Ben and Lund were asked to find the area of a circle with a diameter of 9 cm.
Use their work to answer questionis 1-2.
AREA OF A CIRCLE
BEN
A-3.14 (92)
A-3.14 (81)
A-254.34 cm²
1 Describe Ben’s mistake in solving for the area of the circle.
LUNA
A-3.14 (45²)
A-3.14 (9)
A-28.26 cm²
- Describe Luna’s mistake in solving for the area of the circle.
- Affer identifying each of the mistakes that Ben and Luna made, find the correct area of the circle. Round your answer to the nearest hundredth.
The Correct Answer and Explanation is:
Let’s break down Ben’s and Luna’s mistakes in solving for the area of the circle and identify the correct calculation.
Ben’s Mistake:
Ben uses the formula for the area of a circle:
[
A = \pi r^2
]
where (r) is the radius of the circle, and (\pi \approx 3.14).
Ben was given the diameter of the circle, which is 9 cm. The radius is half the diameter, so:
[
r = \frac{9}{2} = 4.5 \text{ cm}
]
However, in Ben’s work, he incorrectly used 9 as the radius, which is the diameter of the circle. He should have used 4.5 as the radius. His equation was:
[
A = 3.14 \times (9^2) = 3.14 \times 81 = 254.34 \, \text{cm}^2
]
This is incorrect because Ben used the diameter instead of the radius in his calculation.
Luna’s Mistake:
Luna also uses the correct formula for the area of a circle:
[
A = \pi r^2
]
She correctly recognizes that the diameter is 9 cm, so the radius should be 4.5 cm. However, in Luna’s work, she mistakenly calculates the radius as 9 cm again (instead of 4.5 cm), which leads to an error in her calculation. Luna writes:
[
A = 3.14 \times (4.5^2) = 3.14 \times 20.25 = 28.26 \, \text{cm}^2
]
While this answer is correct, the error in Luna’s steps is the value she incorrectly wrote for the radius — it should be 4.5 cm, but Luna wrote 9 cm. Once she correctly recognizes the radius, the calculation of the area becomes accurate.
Correct Calculation:
Now that we’ve identified Ben and Luna’s mistakes, let’s correctly calculate the area of the circle. The radius of the circle is:
[
r = \frac{9}{2} = 4.5 \, \text{cm}
]
The formula for the area of the circle is:
[
A = \pi r^2 = 3.14 \times (4.5^2) = 3.14 \times 20.25 = 63.585 \, \text{cm}^2
]
Final Answer:
The correct area of the circle, rounded to the nearest hundredth, is:
[
A \approx 63.59 \, \text{cm}^2
]
Explanation:
The key here is recognizing the difference between the diameter and the radius. The diameter of the circle is given as 9 cm, but the formula for the area of a circle requires the radius, which is half of the diameter. Ben mistakenly used the diameter as the radius, leading to an inflated area, while Luna mistakenly calculated using the radius, but the value she used for the radius was incorrectly written down. The correct radius, (r = 4.5 \, \text{cm}), should be substituted into the formula for the area of a circle:
[
A = \pi r^2 = 3.14 \times (4.5)^2 = 63.59 \, \text{cm}^2
]
This is the correct solution.