Calculate the area enclosed by shown ellipse

Options

A. `2 pi`
B. `3 pi`
C. `6 pi`
D. `3/2 pi`

Explanation

To calculate the area enclosed by the given ellipse, we will follow these steps:

### Step 1: Identify the semi-major and semi-minor axes
- The semi-major axis (a) is the longest radius of the ellipse, while the semi-minor axis (b) is the shortest radius.
- From the problem, we know that the semi-major axis is represented by the distance from the center of the ellipse to the edge along the x-axis, and the semi-minor axis is represented by the distance from the center to the edge along the y-axis.

### Step 2: Determine the lengths of the semi-major and semi-minor axes
- The length of the semi-major axis \( a \) can be calculated as the difference between the maximum and minimum x-coordinates of the ellipse.
- Given that the x-coordinates range from 4 to 6, we calculate:
\[ a = \frac{6 - 4}{2} = 1 \text{ unit} \]
- The length of the semi-minor axis \( b \) can be calculated as the difference between the maximum and minimum y-coordinates of the ellipse.
- Given that the y-coordinates range from 3 to 4, we calculate:
\[ b = \frac{4 - 3}{2} = 0.5 \text{ unit} \]

### Step 3: Use the formula for the area of an ellipse
- The area \( A \) of an ellipse is given by the formula:
\[ A = \pi \times a \times b \]
- Substituting the values of \( a \) and \( b \):
\[ A = \pi \times 1 \times 0.5 = 0.5\pi \text{ square units} \]

### Final Answer
The area enclosed by the ellipse is \( 0.5\pi \) square units.

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