Optimization Calculus question! Please help!!

Question

Sketch the first quadrant region bounded by the axes and the graph of g(x)= 9−x/2+x. Consider various rectangles: each with one corner at the origin, adjacent sides along the axes, and opposite corner on the graph of g. Among this collection of rectangles, find the one with maximal area.

 

Width of the largest rectangle is: ?

Area of the largest rectangle is: ?

Answer 1

OK. There is a chance that g(x) is written incorrectly. I am assuming that you mean

 

g(x)= (9−x)/(2+x)

 

Because as you wrote it, g(x) is a straight line.

 

Anyway, the area of a rectangle is LW. I'll assume W to be the horizontal dimension so W = x.

If L is the vertical, then L = g(x).

And area A = x g(x).

 

To maximize area, find its derivative with respect to x, set that to zero and solve for x.

 

A = x (9 - x) / (2 + x)

 

dA/dx = -(x² + 4x - 18)/(x + 2)²

 

This derivative is zero when the numerator is zero.

 

Or when x=√22 - 2 = 2.69. That's your width. Your length is 1.35.

Area is 3.63.

 

Sanity check: try a couple of nearby values.

if x=1 g(x)=8/3 and area is 2.67.

if x=2 g(x)=7/4 and area is 3.5.

if x=3 g(x)=6/5 and area is 3.6.

 

Closing in, but you have the dimensions with the largest area. Cheers!