Question

The measurement of the circumference of a circle is found to be 64 centimeters, with a possible error of 0.9 centimeter. (a) Approximate the percent error in computing the area of the circle. (Round your answer to two decimal places.) 2.81 Correct: Your answer is correct. % (b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 1%. (Round your answer to one decimal place.)

Answer 1

Answer:

(a) 2.81%

(b) 0.5%

Step-by-step explanation:

We have the following information from the statement:

P = 64 + - 0.9

(a) We know that the perimeter is:

P = 2 * pi * r

if we solve for r, we have to:

r = P / 2 * pi

We have that the formula of the area is:

A = pi * r ^ 2

we replace r and we are left with:

A = pi * (P / 2 * pi) ^ 2

A = (P ^ 2) / (4 * pi)

We derive with respect to P, and we are left with:

dA = 2 * P / 4 * pi * dP

We know that P = 64 and dP = 0.9, we replace:

dA = 2 * 64/4 * 3.14 * 0.9

dA = 9.17

The error would come being:

dA / A = 9.17 / (64 ^ 2/4 * 3.14) = 0.02811

In other words, the error would be 2.81%

(b) tell us that dA / A <= 0.01

we replace:

[P * dP / 2 * pi] / [P ^ 2/4 * pi] <= 0.01

solving we have:

2 * dP / P <= 0.01

dP / P <= 0.01 / 2

dP / P <= 0.005

Which means that the answer is 0.5%