Question

lesson 2 cool-down the side lengths of rectangles

1. what are all of the possible side lengths of a rectangle with area 21 square units?
2. what are all of the possible side lengths of a rectangle with area 50 square units?

Explanation:

Question 1

Step1: Recall the area formula of a rectangle

The area of a rectangle is given by \( A = l \times w \), where \( l \) is the length and \( w \) is the width. We need to find all pairs of positive integers \( (l, w) \) such that \( l \times w = 21 \).

Step2: Find the factor pairs of 21

We find the positive integer factors of 21. The factors of 21 are 1, 3, 7, and 21. So the factor pairs (length, width) are:

• When \( l = 1 \), \( w = \frac{21}{1}=21 \)
• When \( l = 3 \), \( w = \frac{21}{3} = 7 \)
• When \( l = 7 \), \( w = \frac{21}{7}=3 \) (but this is the same as the previous pair with length and width swapped)
• When \( l = 21 \), \( w=\frac{21}{21} = 1 \) (also same as the first pair with length and width swapped)

Step1: Recall the area formula of a rectangle

The area of a rectangle is \( A=l\times w \), where \( l \) is the length and \( w \) is the width. We want to find all pairs of positive integers \( (l, w) \) such that \( l\times w = 50 \).

Step2: Find the factor pairs of 50

The factors of 50 are 1, 2, 5, 10, 25, 50. The factor pairs (length, width) are:

• When \( l = 1 \), \( w=\frac{50}{1}=50 \)
• When \( l = 2 \), \( w=\frac{50}{2}=25 \)
• When \( l = 5 \), \( w=\frac{50}{5}=10 \)
• When \( l = 10 \), \( w=\frac{50}{10}=5 \) (same as the previous pair with length and width swapped)
• When \( l = 25 \), \( w=\frac{50}{25}=2 \) (same as the pair with \( l = 2 \) and \( w = 25 \) with length and width swapped)
• When \( l = 50 \), \( w=\frac{50}{50}=1 \) (same as the pair with \( l = 1 \) and \( w = 50 \) with length and width swapped)

Answer:

The possible side - length pairs (length, width) of the rectangle with an area of 21 square units are \( (1, 21) \), \( (3, 7) \), \( (7, 3) \) and \( (21, 1) \). If we consider non - ordered pairs of side lengths (treating length and width as just two sides), the possible side lengths are 1 and 21, 3 and 7.

Question 2