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.

i have never gotten one straightforward correct answer from this guy steve

I did a(a+2a+a)/2=18

and got 3 but idk if that's right

Look at the graph and you'll see the triangle and rectangle. It is baseline geometry problem. The answer is 3

the answer choices are

A. 3
B. 3.772
C. 4.242
D. 9

Where did steve get the triangle parts from?? :(

To find the value of a such that the area of region R is 18 square units, we first need to determine the points where the curve y = 2x + a intersects the x-axis and the line x = a.

For the curve y = 2x + a to intersect the x-axis, the y-coordinate must be zero. Substituting y = 0 into the equation, we get:

0 = 2x + a

Solving for x, we have:

2x = -a
x = -a/2

So one of the points where the curve intersects the x-axis is (-a/2, 0).

Next, we need to find the point of intersection between the curve y = 2x + a and the line x = a. Since x is already given to be a, substituting this value into the equation, we get:

y = 2a + a
y = 3a

So the point of intersection is (a, 3a).

Now, we have two points that define the base of the region R. The length of the base is the difference in x-coordinates between these two points, which is:

length of base = a - (-a/2) = 3a/2

To find the height of the region, we need to determine the difference in y-coordinates between these two points. The y-coordinate of the first point is 0, and the y-coordinate of the second point is 3a:

height = 3a - 0 = 3a

The formula for calculating the area of a triangle is:

Area = (base * height) / 2

Substituting the values we have:

18 = (3a/2 * 3a) / 2

Multiplying both sides by 2 and rearranging the equation, we get:

36 = 9a^2

Dividing both sides by 9:

4 = a^2

Taking the square root of both sides:

2 = a

Therefore, the value of a that makes the area of region R equal to 18 square units is a = 2.

even without calculus, this one is easy. The x-intercept of y=2x+a is x = -a/2

So, you have a triangle with base 3a/2 and height 3a, so the area is 9a^2/4.

If 9a^2/4 = 18, a = √8

check:

∫[-a/2,a] 2x+a dx = 9a^2/4