- Determine the radius of the circle from the diameter: \(r = \frac{14}{2} = 7 \text{ m}\).
- Use the formula for the area of a circle: \(A = \pi r^2\).
- Substitute the radius to find the area: \(A = \pi(7)^2 = 49\pi \text{ m}^2\).
- Select the matching option: \(\boxed{49 \pi \text{ m}^2}\).
Explanation
- Calculate the radius of the circle
The problem asks for the area of a circle with a given diameter. The diameter \(d\) of the circle is \(14\) meters. The radius \(r\) of a circle is defined as half of its diameter:
\[r = \frac{d}{2} = \frac{14 \text{ m}}{2} = 7 \text{ m}\]
- Calculate the area of the circle
The formula for the area \(A\) of a circle with radius \(r\) is given by:
\[A = \pi r^2\]
Substituting the radius \(r = 7 \text{ m}\) into the formula:
\[A = \pi (7 \text{ m})^2 = \pi \times 49 \text{ m}^2 = 49 \pi \text{ m}^2\]
- Compare with options and conclude
Comparing the calculated area of \(49 \pi \text{ m}^2\) with the provided options:
- \(7 \pi \text{ m}^2\)
- \(14 \pi \text{ m}^2\)
- \(28 \pi \text{ m}^2\)
- \(49 \pi \text{ m}^2\)
The correct option is \(49 \pi \text{ m}^2\). The final answer is \(\boxed{49 \pi \text{ m}^2}\).
Examples
In landscape architecture, calculating the area of a circular garden bed is essential for determining how much mulch or fertilizer is needed. For a circular garden with a diameter of \(14\) meters, you would calculate the radius (\(r = 7 \text{ m}\)) and then the total area (\(A = \pi r^2 = 49\pi \approx 153.94 \text{ m}^2\)) to ensure you purchase the correct amount of supplies for your project. Understanding this relationship between diameter and area helps in accurately estimating resources for any circular space.