Problem 32 - Intermediate Algebra

Question

Find the area of each circle to the nearest tenth. See Example 3. (Answers may vary slightly depending on which approximation of is used.) A circle with radius \(5 \frac{3}{4} \mathrm{cm}\)

Step-by-Step Solution

Answer

The area of the circle is approximately 103.8 square centimeters.

Step 1: Convert Mixed Number to an Improper Fraction

First, convert the mixed number radius of the circle from a mixed number to an improper fraction. The radius given is \(5 \frac{3}{4}\) cm. Multiply the whole number (5) by the denominator (4) and add the numerator (3) to get \(23/4\). Thus, the radius as an improper fraction is \(\frac{23}{4}\) cm.

Step 2: Use the Formula for the Area of a Circle

Recall the formula for the area of a circle, which is \(A = \pi r^2\). Here, \(r\) is the radius of the circle which we have as \(\frac{23}{4}\) cm.

Step 3: Square the Radius

Square the radius by multiplying \(\frac{23}{4}\) by itself. \[\left(\frac{23}{4}\right)^2 = \frac{23 \times 23}{4 \times 4} = \frac{529}{16}.\] Thus, \(r^2 = \frac{529}{16}\).

Step 4: Calculate the Area Using \( \pi \)

Now, calculate the area using the formula \(A = \pi \left(\frac{529}{16}\right)\). Use the approximation of \(\pi \approx 3.14\) for this calculation. \[A \approx 3.14 \times \frac{529}{16}\]Multiply \(3.14\) by \(529\) and then divide the result by \(16\): \[A \approx \frac{1660.06}{16} \approx 103.75\] So, the area is approximately \(103.8\) square centimeters to the nearest tenth.

Key Concepts

Mixed Number Conversion

When dealing with measurements in mathematics, you may encounter numbers in mixed form, like \(5 \frac{3}{4}\). Mixed numbers are composed of a whole number and a fraction. Converting these into improper fractions is essential for many calculations, including finding the area of a circle. An improper fraction has a numerator larger than its denominator, making it easier to use in mathematical operations.

To convert a mixed number into an improper fraction, follow these simple steps:

• Multiply the whole number by the fraction's denominator.
• Add the numerator of the fraction to this product.
• Place the sum over the original denominator.

For example, with the mixed number \(5 \frac{3}{4}\):

• Multiply 5 (whole number) by 4 (denominator) to get 20.
• Add the numerator 3, resulting in 23.
• Place 23 over 4 to get the improper fraction \(\frac{23}{4}\).

Now you're ready to tackle further calculations using this improper fraction!

Improper Fraction

Improper fractions, like \(\frac{23}{4}\), are invaluable in precise calculations. Unlike mixed numbers, improper fractions provide a straightforward way to perform arithmetic operations. This characteristic is particularly advantageous when applying mathematical formulas to solve problems, such as calculating the area of a circle.

In the context of our example, we converted the mixed number radius of a circle into an improper fraction \(\frac{23}{4}\) cm. This step simplifies subsequent mathematical processes, allowing us to work seamlessly with the formula for the area.

Using improper fractions ensures accuracy and removes complexity from calculations that involve multiple steps, like squaring or using specific mathematical constants such as \(\pi\). By working with improper fractions, you set a solid foundation for error-free calculations.

Formula for Area of a Circle

Understanding the formula for calculating the area of a circle is crucial. The formula is represented as \(A = \pi r^2\), where \(A\) is the area, \(r\) is the radius, and \(\pi\) is a mathematical constant approximate to 3.14.

This formula requires you to have the radius either in a whole number form or as a simplified improper fraction, like we've done with \(\frac{23}{4}\) cm in our example.

Once the radius is squared (multiplied by itself) and the value of \(\pi\) is applied, you can quickly determine the area. Understanding and correctly applying this formula allows you to find the area precisely. Remember always to double-check your conversion and calculations to ensure accuracy.

Squaring Fractions

Squaring a fraction involves multiplying the fraction by itself. It is an essential skill for solving mathematical problems involving circles and other geometric figures. When you square a fraction, you should square both the numerator and the denominator separately.

For example, when squaring \(\frac{23}{4}\), you multiply the numerator 23 by itself to get 529 and the denominator 4 by itself to get 16, resulting in \(\frac{529}{16}\).

This process is straightforward:

• Identify the fraction you wish to square.
• Multiply the numerator by itself.
• Multiply the denominator by itself.

Squaring fractions is a significant step in accurately applying the formula for the area of a circle. It is crucial to perform each multiplication carefully and systematically to get the correct result, ensuring your final calculations are accurate and reliable.