Question

A rectangular room has a width of n feet and a length of m, and n does not equal m. If they are both integers and the perimeter is 36 feet, which of the following cannot be the area of the room?

1) 45
2) 56
3) 72
4) 77
5) 81

Answer 1

The perimeter of a rectangular room with a width of n feet and a length of m is 36 feet. The area of the room can be calculated by multiplying the width and length. Among the options given, the area 81 cannot be the area of the room.

Step-by-step explanation:

The perimeter of a rectangular room is the sum of the lengths of all its sides. In this case, the perimeter is given as 36 feet. We know that the sum of the lengths of two adjacent sides of a rectangle is equal to twice the sum of its width and length. So, we can write the equation 2(n + m) = 36, where n is the width and m is the length. Rearranging the equation, we get n + m = 18.

From the given information, we know that n does not equal m. So, we need to find two integers that add up to 18, where one of them could be the width and the other could be the length. The possible pairs are (1, 17), (2, 16), (3, 15), (4, 14), (5, 13), (6, 12), (7, 11), and (8, 10).

To find the area of the room, we multiply the width and length. Calculating for each possible pair, we find that the areas are 17, 32, 45, 56, 65, 72, 77, and 80. Comparing these areas to the options given, the area 81 cannot be the area of the room.