Problem 28 - Calculus of a Single Variable

Question

Circumference 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. (b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed \(3 \% .\)

Step-by-Step Solution

Answer

The approximate percent error in computing the area of the circle is equivalent to the value found in Step 3. For part (b), the maximum allowable percent error in measuring the circumference to ensure that the error in computing the area does not exceed 3% is equivalent to the value found in Step 5.

Step 1: Convert the Circumference into Radius

Using the formula for the circumference of a circle \(C=2 \pi r\), the radius \(r\) can be determined. So, rearrange the equation to solve for \(r\): \(r=C/(2 \pi)\). Given that the circumference \(C\) is 64 cm, the radius is \(r = 64/(2 \pi) cm.\)

Step 2: Calculate the differential of Area with respect to radius \(r\)

Because \(A = \pi r^2\), we use the Powers Rule to find: \(dA/dr = 2 \pi r\). To calculate the differential \(dA\), multiply the derivative with the given change in \(r\), in this case, half of the change in the circumference, which is \(0.9/2 = 0.45 cm\). Hence, \(dA = 2 \pi r * 0.45 cm\).

Step 3: Approximate the percent error in computing the area of the circle

The error in computing the area is given by the differential \(dA\). The percent error is given by the ratio of this error to the total area, multiplied by 100. The total area is \(\pi r^2\). So the percent error is \((dA / \pi r^2) * 100\). Find this value.

Step 4: Calculate the maximum allowable error in the radius

Rearrange the percent error formula from Step 3 to solve for an allowable difference in radius \(dr\)\. Using 3% as the maximum allowable error in area, the equation becomes \(0.03 = (2 \pi r * dr) / \pi r^2\). Solve for \(dr\).

Step 5: Convert the radius error to percent error in circumference

The radius error \(dr\), calculated in step 4, can be converted to a circumference error by multiplying by \(2 \pi\) (since \(2 \pi dr\) yields the differential in circumference \(dC\)). To find the percentage error, divide by the total circumference and multiply by 100. The equation becomes \((2 \pi dr / C) * 100\). Calculate this value for the solution to part (b).

Key Concepts

Circumference of a Circle

The circumference of a circle is a measure of the distance around the circle. It's like measuring the length of a piece of string needed to wrap around the circle. The formula used to calculate the circumference is directly connected to the circle's radius, using the well-known formula:

• \[ C = 2 \pi r \]

In this formula, \( C \) is the circumference, \( \pi \) (pi) is approximately 3.14159, and \( r \) is the radius of the circle. Knowing the circumference allows us to solve for the radius, which is particularly useful if we're measuring the dimensions of a circular object.
Divide the circumference by \( 2\pi \) to find the radius:

• \[ r = \frac{C}{2 \pi} \]

Once you have the radius, you can easily compute other properties of the circle, such as its area, using the radius in the appropriate formulas.

Differential Calculus

Differential calculus is a field of mathematics that deals with the rate at which quantities change. It's particularly useful when we want to approximate the error or change in a particular measurement. Here, it helps us understand how a small change in the radius of a circle can lead to a change in the area of that circle.
At its core, differential calculus looks at the derivative of a function. For instance, to understand how the area \( A \) of the circle (

• \( A = \pi r^2 \)

) changes with radius, one finds the derivative \( dA/dr \). This gives

• \[ \frac{dA}{dr} = 2\pi r \]

This derivative tells us that the area change is twice the product of \( \pi \) and the radius, for a small change in radius. This formula helps us calculate how much the area will change with a slight increase or decrease in the radius and helps approximate error when radius measurements might not be exact.

Area of a Circle

The area of a circle represents the space contained within its circumference. To calculate this area, use the radius obtained from the circumference or directly from a measurement:

• \[ A = \pi r^2 \]

This formula means the area increases with the square of the radius, meaning a small error in measuring the radius can significantly affect the area computation. When considering errors, differential calculus allows the calculation of potential changes in the area due to slight inaccuracies in the radius measurement. For instance, if the radius changes slightly, the error in area can be approximated using

• \[ dA = 2\pi r \cdot dr \]

where \( dr \) is the small change in radius. Calculating the percent error helps express this change as a percentage of the total area, indicating how much the computed area could deviate from the actual area due to measuring inaccuracies. Understanding the relationship between the radius and area is crucial for minimizing potential errors in calculating real-world circle areas.