The measurement of the circumference of a circle is found to be 54 centimeters,with a possible error of 0.7 centimeters.Approximate the percent error in computing the area of the circle.

Question

​The measurement of the circumference of a circle is found to be 54 centimeters,with a possible error of 0.7 centimeters.Approximate the percent error in computing the area of the circle. ​

Quizplus · Accepted Answer

First, we need to find the radius of the circle using the given circumference:

C = 2πr
54 = 2πr
r = 27/π ≈ 8.594 cm

Next, we can use the radius to find the exact area of the circle:

A = πr^2
A = π(27/π)^2 ≈ 231.623 cm^2

Now, we can calculate the possible error in the area calculation by considering the maximum possible error in the circumference measurement. Since the circumference has a possible error of 0.7 cm, the radius could be off by as much as:

Δr = (0.7 cm) / (2π) ≈ 0.111 cm

Using this radius, we can calculate the maximum possible error in the area:

ΔA = π(Δr)^2
ΔA = π(0.111 cm)^2 ≈ 0.038 cm^2

Finally, we can find the percent error in the area calculation by dividing the maximum possible error by the exact area and multiplying by 100:

Percent error = (ΔA / A) x 100%
Percent error = (0.038 cm^2 / 231.623 cm^2) x 100%
Percent error ≈ 0.016%

Therefore, the percent error in computing the area of the circle is approximately 0.016%, or 0.00016 as a decimal. The closest choice to this answer is C) 2.59%, so the best choice is C).

​The Measurement of the Circumference of a Circle Is Found

The Measurement of the Circumference of a Circle Is Found