Question

1. complete a table of values for the width and length of a rectangle with an area of 12 square inches. then complete the graph.

table: width of rectangle (in.) rows: 1, 8, (handwritten: 4), (handwritten: 1.5), (handwritten: 16), 20; length of rectangle (in.) rows: (handwritten), (handwritten), 3, 8, 0.75 (in.), (handwritten: 0.6).

1. describe how the width of the rectangle affects the length of the rectangle. handwritten: \the width and length of the rectangle are inversely proportional.\
2. do the width and length of a rectangle with an area of 12 square inches form a proportional relationship? explain your reasoning.

Explanation:

Question 6: Completing the Table (Using Area Formula of Rectangle: \( A = \text{width} \times \text{length} \), \( A = 12 \))

Step 1: Recall the area formula

The area \( A \) of a rectangle is given by \( A = w \times l \), where \( w \) is the width and \( l \) is the length. Here, \( A = 12 \), so \( l=\frac{12}{w} \) (or \( w=\frac{12}{l} \)).

Step 2: Calculate for \( w = 1 \)

Substitute \( w = 1 \) into \( l=\frac{12}{w} \):
\( l=\frac{12}{1}=12 \).

Step 3: Calculate for \( w = 8 \)

Substitute \( w = 8 \) into \( l=\frac{12}{w} \):
\( l=\frac{12}{8}=\frac{3}{2}=1.5 \).

Step 4: Calculate for \( l = 3 \)

Substitute \( l = 3 \) into \( w=\frac{12}{l} \):
\( w=\frac{12}{3}=4 \).

Step 5: Calculate for \( l = 8 \)

Substitute \( l = 8 \) into \( w=\frac{12}{l} \):
\( w=\frac{12}{8}=\frac{3}{2}=1.5 \).

Step 6: Calculate for \( l = 0.75 \)

Substitute \( l = 0.75 \) into \( w=\frac{12}{l} \):
\( w=\frac{12}{0.75}=16 \).

Step 7: Calculate for \( w = 20 \)

Substitute \( w = 20 \) into \( l=\frac{12}{w} \):
\( l=\frac{12}{20}=0.6 \).

Question 7: How Width Affects Length

For a rectangle with a constant area (12 square inches), the length and width are related by \( l=\frac{12}{w} \). As the width (\( w \)) increases, the length (\( l \)) decreases, and vice versa. This is an inverse relationship (when one quantity increases, the other decreases proportionally to maintain the constant product \( w \times l = 12 \)).

A proportional relationship (direct proportion) is of the form \( y = kx \) (constant ratio \( \frac{y}{x}=k \)), while an inverse proportion is \( y=\frac{k}{x} \) (constant product \( xy = k \)). For the rectangle, \( w \times l = 12 \) (constant product), not a constant ratio. For example:

• When \( w = 1 \), \( l = 12 \), ratio \( \frac{l}{w}=12 \).
• When \( w = 4 \), \( l = 3 \), ratio \( \frac{l}{w}=\frac{3}{4}=0.75 \).

The ratio \( \frac{l}{w} \) is not constant, so width and length do not form a direct proportional relationship. (They form an inverse proportional relationship, but the question asks about "proportional" which typically means direct unless specified otherwise.)

Answer:

As the width of the rectangle increases, the length decreases (and vice versa) because their product (the area) remains constant at 12 square inches. They have an inverse relationship.

Question 8: Proportional Relationship?