area and graph question

Question

let p (x,y) be a point on the graph y=2x+3. connect p with q(8,0). let a perpendicular from p to the x axis intersect the x axisat the point r. restricting the domain of p to 0<x<8. find the maximum area of triangle prq

Answer 1

For this problem we are attempting to find the maximum area of triangle PRQ. A=(1/2)bh and for this triangle the base b = 8-x where x is the x coordinate of point P. The height of PRQ is given by h=2x+3, thus the area is given as A=(1/2)(8-x)(2x+3), this is a quadratic equation which we are being asked to maximize which is equivalent to finding the y coordinate of the vertex. This can be done in various ways some of which you likely covered, you can foil then complete the square, or alternatively find the roots(which is convenient as our equation is in intercept form) and use them to find the vertex. Regardless of the path you choose, you will find the vertex is (13/4, 361/16) thus our maximum area is 361/16 or 22.5625.

Answer 2

I checked the scale on the graph. The correct area is (16*8)/2=64.