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?

Answer & Explanation

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

$(5-r{)}^{2}+(8-r{)}^{2}={r}^{2}$

$25-10r+{r}^{2}+64-16r+{r}^{2}={r}^{2}$

${r}^{2}-26r+89=0$

If we use the quadratic formula, we get solutions of

$r=13\pm 4\sqrt{5}$

One of these solutions, $13-4\sqrt{5}$, 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

$r=13+4\sqrt{5}$

which consequently means

$d=2r=2(13+4\sqrt{5})=26+8\sqrt{5}$

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