Really really need assistance and help. Thank you in advance..

Question

R is the first quadrant region enclosed by the x-axis, the curve y = 2x + a, and the line x = a, where a > 0. Find the value of a so that the area of the region R is 18 square units.

Options:

1)	3
2)	3.772
3)	4.242
4)	9

Answer 1

Let's first find a formula for the area of the region in terms of a:

If you plot the region, you will find that it is a trapezoid with height a (along the x-axis) and base lengths a and 3a (parallel to y-axis). Thus, the area is

A = 0.5*h*(b1+b2) = 0.5*a*(a+3a) = 0.5*a*4a = 2a^2.

Thus, setting up 2a^2 = 18, it follows that a = 3.