Question

Find the surface area of the cone. Leave answer in terms of π (1 point) Responses 35.2 π 35.2 pi 37.5 π 37.5 pi 65.2 π 65.2 pi 28.5 π (height is 9mm. Bottom is 3mm)

Answer 1

To find the surface area of a cone, we can use the formula:

\[
\text{Surface Area} = \pi r (r + l)
\]

where \( r \) is the radius of the base, and \( l \) is the slant height of the cone.

Given:
- Height (\( h \)) = 9 mm
- Bottom radius (\( r \)) = 3 mm

First, we need to find the slant height (\( l \)). The slant height can be calculated using the Pythagorean theorem:

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

Substituting the values:

\[
l = \sqrt{(3)^2 + (9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
\]

Now we can substitute \( r \) and \( l \) back into the surface area formula:

\[
\text{Surface Area} = \pi r (r + l) = \pi (3) (3 + 3\sqrt{10})
\]

Calculating this:

\[
= \pi (3) (3 + 3\sqrt{10}) = 3\pi (3 + 3\sqrt{10}) = 9\pi + 9\sqrt{10}\pi
\]

Now to see if this matches with the provided options. However, let's calculate it step-by-step numerically.

1. **Estimate \( 3\sqrt{10} \)**:
Since \( \sqrt{10} \) is approximately 3.162, we can approximate:

\[
3\sqrt{10} \approx 3 \times 3.162 \approx 9.486
\]

2. **Now calculate \( 3 + 3\sqrt{10} \)**:

\[
3 + 3\sqrt{10} \approx 3 + 9.486 \approx 12.486
\]

3. **Finally, calculate the surface area**:

\[
\text{Surface Area} \approx 3 \times 12.486 \approx 37.458 \text{ mm}^2
\]

This value, when rounded, corresponds to \( 37.5 \pi \).

Thus, the surface area of the cone, in terms of \(\pi\), is:

\[
\boxed{37.5 \pi}
\]