Solve each problem. The speed of a pulley varies inversely as its...

Question

Solve each problem. The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

Accepted Answer

Hey, everyone in this problem, we're told that the speed at which the shaft of a conveyor system rotates varies inversely as its diameter. On one end of the conveyor system, the shaft has a diameter of 50 millimeters and rotates at revolutions per minute. We're asked to calculate the speed and revolutions per minute at which the shaft rotates on the opposite end where the diameter is 125 millimeters. We're given four answer choices. Option A 70 R PM, option B 35 R PM, option C 50 R PM and option D 40 R PM. Now we're told that the speed of this shaft which we're gonna call V is varies inversely as its diameter. What that tells us is that it's inversely proportional. And so the speed V is proportional to one divided by the diameter D OK. If it's inversely proportional, that means it's proportional to one divided by the diameter. So we have this proportionality and we want to turn that into an equation and we can do that by replacing that proportionality symbol with an equal sign if we introduce a constant K and we have that constant of variation. And so we can write that V is equal to some constant K divided by the diameter D. Now, what we want to find is a speed V for a particular diameter of 125 millimeters. But we can't calculate V without knowing K. So what we wanna do is use the other information we were given in the problem. OK, where we have a particular diameter and a particular speed to calculate K, then we can use that K value in the diameter we're interested in to find the speed. So let's start with the information we were given, we have a diameter of 50 millimeters and a speed V of 100 R pm. So substituting these into our equation, we have the, the speed V is equal to our constant K divided by the diameter D that tells us that R PM is equal to K divided by millimeters. So our constant K is going to be equal to multiplied by 50 which gives us in the unit. We have R PM multiplied by millimeter. And we wanna include these units, we wanna keep these units so that when we work out the rest of our work, we can double check and make sure the units match up and we've made no mistakes. So now we have a K value that K value means that we can write our equation as V is equal to R PM multiplied by millimeters divided by the diameter D. Now, we're looking for the speed V, we're given the diameter D is equal to millimeters. So we can substitute that in now and we only have one unknown V which is what we're solving for. So we get that V is equal to R PM multiplied by millimeters divided by 125 millimeters. And the unit of millimeter will divide out and we're gonna be left with just the unit of R PM, which is the unit that the question asks us to solve this in which is great. We're gonna have the correct unit for our final answer. We have 5000 divided by 125 which gives us 40. And so we have that the speed V is equal to 40 R PM. OK? So the speed of the shaft at the opposite end where the diameter was 125 millimeters is 40 R PM, which corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

Key Concepts

Inverse Variation

Proportional Relationships

Setting Up Equations

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