Expert: Scotto - 4/21/2008

Question
QUESTION: A rectangle PQRS is inscribed in the region between the x-axis and the part of the graph y= cos 4x specified by -pi/8 < x< pi/8. ( the < sign is less than or equal to). Determine the coordinates of P for which the perimeter of PQRS is a maximum.

Im not too sure where to begin with this one, all i can think of is trying to come up with a perimeter eqn. and have only the variable x in it but then it gets too complicated. Not sure thats the right way to go...
Thanks in advance for any help

ANSWER: The cos(x) curve has a maximum of 1 at x=0, which means length of the base is x - -x = 2x.  The height is then cos(x).  The equation is then base * height.  Base is 2x, height is cos(x).  Use the product rule, differentiate, and sove for x.

Once this has been done, check the end points and see what the value is there, or you can take the second derivative and put in the value you got for x.  A negative result means that it is a maximum.

You can right back for more explanation on this or any other question.

Have a great day.

---------- FOLLOW-UP ----------

QUESTION: Why is the eqn h*b?.. shouldnt it be P= 2b + 2h where b would be 2x and h would be cos 4x?

Answer
What I did was an error - I was starting to think that the area was somehow involved.  After reading the problem again, I realize that this is not so.

You're right - set P to be 2b plus 2w.  The base would be 2x and the height would be cos(4x).  Take the derivative of this function, set it equal to zero, and solve.  Also, check the endpoints of this function by setting x equal to pi/8.

Thanks for correcting me and have a great day!  I hope this clarifies the problem.