Rectangle in a circle

• Jun 8th 2012, 02:14 PM
srh
Rectangle in a circle
Hi guys, I'm not sure if this is the right place for this question but here goes..

A washer is made from a circular disc of radius 5.5cm by removing a rectangular area of 18cm2. If a line is drawn from the centre of the washer through a corner of the rectangular area to the outer radius of the washer, then the distance from the edge to the outer radius of the washer is 0.65cm.

Determine the dimensions of the rectangular area removed.

I need to know the answer but more importantly I want to know how/why the method used works.

Thanks :)
• Jun 9th 2012, 11:22 AM
Soroban
Re: Rectangle in a circle
Hello, srh!

Did you make a sketch?

Quote:

A washer is made from a circular disc of radius 5.5 cm by removing a rectangular area of 18 cm2.
If a line is drawn from the centre of the washer through a corner of the rectangular area to the outer radius
of the washer, then the distance from the edge to the outer radius of the washer is 0.65cm.

Determine the dimensions of the rectangular area removed.

Code:

              * * *           *          *         *              *       *                *             *-------*       *    |    x |    *       *    |  O*---+A    *       *    |    * |y    *             *-------*       *            B *  *         *              *           *          *  P               * * *
The center of the circle is $O.$
The radius is $OP = 5.5$
Distance $BP = 0.65$
Hence:. $OB = 4.85$
Let $x = OA,\;y = AB.$

In right triangle $OAB$, we have: . $x^2 + y^2 \:=\:(4.85)^2$ .[1]

The area is 18: . $(2x)(2y) \:=\:18 \quad\Rightarrow\quad y \:=\:\tfrac{9}{2x}$

Substitute into [1]: . $x^2 + \left(\tfrac{9}{2x}\right)^2 \:=\:(4.85)^2 \quad\Rightarrow\quad 4x^4 - 94.09x^2 + 81 \:=\:0$

Quadratic Formula: . $x^2 \;=\;\frac{94.09 \pm \sqrt{7556.9281}}{8} \;=\;\begin{Bmatrix}22.6275742 \\ 0.894925803\end{array}$

Then: . $x \;=\;\begin{Bmatrix}4.756844984 & \text{too big} \\ 0.946005181 \end{array}$

The dimensions of the rectangle are: . $\begin{Bmatrix}2x &\approx&1.892 \\ 2y &\approx& 9.514 \end{Bmatrix}$