# Area of a Rectangle and the Uniform Distribution

This is a video on the area of a rectangle. The question states, Suppose that elementary students’ ages are uniformly distributed from 5 to 11 years old. The rectangle that depicts the probability that a randomly selected child will be between 6.5 and 8.6 years old has base from 6.5 to 8.6 and the same height as the larger rectangle with base 5 to 11. Find the area of the smaller rectangle given that the larger rectangle has area 1. This is a long question to read. There is no way we will just be able to quickly figure this out unless we draw the picture. This picture will be a larger rectangle with a smaller rectangle inside. But the heights of the two rectangles will be the same. Here are our rectangles. Notice, the larger rectangle goes from 5 to 11. It has height h. The smaller rectangle goes from 6.5 to 8.6 and it also has that same height. I will use the fact that the area of the larger rectangle. we were told was equal to 1. I will use the area formula. I can write that 1 is equal to the area. The formula for area is base times the height. That is equal to the base which is, well, it goes from 5 to 11, so to find the base, we subtract 11 – 5 times the height, which we don’t know, so I just write h. Notice 11 – 5=6. So this equals 6h. We can say that 1 is equal to 6h. If 1=6h, then h must be 1/6. So that 6 times 1/6 is 1. Now I will use the same area formula for the smaller rectangle. We just have a different base. We know now that the area of the smaller rectangle is still the base times the height, but now the base of the smaller rectangle goes from 6.5 to 8.6. So the base is 8.6 – 6.5 times the height, which is the same as the height of the larger rectangle or 1/6. I put this into a calculator. I have to remember the parentheses, because if you forget the parentheses, you get the wrong answer. If I do 8.6 – 6.5, do that calculation, or put it in parentheses, and then multiply by 1/6, I get 0.35. That’s the answer. The probability that a child selected at random, will be between 6.5 and 8.6 years old is 0.35. I’m done with the problem. Thank you for watching this video.