Question Statement

The side of a square is measured to be $10\text{ cm}$ with a possible error of $\pm 0.3\text{ cm}$. Use differentials to find:

1. An approximation to the maximum error in the area.
2. The approximate relative error.
3. The approximate percentage error.

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Background and Explanation

Differentials are used to estimate how a small change (or error) in an independent variable affects a calculated dependent variable. For a function $y=f(x)$, the differential $dy=f^{\prime }(x)dx$ approximates the actual change $\Delta y$ when $x$ changes by a small amount $dx$.

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Solution

Part (a): Maximum Error in Area

Let $s$ represent the side of the square and $A$ represent its area. The formula for the area of a square is: $A=s^2$

To find the differential $dA$, we differentiate $A$ with respect to $s$: $\frac{dA}{ds}=2s⟹dA=2sds$

Given:

• Measured side $s=10\text{ cm}$
• Possible error in measurement $ds=\pm 0.3\text{ cm}$

Substitute these values into the differential equation: $dA=2(10)(\pm 0.3)$ $dA=\pm 6{\text{ cm}}^2$

The approximate maximum error in the area is $\pm 6{\text{ cm}}^2$.

Part (b): Relative Error

The relative error is defined as the ratio of the error in the area ($dA$) to the total area ($A$).

First, calculate the area $A$ at the measured value: $A=s^2=(10{)}^2=100{\text{ cm}}^2$

Now, calculate the relative error: $\text{Relative Error}=\frac{dA}{A}$ $\text{Relative Error}=\frac{\pm 6}{100}=\pm 0.06$

The approximate relative error is $\pm 0.06$.

Part (c): Percentage Error

The percentage error is the relative error expressed as a percentage.

$\text{Percentage Error}=\frac{dA}{A}\times 100\%$ $\text{Percentage Error}=\pm 0.06\times 100\%=\pm 6\%$

The approximate percentage error is $\pm 6\%$.

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Key Formulas or Methods Used

• Area of a square: $A=s^2$
• Differential of a power function: $d(s^n)=n{s}^{n-1}ds$
• Relative Error: $\frac{dA}{A}$
• Percentage Error: $\frac{dA}{A}\times 100\%$

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Summary of Steps

1. Define the area formula $A=s^2$ and find its derivative to obtain the differential $dA=2sds$.
2. Substitute the measured side length ($s=10$) and the measurement error ($ds=0.3$) to calculate the absolute error in area.
3. Divide the absolute error by the calculated area ($s^2$) to find the relative error.
4. Multiply the relative error by 100 to determine the percentage error.