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

The volume of metal used is 141π cubic inches.

The total surface area to be powder-coated is 1,266.13π square inches.

Explanation:

The pipe is formed by two cylinder.

We know that the height of the pipe is 4 feet, which is equivalent to 48 inches, because 1 feet equals 12 inches. So, the height of both cylinders is 48 inches.

Now, the volume of a circular cylinder is

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

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

For the outside cylinder, we have

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

For the inner cylinder, we have

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

Notice that the volume of the pipe is difference between the outside cylinder and the inside cylinder

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

Therefore, the volume of metal used is 141π cubic inches.

Now, the total surface is sum of the surface of both cylinders.

`S_(total)=(2 \pi r_(out) ^(2)+2 \pi r_(out) h)+(2 \pi r_(in) ^(2)+2 \pi r_(in) h)\\S_(total)=(2 \pi (6in)^(2)+2\pi (6in)(48in) )+(2 \pi (5.75in)^(2)+2\pi (5.75in)(48in)\\S_(total)=(72\pi in^(2) +576 \pi in^(2) )+(66.13 \pi in^(2) +552 \pi in^(2))\\ S_(total)=1,266.13 \pi in^(2)`

Therefore, the total surface area to be powder-coated is 1,266.13π square inches.