Basic Trig Problem

So I have this problem I'm stuck on in my math class, and we've just started trig (11th grader, 12th grade math). I feel like the problem is much easier than what I'm looking at, but here's a crude drawing of the image the text book gives for the problem:



The straight lines making the 'x' shape are crossing belts (I think the textbook called them) and x is the angle measure. I messed up the motor's x and it should just be the angle of the top triangle.

So far I looked at my trig identities reference sheet, thought of using the angular speed formula (but couldn't figure out how to deal with the answer I got from it), and asked kids in the other/higher math classes, and none of them can figure it out either. Knowing the correct answer isn't necessary due to my teacher only checking completeness, but it would be nice if I knew how to do this type of problem.

I don't see how the length of the belt matters at all. Maybe you could take a snapshot of the problem with your phone's camera or something?

I have a feeling this is too simple to be right, but anyway, I'll work more on it.


The total length of the belt L1 is the part that does not touch any wheel, L2 is the part around the big wheel, and L3 small wheel:

I don't know if there are any easy ways to invert this to find x with L, R and r, though.

The RPM has nothing to do with the length of the belt.
My simplistic guess would be that since the saw's circumference is twice the motor's, it will be turning half as fast. So 850 RPM.

As for the angle X, of course it depends on the belt length. If the two circles are infinitely far apart, the angle will be zero since the belt will be flattened. If they are touching, the angle will be 180.
I've forgotten way too much of the trig I never knew to actually put it down in a formula, though.

Alright, I have my textbook in front of me now. It has what I drew, and it wants me to find the following:

a) Determine the rpm of the saw
b) Let L be the total length of the belt. Write L as a function of x, where x is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley)
c) Use the function from b to finish the table (Don't need help for this so long as there's an equation to use)
d) As x increases, do the lengths of the straight sections of the belt change faster or slower than the lengths of the belts around each pulley?

So apparently the value of x is unnecessary, but still needed as a group of values. Still unsure of what to do there.

Alright, I have my textbook in front of me now. It has what I drew, and it wants me to find the following:

a) Determine the rpm of the saw
b) Let L be the total length of the belt. Write L as a function of x, where x is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley)
c) Use the function from b to finish the table (Don't need help for this so long as there's an equation to use)
d) As x increases, do the lengths of the straight sections of the belt change faster or slower than the lengths of the belts around each pulley?

So apparently the value of x is unnecessary, but still needed as a group of values. Still unsure of what to do there.

Miraculously, I managed to get all that is necessary for the problem.

b/ L as a function of x can be seen up there. Domain:


d/

Then you just need to compare those derivatives.

Oh wait, this is grade 12 math, are you allowed to use derivatives at all? I'm kind of lost on how to compare rates of change without using derivatives. Maybe you could set the angle to 2x and find respective L's, then compare the deltaL's for a given deltaX = x.

Most of the Trigonometry in this problem is in section b.

So I have this problem I'm stuck on in my math class, and we've just started trig (11th grader, 12th grade math). I feel like the problem is much easier than what I'm looking at, but here's a crude drawing of the image the text book gives for the problem:



The straight lines making the 'x' shape are crossing belts (I think the textbook called them) and x is the angle measure. I messed up the motor's x and it should just be the angle of the top triangle.

So far I looked at my trig identities reference sheet, thought of using the angular speed formula (but couldn't figure out how to deal with the answer I got from it), and asked kids in the other/higher math classes, and none of them can figure it out either. Knowing the correct answer isn't necessary due to my teacher only checking completeness, but it would be nice if I knew how to do this type of problem.

Alright, I have my textbook in front of me now. It has what I drew, and it wants me to find the following:

a) Determine the rpm of the saw
b) Let L be the total length of the belt. Write L as a function of x, where x is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley)
c) Use the function from b to finish the table (Don't need help for this so long as there's an equation to use)
d) As x increases, do the lengths of the straight sections of the belt change faster or slower than the lengths of the belts around each pulley?

So apparently the value of x is unnecessary, but still needed as a group of values. Still unsure of what to do there.

B) The amount that the belt goes around the circle is (2pi-[180-2x])r, where r is the radius of the circle. The belt travels a distance from circle to circle of (r+R)*cot[x]. So, the combined function L(x)=(2pi-[180-2x]+2cot[x])(r+R).

C) Just follow from B, but I can explain B) better if you want later.

D) For this, if you haven't learned derivatives yet, you can segment your table into the functions (r+R)*cot[x] and (2pi-180+2x)(r+R) then use the formula [f(x1)-f(x2)]/[x1-x2], or just learn derivatives. Since there is a value of x that cannot be passed (when the saw and motor touch, for instance), you can just do it for that range.

Edit:

All correct, except that d/dx of L2 and L3 are postitives, not negatives.

All correct, except that d/dx of L2 and L3 are postitives, not negatives.

Yeah, I kind of messed things up in my head.
suspended_angle = 360 - 2*(90-x) = 180 + 2x
> d/dx L2 = 2R
> d/dx L3 = 2r