Question

A rectangle has its two lower corners on the x-axis and its two upper corners on the curve y=25-x^2 . For all such rectangles, what are the dimensions of the one with the largest area?

Answer 1

For a varying y0 the area of the rectangle is \[A=2\cdot\sqrt{25-y_0}\cdot y_0\]
To find the critical points of the area function, differentiate and set to 0:
\[\frac{dA}{dy_0}=\frac{50y_0-3y_0^2}{\sqrt{25y_0^2-y_0^3}}=0\rightarrow y_{01}=0, y_{02}=\frac{50}{3}\]
whereby y02 corresponds to a maximum.
The dimensions of the largest rectangle are therefore: \[\frac{10}{\sqrt{3}}\times\frac{50}{3}\]