Question

performance task
answer the questions and show all your work on separate paper.
cody is having a rectangular wooden fence built. the fencing company just built a rectangular fence for cody’s neighbor and can adjust its plans for cody’s fence. cody knows the following information about the fences and the company’s plans.

• cody’s fence must have sides that are twice as long as the sides of his neighbor’s fence.
• his neighbor’s fence contains an area of 887.5 ft².
• the length of his neighbor’s fence is 12.5 ft.
• the fencing company can use boards that have widths of either 0.33 ft, 0.5 ft, or 0.75 ft. all the boards in the fence must be the same size, and only a whole number of boards can be used.
• the fencing company charges $2.00 per board for the first 100 boards and $1.50 for each additional board.

help cody understand the plans for his fence. first, determine the length and width of cody’s fence. then choose a board width so that the fencing company will use a whole number of boards. finally, find out how much his fence costs. write an equation that relates the number of boards cody needs for his fence, b, to the total cost in dollars, c.
reflect
use mathematical practices after you complete the task, choose one of the following questions to answer.

• make sense of the problem how did you use the information about the neighbor’s fence to determine length and width of cody’s fence?
• argue and critique how did you check to make sure that cody chose the best width for the boards in his fence?

Explanation:

Step1: Find neighbor's width

The area of a rectangle is \( A = l \times w \). For the neighbor's fence, \( A = 887.5\space\text{ft}^2 \) and \( l = 12.5\space\text{ft} \). So, \( w=\frac{A}{l}=\frac{887.5}{12.5}=70\space\text{ft} \).

Step2: Find Cody's length and width

Cody's sides are twice the neighbor's. So, Cody's length \( L = 2\times12.5 = 25\space\text{ft} \), and width \( W = 2\times70 = 140\space\text{ft} \).

Step3: Calculate perimeter of Cody's fence

Perimeter of a rectangle is \( P = 2(L + W) \). So, \( P = 2(25 + 140)=2\times165 = 330\space\text{ft} \).

Step4: Choose board width (e.g., 0.5 ft)

For board width \( 0.5\space\text{ft} \), number of boards \( b=\frac{330}{0.5}=660 \).

Step5: Calculate cost

First 100 boards: \( 100\times2 = 200 \). Additional boards: \( 660 - 100 = 560 \), cost for additional: \( 560\times1.5 = 840 \). Total cost \( c = 200 + 840 = 1040 \).

Step6: Write cost equation

If \( b\leq100 \), \( c = 2b \). If \( b > 100 \), \( c = 2\times100+1.5(b - 100)=200 + 1.5b - 150 = 1.5b + 50 \).

Answer:

• Length of Cody's fence: \( 25\space\text{ft} \), Width: \( 140\space\text{ft} \)
• For board width \( 0.5\space\text{ft} \), number of boards \( b = 660 \), cost \( c=\$1040 \)
• Cost equation: \( c=

$$\begin{cases}2b, & b\leq100 \\ 1.5b + 50, & b > 100\end{cases}$$

\) (example with \( 0.5\space\text{ft} \) board width)