Question

Susie called the post office to find out about the shipping tubes that she could buy there. The person helping her told her that the tubes had a surface area of 424 in2 and a radius of 4 inches.

What is the height of the tube?

Hint: Shipping tubes are shaped like cylinders. Use the formula for the surface area of a cylinder.

SA = 2 × × radius2 + 2 × height × radius ×

Answer 1

To find the height of a shipping tube shaped like a cylinder, we can use the formula for the surface area (SA) of a cylinder:

\[
SA = 2\pi r^2 + 2\pi rh
\]

Where:
- \(SA\) is the surface area,
- \(r\) is the radius,
- \(h\) is the height.

Given:
- The surface area \(SA = 424 \, \text{in}^2\)
- The radius \(r = 4 \, \text{in}\)

Plugging the radius into the surface area formula, we can solve for the height \(h\):

1. Substitute the values into the formula:

\[
424 = 2\pi (4^2) + 2\pi (4)h
\]

2. Calculate \(2\pi (4^2)\):

\[
4^2 = 16 \quad \Rightarrow \quad 2\pi (16) = 32\pi
\]

So the equation now is:

\[
424 = 32\pi + 8\pi h
\]

3. To isolate \(h\), first rearrange the equation:

\[
424 - 32\pi = 8\pi h
\]

4. Divide both sides by \(8\pi\):

\[
h = \frac{424 - 32\pi}{8\pi}
\]

5. Calculate \(32\pi\):

Using \(\pi \approx 3.14\):

\[
32\pi \approx 32 \times 3.14 \approx 100.48
\]

So,

\[
424 - 32\pi \approx 424 - 100.48 \approx 323.52
\]

6. Now substitute that back into the height formula:

\[
h \approx \frac{323.52}{8 \times 3.14}
\]

7. Calculate:

\[
8 \times 3.14 \approx 25.12
\]

Now, divide:

\[
h \approx \frac{323.52}{25.12} \approx 12.86
\]

Thus, the height of the shipping tube is approximately:

\[
h \approx 12.86 \, \text{inches}
\]