Problem 22 - Precalculus Student Solutions Manual 5th

Question

A landscape architect has included a rectangular flower bed measuring 9 ft by 5 ft in her plans for a new building. She wants to use two colors of flowers in the bed, one in the center and the other for a border of the same width on all four sides. If she has enough plants to cover \(24 \mathrm{ft}^{2}\) for the border, how wide can the border be?

Step-by-Step Solution

Answer

The border width is 1 foot.

Step 1: Understand the Problem

The problem describes a rectangular flower bed, and we need to find the width of the border that takes up 24 square feet.

Step 2: Set Up Equations

Let's denote the width of the border by x feet. The inner rectangle (center) dimensions will be reduced by 2x (both length and width) because the border is on all four sides. Thus, the dimensions of the inner rectangle will be (9 - 2x) by (5 - 2x).

Step 3: Calculate the Area of the Inner Rectangle

The area of the inner rectangle is given by the formula: Area_{inner} = (9 - 2x)(5 - 2x)

Step 4: Total Area Covered by Flowers

The total area of the entire flower bed is 45 square feet (since 9*5 = 45). The area covered by the inner rectangle will be Area_{inner} = (9 - 2x)(5 - 2x)

Step 5: Area of the Border

The area of the border will be the total area of the bed minus the area of the inner rectangle, and this should equal 24 square feet: Area_{border} = 45 - (9 - 2x)(5 - 2x) = 24

Step 6: Solve the Equation

45 - (45 - 18x - 10x + 4x^2 ) = 24 45 - (45 - 28x + 4x^2) = 24 45 - 45 + 28x - 4x^2 = 24 28x - 4x^2 = 24

Step 7: Simplify and Solve for x

Rearrange the equation: 4x^2 - 28x + 24 = 0 Divide the whole equation by 4: x^2 - 7x + 6 = 0 By factoring: (x - 6)(x - 1) = 0 So x = 6 or x = 1

Step 8: Determine the Feasible Solution

Since the width of the border cannot exceed the smallest dimension of the bed (5 ft), x = 6 is not feasible. Thus, x = 1 is the only solution.

Key Concepts

rectangular flower bed

A rectangular flower bed is a common design element used by landscape architects. In this problem, the flower bed measures 9 feet in length and 5 feet in width. It is important to understand the total area initially, which can be calculated by multiplying the length and width. When working with rectangular flower beds, you should always remember:

• Total Area = Length × Width
• Dimensions are usually given in feet or meters

Remember, the dimensions provide a starting point for further calculations, especially when additional elements like borders are added.

border width calculation

To find the border width, you need to determine how the dimensions change due to the border. In this case, the border has the same width on all four sides. Let's denote this width by x feet. This means the flower bed's inner rectangle, which remains after accounting for the border, has its dimensions reduced by 2x (because the border is on both sides of each dimension).
Follow these steps for border width calculation:

• Use x to represent the unknown border width
• Subtract 2x from both original dimensions
• New dimensions: (Original Length - 2x) and (Original Width - 2x)

Here, the new dimensions become (9 - 2x) feet and (5 - 2x) feet.

area subtraction method

The area subtraction method is used to find the area of a designated part of a larger shape. In this problem, we want to find out how much area the border occupies. Here's a step-by-step guide:

• Calculate the total area of the flower bed without any border
• Calculate the area of the inner rectangle (the part inside the border)
• Subtract the area of the inner rectangle from the total area to find the border's area

So, for our flower bed, the total area is 45 square feet, the inner rectangle's area is (9 - 2x)(5 - 2x) square feet, and the border's area can be found using:
Area_{border} = Total Area - Inner Area
45 - (9 - 2x)(5 - 2x)

quadratic equations

Quadratic equations are essential when dealing with areas that involve squares. In this problem, after setting up our area equation, we get a quadratic equation in terms of x. Here’s how to handle it:

• Start with the equation derived from the area subtraction method: 45 - (9 - 2x)(5 - 2x) = 24
• Simplify this equation to standard quadratic form: 4x^2 - 28x + 24 = 0
• Reduce it: x^2 - 7x + 6 = 0
• Factor it: (x - 6)(x - 1) = 0, giving solutions x = 6 or x = 1

Among the solutions, you need to select the feasible one based on physical constraints (e.g., the border cannot exceed the smallest dimension of the flower bed). In this case, we accept x = 1 as the valid border width.