Problem 94 - Prealgebra

Question

A circle has a diameter of \(15.49\) inches. Using \(\pi \approx 3.14\), find the area of the circle, correct to the nearest hundredth of a square inch.

Step-by-Step Solution

Answer

The area of the circle is approximately 188.38 square inches.

Step 1: Find the Radius from the Diameter

The diameter of the circle is given as 15.49 inches. The radius is half of the diameter. Calculate the radius using the formula: \[\text{Radius} = \frac{\text{Diameter}}{2} = \frac{15.49}{2}\] The radius is therefore 7.745 inches.

Step 2: Calculate the Area of the Circle

The formula to find the area of a circle is \(A = \pi r^2\), where \(A\) is the area and \(r\) is the radius. Substitute \(r = 7.745\) and \(\pi = 3.14\) into the formula: \[A = 3.14 \times (7.745)^2\] Calculate \(7.745^2\) which equals approximately 59.992025.

Step 3: Solve for the Area

Substitute \(59.992025\) back into the area formula: \[A = 3.14 \times 59.992025\] Calculate the product to find the area. The resulting area is approximately 188.37797 square inches.

Step 4: Round the Answer

Round the calculated area 188.37797 to the nearest hundredth. The value rounds to 188.38 square inches.

Key Concepts

Diameter and Radius Relationship

Understanding the relationship between the diameter and the radius of a circle is key to calculating various properties of the circle. The diameter is the longest straight line that passes through the center of the circle, connecting two points on its boundary. Meanwhile, the radius is half the diameter. It runs from the center to any point on the circle's edge. Thus, the formula to find the radius \(\text{Radius} = \frac{\text{Diameter}}{2}\) helps you easily convert between these two measurements.
For example, if a circle has a diameter of 15.49 inches, its radius can be derived as \(\frac{15.49}{2}\), which equals 7.745 inches. This calculation is crucial because many circle-related formulas, like the area or circumference, require the radius.
Remember:

• Diameter = 2 x Radius
• Radius = Diameter / 2

Area of a Circle Formula

The area of a circle tells you how much space is enclosed within its boundary. To calculate this area, you need the radius of the circle and the constant \(\pi\). Pi (\(\pi\) is approximately equal to 3.14, though it can be more accurately represented with more decimal places in advanced calculations.
The formula used is: \[A = \pi r^2\]where \(A\) is the area and \(r\) is the radius. This formula means you'll multiply \(\pi\) by the square of the radius. For a circle with a radius of 7.745 inches, the calculation is: \(3.14 \times (7.745)^2\).
Compute \((7.745)^2\), which is approximately 59.992025. Next, multiply by \(3.14\) for an area of about 188.37797 square inches.

• Always square the radius first.
• Multiply by \(\pi\) next.

Rounding Numbers

Rounding numbers simplifies them, making them easier to work with. When you round, you decide which digit will be your last and adjust it accordingly. Rounding to the nearest hundredth means you'll keep only two decimal places. If the number in the third decimal place is 5 or more, round up. Otherwise, round down or keep it the same.
For the circle's area that we calculated as 188.37797 square inches, the third decimal is 7. This is why you round the area to 188.38 square inches.

• Always check the digit immediately after your desired decimal place.
• If it's 5 or higher, increase the last kept digit by one.
• Otherwise, retain your rounded value.

Rounding is especially helpful when dealing with numbers in measurements or computations, ensuring that results are easy to read and understandable.