Problem 90 - Elementary Algebra

Question

Grease blocked a 12-in.-diameter sewer pipe in Inglewood, California, resulting in a spill of about 11,200 gal of sewage. Find the area of a cross- section of the pipe. \((\pi \approx 3.14\).) Round to the nearest tenth.

Step-by-Step Solution

Answer

113.0 square inches

Step 1: Understand the Problem

You're asked to find the area of the cross-section of a sewer pipe with a diameter of 12 inches.

Step 2: Find the Radius

The diameter of the pipe is given as 12 inches. The radius is half of the diameter. So, the radius is \[ radius = \frac{12}{2} = 6 \text{ inches} \].

Step 3: Use the Area Formula for a Circle

To find the area of a circle, use the formula \[ Area = \pi \times radius^2 \].

Step 4: Calculate the Area

Substitute the radius and \[ \pi \approx 3.14 \] into the formula: \[ Area = 3.14 \times 6^2 \. \] Simplify the calculation to get \[ Area = 3.14 \times 36 = 113.04 \text{ square inches} \].

Step 5: Round to the Nearest Tenth

Round the result to the nearest tenth to get \[ Area \approx 113.0 \text{ square inches} \].

Key Concepts

radius and diameter relationship

Understanding the relationship between the radius and the diameter of a circle is essential for solving problems like finding the area. Think of a circle as a perfect round shape. The distance across the widest part of this shape is called the diameter. The radius, on the other hand, is half of this distance. If you know the diameter, you just divide it by two to get the radius. For instance, in the problem of the blocked sewer pipe, the diameter is 12 inches. Dividing 12 by 2 gives us a radius of 6 inches. Remembering this simple relationship helps ensure you always have the correct value for your calculations.

circle area formula

To find the area of a circle, you need a specific formula: \( Area = \pi \times radius^2 \). This formula essentially tells you to multiply the value of pi ( which is approximately 3.14 for most calculations ) by the square of the radius. Squaring the radius means multiplying the radius by itself. For example, if the radius is 6 inches, then squaring it gives you \( 6 \times 6 = 36 \). Next, you multiply this result by pi (3.14) to get the area. \( Area = 3.14 \times 36 = 113.04 \) square inches.

rounding numbers

When dealing with arithmetic, sometimes you need to round your answer to make it simpler or to meet a problem's requirements. In this case, you are asked to round the area of the sewer pipe's cross-section to the nearest tenth. Rounding involves looking at the digits in your number. For the number 113.04, you look at the second digit after the decimal point. Here it's a 4, which is less than 5, so you round down and keep the number as it is. Therefore 113.04 rounded to the nearest tenth is 113.0. This method ensures your answer is as accurate as needed without being overly complex.