Problem 601 - Elementary Algebra

Question

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify \(242^{\frac{1}{2}}\) to find the length of each side of his patio.Round to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The length of each side of the patio is approximately 15.6 feet.

Step 1: Understand the problem

Elliott wants to make a square patio. The area of the square patio is 242 square feet. To find the side length of the patio, we need to find the square root of the area.

Step 2: Use the formula for the square root

The side length of a square is the square root of its area. The side length can be found using the formula \[s = \text{area}^{\frac{1}{2}}\]. Here, the area is 242, so we need to find \[242^{\frac{1}{2}}\].

Step 3: Calculate the square root

Calculate \[242^{\frac{1}{2}}\] by using a calculator. The value of \[242^{\frac{1}{2}} \approx 15.5563\].

Step 4: Round to the nearest tenth

Round the calculated value 15.5563 to the nearest tenth. The nearest tenth of 15.5563 is 15.6.

Key Concepts

Finding Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 * 5 = 25. In mathematical notation, the square root of a number 'n' is written as \(^{\frac{1}{2}}\).

In the context of the given problem, Elliott needs to find the length of each side of a square patio with an area of 242 square feet. The side length is the square root of the area, given by the formula:

\(s = \text{area}^{\frac{1}{2}}\)

To find \( = 242^{\frac{1}{2}}\), you can use a calculator to get approximately 15.5563.

Area of a Square

The area of a square is calculated using the formula: \( = s^2\), where 's' is the length of one side of the square. This formula gives the space enclosed within the four sides of the square.

In Elliott's case, the area is given as 242 square feet. To reverse this process and find the side length from the area, we need the square root of 242. The side length 's' is: \(s = 242^{\frac{1}{2}}\). Once you calculate it, you find approximately 15.5563 feet.

Rounding Numbers

Rounding numbers makes them easier to work with by reducing the number of digits while retaining a value close to the original. The general rule for rounding is to look at the digit immediately after the place value you are rounding to:

• If it is 5 or greater, round up.
• If it is less than 5, round down.

For Elliott's side length calculation, we have approximately 15.5563 feet.

To round this number to the nearest tenth, look at the hundredths place (second digit after the decimal point):

• The number is 5, so we round up.

Therefore, 15.5563 rounded to the nearest tenth is 15.6 feet.