I need some help with this...

[Kantonar]

ello everyone.

Ok, so I got this take home quiz for my math class (honors geometry -- I'm a freshman). This one problem ends up counting for more than 50% of the grade. Could someone help me out?

ok, here it is.

Two pulleys are connected by a belt.
The radii of the pulleys are 3cm and 15cm.
The distance between the two centers is 24cm.
Find the total length of the belt.

sorry for the poor quality pic, but I'm on a laptop and had to draw it in paint. Thanks in advance for your help...

 

[tkhunny]

Due to your excellent drawing, the straight pieces are easy.

24^2 = (Straight)^2 + (15-3)^2

Now what?

 

[soroban]

Hello, Kantonar!

As tkhunny said, it's an excellent diagram . . . Thank you!

Two pulleys are connected by a belt.
The radii of the pulleys are 3cm and 15cm.
The distance between the two centers is 24cm.
Find the total length of the belt.

              * * *
          *           *
        *               *
       *                 *
                                    * * *
      *         A         *       *       *
      *         *         * 24
      *         :         *     *     B     *
               C+ - - - - - - - * - - *     *
       *      15:        *      *     :     *
        *       :       *             :3
          *     :     *           *   :   *
              * * * - - - - - - - - * * *
                E                     D

Draw the line of centers \(\displaystyle AB\,=\,24\)

Draw a horizontal segment \(\displaystyle BC\).

In right triangle \(\displaystyle ABC:\,AB\,=24,\;AC\,=\,12\)
\(\displaystyle \;\;\)Hence: \(\displaystyle \,BC\,=\,12\sqrt{3}\,=\,DE\)
\(\displaystyle \;\;\) Moreover: \(\displaystyle \,\angle ABC\,=\,30^o\)

Extend \(\displaystyle AB\) to \(\displaystyle F\) on circle \(\displaystyle B\)
We find that: \(\displaystyle \angle FBD\,=\,60^o\)

Extend \(\displaystyle BA\) to \(\displaystyle G\) on circle \(\displaystyle A\).
We find that: \(\displaystyle \angle GAE\,=\,120^o\)

And that is enough to determine the curved portions of the belt.