Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 10 if one side of the rectangle lies on the base of the triangle.
width=?
height=?
Thanks
The best thing to do here is DEFINITELY draw a picture.
Thats the extent of my paint skills, but you can see that if we have an equilateral triangle, we can imagine one half of that triangle, with the base positioned on the X-axis. What we want are the dimensions of that rectangle that will give us the largest area. We know that to be the optimization of the formula A=(length)(width), but what the heck function are we using to define these? For our length it is clear that it is going to be some value between 0 and 5 non-inclusive (because then our area is just 0). That means our length should be:
$\displaystyle L(x)=5-x$
But what should our Y be? Well look at one of the sides of that triangle. Doesn't that look like a line? A line that has points $\displaystyle (0,5\sqrt{5})$ and $\displaystyle (5,0)$. From this we can form an equation that defines that line, which will allow you to define the height of your rectangle in terms of X. From there, you will be able to get your equation for the area of a rectangle, and you should then be able to optimize it.
Do you think you can take it from there?
My apologies for not being clear. I split your triangle in two to make it easier to visualize. Therefore when you calculate the area of the FULL rectangle (since you only have half of it here), you will use A=(2length)(height). This will be the same area for the rectangle inscribed in the triangle, if you used the full triangle. Splitting it in two makes it easier to work with.
I also edited my response in the beginning to be more clear as I referred to a triangle when I meant rectangle.