Problem 11 - Intermediate Algebra

Question

Employees at Walmart constantly reorganize and reshelve merchandise. In doing so, they calculate floor space needed for displays The algebraic expression \(l \cdot w\) gives the floor space needed in square units for a display that measures length \(l\) units and width \(w\) units. Calculate the floor space needed for a display whose length is 5.1 feet and whose width is 4 feet.

Step-by-Step Solution

Answer

The floor space needed is 20.4 square feet.

Step 1: Identify the given values

The problem provides two measurements: length \( l = 5.1 \) feet and width \( w = 4 \) feet. These values correspond to the dimensions of the display.

Step 2: Understand the formula for floor space

The formula given for calculating the floor space needed is \( l \cdot w \), where \( l \) is the length and \( w \) is the width of the display.

Step 3: Substitute the given values into the formula

Substitute length (\( l = 5.1 \)) and width (\( w = 4 \)) into the floor space formula: \( 5.1 \cdot 4 \).

Step 4: Perform the multiplication

Calculate the floor space by multiplying 5.1 by 4: \( 5.1 \times 4 = 20.4 \).

Step 5: State the result

The floor space needed for the display is 20.4 square feet.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics used to represent numbers and operations concisely. In our problem, the algebraic expression is given as \( l \cdot w \). Here, \( l \) and \( w \) stand for the length and width of a display. These symbols act as placeholders for values, allowing us to use the same expression for different numbers of length and width.
In general, algebraic expressions can include variables (like \( l \) and \( w \)), constants, and operators such as addition, subtraction, multiplication, and division. This flexibility is what makes algebra so useful in solving a wide variety of problems. By understanding and manipulating these expressions, we can solve for unknowns or calculate specific values. It's like working with a mathematical formula that's tailored to adjust as per the given inputs.

Area Calculation

The calculation of area is essential in numerous practical scenarios including the organization of space, as observed in our exercise. Area is defined as the amount of space inside a two-dimensional shape, such as a rectangle, triangle, or circle. For rectangles, the area can be calculated using the simple formula: \( \text{Area} = \text{Length} \times \text{Width} \).
In our example, we're calculating the floor space needed for a display, essentially the area of a rectangle. The given measurements: length \( = 5.1 \) feet and width \( = 4 \) feet, when plugged into the formula \( l \cdot w \), give us the area. Understanding how these measurements interact through multiplication allows us to determine the total space required for any given rectangle, whether it's a room, garden, or as in our case, a merchandise display.

Multiplication of Decimals

Multiplying decimals can initially seem tricky, but it follows the same principles as multiplying whole numbers. The key is to pay attention to the placement of the decimal point in both the original numbers and the product.
In our problem, we need to multiply 5.1 by 4. To perform this operation, consider:

• Ignore the decimal and multiply as if dealing with whole numbers, which gives \( 51 \times 4 = 204 \).
• Count the total number of decimal places in both the original numbers. Here, 5.1 has one decimal place.
• Place the decimal in the product to have the same number of decimal places: 20.4 (one place).

Practicing multiplying decimals helps to reinforce basic number manipulation and prepares you for more complex calculations. This skill is highly applicable in real-world situations where measurements are not always whole numbers.

Problem Solving in Mathematics

Problem solving in mathematics involves understanding the given problem, identifying key information, applying appropriate formulas, and computing results accurately. It requires a structured approach to ensure nothing is overlooked.
In our display area problem, we followed these steps:

• Understand the Problem: Identify that we need to calculate area based on given length and width.
• Identify Key Information: We have the measurements \( l = 5.1 \) and \( w = 4 \).
• Apply Suitable Formula: Use the area calculation formula \( l \times w \).
• Compute the Result: Multiply 5.1 by 4 to find the area (20.4 square feet).

Consistently using such a methodical approach develops mathematical thinking skills and enhances problem-solving capabilities. As you practice, this systematic method of breaking down complex problems becomes second nature, making mathematical calculations more manageable.