bergvolk0k

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?

Do you have a similar question?

Spielgutq1

Expert

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
$\left(x-r{\right)}^{2}+\left(y-r{\right)}^{2}={r}^{2}$
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
$\left(5-r{\right)}^{2}+\left(8-r{\right)}^{2}={r}^{2}$
$25-10r+{r}^{2}+64-16r+{r}^{2}={r}^{2}$
${r}^{2}-26r+89=0$
If we use the quadratic formula, we get solutions of
$r=13±4\sqrt{5}$
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
$d=2r=2\left(13+4\sqrt{5}\right)=26+8\sqrt{5}$

Still Have Questions?

Free Math Solver