Two curves, y = px2 and y = qx2 intersect rectangle ABCD at its four corners. Which of the following expressions gives the area of the rectangle?
(A) 4 · (p + q)
(B) 8 · (p + q)
(C) 16 · (p + q)
(D) 8 · (p – q)
(E) 16 · (p – q)
Notice that side BC crosses the x-axis at (2,0). Since this is a parabola and parabolas are always symmetrical, we know that line AD must cross the x-axis at (-2,0). This means that the width of the rectangle is 4 (there are four units between -2 and 2). Now all we need to do is figure out the length of either BC or AD, and multiply this by 4 (remember, the area of a rectangle is determined by the length times the width). Here, we will use line BC. If we can identify the y-value of the coordinates for point B and point C, then we can figure out the length of BC. The way to do this is plug in the x value (notice that the x value for all coordinates on line BC must be 2).
For point C, we need to plug in 2 for x in the equation y = px2 . Doing so gives us y = 4p. For point B, we need to plug in 2 for x in the equation y = qx2 . This gives us y = 4q. The length of line BC is therefore going to be 4p – 4q. (Why? Well, let’s say the y value point C is 5 and the y value of point B is -3. The length of the line would be 5 – (-3), or 5 + 3, which is 8.)
Now all we have to do is multiply the width (4) by the length (4p – 4q). That gives us 4(4p – 4q), which would be the same as 16(p – q). The answer is E.
source: crushthetest.com
