Problem 69 - Algebra 2

Question

GARDENS Helene Jonson has a rectangular garden 25 feet by 50 feet. She wants to increase the garden on all sides by an equal amount. If the area of the garden is to be increased by 400 square feet, by how much should each dimension be increased?

Step-by-Step Solution

Answer

Each dimension should be increased by 2.5 feet.

Step 1: Understand the Problem

The original garden is rectangular with dimensions of 25 feet by 50 feet. We need to increase each side by the same length so the total area of the garden increases by 400 square feet. We want to find this length.

Step 2: Determine the Equation

Let the increase in each dimension be represented by \( x \). The new dimensions of the garden will be \( 25 + 2x \) and \( 50 + 2x \). The area of the new garden will be \((25 + 2x)(50 + 2x)\).

Step 3: Set Up the Area Equation

The new area minus the original area (1250 square feet) should equal 400 square feet. Therefore, the equation is: \((25 + 2x)(50 + 2x) = 1250 + 400\).

Step 4: Simplify the Equation

First, solve \( (25 + 2x)(50 + 2x) = 1650 \). Expand this to get:\( 1250 + 50 \times 2x + 25 \times 2x + 4x^2 = 1650 \).This simplifies to \( 1250 + 150x + 4x^2 = 1650 \).

Step 5: Rearrange and Solve for 'x'

Subtract 1250 from both sides to simplify the equation:\( 150x + 4x^2 = 400 \).Rearrange it to form a standard quadratic equation: \( 4x^2 + 150x - 400 = 0 \).

Step 6: Solve the Quadratic Equation

Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), to solve where \( a = 4 \), \( b = 150 \), \( c = -400 \).Calculate:\( b^2 - 4ac = 150^2 - 4 \times 4 \times (-400) = 22500 + 6400 = 28900 \).So \( x = \frac{-150 \pm \,17}{8} \).

Step 7: Calculate the Values of 'x'

The possible solutions for \( x \) are:\( x = \frac{-150 + 170}{8} = \frac{20}{8} = 2.5 \)\( x = \frac{-150 - 170}{8} = \frac{-320}{8} = -40 \).

Step 8: Select the Physical Solution

Since an increase in physical dimensions cannot be negative, we discard \( x = -40 \). Therefore, \( x = 2.5 \) is the correct increase amount.

Key Concepts

Rectangular Garden

A rectangular garden is simply a garden plot that is shaped like a rectangle. Rectangle means it has four sides with opposite sides being equal in length. In this exercise, the original garden owned by Helene Jonson has dimensions of 25 feet by 50 feet. This means the garden is not a square, but an elongated rectangle. Rectangles are basic geometric shapes that make calculating areas and perimeters straightforward with clear formulas.

• Area is calculated as width times length.
• Perimeter is the sum of all the sides.

Knowing the original dimensions helps you start solving problems related to changes in size or other geometric properties.

Area Increase

An increase in area means that the size of the space within the garden’s boundaries is made larger. Helene wishes to expand her rectangular garden by increasing its dimensions uniformly on all sides. In this problem, the new area should be 400 square feet larger than the current area.

• The current area of the garden is calculated as 25 feet multiplied by 50 feet, which equals 1250 square feet.
• The aim is to add 400 square feet, leading to a total new area of 1650 square feet.

Increasing area usually involves stretching both the length and the width, which is why uniform adjustment is needed for geometry problems like this.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In Helene's garden problem, we are tasked with determining how much each dimension of the garden should increase by a certain amount. To do this, we need to form an equation based on our understanding of the garden's dimensions. In this problem, we have applied algebraic methods to represent the problem:

• Let the increase in each side be represented by the variable \( x \).
• The new dimensions become \( 25 + 2x \) and \( 50 + 2x \).

By substituting these dimensions into the formula for area, you get a new equation. Solving it requires careful expansion and simplification of terms to isolate \( x \).

Quadratic Formula

The quadratic formula is a method for finding the roots of a quadratic equation, which is any equation that can be rearranged into the standard format \( ax^2 + bx + c = 0 \). In this exercise, the equation formed is \( 4x^2 + 150x - 400 = 0 \). Each of the terms represents parts of this garden expansion problem:

• \( a = 4 \)
• \( b = 150 \)
• \( c = -400 \)

To find \( x \), you use the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Plugging in these values helps calculate \( x \). The discriminant \( b^2 - 4ac \) tells us whether there are real roots. Here it’s positive, so two solutions exist. Due to physical constraints (we can't increase dimensions by negative lengths), we select the positive root, leaving \( x = 2.5 \) as the sensible choice.