Question

Paper water cups in the shape of a cone have a radius of 7 centimeters and a height of 12 centimeters. Cups come in packs of 12. What volume of water will a pack of 12 cups hold, rounded to the nearest hundredth?

Answer 1

To determine the total volume of water that a pack of 12 cone-shaped paper cups can hold, we first need to find the volume of a single cone.

The formula for the volume \( V \) of a cone is given by:

\[
V = \frac{1}{3} \pi r^2 h
\]

where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

For the cups:
- The radius \( r = 7 \) cm,
- The height \( h = 12 \) cm.

Substituting the values into the volume formula:

\[
V = \frac{1}{3} \pi (7)^2 (12)
\]

Calculating \( 7^2 \):

\[
7^2 = 49
\]

Now, substituting back:

\[
V = \frac{1}{3} \pi (49) (12)
\]

Calculating \( 49 \times 12 \):

\[
49 \times 12 = 588
\]

Now we have:

\[
V = \frac{1}{3} \pi (588)
\]

Calculating \( \frac{588}{3} \):

\[
\frac{588}{3} = 196
\]

So, the volume of one cup is:

\[
V = 196 \pi
\]

Next, we calculate the total volume for a pack of 12 cups:

\[
V_{\text{total}} = 12 \times V = 12 \times 196 \pi
\]

Calculating \( 12 \times 196 \):

\[
12 \times 196 = 2352
\]

Thus, the total volume is:

\[
V_{\text{total}} = 2352 \pi
\]

Now, we can use the approximate value of \( \pi \approx 3.14159 \) to calculate the numerical value:

\[
V_{\text{total}} \approx 2352 \times 3.14159 \approx 7390.14568
\]

Rounding to the nearest hundredth:

\[
V_{\text{total}} \approx 7390.15
\]

Therefore, the total volume of water that a pack of 12 cups can hold is:

\[
\boxed{7390.15} \text{ cm}^3
\]