A steel drum in the shape of a right circular cylinder is required to have a volume of 100 cubic feet. express the amount of material required to make the drum as a function of the radius r of the cylinder. also how much material is required if the drum is of radius 3 feet?

V = pi r^2 h

h = V/pi*r^2 = 100/(pi*r^2)
A = 2pi*rh + 2pir^2
A = 2*pi*r*100/(pi*r^2) + 2pi*r^2
= 200/r + 2pi*r^2

A(3) = 200/3 + 2pi*9 = 123.2 ft^2

Well, to determine the amount of material required to make the drum, we need to find the surface area of the drum.

The formula for the surface area of a right circular cylinder is given by:

Surface Area = 2πr^2 + 2πrh

Since we know the volume of the drum is 100 cubic feet, we can express the height (h) in terms of the radius (r) using the formula for the volume of a cylinder:

Volume = πr^2h

100 = πr^2h

Solving for h, we have:

h = 100/(πr^2)

Now, substituting the value of h into the surface area formula, we get:

Surface Area = 2πr^2 + 2πr(100/(πr^2))
= 2πr^2 + 200/r

Therefore, the amount of material required to make the drum, as a function of the radius r, is given by 2πr^2 + 200/r.

If the radius of the drum is 3 feet, we can substitute r = 3 into the formula:

Amount of Material Required = 2π(3)^2 + 200/3
= 18π + 200/3

So, if the drum has a radius of 3 feet, approximately 18π + 66.67 feet of material would be required. (approximations can be fun too, right?)

To find the amount of material required to make the drum as a function of the radius, we need to determine the surface area of the drum.

The surface area of a cylinder can be calculated using the formula:

Surface Area = 2πr^2 + 2πrh

Where r is the radius of the cylinder and h is the height (or length) of the cylinder.

Since the drum is in the shape of a right circular cylinder, we can express the height in terms of the volume.

The formula for the volume of a cylinder is:

Volume = πr^2h

Rearranging the formula, we can solve for h:

h = Volume / (πr^2)

Now, let's substitute the expression for h in the surface area formula:

Surface Area = 2πr^2 + 2πr(Volume / (πr^2))

Simplifying, we get:

Surface Area = 2πr^2 + 2(Volume / r)

Therefore, the amount of material required to make the drum as a function of the radius r is given by the formula:

Material Required = 2πr^2 + 2(100 / r)

Now, let's calculate the amount of material required if the drum has a radius of 3 feet:

Material Required = 2π(3)^2 + 2(100 / 3)
= 2π(9) + 2(33.33)
= 18π + 66.66
≈ 18(3.14) + 66.66
≈ 56.52 + 66.66
≈ 123.18 square feet

Therefore, if the drum has a radius of 3 feet, approximately 123.18 square feet of material is required.

To find the amount of material required to make the drum, we need to calculate the surface area of the cylinder.

The volume of a right circular cylinder is given by the formula:

V = π * r^2 * h

Where V is the volume, r is the radius, and h is the height of the cylinder. In this case, the volume is given as 100 cubic feet.

Since we are given the volume and want to express the amount of material required as a function of the radius, we can rearrange the formula to solve for the height:

h = V / (π * r^2)

Now, let's calculate the surface area of the cylinder. The surface area consists of the top and bottom circle and the curved side.

The top and bottom circles have the same area, given by the formula:

A_circle = π * r^2

The curved side has an area equal to the circumference of the circle multiplied by the height:

A_side = 2π * r * h

Therefore, the total surface area of the cylinder is given by:

A_total = 2 * A_circle + A_side
= 2 * π * r^2 + 2π * r * h
= 2π * r^2 + 2π * r * (V / (π * r^2))
= 2π * r^2 + (2V) / r

Now, we have the amount of material required expressed as a function of the radius:

A_total = 2π * r^2 + (2V) / r

To find the amount of material required if the drum has a radius of 3 feet, we substitute r = 3 into the equation:

A_total = 2π * (3^2) + (2 * 100) / 3
= 18π + 200/3
≈ 57.13 square feet

Therefore, if the drum has a radius of 3 feet, approximately 57.13 square feet of material would be required.