Problem 71 - Elementary Algebra

Question

. The length of a side of a square is \(s\) yards. What is the area of the square in square feet?

Step-by-Step Solution

Answer

The area of the square is \(9s^2\) square feet.

Step 1: Formula for Area of a Square

The formula to calculate the area of a square is given by the square of its side length: \(A = s^2\). Here, \(s\) is the side length in yards.

Step 2: Convert Yards to Feet

Since the problem asks for the area in square feet, we first need to convert the side length from yards to feet. There are 3 feet in a yard, so one side of the square is \(3s\) feet.

Step 3: Calculate the Area in Square Feet

Now that we have the side length in feet, we calculate the area in square feet using the formula: \(A = (3s)^2 = 9s^2\).

Key Concepts

Length Conversion

Length conversion is a crucial step when working on problems involving measurements. In this exercise, the side of the square is initially given in yards. To find the area in square feet, we need to convert the side length to feet. Knowing the conversion factor is essential:

• 1 yard equals 3 feet.

This means when you have a measurement in yards, you multiply by 3 to convert it to feet. For instance, if the side length of a square is given as \(s\) yards, then the length in feet would be \(3s\) feet. Remember, understanding and accurately applying length conversion is key in ensuring your calculations are correct and your final answer is in the desired unit.

Square Yards

When talking about the area of a square, it's usually measured in square units, such as square yards or square feet. In this problem, the initial measurement is provided in square yards and needs to be converted into square feet. Here are some essential points about these measurements:

• Square yards measure area when each side is in yards.
• To convert an area from square yards to square feet, each yard must be expressed in feet, essentially squaring the conversion factor.
• If one side of the square is \(s\) yards, the area directly in square yards would be \(s^2\) square yards.

Realizing that for the area in square feet, the conversion involves squaring the length conversion factor is crucial since you must compute the area based on the side lengths in feet.

Algebraic Formulas

Algebraic formulas are vital tools in solving mathematical problems. In this exercise, we use the formula for the area of a square. The algebraic expression \(A = s^2\) is the standard formula, where \(A\) is the area and \(s\) is the length of the side. However, to convert the area into square feet, we need to adjust the formula:

• When the side is converted to feet, it becomes \(3s\) feet.
• Thus, the area formula in feet becomes \(A = (3s)^2 = 9s^2\).

The use of algebraic manipulation here helps to transition from one unit of measurement to another and helps highlight the importance of correctly applying formulas to solve area and conversion problems efficiently.