Problem 54 - Elementary 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 foet. 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 width is 14 feet, and the length is 18 feet.

Step 1: Define Variables

Let the original width of the room be \( w \) feet, and the original length of the room be \( l \) feet.

Step 2: Set Up First Equation

From the problem statement, if the width increases by 2 feet, the area increases by 36 square feet. The equation becomes: \[(l)(w + 2) - lw = 36\]This simplifies to: \[2l = 36\] which gives the first equation: \[l = 18\]

Step 3: Set Up Second Equation

If the width is increased by 1 foot and the length by 2 feet, the area increases by 48 square feet. The equation is: \[(l + 2)(w + 1) - lw = 48\]Expanding and simplifying gives: \[2w + l + 2 = 48\]Since \(l = 18\), substitute in to get:\[2w + 18 + 2 = 48\]This simplifies to:\[2w + 20 = 48\]

Step 4: Solve for Width

We simplify the equation \(2w + 20 = 48\) by subtracting 20 from both sides:\[2w = 28\]Next, divide by 2 to find \(w\):\[w = 14\]

Step 5: Verify Solution with Length

Using the value \(l = 18\) from our earlier equation, we verify it against the conditions described. Both modified scenarios fit the increases in area as given in the problem statement.

Key Concepts

Rectangular Area Calculation

Understanding how to calculate the area of a rectangle is foundational in geometry. The area tells us how much space is inside the rectangle.
It is calculated by multiplying the width by the length.For example, if a room has a width of 5 feet and a length of 10 feet, then the area would be:

• Area = width \( \times \) length
• Area = 5 \( \times \) 10 = 50 \, \text{square feet}

In problems where dimensions are altered, such as increasing the width or length, it's crucial to adjust your calculations. The task is to figure out the new area and see how it changes compared to the original one.
Let's imagine you're given an exercise where the width increases by 2 feet, and the area increases by a certain square footage. You can set up an equation with this information to understand how much extra space is gained by the increase. Being able to do this kind of calculation helps solve real-world problems, like determining paint needed to cover walls!

Linear Equations

Linear equations are mathematical statements that show the relationship between two variables.
They are expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. These equations form straight lines when graphed.In the context of the exercise, linear equations help describe how the area changes with the alterations in dimensions. If you change the width of a rectangle, while keeping the length constant, this change can be described using a linear equation. For instance, the equation for the width increasing by 2 feet, creating an extra 36 square feet of area, can be simplified to focus on the length:

• \( 2l = 36 \)
• Resulting in \( l = 18 \)

This solves for the length when only one aspect of the rectangle (width or length) changes. By breaking the problem into simple, linear calculations, you can easily manage and evaluate adjustments in the room's dimensions.

System of Equations

A system of equations is a set of two or more equations with the same variables.
The goal is to find a common solution that satisfies all equations simultaneously.In the original problem, two scenarios are provided that lead to two different equations. The first equation is derived from the width increasing by 2 feet, while the length remains constant. The second equation considers both the width and length increasing:

• First equation: \( 2l = 36 \)
• Second equation (after simplification): \( 2w + 20 = 48 \)

By solving these equations together, you'll find values for both the width \( w \) and the length \( l \). This coordinated approach ensures you consider all conditions and adjustments described in the problem, providing a comprehensive solution. Using systems of equations is particularly useful when dealing with complex scenarios where multiple variables interact.