Final answer:
To find the area of the triangle formed by the origin and the points of intersection of the parabolas, we find the points of intersection, (±3, -7) and (-3, 5). Using the formula for the area of a triangle, the area is 0 units squared.
Step-by-step explanation:
To find the area of the triangle formed by the origin and the points of intersection of the parabolas, we first need to find the points of intersection. This is done by setting the two parabolas equal to each other and solving for x. So, -3x^2 + 20 = x^2 - 16.
Combining like terms, we get 4x^2 = 36.
Taking the square root of both sides, we get x = ±3.
Now substitute these x-values back into either equation to find the corresponding y-values.
For x = -3, y = -3(-3)^2 + 20 = 5.
For x = 3, y = 3^2 - 16 = -7.
Now we have the three points: (0,0), (-3,5), and (3,-7). To find the area of the triangle, we can use the shoelace formula or the formula for the area of a triangle given its vertices. Using the formula for the area of a triangle, the base is the distance between (-3,5) and (3,-7), which is 6 units, and the height is the y-coordinate of the origin, which is 0. So the area of the triangle is (1/2) * 6 * 0 = 0 units squared.