Question

A cylindrical metal can is to have no lid. It is to have a volume of 125π in3. What height minimizes the amount of metal used?

Answer 1

Answer:

Minimum height of metal = 5 inches

Explanation:

Volume of the cylindrical metal = πR²H = 125π

cancelling out π on both sides

R²H = 125

Hence it can be deduced that R² = 25 and H = 5

Hence minimum height of metal = 5 inches

Answer 2

The height which minimizes the amount of metal used is 5 inches.

Given the following data:

• Volume of cylindrical metal = 125π $in^3$

To determine the height which minimizes the amount of metal used:

Mathematically, the total surface area (TSA) of a cylinder is given by the formula:

$TSA = 2\pi rh + \pi r^2$   ....equation 1.

Mathematically, the volume of a cylinder is given by the formula:

$Volume = \pi r^2h$   ....equation 2.

Substituting the given volume into eqn. 2, we have:

$125\pi = \pi r^2h\\\\h=\frac{125}{r^2}$  ....equation 3.

Substituting eqn. 3 into eqn. 1, we have:

$TSA = 2\pi r(\frac{125}{r^2} ) + \pi r^2\\\\TSA = \frac{250}{r} + \pi r^2$

In order to determine the minimal height, we would solve for the radius of the cylinder by taking the first derivative of the total surface area (TSA) and equate it to 0 as follows:

$\frac{-250\pi}{r^2} + 2 \pi r = 0\\\\2 \pi r = \frac{250\pi}{r^2} \\\\2 \pi r^3 =250\pi\\\\r^3=125\\\\r=\sqrt[3]{125}$

r = 5 inches

Now, we can determine the minimal height of the cylinder:

From eqn. 3, we have;

$h=\frac{125}{r^2}\\\\h=\frac{125}{5^2}\\\\h=\frac{125}{25}$

Height, h = 5 inches.

Read more on the total surface area (TSA) of a cylinder here: https://brainly.com/question/22497109