / r

To find minimum area, differentiate with respect to \"r\" and equate to zero

dA/dr = d/dr(pi * r² + 847 / r)

dA/dr = d/dr(pi * r² + 847 r^-1)

dA/dr = 2*pi*r -847/ r²

equating this to zero

2*pi*r -847/ r² = 0

(44 /7) r³ - 847 = 0 // multiplying both sides by r²

44 r³ - 5929 = 0

44r³ = 5929

r³ = 5929/44

r = 5.13 cm

from (1)

h = 847/2r²

h = 16.11 cm

radisu = 5.13 cm and height = 16.11 cm