A rectangle has a perimeter of 36 m. If the length of the rectangle is randomly chosen, what is the expected area?
To find the expected area of a randomly chosen length for a rectangle with a perimeter of 36 m, we need to consider the relationship between the length and the width of the rectangle.
Let's assume the length of the rectangle is l meters. Since the perimeter of a rectangle is given by the formula P = 2l + 2w, where P is the perimeter, l is the length, and w is the width, in this case, we have:
36 = 2l + 2w
We want to find the expected area, which is the average area of all possible rectangles with a perimeter of 36 m. The formula for calculating the area of a rectangle is A = l * w. Since we randomly choose the length, we need to express the width in terms of the length.
Rearranging the perimeter formula, we get:
2w = 36 - 2l
w = (36 - 2l) / 2
w = 18 - l
Substituting the width into the formula for the area, we have:
A = l * (18 - l)
A = 18l - l^2
Now, to calculate the expected area, we need to find the average of the area across all possible lengths. Since the length can vary from 0 to 18, we integrate the area formula over this interval and divide by the total length:
Expected area = (1 / (18 - 0)) * ∫[0, 18] (18l - l^2) dl
Integrating the above expression, we get:
Expected area = (1 / 18) * [9l^2 - (l^3 / 3)] evaluated from l = 0 to l = 18
Expected area = (1 / 18) * [(9 * 18^2 - (18^3 / 3)) - (9 * 0^2 - (0^3 / 3))]
Simplifying further, we find:
Expected area = (1 / 18) * (2916 - 216)
Expected area = (1 / 18) * 2700
Expected area = 150
Therefore, the expected area of a rectangle with a perimeter of 36 m, when the length is randomly chosen, is 150 square meters.