Question

A hollow cylindrical copper pipe is 1.70 m long and has an outside diameter of 3.90 cm and an inside diameter of 2.40 cm . How much does it weigh?

Answer 1

Approximately
`45.2\; {\rm kg}` (assuming that the density of the copper is
`8.96\; {\rm g \cdot cm^(-3)}`.)

Step-by-step explanation:

Convert length
`h` of the pipe to centimeters:

`h = 1.70\; {\rm m} = 170\; {\rm cm}`.

If a cylinder is of radius
`r` and height
`h`, the volume of that cylinder will be
`\pi\, r^(2)\, h`.

Let
`r` denote the inner diameter of this pipe, and let
`R` denote the outer diameter of the pipe.

If the pipe is filled with copper, volume of the entire pipe cylinder will be
`V = \pi\, R^(2)\, h`. In reality, the inside of the pipe is a hollow cylinder of radius
`r` and volume
`\pi\, r^(2)\, h`.

To find the volume of copper in this pipe, subtract the volume of the hollow cylinder
`\pi\, r^(2)\, h` from the volume of the entire pipe cylinder
`\pi\, R^(2)\, h`:

`V = \pi\, R^(2) \, h - \pi\, r^(2)\, h = (R^(2) - r^(2))\, \pi\, h`.

Substitute in
`R = 3.90\; {\rm cm}`,
`r = 2.40\; {\rm cm}`, and
`h = 170\; {\rm cm}` to find the volume of this pipe cylinder:

`\begin{aligned}V &= (R^(2) - r^(2))\, \pi\, h \\ &= ((3.90\; {\rm cm})^(2) - (2.40\; {\rm cm})^(2))\, \pi\, (170\; {\rm m}) \\ &\approx 5047.0\; {\rm cm^(3)}\end{aligned}`.

Multiply the volume
`V` of copper in this pipe by density
`\rho` to find mass
`m`:

`\begin{aligned}m &= \rho\, V \\ &\approx (8.96\; {\rm g \cdot cm^(-3)})\, (5047.0\; {\rm cm^(3)) \\ &\approx 45220\; {\rm g} \\ &= 45.2\; {\rm kg}\end{aligned}`.