# TrigonometryTrigonometry problem-belt driven pulley

#### paulmdrdo

##### Active member
A pulley with a radius of 10 inches uses a belt to drive a pulley with a
radius of 6 inches. Find the angle through which the smaller pulley turns
and also in degrees.

can you rephrase the question for me. thanks!

#### MarkFL

Staff member
Re: Trigonometry problem.

Through what angle must a point on the circumference of a circle having a radius of 6 units travel, to equal the distance traveled by a point on the circumference of a circle having a radius of 10 units which makes one complete circuit?

Can you set up an appropriate proportion, or use the arc-length formula?

#### paulmdrdo

##### Active member
Re: Trigonometry problem.

i would use this,

$\displaystyle \frac{10}{\theta_{1}}=\frac{6}{\theta_{2}}$

then,

$\displaystyle \theta_{2}=\frac{6\theta_{1}}{10}$

but i don't know what $\theta_{1}$ is.

#### MarkFL

Staff member
Re: Trigonometry problem.

The ratio of the radius of the larger pulley to the angle through which the smaller pulley turns is equal to the ratio of the radius of the smaller pulley to the angle through which the larger pulley moves.

If this seems counter-intuitive, look at the arc-length formula:

$$\displaystyle s=r\theta$$

Since $s$ is the same for both pulleys, you may write:

$$\displaystyle s=r_1\theta_1=r_2\theta_2$$

If $r_1=10$, then what is $\theta_1$, recalling that this pulley moves through one complete revolution?

#### paulmdrdo

##### Active member
Re: Trigonometry problem.

do you mean this ratio

$\displaystyle \frac{10}{\theta_{2}}=\frac{6}{\theta_{1}}$?

#### MarkFL

Staff member
Re: Trigonometry problem.

do you mean this ratio

$\displaystyle \frac{10}{\theta_{2}}=\frac{6}{\theta_{1}}$?
Yes, what is $\theta_1$, since it represents one complete revolution?

#### paulmdrdo

##### Active member
Re: Trigonometry problem.

it would be $2\pi$

but why do we compare the radius of the larger pulley to the angle of the smaller?

i calculated the final answer to be $\displaystyle \frac{10}{3}\pi$

but i'm still confused. how do you know that ratio?

#### MarkFL

Staff member
Re: Trigonometry problem.

I find the arc-length method much more intuitive. We know the belt will cause the smaller pulley to turn more rapidly than the larger one since the circumference of the larger pulley is greater than that of the smaller pulley.