Question

Find the dimensions of the rectangle of largest area that can be inscribed in an isosceles trapezoid of height 6 cm, with bases 8 cm and 12 cm.

Answer 1

I split the trapezoid in half (since it is symmetrical) and put the origin of an XY plane at the midpoint of the longest base.

Answer 2

This means that the slanting side is on the line y=-3x+18.
So the upper-right vertex of the rectangle is at the point (x , -3x+18)
Those coordinates represent the height and half the base of the rectangle you are trying to optimize. See if you can get it from there.

Answer 3

I'm not sure if that is the calculus way to get it though. Some of what you said seems pretty foreign to me.

Answer 4

It's analytic geometry (algebra really). It is a prerequisite for calculus.
That just gets you the equation, then you can use calculus to find the maximum of the function.
Though, since the function is only 2nd degree, you can do that with algebra too.

Answer 5

How did you get y=-3x+18?

Answer 6

From the placement of my coordinate plane. Placing the short base at the top, going down a height of 6 happens while going to the right 2 to get down to the longer base. That's a slope of -3, then I solved for the y intercept.