Problem 431 - Elementary Algebra

Question

In the following exercises, solve. Round approximations to one decimal place. Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

Step-by-Step Solution

Answer

Each side of Vince's patio is approximately 11.4 feet.

Step 1: Understand the Problem

Vince wants to create a square patio, so the area he needs is 130 square feet. The side of the square patio can be found using the formula for the side of a square given its area: \( s = \sqrt{A} \).

Step 2: Substitute the Area

Substitute the value of the area into the formula. Here, the area \( A = 130 \) square feet. The formula becomes: \( s = \sqrt{130} \).

Step 3: Calculate the Square Root

Find the value of \( \sqrt{130} \) using a calculator. \( \sqrt{130} \approx 11.4018 \).

Step 4: Round to One Decimal Place

Round the value to one decimal place as required. \( 11.4018 \approx 11.4 \).

Key Concepts

Solving Square Roots

Square roots are mathematical operations that find a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. This is written as \( \sqrt{16} = 4 \) .

To solve square roots:

• Identify the number you need the square root of, in this case, 130.
• Use a calculator for precise values. \sqrt{130} \approx 11.4018.

Knowing how to solve square roots is essential for dealing with geometric problems and equations involving areas and lengths.

Area of a Square

The area of a square is calculated by multiplying the length of one side by itself. The formula is: \[ A = s^2 \]
If you know the area and need to find the length of a side, you rearrange the formula: \ s= \sqrt{A} \ .
In our example:

• The area (A) is 130 square feet.
• The side length (s) is found by calculating the square root, \( s = \sqrt{130} \approx 11.4 \).

This relationship helps in designing spaces and ensuring measurements fit within given dimensions.

Rounding Numbers

Rounding makes numbers simpler and easier to work with, especially after calculations. To round to one decimal place:

• Identify the digit in the first decimal place and the digit right after it.
• If the next digit is 5 or higher, round the first decimal place up.
• If the next digit is below 5, keep the first decimal place as is.

Using our example:

• Original value: 11.4018
• The first decimal place is 4, and the next digit is 0.
• Since 0 is less than 5, 11.4018 rounds to 11.4.

Rounding helps in practical measurements and makes final results more usable.