Question

The short sides of a rectangle are 2 inches. The long sides of the same rectangle are three less than an unknown number of inches. Two students are told that the perimeter of this rectangle is at least 30 inches. The first student says, "The length of the long side must be at least 13 inches". The second student says, "The unknown number must be at least 10 inches". Which of the following statements is true?

Answer 1

The first student was right.

The length of the long side is at least 13 inches.

Explanation

The perimeter of any figure is the distance all round.

Perimeter of a rectangle = 2(l+w). Where l is length and w is the width.

2(l + w) ≥ 30

2{(x-3) + 2} ≥ 30

2(x-3+2) = 30

2(x - 1) ≥ 30

x - 1 ≥ 15

x ≥ 16

When x = 16,

l = 16-3

= 13 inches

The length of the long side is 13 inches. The first student was right.

Answer 2

The true statement is "The length of the long side must be at least 13 inches".

Explanation:

The short sides of a rectangle are 2 inches which will be its width.

The perimeter of the rectangle is known which is 30 inches.

The longest side (length) is said to be 3 less an unknown number, so assuming an unknown number x, it will be x - 3.

We can write an equation for perimeter:

Perimeter = 2l + 2w

Perimeter <= 2(x-3) +2(2)

30 <= 2x - 6 + 4

2x >= 32

x >= 16

Therefore the length x - 3 would be = 16 - 3 = 13 inches

So the first student is right.