You are designing an open-top cylindrical container. The cylinder must have a volume of 81π cm3 . The bottom of the container must be strong and costs $3 per cm2 while the sides cost $1 per cm2 . Find the dimensions of the container that costs the least to make? What is the cost of this container?

Minimum dimensions are r=3cm, h=9cm

Minimum Cost=$254.47

Step-by-step explanation:

Volume of a Cylinder=πr²h

Volume of the Open Top Cylinder=81π cm³.

Therefore:

πr²h=81π

The bottom costs $3 per cm² and the side costs $1 per cm².

Total Surface Area of the open top Cylinder= πr²+2πrh

Cost, C=3πr²+2πrh

As the Volume is fixed.

πr²h=81π

r²h=81

h=81/r²

Modifying C,

We differentiate C with respect to r

C'=

At the minimum cost, C'=0.

Next we solve C'=0 for r

6πr³-162π=0

6πr³=162π

r³=27

r=3

The dimensions of the cylinder at minimum cost are therefore:

r=3 cm

h=81/9=9cm

The minimum cost of the Cylinder

C=3πr²+2πrh

=(3XπX3²)+(2XπX3X9)

=$254.47

correct answers below:

72 is the answer to this problem