Inside a parabola
A triangle of area 64 square units is drawn inside the parabola $y=k^2-x^2$. Find the value of $k$.
Age
14 to 16
Challenge level
Problem
The points where the parabola $y=k^2-x^2$ crosses the coordinate axes are joined to form a triangle.
Image
The area of the triangle is 64 square units. What is the value of $k$?
This problem is adapted from the World Mathematics Championships.
Student Solutions
Answer: $k=4$
Image
$y$-intercept:
$x=0\Rightarrow y=k^2-0^2=k^2$
$x$-intercepts:
$\begin{align} y=0&\Rightarrow k^2-x^2=0\\
&\Rightarrow k^2=x^2\\
&\Rightarrow x=\pm k\end{align}$
Area of the triangle is given by $\frac{1}{2}\times2k\times k^2=k^3$
So $k^3=64\Rightarrow k=4$