Question

8 challenge huan is redesigning his bedroom, which is the shape of a rectangle. a huan knows that the area of his bedroom is 180 square feet. what are all the possible whole number dimensions of huan’s bedroom?

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = l \times w \), where \( l \) is the length and \( w \) is the width. Here, \( A = 180 \) square feet, and we need to find all pairs of positive whole numbers \( (l, w) \) such that \( l \times w=180 \).

Step2: Find the factor pairs of 180

We start by finding all the positive - integer factors of 180.

• When \( l = 1 \), \( w=\frac{180}{1}=180 \) (since \( 1\times180 = 180 \))
• When \( l = 2 \), \( w=\frac{180}{2}=90 \) (since \( 2\times90=180 \))
• When \( l = 3 \), \( w=\frac{180}{3} = 60 \) (since \( 3\times60 = 180 \))
• When \( l = 4 \), \( w=\frac{180}{4}=45 \) (since \( 4\times45=180 \))
• When \( l = 5 \), \( w=\frac{180}{5}=36 \) (since \( 5\times36 = 180 \))
• When \( l = 6 \), \( w=\frac{180}{6}=30 \) (since \( 6\times30=180 \))
• When \( l = 9 \), \( w=\frac{180}{9}=20 \) (since \( 9\times20 = 180 \))
• When \( l = 10 \), \( w=\frac{180}{10}=18 \) (since \( 10\times18=180 \))
• When \( l = 12 \), \( w=\frac{180}{12}=15 \) (since \( 12\times15 = 180 \))

We can also consider the reverse pairs (swapping length and width), but since length and width are just two dimensions of the rectangle, the unique pairs of whole - number dimensions (considering \( l\geq w \) to avoid repetition) are: \( (1,180) \), \( (2,90) \), \( (3,60) \), \( (4,45) \), \( (5,36) \), \( (6,30) \), \( (9,20) \), \( (10,18) \), \( (12,15) \) and their reverse pairs (e.g., \( (180,1) \), \( (90,2) \) etc.). But if we consider the non - ordered pairs (where the order of length and width does not matter), the pairs are as listed above with \( l\) and \( w\) as positive integers.

Answer:

The possible whole - number dimension pairs (length, width) in feet are: \((1, 180)\), \((2, 90)\), \((3, 60)\), \((4, 45)\), \((5, 36)\), \((6, 30)\), \((9, 20)\), \((10, 18)\), \((12, 15)\) and their reverse pairs (e.g., \((180, 1)\), \((90, 2)\) etc.). If we consider unordered pairs (where length and width are not distinguished by order), the pairs are \((1, 180)\), \((2, 90)\), \((3, 60)\), \((4, 45)\), \((5, 36)\), \((6, 30)\), \((9, 20)\), \((10, 18)\), \((12, 15)\)