Question

A 4-foot long steel pipe consists of two concentric cylinders, with the inner cylinder hollowed

out. The radius of the outside of the pipe is 6 inches and the radius of the inside of the pipe
is 5.75 inches.
HINT: The units of measure must be the same! Convert to inches and keep your answer in
terms of π.
A. Determine the volume of metal used to build the pipe.
B. If the pipe is to be powder-coated on the inside and outside surfaces, what is the total
surface area to be powder-coated?

Answer 1

A. volume of the metal used to build the pipe is 141
`\pi` cubic inches.

B. The total surface area to be powder coated is 1122.125 square inches.

Explanation:

The length of the cylinder = 4 feet = 48 inches.

Radius of the outside of the pipe = 6 inches.

Radius of the inside of the pipe = 5.75 inches

A. volume of the metal used to build the pipe = volume of the outside pipe - volume of the inside pipe

volume of a cylinder =
`\pi`
`r^(2)`h

volume of the outside pipe =
`\pi`
`r^(2)`h

=
`\pi` ×
`6^(2)` × 48

= 1728
`\pi` cubic inches

volume of the inside pipe =
`\pi`
`r^(2)`h

=
`\pi` ×
`5.75^(2)` × 48

= 1587
`\pi` cubic inches

volume of the metal used to build the pipe = 1728
`\pi` - 1587
`\pi`

= 141
`\pi` cubic inches

B. Total surface area of a hollow cylinder = 2
`\pi` (
`r_(1)` +
`r_(2)`) (
`r_(2)` -
`r_(1)` + h)

where
`r_(1)` is the inner radius and
`r_(2)` is the outer radius.

= 2
`\pi` (6 + 5.75)(5.75 - 6 + 48)

= 2
`\pi` (11.75 × 47.75)

= 1122.125
`\pi` square inches

The total surface area to be powder coated is 1122.125 square inches.

Answer 2

• The pipe is formed by two concentric cylinders.
• The outside cylinder has 6 inches of radius.
• The inside cylinder has 5.75 inches of radius.

To find the volume of the pipe, we need to subtract the inside cylinder volume from the outside cylinder volume.

Remember that the volume of a circular cylinder is

`V=\pi r^(2) h`

Where
`r` is the radius and
`h` is the height.

Outside cylinder volume.

`V_(outside)=\pi r^(2)h= \pi (6in)^(2) (48in)=1,728 \pi in^(3)`

Inside cylinder volume.

`V_(inside)=\pi r^(2)h= \pi (5.75in)^(2) (48in)=1,587 \pi in^(3)`

Notice that we used the height 4 feet in inches units, that's why the height in the formulas is 48 inches, because each feet is equivalent to 12 inches.

Volume of the pipe.

`V_(pipe)=V_(outside) -V_(inside) =1,728 \pi in^(3)-1,587 \pi in^(3) =141 \pi in^(3)`

(A) Therefore, the volume of metal used to build the pipe is 141π cubic inches.

Now, to know the amount of powder-coat we must use, we need to find the surface area of the pipe, which is basically the sum of the surface area of both cylinders.

Surface area of outside cylinder.

`S_(outside)=2\pi r^(2)+2\pi rh=2 \pi (6in)^(2)+2 \pi (6in)(48in)= 72 \pi in^(2) +576 \pi in^(2) =648 \pi in^(2)`

Surface area of the inside cylinder.

`S_(inside)=2\pi r^(2)+2\pi rh=2 \pi (5.75in)^(2)+ 2 \pi (5.75in) (48in)= 66.13 \pi in^(2) +552 \pi in^(2) =618.13 \pi in^(2)`

The total surface is

`S_(powder)=648 \pi in^(2) + 618.13 \pi in^(2) =1,266.13 \pi in^(2)`

(B) Therefore, we need 1,266.13π sqaure inches of powder to cover the whole pipe.