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Chapter 1: Problem 69

Suppose that you want to build a square fenced-in area for your dog. Fencing is purchased in linear feet. a. Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased. b. If you purchase 100 linear feet, what is the area of your dog pen? c. If you purchase 200 linear feet, what is the area of your dog pen?

Short Answer

Expert verified
a. The composite function is \( A(x) = \frac{x^2}{16} \). b. 625 square feet. c. 2500 square feet.

Step by step solution

01

Understanding the Problem

To solve this problem, we want to find the area of a square pen given a certain amount of fencing in linear feet. Since the pen is a square, the perimeter will be 4 times the side length.
02

Define Functions

Let \( P(x) = x \) be the function representing the perimeter based on the linear feet \( x \). Since the pen is square, the side length \( s \) is \( \frac{x}{4} \). We want to express the area \( A(s) = s^2 \) as a function of \( x \).
03

Create Composite Function

Substitute the expression for \( s \) into the area function \( A \). The composite function will be \( A(x) = \left(\frac{x}{4}\right)^2 \). Simplifying, we get \( A(x) = \frac{x^2}{16} \).
04

Area with 100 Linear Feet

Plug \( x = 100 \) into the composite function \( A(x) = \frac{x^2}{16} \). Calculate \( A(100) = \frac{100^2}{16} = \frac{10000}{16} = 625 \). The area is 625 square feet.
05

Area with 200 Linear Feet

Plug \( x = 200 \) into the composite function \( A(x) = \frac{x^2}{16} \). Calculate \( A(200) = \frac{200^2}{16} = \frac{40000}{16} = 2500 \). The area is 2500 square feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

perimeter
When thinking about building a fenced-in area, the perimeter is a crucial concept to understand. The perimeter is the total distance around the edges of a two-dimensional shape. In this case, we are dealing with a square dog pen.
Remember that a square has four equal sides. Therefore, the perimeter of a square is simply four times the length of one side. So, if the length of one side is \( s \), the perimeter \( P \) can be calculated as follows:
  • \( P = 4s \)
If you know the length of one side, multiplying it by four gives you the perimeter. Conversely, if the total perimeter is known, you can find the side length by dividing the perimeter by four:
  • \( s = \frac{P}{4} \)
This relationship is essential for figuring out how much fencing you'll need and for calculating the area enclosed by the fence.
area of a square
The area of a square refers to the amount of two-dimensional space it covers. Determining the area of your dog pen is key to understanding how much room your furry friend will have to roam.
The area is calculated using the side length of the square. If \( s \) represents the length of a side, the formula for the area \( A \) is:
  • \( A = s^2 \)
This means you multiply the side length by itself. For example, if each side of the square is 5 feet, the area would be \( 5 \times 5 = 25 \) square feet.
In the case of purchasing linear feet for the perimeter, you find out the side length first by dividing the perimeter by four, then use this side length to compute the area using the formula mentioned.
This step-by-step approach ensures accurate calculation of space within the fenced area.
linear feet
Linear feet measure length in a straight line, without considering width, height, or thickness. It鈥檚 a simple unit of measure used for materials like fencing, molding, or flooring, which are often sold by the foot.
In this exercise, when you purchase fencing in linear feet, you're buying the total length of all the sides of your square dog pen combined.
Linear feet help you plan how much fencing material you need. When building your square pen, knowing the total linear feet aids in determining the perimeter, and thus the side length:
  • Side length of the square = Total Linear Feet Purchased / 4
This conversion ensures all purchased materials are used effectively to maximize the fenced area your dog gets to enjoy. Keep in mind, more linear feet mean a larger perimeter and, consequently, a larger area.

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