Problem 25 - Elementary Algebra

Question

The width of a rectangle is 7 feet less than its length. If the area of the rectangle is 170 square feet, then find the length and width.

Step-by-Step Solution

Answer

The length is 17 feet and the width is 10 feet.

Step 1: Understand the Problem

We are given a rectangle where the width is 7 feet less than the length. The area of the rectangle is 170 square feet. We need to find both the length and width.

Step 2: Define the Variables

Let the length of the rectangle be \( l \) feet. Therefore, the width will be \( l - 7 \) feet.

Step 3: Write the Equation for Area

The area of a rectangle is given by the formula \( \text{Area} = \text{length} \times \text{width} \). So, we write the equation: \[ l \times (l - 7) = 170 \]

Step 4: Simplify the Equation

Expand the equation: \[ l^2 - 7l = 170 \]

Step 5: Solve the Quadratic Equation

Rearrange the equation as \( l^2 - 7l - 170 = 0 \) and solve for \( l \) using the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -7 \), \( c = -170 \).

Step 6: Calculate Discriminant and Solves

The discriminant is \( b^2 - 4ac = (-7)^2 - 4 \times 1 \times (-170) = 49 + 680 = 729 \). Calculate \( l \): \[ l = \frac{7 \pm \sqrt{729}}{2} \] \( \sqrt{729} = 27\), so \( l = \frac{7 \pm 27}{2} \).

Step 7: Find Possible Lengths

Calculate the possible values of \( l \): \( l = \frac{7 + 27}{2} = 17 \) and \( l = \frac{7 - 27}{2} = -10 \). Since the length cannot be negative, \( l = 17 \) is the solution.

Step 8: Determine the Width

Substitute \( l = 17 \) back to find the width: \[ \text{Width} = l - 7 = 17 - 7 = 10 \] feet.

Key Concepts

Rectangle Area

The area of a rectangle is found by multiplying its length by its width. This is expressed in the formula:

• Area = Length × Width

In our problem, the width is defined as 7 feet less than the length. If we let the length be represented by the variable \( l \), the width then becomes \( l - 7 \).
With an area of 170 square feet given, the equation to determine these dimensions becomes \( l \times (l - 7) = 170 \). Solving this equation helps us find the dimensions of the rectangle. Understanding this relationship between area, length, and width is crucial for approaching similar problems.

Solving Equations

Solving the problem involves working with a quadratic equation. The equation derived is \( l^2 - 7l = 170 \).
Step by step, we rearrange this to form:

• \( l^2 - 7l - 170 = 0 \)

This is a quadratic equation, recognizable by its \( l^2 \) term. Quadratic equations can often be solved using the quadratic formula:

• \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = 1 \), \( b = -7 \), and \( c = -170 \). Calculating the discriminant (\( b^2 - 4ac \)) reveals the nature of the solutions.
The discriminant here was \( 729 \), which is a perfect square (\( 27^2 \)). This confirms two real solutions, but only one can be physically real in this context since length cannot be negative.

Algebraic Problem-Solving

This problem highlights critical algebraic problem-solving skills. It involves setting up an equation, simplifying, and strategically solving it using algebraic methods.
We followed a clear process:

• Defining variables based on the problem description.
• Setting up the equation using given conditions.
• Simplifying the equation to a recognizable form.
• Applying the quadratic formula to find possible solutions.
• Interpreting results logically, rejecting negative lengths.

Eventually, we determined that the length of the rectangle is 17 feet and the width 10 feet. By understanding and practicing these steps, you can tackle various algebraic challenges with confidence.