Question

Consider all rectangles such that the rectangle's length is greater than the rectangle's width and the length and width are whole numbers of inches. Which of the following perimeters, in inches, is NOT possible for such a rectangle with an area of 144 square inches?​ A. 48 B. 60 C. 80 D. 102

Answer 1

By calculating the potential perimeters for all whole number pairs of length and width that multiply to an area of 144 square inches, it is determined that the perimeter of 48 inches is not possible, making it the correct answer.

Step-by-step explanation:

To determine which perimeter is not possible for a rectangle with an area of 144 square inches, where the length is greater than the width and both are whole numbers, we'll approach the problem systematically.

The area of a rectangle is found by multiplying its length by its width (l × w). Since the area is 144 square inches, the possible pairs of whole numbers for length and width are (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), and (12, 12). However, since the length must be greater than the width, the pair (12, 12) is eliminated.

The perimeter of a rectangle is found by adding together twice the width and twice the length (2l + 2w). Therefore, we calculate the perimeter for each pair:

• (1, 144): 2(1) + 2(144) = 2 + 288 = 290 inches
• (2, 72): 2(2) + 2(72) = 4 + 144 = 148 inches
• (3, 48): 2(3) + 2(48) = 6 + 96 = 102 inches
• (4, 36): 2(4) + 2(36) = 8 + 72 = 80 inches
• (6, 24): 2(6) + 2(24) = 12 + 48 = 60 inches
• (8, 18): 2(8) + 2(18) = 16 + 36 = 52 inches
• (9, 16): 2(9) + 2(16) = 18 + 32 = 50 inches

From this list, we see that a perimeter of 48 inches is not possible. Therefore, the correct answer is A. 48 inches.

Answer 2

The impossible perimeter for a rectangle with an area of 144 square inches and whole number dimensions where length is greater than width is 48 inches because it does not correspond to any valid combinations of factors of 144.

Step-by-step explanation:

When considering rectangles with a fixed area of 144 square inches where the length is greater than the width and both dimensions are whole numbers, we are looking for combinations of factors of 144 that meet these criteria. For example, possible pairs could be (1,144), (2,72), (3,48), (4,36), (6,24), (8,18), (9,16), and (12,12). Only the pairs where the first number is less than the second number are valid, as the length must be greater than the width. From these pairs, we can calculate their perimeters by using the formula P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

For all the valid dimension pairs:

(1,144): P = 2(1+144) = 290 inches (Not listed)

(2,72): P = 2(2+72) = 148 inches (Not listed)

(3,48): P = 2(3+48) = 102 inches (Listed)

(4,36): P = 2(4+36) = 80 inches (Listed)

(6,24): P = 2(6+24) = 60 inches (Listed)

(8,18): P = 2(8+18) = 52 inches (Not listed)

(9,16): P = 2(9+16) = 50 inches (Not listed)

Therefore, the only perimeter option listed that is not possible for a rectangle with the given criteria is 48 inches, as it does not correspond to any of the valid dimension pairs' perimeters. The answer to the question is option A: 48, which is not a possible perimeter for a rectangle with an area of 144 square inches where length is greater than width.