Question

A firework rocket consists of a cone stacked on top of a cylinder, where the radii of the cone and the cylinder are equal. The diameter of the cylindrical base of the rocket is 8 in and the height of the cylinder is 5 in, while the height of the cone is 3 in. Calculate the surface area of the rocket. Leave your answer in terms of π.

184π sq. in.
76π sq. in.
168π sq. in.
88π sq. in.

Answer 1

To find the surface area of the firework rocket, which consists of a cylinder and a cone, we will calculate the surface area of each component separately and then combine them.

### 1. Surface Area of the Cylinder

The formula for the lateral surface area of a cylinder is:

\[
A_{\text{cylinder}} = 2\pi r h
\]

Where:
- \( r \) is the radius of the cylinder.
- \( h \) is the height of the cylinder.

Given:
- The diameter of the cylindrical base is 8 inches, so the radius \( r \) is:

\[
r = \frac{8}{2} = 4 \text{ in}
\]

- The height of the cylinder \( h \) is 5 in.

Plugging in the values:

\[
A_{\text{cylinder}} = 2\pi(4)(5) = 40\pi \text{ sq. in.}
\]

Since the cylinder has a top and doesn't have a bottom (as it is connected to the cone), we also need to add the area of the top (the circle) of the cylinder:

\[
A_{\text{top}} = \pi r^2 = \pi(4^2) = 16\pi \text{ sq. in.}
\]

So the total surface area of the cylinder including the top is:

\[
A_{\text{cylinder, total}} = 40\pi + 16\pi = 56\pi \text{ sq. in.}
\]

### 2. Surface Area of the Cone

For the surface area of the cone, we will use the formula for lateral surface area:

\[
A_{\text{cone}} = \pi r l
\]

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

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

Given:
- The radius \( r \) of the cone is also 4 in (the same as the cylinder).
- The height \( h \) of the cone is 3 in.

Calculating the slant height:

\[
l = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ in}
\]

Now we calculate the lateral surface area of the cone:

\[
A_{\text{cone}} = \pi(4)(5) = 20\pi \text{ sq. in.}
\]

### 3. Adding the Surface Areas

Now, we combine the surface area of the cylinder (without the bottom) and the lateral surface area of the cone:

\[
\text{Total Surface Area} = A_{\text{cylinder}} + A_{\text{top}} + A_{\text{cone}}
\]

\[
\text{Total Surface Area} = 56\pi + 20\pi = 76\pi \text{ sq. in.}
\]

The total surface area of the firework rocket is:

\[
\boxed{76\pi} \text{ sq. in.}
\]