1. ## Word Problem

1) The graphs of y=radical of x, x=9, and y=0 bound a region in the first quadrant. Find the dimensions of the rectangle of maximum area and the dimensions of the rectangle of maximum perimeter that can be inscribed in this region. (the sides of the rectangles should be parallel to the axes)

-I was able to get the Area as (9-x)(the radical of x) and the perimeter as
2(9-x)+2(the radical of x). However, I don't know what i need to do next in order to solve the problem.

-M

2. Now differentiate your problems, this will give you the slope of the area, and the slope of the perimeter, which basically tells you the rate at which they change.

That means when the derivative is increasing, the area/perimeter is getting larger, and when the slope is decreasing, they are getting smaller. Obviously, as it is increasing, it is getting larger, which is what you want. So if we were looking at the graph of the area or perimeter, we would keep letting it increase until it hit a point where it started to decrease again. That point would be the largest area, or the largest perimeter. And at that point, it is changing from increasing to decreasing. Because it is changing, it's tangent line is zero. And since the derivative graphs the slope, wherever the derivative equals zero is a potential maximum or minimum.

So find all the zeros of your derivative, rule out any that are not in the domain, and then test points on either side to make sure that the left side is positive (meaning the function is increasing up to the point) and the right side is negative (meaning the function is decreasing after the point)

This would mean that you had found a maximum.

Recap:
Find the zeroes of the derivatives.
Check points on either side to make sure they are maximums.
If they are, then that is the x value of your equation, plug it into the function to find the actual area and actual perimeter.

3. Originally Posted by blurain
1) The graphs of y=radical of x, x=9, and y=0 bound a region in the first quadrant. Find the dimensions of the rectangle of maximum area and the dimensions of the rectangle of maximum perimeter that can be inscribed in this region. (the sides of the rectangles should be parallel to the axes)

-I was able to get the Area as (9-x)(the radical of x) and the perimeter as
2(9-x)+2(the radical of x). However, I don't know what i need to do next in order to solve the problem.

-M
why didn't you make this comment in the original post?

you are correct. now you want to maximize the function. find the derivative, set it to zero and solve for x