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Found in: Page 139

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# The radius of a circular disk is given as $${\rm{24\;cm}}$$ with a maximum error in measurement of $${\rm{0}}{\rm{.2\;cm}}$$.$$({\rm{a}})$$ Use differentials to estimate the maximum error in the calculated area of the disk.$$({\rm{b}})$$ What is the relative error? What is the percentage error?

Part $$(a)$$ The maximum error in the calculated area of the disk is $$9.6\pi c{m^2}$$.

Part $$(b)$$ The relative error is $$0.0167$$ and the percentage error is $$1.67\%$$.

See the step by step solution

## Step 1: Given Information.

The radius of a circular disk is given as $$24\;cm$$ with a maximum error in measurement of $$0.2\;cm$$.

## Step 2: Meaning of the Derivatives.

Calculus relies heavily on derivatives. The derivative of a function of a real variable measures a quantity's sensitivity to change as defined by another quantity.

## Step 3: Part $$\left( {\rm{a}} \right)$$ Find the maximum error in the calculated area of the disk

Apply the formula of the surface area of the disk

$$\begin{array}{c}S = \pi {r^2}{\rm{ }}\\{S^\prime }(r) = 2\pi r\end{array}$$

Calculate the maximum error in $$S$$

$$\begin{array}{c}\Delta S \approx {S^\prime }(r)\Delta r\\ = 2\pi r\Delta r\\ = 2\pi (24)0.2\\ = 9.6\pi \end{array}$$

## Step 4: Part $$\left( {\rm{b}} \right)$$ Find the relative error and the percentage error

Apply the formula of the area of the circle

$$\begin{array}{c}A = \pi {r^2}\\ = \pi {(24\;cm)^2}\\ = 576\pi c{m^2}\end{array}$$

By definition, calculate the relative error

$$\begin{array}{c}\frac{{\Delta A}}{A} = \frac{{9.6\pi c{m^2}}}{{576\pi c{m^2}}}\\ = \frac{1}{{60}}\\ \approx 0.0167\end{array}$$

The percentage error is simply the relative error, which is rewritten in the percentage form. To rewrite a number in the percentage form, multiply that number by $$100$$ Therefore

$$\begin{array}{c}0.0167 = 0.0167 \cdot 100\% \\ = 1.67\% \end{array}$$

Part $$(a)$$ The maximum error in the calculated area of the disk is $$9.6\pi c{m^2}$$.

Part $$(b)$$ The relative error is $$0.0167$$ and the percentage error is $$1.67\%$$.