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
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
and got 3 but idk if that's right
A. 3
B. 3.772
C. 4.242
D. 9
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.