Question

Paper water cups in the shape of a cone have a radius of 7 centimeters and a height of 12 centimeters

Answer 1

To calculate the volume and surface area of a cone-shaped paper water cup with a radius of 7 centimeters and a height of 12 centimeters, we can use the following formulas:

### Volume of a 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 and \( h \) is the height.

Plugging in the values:

- \( r = 7 \) cm
- \( h = 12 \) cm

\[
V = \frac{1}{3} \pi (7)^2 (12)
\]
\[
V = \frac{1}{3} \pi (49) (12)
\]
\[
V = \frac{1}{3} \pi (588)
\]
\[
V = 196\pi \approx 615.75 \text{ cm}^3
\]

### Surface Area of a Cone
The formula for the surface area \( A \) of a cone (excluding the base) is:

\[
A = \pi r l
\]

where \( l \) is the slant height of the cone, which can be calculated using the Pythagorean theorem:

\[
l = \sqrt{r^2 + h^2}
\]

Calculating the slant height:

\[
l = \sqrt{(7)^2 + (12)^2}
\]
\[
l = \sqrt{49 + 144}
\]
\[
l = \sqrt{193} \approx 13.89 \text{ cm}
\]

Now we can calculate the lateral surface area:

\[
A = \pi (7)(13.89)
\]
\[
A \approx 305.84 \text{ cm}^2
\]

### Summary
- **Volume:** \( \approx 615.75 \) cm³
- **Lateral Surface Area:** \( \approx 305.84 \) cm²

So, the paper water cup has a volume of approximately 615.75 cm³ and a lateral surface area of approximately 305.84 cm².