Problem 60 - Intermediate Algebra

Question

A blueprint indicates a master bedroom in the shape of a rectangle. If the width is increased by 2 feet and the length remains the same, then the area is increased by 36 square feet. However, if the width is increased by 1 foot and the length is increased by 2 feet, then the area is increased by 48 square feet. Find the dimensions of the room as indicated on the blueprint.

Step-by-Step Solution

Answer

The dimensions are 14 feet by 18 feet.

Step 1: Identify Variables

Let's assume the original width of the master bedroom is \( w \) feet and the original length is \( l \) feet. The area of the room can be described as \( A = w \times l \).

Step 2: Create Equations for Changed Area

Given the information:1. When the width is increased by 2 feet, the area becomes \((w + 2) \times l = wl + 36\).2. When the width is increased by 1 foot and length increased by 2 feet, the area becomes \((w + 1) \times (l + 2) = wl + 48\).

Step 3: Formulate the Equations

From the information given:1. \((w + 2)l = wl + 36\) simplifies to \(2l = 36\), so \(l = 18\).2. \((w + 1)(l + 2) = wl + 48\) simplifies to \(2w + l + 2 = 48\).Now, substituting \(l = 18\) into the second equation gives:\(2w + 18 + 2 = 48\).

Step 4: Solve for Width

Substitute \(l = 18\) into the simplified equation from Step 3:\(2w + 20 = 48\).Solving for \(w\), we get:\(2w = 28\),\(w = 14\).

Step 5: Verify the Solution

With the dimensions \(w = 14\) and \(l = 18\), check the conditions:- If the width is increased by 2: - \((14 + 2) \times 18 = 16 \times 18 = 288\). - \(288 - 252 = 36\), matches the condition.- If the width is increased by 1 and the length by 2: - \((14 + 1)(18 + 2) = 15 \times 20 = 300\). - \(300 - 252 = 48\), matches the condition.Thus, the original width and length satisfy both conditions.

Key Concepts

Geometry Problems

When dealing with geometry problems, the key is to understand shapes and areas. In this particular scenario, we are asked to investigate a rectangle. A rectangle's area is calculated by multiplying its width by its length. Here, the problem involves changes to these dimensions, leading to different area outcomes.

As students approach geometry problems like this, they need to carefully interpret what is being changed (like an increase in the width or length). Each alteration will affect the rectangle's area, giving us the basis to form equations. Mastering these types of problems helps build a strong foundation in geometry, enhancing not only math skills but also spatial reasoning. So, always pay attention to how adjustments impact the dimensions, as this is where the solution path begins.

Variable Identification

Identifying variables is crucial in solving algebraic problems, especially when given a scenario with unknowns. In this exercise, the situation involves two fundamental variables: the width and the length of the room. These are the initial unknowns denoted by letters, usually the first step in algebraic problem-solving.

By denoting the width as \( w \) and the length as \( l \), we translate the word problem into a mathematical framework. This enables us to write equations that represent the relationships described. If variables are well-identified, they make setting up equations to solve the problem much more straightforward.

• Identify what you don't know (here, it's the width and the length).
• Use symbols to represent these unknowns (e.g., \( w \) for width).
• Understand how changes to these variables affect the problem (an increase in width and length affects area).

Solving Equations

Once variables are established and equations are formulated, solving these equations is the next step. In the original problem, we created two equations based on changes to the rectangle's dimensions. This step involves both setting up and simplifying these equations to find the values of the variables.

To solve, substitute known values where possible, and use algebraic methods like combining terms or isolating variables.

• Write down what each equation represents.
• Substitute any known values into your equations to simplify.
• Use inverse operations to solve for the variables. This can include operations like adding, subtracting, multiplying, or dividing both sides of an equation.
• Verify your solutions by plugging them back into the original problem conditions to ensure they satisfy all aspects.

Thus, through solving, we confirm that the width is \( 14 \) feet and the length is \( 18 \) feet, making the logical connections from the problem to the solution crystal clear.