Francesca plays a bass drum. After measuring it, she calculates that it has a circumference of 81.64 centimeters. What is the drum’s area?
Use 3.14 for .
The Correct Answer and Explanation is:
To calculate the area of the bass drum, we need to use the formula for the area of a circle, which is:
[
A = \pi r^2
]
Where:
- (A) is the area of the circle.
- (\pi) is a mathematical constant (approximately 3.14 in this case).
- (r) is the radius of the circle.
We are given the circumference of the drum, which is 81.64 cm. The circumference of a circle is related to its radius by the formula:
[
C = 2 \pi r
]
Where:
- (C) is the circumference.
- (\pi) is approximately 3.14.
- (r) is the radius.
Step 1: Find the radius using the circumference formula
We are given the circumference (C = 81.64 \, \text{cm}), so we can rearrange the formula to solve for (r):
[
r = \frac{C}{2\pi}
]
Substitute the known values:
[
r = \frac{81.64}{2 \times 3.14}
]
[
r = \frac{81.64}{6.28} \approx 13 \, \text{cm}
]
So, the radius of the bass drum is approximately 13 cm.
Step 2: Calculate the area
Now that we know the radius ((r = 13 \, \text{cm})), we can use the area formula to calculate the drum’s area:
[
A = \pi r^2
]
Substitute the known values:
[
A = 3.14 \times (13)^2
]
[
A = 3.14 \times 169
]
[
A \approx 530.66 \, \text{cm}^2
]
Final Answer:
The area of the bass drum is approximately 530.66 cm².
Explanation:
We used the given circumference to first find the radius of the bass drum. After that, we applied the formula for the area of a circle, (\pi r^2), to find the area. By using 3.14 for (\pi), we calculated the area as approximately 530.66 square centimeters. This approach is based on the fundamental geometric relationships between a circle’s circumference, radius, and area.