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:
- Lateral surface area:
\[ 2\pi R h = 2\pi (0.977)(2) \approx 2\pi (1.954) \approx 12.26 , \text{ft}^2 \]
- 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:
\[ \boxed{18.26 , \text{ft}^2} \]
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