Higher Derivatives & Applications - Maximum & Minimum Values

[Zain Qasmi]

In an upright triangular prism, the triangular base ABC is right-angled at B, AB=5x cm and BC=12x cm. The sum of the lengths of all its edges is 180cm.

(a) Show that the volume, V cm^3 is given by V= 1800x^2 - 600x^3
(b) Find the value of x for which V has a maximum value.

It would be nice if anyone could atleast provide me with a sketch of the prism !

 

[galactus]

By Pythagoras, the hypoteneuse of the base (and top ) is \(\displaystyle \sqrt{(5x)^{2}+(12x)^{2}}=13x\)

Let the height be y. There are 3 edges along the vertical and 3 along the base.

The volume of the prism \(\displaystyle \text{(area of base)( height)}\).

Thus, we have \(\displaystyle V=y\cdot \frac{12x\cdot 5x}{2}=30x^{2}y\)

But, the sum of all the edges is 180:

\(\displaystyle 3y+26x+10x+24x=180\)

Solve for y and sub into the volume equation.