Expert: Paul Klarreich - 4/11/2006

Question
Hi,
A rectangle with one side on the x-axis has its upper vertices on the graph of y=cosx .What is the minimum area of the shaded region?
They give a figure with this question that I will try to describe to you. The rectangle is on the part of cosx by y=0 and coming out from both sides of the x-axis horizontally. The shaded region is everything,but that rectangle. They don't give a whole cos graph. They only give the y=-x^2 portion of it till it hits the x-axis.
[If you go to http://apcentral.collegeboard.com/repository/05836apcoursdesccalc0_4313.pdf
, you should be able to get the graph by scrolling down to the sample BC questions.]
The answer to this question is .878, but I can't get that.Please help.

Answer
Hi, Jeff,

Subject:  area
Question:  Hi,
A rectangle with one side on the x-axis has its upper vertices on the graph of y=cosx.  What is the minimum area of the shaded region?

They give a figure with this question that I will try to describe to you. The rectangle is on the part of cosx by y=0 and coming out from both sides of the x-axis horizontally. The shaded region is everything,but that rectangle. They don't give a whole cos graph. They only give the y=-x^2 portion of it till it hits the x-axis.
[If you go to http://apcentral.collegeboard.com/repository/05836apcoursdesccalc0_4313.pdf
, you should be able to get the graph by scrolling down to the sample BC questions.]
The answer to this question is .878, but I can't get that.Please help.
--------------------------------------
OK, I see the question.  The area should be:

The area under  y = cos x  from -pi/2 to pi/2  MINUS the area of the largest rectangle that can be inscribed in it.

The first area is obtained by DOUBLING:

{pi/2
|    cos x dx =
}0

sin x from 0 to pi/2 = sin pi/2 - sin 0 = 1

So this area is 2.

Now you have a standard maximization problem.  Assuming your rectangle has width 2x, and height y = cos x, the area is:

A = 2x cos x

dA/dx = 2 cos x - 2x sin x

Set that equal to zero:

cos x - x sin x = 0
cos x = x sin x

x = cot x

I don't know any good way to solve this, except numerically.  I see you are allowed to use a graphical calculator on this part. (What HAS the world come to?)  So I think you obtain an approximate solution to this, which MY crude calculator gives as  x ~~ 0.862

Now the area of the rectangle is  2x cos x = 2(0.862) cos(0.862) = 1.122

and subtracting that from 2.000 gives  0.878