A two-dimmensional, silo-shapped figure is formed by placing a semicircle of diameter 1 on top of a unit square, with the diameter coinciding with the top square. What is the radius of the smallest circle that contains this figure?
Draw a unit square (each side is 1 unit). Then draw a semicircle of diameter 1 on top of this, if you do this right the centerpoint of the circle will be at the midpoint of the top line of the square. Now draw a circle that goes thru the two lowest corners of the square the top of your silo, this is the smallest circle that contains it. Remember this doesn't have to be perfect to follow my explaination. Starting at the very top of the circle, draw a line straight down thru the top line of the square, but stop at the bottom line of the square. Label the top point of this line E, the mid point of this line F the bottom of the line G. If you've done everything right, the length of EF is 1/2 (the radius of the semicircle) and the length of FG is 1 (same length as the sides of the square). Pick a point on this line that looks like it's at the center of the circle (this should be between F G) draw a line out to one of the lower corners of the square. Label the end of this line at the center of the circle C the other end at the corner of the square H. Now we are done drawing. Since C is the center of the circle, we can see that CE is the radius of the circle. And CH is also the radius of the circle. So CE = CH. We can write an equation for each set them equal to each other. Lets say the distance from C to G is x, write an x next to it. Since FG = 1, FC is 1 - x, write that next to FG. So the length of EC = EF + FC = 1/2 + 1 - x = 3/2 - x. Now find the length of CH. Using the pythagorean theorum, we know square root of HG squared + CG squared = CH or CH = (HG^2 + CG^2)^1/2 = (1/2^2 + x^2)^1/2 = (1/4 + x^2)^1/2 EC = CG, so 3/2 - x = (1/4 + x^2)^1/2 Square both sides to get (3/2 - x)^2 = 1/4 + x^2 or 3/2^2 - 3/2x - 3/2x + x^2 = 1/4 + x^2 or 9/4 -3x + x^2 = 1/4 + x^2 or 9/4 - 1/4 = 3x or 2 = 3x, so x = 2/3 EC = 1/2 + 1 - x = 1/2 + 1 - 2/3 = 3/2 - 2/3 = 9/6 - 4/6 = 5/6. Your radius is 5/6 or 0.833333.