Question

.Find the area of the triangle formed by the origin and the points of intersection of the parabolas with equations y=−3x^2+20 and y=x^2−16 .

Answer 1

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.

Answer 2

The area of the triangle formed by the origin and the points of intersection of the given parabolas is 21 square units. The points of intersection are calculated by setting the equations equal to each other, finding the x-coordinates, and substituting them back to find the y-coordinates.

Step-by-step explanation:

To find the area of the triangle formed by the origin and the points of intersection of the parabolas with equations y = -3x^2 + 20 and y = x^2 - 16, we must first determine the points of intersection. To do this, we set the two equations equal to each other:

-3x^2 + 20 = x^2 - 16

Combining like terms:

-4x^2 + 36 = 0

Dividing by -4 gives:

x^2 - 9 = 0

Which factors to:

(x + 3)(x - 3) = 0

So, the x-coordinates of the points of intersection are x = -3 and x = 3. Substituting back into one of the original equations to find the corresponding y-values, we have:

For x = -3:

y = (-3)^2 - 16

y = 9 - 16

y = -7

For x = 3:

y = (3)^2 - 16

y = 9 - 16

y = -7

The points of intersection are (-3, -7) and (3, -7). The triangle formed by these points and the origin is a right triangle with base 6 units and height 7 units. The area of a triangle is given by the formula:

Area = 1/2 × base × height

Applying the formula to our triangle:

Area = 1/2 × 6 × 7

Area = 21 square units

The area of the triangle is 21 square units.