Question

You are hired by a company to design a right cylindrical can without a top
(they will use a different material for the top). If the can must hold 3007 cm3
Round all answers to four decimal places.
a. What dimensions will minimize the cost of making this can?
radius =
cm height =
cm
b. What is the amount of material needed to make the can described above?
cm2​

Answer 1

Answer:

a.

radius = 9.8551 cm

height = 9.8551 cm

b.

area = 915.3633 cm^2

Step-by-step explanation:

a.

Let the radius of the cylinder be r, and let the height of the cylinder be h.

Volume of the cylinder:

$V = \pi r^2h$

$\pi r^2 h = 3007$

Solve for h:

$h = \dfrac{3007}{\pi r^2}$

Surface area of the cylinder (lateral are + 1 base only):

$A = \pi r^2 + 2 \pi rh$

Substitute h found above:

$A = \pi r^2 + 2 \pi r \times \dfrac{3007}{\pi r^2}$

$A = \pi r^2 + \dfrac{6014}{r}$

Take the first derivative of the area with respect to the radius.

$A = \pi r^2 + 6014r^{-1}$

$\dfrac{dA}{dr} = 2 \pi r + (-1)6014r^{-2}$

$\dfrac{dA}{dr} = 2 \pi r - \dfrac{6014}{r^2}$

Set the derivative equal to zero and solve for r.

$(r^2)2 \pi r - (r^2)\dfrac{6014}{r^2} = 0$

$2 \pi r^3 - 6014 = 0$

$\pi r^3 - 3007 = 0$

$r^3 = \dfrac{3007}{\pi}$

$r = \sqrt[3]{\dfrac{3007}{\pi}}$

$r = 9.8551$

$h = \dfrac{3007}{\pi r^2}$

$h = \dfrac{3007}{\pi (9.8551)^2}$

$h = 9.8551$

radius = 9.8551 cm

height = 9.8551 cm

b.

$A = \pi r^2 + \dfrac{6014}{r}$

$A = \pi (9.8551)^2 + \dfrac{6014}{9.8551}$

$A = 305.1211 + 641.2422$

$A = 915.3633$

area = 915.3633 cm^2