The radius r of a circle is measured with an error of at most ...

Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

a. circumference?

Accepted Answer

Welcome back, everyone. What is the maximum corresponding percentage error in computing the sphere's volume if it is calculated using its radius? The radius has a measurement error of up to 3%. A says 3%, B 9%, C 12%, and D 27%. For this problem, first of all, let's define the volume function. It is equal to 4/3 by our cubed, and we're going to use differentials. Let's recall that the differential dV equals the derivative of V multiplied by dr. So let's differentiate the volume function. V equals 4/3 pi multiplied by. The derivative of R cubed, and the derivative of R cubed is 3R2, right? So we get 3 pi multiplied by 3R2, which gives us 4 pi R squad. And now we can use the relative error formula, which says that if we take the absolute value of DV divided by V. This is going to be exactly what we're looking for, right, in a decimal form. So what we want to do is simply substitute for DV, which is V. That's for pi R squared multiplied by the R, and then we are dividing by V, which is our volume function. So that's 4/3IRQ. What we're going to do from here is simplify. So we have 4 divided by 4, when we divide by 1/3, well, basically we get 3 in the numerator, we can also take it out from the absolute value function, right, because it's a positive number, and then we can see that we can cancel out the pi terms, we can cancel out R squared, which gives us R in the denominator. So we get 3 multiplied by DR divided by R. So we can see that The absolute value of DV divided by V is equal to 3, multiplied by the absolute value of DR divided by R, and we know that. The absolute value of the R divided by R. is less than or equal to 3%, which is 0.03. So we can substitute this value in. And we can show that. The absolute value of DV divided by V should be less than or equal to 0.09, right? So now we can conclude our findings and we can show that the absolute value of the divided by V. Should be less than or equal to 9% if we convert into the percentage, right? So our final answer corresponds to the answer choice B, 9%. Thank you for watching.

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

a. circumference?

Key Concepts

Differential Error Analysis

Circumference of a Circle

Percentage Error

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