Question

2 a roll of paper that is 3 feet wide can be cut to any length.
a. if we cut a length of 2.5 feet, what is the perimeter of the paper?
b. complete the table with the perimeter for each given side length.

side length (feet)	perimeter (feet)
2
6.3
11
ℓ

then, write a rule for a function, p, that gives the perimeter of the paper in feet when the side length in feet is ℓ. use function notation.
c. what does p(11) represent in this situation? what is its value?
d. on the coordinate plane, sketch a graph of this function.
graph with y-axis labeled perimeter (feet) from 0 to 25 and x-axis labeled side length (feet) from 0 to 8

Explanation:

Part (a)

Step1: Identify the shape and formula

The paper is a rectangle with width \( w = 3 \) feet and length \( l = 2.5 \) feet. The perimeter \( P \) of a rectangle is given by the formula \( P=2(l + w) \).

Step2: Substitute the values

Substitute \( l = 2.5 \) and \( w = 3 \) into the formula:
\( P=2(2.5 + 3) \)

Step3: Calculate the sum inside the parentheses

First, calculate \( 2.5+3 = 5.5 \)

Step4: Multiply by 2

Then, \( P = 2\times5.5=11 \)

Step1: Substitute \( l = 1 \) into \( P = 2l + 6 \)

\( P=2(1)+6 \)

Step2: Calculate

\( 2 + 6=8 \)

Step1: Substitute \( l = 2 \) into \( P = 2l + 6 \)

\( P=2(2)+6 \)

Step2: Calculate

\( 4 + 6 = 10 \)

Answer:

The perimeter of the paper is 11 feet.

Part (b)

The perimeter formula for a rectangle is \( P = 2(l + 3)=2l+6 \) (since \( w = 3 \)). We will use this formula to find the perimeter for each side length.

• For \( l = 1 \):