The area of a rectangle is 32 square units. Its sides have integral lengths. Find the probability that its perimeter is greater than 30 units.

The only possible integral dimensions are

1x32
2x16
4x8

Of the three choices, two have perimeter > 30

To find the probability, we need to determine the total number of possible rectangles with integral side lengths that have an area of 32 square units.

Let's start by listing down all the possible combinations for the side lengths that would result in an area of 32 square units. Since the sides have integral lengths, the combinations would include pairs of factors of 32.

The factors of 32 are: 1, 2, 4, 8, 16, 32.

Therefore, the possible combinations for the side lengths are:
1 x 32, 2 x 16, 4 x 8, 8 x 4, 16 x 2, 32 x 1.

Next, we need to calculate the perimeter for each rectangle. The perimeter of a rectangle is given by the formula:

perimeter = 2(length + width)

Calculating the perimeter for each combination:

For the combination 1 x 32:
perimeter = 2(1 + 32) = 2(33) = 66 units

For the combination 2 x 16:
perimeter = 2(2 + 16) = 2(18) = 36 units

For the combination 4 x 8:
perimeter = 2(4 + 8) = 2(12) = 24 units

For the combination 8 x 4:
perimeter = 2(8 + 4) = 2(12) = 24 units

For the combination 16 x 2:
perimeter = 2(16 + 2) = 2(18) = 36 units

For the combination 32 x 1:
perimeter = 2(32 + 1) = 2(33) = 66 units

Now, let's determine the number of rectangles with a perimeter greater than 30 units. From the above calculations, we can see that there are 2 rectangles with a perimeter greater than 30 units (perimeter = 36 units or 66 units).

Since the total number of possible rectangles is 6, the probability can be calculated as:
probability = (number of rectangles with perimeter > 30) / (total number of possible rectangles)
probability = 2 / 6
probability = 1/3

Therefore, the probability that the perimeter of a rectangle with an area of 32 square units is greater than 30 units is 1/3.