Question

A rectangle has its base on the x-axis and its upper two vertices on the parabola y=12-x^2.


Write a formula for the area as a function of x in the form A=a+bx+cx^2+dx^3 where a, b, c and d are integers (some may be zero).

Answer 1

Diagram will look something like the above figure.

So, Let the co-ordinates of corners of rectangle on x axis be (x,0) and (-x,0).

and Co-ordinates of corners on the parabola be ($x,12-x^{2}$ ) and ($-x, 12 - x^{2}$ )

Now sides of the rectangles are $2x$ and $12 - x^{2}$

By Geometry, Area of the rectangle = length and width

so, area of rectangle = $2x* (12 - x^{2})$ = $24x - 2x^{3}$