Question

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?

Answer 1

Answer:

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,

$C=3\pi r^{2}+2 \pi r \frac{81}{r^{2}}$

$C=3\pi r^{2}+ \frac{162 \pi}{r}$

We differentiate C with respect to r

C'=$6\pi r -\frac{162 \pi}{r^2}$

At the minimum cost, C'=0.

Next we solve C'=0 for r

$6\pi r -\frac{162 \pi}{r^2}=0$

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

Answer 2

Answer:

The sides of the container cost the least to make.

The cost of the container = cost of circular part + cost of the sides = 763.5 + 56.56 = $820.06

Step-by-step explanation:

The volume of cylinder is given by the formula V = πr²h.............Equation 1

where r is the radius of the cylinder and h is the height of the cylinder.

from the question 81π = πr²h ...........................................Equation 2

divide both sides of the equation by π we have

81 =r²h  , therefor h = 81/r² =9/r, substitute the value of h into equation 2

81π = πr²*9/r

81 = 9r

r - 9

if r = 9, then h = 9/9 = 1

the Area of the bottom of the container is πr² =81π cm²

cost of the bottom of the container = 3*81π = $763.5 (π = 3.142)

The Area of the side is 2πr*h = 18π cm²

Cost of the sides =1* 18π = $56.56  (π=3.142)

The container is made up of a circle and a rectangle

The dimension of the circular part is 2πr = 18π

The dimension of the rectangular part = 2( L + B) = 2(2πr + h) = 36π + 2

The sides of the container cost the least to make.

The cost of the container = cost of circular part + cost of the sides = 763.5 + 56.56 = $820.06