Problem 93 - Elementary Algebra

Question

Problem: The diameter of a circle is \(10 \mathrm{in}\). Using \(\pi \approx 3.14\), find the area of the circle. Incorrect Answer: \(A=\pi r^{2}\) $$ \begin{aligned} A &=(3.14)(10 \text { in. })^{2} \\ A &=(3.14)\left(100 \mathrm{in}^{2}\right) \\ A &=314 \mathrm{in}^{2} \end{aligned} $$

Step-by-Step Solution

Answer

The area of the circle is 78.5 square inches.

Step 1: Understand the formula

The formula to find the area of a circle is given by \[ A = \pi r^{2} \] where \(A\) is the area and \(r\) is the radius of the circle.

Step 2: Identify given values

The diameter of the circle is \(10 \mathrm{in} \). The radius \(r\) is half of the diameter. Therefore, \[ r = \frac{10 \mathrm{in}}{2} = 5 \mathrm{in} \]

Step 3: Plug values into the formula

Substitute \(r = 5 \mathrm{in} \) and \(\pi \approx 3.14\) into the formula to find the area: \[ A = 3.14 \times (5 \mathrm{in})^{2} \]

Step 4: Simplify the equation

Compute the square of the radius: \[ (5 \mathrm{in})^{2} = 25 \mathrm{in}^{2} \] Then multiply by \( \pi \): \[ A = 3.14 \times 25 \mathrm{in}^{2} = 78.5 \mathrm{in}^{2} \]

Key Concepts

Geometry

Geometry is a branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects and spaces. It is an essential part of everyday life, from architecture to engineering. When talking about circles, geometry helps us explore their various properties, like radius, diameter, and area. Understanding these concepts is crucial because they help us solve real-world problems involving circles. For example, the area of a circle can be important in designing a garden, a round table, or even in engineering applications like creating gears and wheels.

Circle Area Formula

The area of a circle is the measure of the space inside its boundary. To calculate this area, we use the circle area formula: \[ A = \pi r^2 \] Where:

• \(A\) is the area of the circle
• \(\pi\) is a constant approximately equal to 3.14
• \(r\) is the radius of the circle

This formula tells us that the area depends on the square of the radius, meaning that if the radius doubles, the area increases by a factor of four. Knowing this formula is essential when solving problems that ask for the area of a circle, as it directly relates the radius to the area.

Radius and Diameter

To find the area of a circle, you must understand two key terms: radius and diameter. The radius (\(r\)) is the distance from the center of the circle to any point on its boundary. The diameter (\(d\)) is twice the radius and is the distance across the circle, passing through the center. In mathematical terms:

• \(d = 2r\)
• \(r = \frac{d}{2}\)

Given the diameter, you can easily find the radius and then use the circle area formula to calculate the area. In our exercise, the diameter is 10 inches, so the radius is \( \frac{10}{2} = 5 \) inches. Plugging this into the formula, \( A = \pi \times (5)^2 = 3.14 \times 25 = 78.5 \) square inches, you get the area of the circle.