A vertical line divides a region into two pieces. Find the value of the coordinate x that maximizes the product of the two areas. Optimization lol Image: http://puu.sh/5Bn0j.png

Question

mathmale · Answer

I see that the two pieces (smaller regions) are first, a triangle, and second, a trapezoid.  Why not start by developing formulas (in terms of x and the slope of the hypotenuse) for each the triangle and the trapezoid?  Once you have those two area formulas, multiply them together (that constitutes a product), find the derivative, equate the derivative to zero, and solve for the critical value (or values).

anonymous · Answer

how do you put the trapezoid in terms of  x?

mathmale · Answer

Please see whether you agree with me:  the slope of the hypotenuse in your figure is 2.

Thus, if the base of the triangle is x, its height is 2x.  This side, 2x, is also the left side of the trapezoid, whose right side is 2 (a constant).

In general, the area of a trapezoid with vertical sides a and b is [ (a+b)/2 ]*w, where w is the (horizontal width of the trapezoid).  The base, w, of the trap. can be represented by 1-x.  Can you now write an expression for the area of the trapezoid?

anonymous · Answer

thank you I figured the problem out