find the dimensions of the largest rectangle that can be...

[wind]

a)Find the dimensions of the largest rectangle that can be inscribed in a circle of radius k.
b) Find the dimensions of the largest rectangle that can be inscribed in an equilateral triangle with side length k of one side of the rectangle lies on one side of the triangle.
c) Find the dimensions of the larges right-circular cylinder that can be inscribed in a sphere of radius k.

I don't know where to start...I drew a picture thats about all..

 

[galactus]

Re: find the dimensions of the largest rectangle that can be

a)Find the dimensions of the largest rectangle that can be inscribed in a circle of radius k.

If you center your circle at the origin and center your rectangle there, you can break your rectangle up into 4 smaller rectangles of area xy.
This gives the area of the entire rectangle: 4xy.

The equation of the circle is \(\displaystyle \L\\y=\sqrt{k^{2}-x^{2}}\)

Now, take it from there?.

b) Find the dimensions of the largest rectangle that can be inscribed in an equilateral triangle with side length k of one side of the rectangle lies on one side of the triangle.

Same premise as #1, Let two vertices of the triangle lie on the x-axis equal distance on either side of the y-axis. That way the coordinates of the triangle will be \(\displaystyle (-a,0), \;\ (a,0), \;\ (0,\sqrt{3}a)\)

This gives a line equation of a side of the triangle as \(\displaystyle y=\sqrt{3}x+\sqrt{3}a\)

The rectangle has area 2xy. Put 'er together.

c) Find the dimensions of the larges right-circular cylinder that can be inscribed in a sphere of radius k.

This is a classic max/min problem. Search through the forum and you'll find it. A Google will turn something up too.

I don't know where to start...I drew a picture thats about all..[/quote]