Expert: Scotto - 5/13/2009

Question
how do I do this, using derivatives?

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 6 if one side of the rectangle lies on the base of the triangle.

height?
width?




and also..

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 26 feet?

Answer
Lets put the bottom side of this triangle on the x axis.
In fact, let's position it so the center is on the y axis.
The left point at the base is given by the -width over two,
and the right point in the base is given by width/2.

This means that the base has points (-3,0) and (3,0).
Lets look at the triangle where x and y are both positive.
We know that the base has length 3, and the hypoteneuse has length 6.
This tells us that the other side is √(6²-3²) = √(36-9) = √27 = 3√2.

What this does is tell us that two points are (0, 3√2) and (3,0).
The slope from two points is given by (y2-y1)/(x2-x1).
That is, (0 - 3√2)/(3 - 0) = -3√2/3 = -√2.
From this, we can see the line is y - 0 = -√2(x - 3),
so we just have y = -√2(x-3).

On the other side, the line would have the negative sloe of this one and cross the axis at -3, so the equation is y = √2(x+3).

Given that we are trying to draw a rectangle, lets say it crosses the x-axis at x = x1.  This means that the width of the base is 2*x1.
This also means the height is y1, and we know y1 = -√2(x1 - 3).
If we multiply the height by the width we get 2*x1*(-√2(x1 - 3)).

Multiplying this out gives (-2√2)x1² + 6*x1.
Taking the derivative gives us (-4√2)x + 6.
Set this to 0, solve for x.  This will be x1: this is the width.
Solve for y1 using  y = -√2(x-3): this is the height.
Multiply x1*y1: this is the area.



Compute the perimiter of this window.  It is known to be 26, but that includes two sides of height H, a base of length B, and a top of length πB.  The equation is the 2H + (1+π)B = 26.

The area is the area of the rectange, which is BH, plus the area of the semicircle.  Since the semicircle is known to have radius H/2 and the area is known to be half of a circle's area, the area of this portion of the window is πr²/2 = π(H/2)²/2 = πH²/8.
All total, the area of the window is BH + πH²/8.

Using 2H + (1+π)B = 26, B can be solved for.   This can be put in
the equation BH + πH²/8 for B.  The derivative can be taken with respect to H, set equal to 0, and then H can be found.  Using this value of H, use the perimeter equation to find B.  Using both H and B, the area A can be found.