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

b. area?

Accepted Answer

Welcome back, everyone. What is the maximum corresponding percentage error in computing the sphere's volume if the diameter D of the sphere is measured with an error of at most 1%? A says 1%, B 2%, C 3%, and D 4%. 1st of all, let's define the volume of a sphere. It is equal to 4/3 by R cubed where R is radius. And we know that radius is half of the diameter D, so we get 4/3 by multiplied by D divided by 2 cubed. Now we're going to simplify. This gives us 4/3 by the cubed divided by 8, right? So if we combine our terms, this gives us 4 pi the cubed divided by. 24. And finally, this is equal to pi the cubed divided by 6. We're going to use the relative error formula, the absolute value of DV divided by V. This is what we're looking for. And as we understand, we need to identify the differential form dV equals the derivative of volume multiplied by DD, right? Our independent variable is the diameter. So let's identify the derivative V. We can factor out the constant which is pi divided by 6 and differentiate the cubed. The derivative of the cubed according to the power rule is 3D squared. So now when we multiply those terms, we get. P I D squared divided by 2. And now let's substitute this into our formula. So in the numerator we have DV, which is pid squared divided by 2. Multiplied by DD. All of that is divided by volume. Which is PIDQ. Divided by 6. And now we can simply cancel out the pi terms. We can also cancel out D squared, which gives us D in the denominator, and now we have 1/2 divided by 1/6, which gives us 3, right? So we can take out 3 because it's positive, and overall we get DD. Divided by D within the absolute value function. Now what do we know? Well, we know that the absolute value of DD divided by D. It is at most 1%, so it is less than or equal to 0.01 in a decimal form. This means that the maximum error in volume. DV divided by V is going to be equal to 3 multiplied by 0.01, which is 0.03. That's the maximum error. In general, we can say that the absolute value of DV divided by V is less than or equal to 3%. So the maximum error is 3%, which corresponds to the answer choice C. 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

b. area?

Key Concepts

Differential Approximation

Derivative of Area with Respect to Radius

Percentage Error Calculation

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