Answer to Question #121359 in Calculus for Jin

Question

An equilateral triangle with side length 8cm has its base on the x-axis and top vertex on the y-axis. A rectangle is inscribed in the triangle. Determine the largest area of the rectangle.

Answer 1

The area of the rectangle:

$S=2xy$

We have:

$y=h-xtan60\degree$

where h is height of triangle

$y=a\sqrt{3}/2-x\sqrt{3}$

where a is side length

Then:

$S=2x(a\sqrt{3}/2-x\sqrt{3})$

$\frac {dS}{dx}=a\sqrt{3}-4x\sqrt{3}=0$

$x=a/4$

$S_{max}=2\frac{a}{4}(\frac{a\sqrt{3}}{2}-\frac{a\sqrt{3}}{4})=\frac{a^2\sqrt{3}}{8}$

Answer:

$S_{max}=\frac{8^2\sqrt{3}}{8}=8\sqrt{3}$ cm