Question 549599: This is an assigned question I do not understand. Answer choices b - e are divided by 2. Please help!!!!
1. Which of the following represents the area of the region in Quadrant I bounded by the lines x = 0, y = 0, x = c, and y = kx + b, where a, b, c, and k are all positive numbers?
a. ckx + cb
b. cb + kc / 2
c. 2bc + kc /2
d. 2bc + kc2 /2
e. 2b + kc2 / 2
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
 is the  -axis.  is the  -axis.  is a vertical line through the point ) which must be to the right of the -axis because  is constrained to the positive numbers.  is a line with a slope of  and a -intercept of ) . The line must slope upward to the right and must be above the -axis because both and  are constrained to the positive numbers.
In sum, these four lines bound a quadrilateral with vertices at ) , , ) , and . Note:  comes from substituting the value for in the equation yielding 
Since is a vertical line it must be parallel to the -axis. Furthermore, since is presumed to be non-zero, the line described by is NOT parallel to the -axis. Therefore, two parallel sides and two not parallel sides -- we have a trapezoid ladies and gents; trapezium if you are in the UK.
The area of a trapezoid is the average of the measures of the two parallel sides times the measure of the altitude of the trapezoid, i.e. the distance between the two parallel sides. Since the side with endpoints at the origin and at is perpendicular to both of the parallel sides, the distance between the two parallel sides is simply the distance from to , which is simply (You can verify that with the distance formula if you like).
The measure of the shorter parallel side is the distance from to , i.e. . The measure of the longer side is the distance from to . Here's the distance formula for that one if the answer isn't obvious.
^2\ +\ (y_1\ -\ y_2)^2})
^2\ +\ (0\ -\ (kc+b))^2}\ =\ kc\ +\ b)
So the average of the measures of the two parallel sides is:
}{2})
Multiply this by the altitude, , to get:
John
My calculator said it, I believe it, that settles it
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