bergvolk0k

bergvolk0k

Answered

2022-10-19

There's a circle of diameter d that is on a wall, and touches a block. Find the value of d.
I'm really unsure how to go about solving this. I wanted to first approach this by using the arc length, but I'm really unsure how to proceed. Anyone have ideas?

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Answer & Explanation

Spielgutq1

Spielgutq1

Expert

2022-10-20Added 17 answers

If appears that the circle is touching a surface to its left and another surface below it.
If we were to graph this circle with these two surfaces representing the x and y axes, the circle would have the equation
( x r ) 2 + ( y r ) 2 = r 2
in order for it to just touch either surface.

If the circle also just touches the block that is 5 units wide and 8 units long, the point (5,8) must be on the circle. If we plug this value into the equation, we get
( 5 r ) 2 + ( 8 r ) 2 = r 2
25 10 r + r 2 + 64 16 r + r 2 = r 2
r 2 26 r + 89 = 0
If we use the quadratic formula, we get solutions of
r = 13 ± 4 5
One of these solutions, 13 4 5 , doesn't work, though, because the radius becomes too small to leave enough room in the bottom left corner for the block.

This means that
r = 13 + 4 5
which consequently means
d = 2 r = 2 ( 13 + 4 5 ) = 26 + 8 5

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