area under a graph problem?

Question

A rectangle is inscribed in the region bounded by the x-aix, the y-axis, and the graph of x=2y-8=0. Write the area as a function of x. Then determine the domain of the function. What are the dimensions of the rectangle that will produce the maximum area?

Unfortunately, I can't draw the diagram of the graph and the shaded box region, but on the x-axis, the width 4 and on the y-axis, the length is 2.

sorry, there's a type on the graph equation! it's x+2y-8=0!

Answer 1

A = (8x-x^2)/2 [0= dA/dx = 2(8-2x)/4
Setting this to ) we get x = 4
So (8*4 -4^2)/2 = 8 = max area
The dimensions are x = 4 y=2.

Answer 2

First, plot the line so you can get and understanding of the problem. Next, you know the area of a rectangle is going to be the x coordinate multiplied by the y coordinate, and y is related to x by y = 1/2*x - 4. Do the multiplication: x*y = x*(1/2*x - 4) = (x^2)/2 - 4x This is the basically the answer. However there are two more points that need to be mentioned. First, area cannot be negative and this rectangle lives in the forth quadrant, and second, you need to put the bounds in. So your final answer should look like: for 0