Problem 13 - Basic Technical Mathematics with Calculus

Question

Find the area of the circle with the given radius or diameter. $$r=0.0952 \mathrm{yd}$$

Step-by-Step Solution

Answer

The area of the circle is approximately 0.0285 square yards.

Step 1: Determine the formula

The formula to find the area of a circle given its radius is \( A = \pi r^2 \), where \( r \) is the radius and \( A \) is the area of the circle.

Step 2: Substitute the radius

Substitute the given radius into the formula. Here, \( r = 0.0952 \text{ yd} \). Thus, the formula becomes \( A = \pi (0.0952)^2 \).

Step 3: Perform the calculation

Calculate the square of the radius: \( 0.0952^2 = 0.00906464 \). Then, calculate the area: \( A \approx 3.14159 \times 0.00906464 \approx 0.028481741376 \).

Step 4: Finalize the answer

Round the answer to a reasonable number of significant figures. The area is \( A \approx 0.0285 \text{ square yards} \).

Key Concepts

Understanding the Radius

The radius of a circle is a crucial concept in geometry. It is the distance from the center of the circle to any point on its perimeter. This measurement is essential in various circle calculations. The radius is half the diameter of the circle, which spans from one edge of the circle directly across to the other through the center.

Calculating the correct radius is important because it's used in several key formulas, including the calculation of a circle's area. If you're given a diameter instead, remember to divide by two to find the radius. For example, if you have a diameter of 2 yards, the radius would be 1 yard. Always ensure your radius is in the correct unit before using it in calculations, as unit discrepancies can lead to significant errors.

The Circle Area Formula

To find the area of a circle, the circle area formula is pivotal: \[ A = \pi r^2 \]where:

• \(A\) is the area of the circle,
• \(\pi\) is a constant approximately equal to 3.14159,
• \(r\) is the radius.

Understanding this formula means you can calculate a circle's area once you know the radius. The formula involves squaring the radius, which means multiplying the radius by itself. Then, this squared number is multiplied by \(\pi\). Having a consistent procedure ensures accuracy.

For instance, if a circle has a radius of 0.0952 yards, squaring this gives 0.00906464. Then multiplying this by \(\pi\) results in the circle area. Always use precise measurements to avoid errors.

Rounding and Significant Figures

Accurate results are essential in mathematics, especially when dealing with measurements. Significant figures aid in expressing the precision of a number in calculations. These are the digits that contribute to the reliability of a measurement, often influencing the final answer.

When calculating the area of a circle, rounding to the appropriate significant figures is important for precision and clarity. This method avoids unnecessary complications by cutting off extra decimal places. In calculations like our exercise, determining how many significant figures are reasonable involves considering how many were present in the initial measurements. For the given radius of 0.0952 yards, which has four significant figures, it's important to balance accuracy with simplicity. Hence, rounding the final answer to match the precision of the input: 0.0285 square yards in this case, reflects this principle.