Question

Area is 3ft. Height is 2ft. We want to wrap this cylinder in gift wrapping paper. what is the total surface area rounded to the nearest hundredth?

Surface area:
PRISM S=Ph. S=ph+2b
Cylinder. SpiRh. S=2piRh+2piR^2

Answer 1

Wrong

Answer 2

To calculate the total surface area of a cylinder, we use the formula:

\[
S = 2\pi R h + 2\pi R^2
\]

where:
- \( S \) is the total surface area,
- \( R \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.

From the information given, the area (which I assume is the base area of the cylinder) is \( 3 \, \text{ft}^2 \), and the height \( h \) is \( 2 \, \text{ft} \).

The area of the base of a cylinder is given by:

\[
\text{Area}_{\text{base}} = \pi R^2
\]

Since we are given that the area of the base is \( 3 \, \text{ft}^2 \), we can set up the equation:

\[
\pi R^2 = 3
\]

To find \( R \), we need to solve for \( R^2 \):

\[
R^2 = \frac{3}{\pi}
\]

Then, we take the square root of both sides:

\[
R = \sqrt{\frac{3}{\pi}}
\]

Now, we can substitute \( R \) and \( h \) into the surface area formula.

First, let's calculate \( R \):

\[
R \approx \sqrt{\frac{3}{3.14159}} \approx \sqrt{0.95493} \approx 0.977
\]

Now, using \( R \approx 0.977 \, \text{ft} \) and \( h = 2 \, \text{ft} \):

Now, we compute the total surface area \( S \):

\[
S = 2\pi R h + 2\pi R^2
\]

Calculating each part:

1. **Lateral surface area**:

\[
2\pi R h = 2\pi (0.977)(2) \approx 2\pi (1.954) \approx 12.26 \, \text{ft}^2
\]

2. **Area of the two bases**:

\[
2\pi R^2 = 2\pi \left(\frac{3}{\pi}\right) = 2 \cdot 3 = 6 \, \text{ft}^2
\]

Adding both components together:

\[
S \approx 12.26 + 6 = 18.26 \, \text{ft}^2
\]

Thus, the total surface area of the cylinder, rounded to the nearest hundredth, is:

\[
\boxed{18.26 \, \text{ft}^2}
\]