Question

Question is asked in the image. Choose answers from the bank below on the image.

Answer 1

We have an isosceles triangle inscribed in the parabola, as seen in the picture.

We know that one of the vertices of the triangle is the origin (0,0).

The other two vertices are reflected points across the vertical axis, of which we don't know the exact coordinates. But, as they are placed on the parabola, they will satisfy the equation of this parabola: y = 9-x².

We can express the area of a triangle as half the base times height.

We can identify the base and the height in this triangle as:

Then, the base is two times the absolute value of x-coordinate that corresponds to the vertex of the triangle:

`b=|x_2-x_1|=|2x|_`

The height will be f(x) for that value of x, as it is the vertical distance from the origin to the y-coordinate of the parabola for that value of x.

Then, we can express it as:

`h=|f(x)-0|=|9-x^2|`

We can then express the area of the triangle as:

`\begin{gathered} A=(1)/(2)bh \\ \\ A=(1)/(2)(2x)(9-x^2) \\ A=x(9-x^2) \\ A=9x-x^3 \end{gathered}`

We can now use this formula to calculate the value of A when x = 2.5 by replacing x with this value:

`\begin{gathered} A=9(2.5)-(2.5)^3 \\ A=22.5-15.625 \\ A=6.875 \end{gathered}`

Answer:

Area of the triangle in function of x: 9x-x³

Area when x = 2.5 = 6.875 un²