 In this video, we provide the solution to question number 16 from the practice final exam for math 1060, in which case, though, we have the tires of a bicycle have a diameter of 26 inches. The tires are turning at a rate of 75 revolutions per minute. And so we're supposed to determine what must be the bicycle speed in miles per hour, all right? And so the basic idea here is we're gonna have to find the linear velocity, which is what we're looking for, V. This is gonna follow from the formula R times omega, where R is the radius of the circle that's here spinning. If your diameter is 26 inches, then the radius would be 13 inches, of course. And then omega here is the angular velocity. How fast is it spinning? We know that we have 75 revolutions per minute, like so. And so that's the basic formula. And so most of this problem is really just about conversion, right? We have to convert into miles per hour here. So in terms of the length, 13 inches. Well, 13 inches, all right? We'd have to switch that to feet, in which case you get one foot for every 12 inches, all right? So we see that the inch unit then cancels out, but we need to be in miles. And so how do you convert from feet to miles? Well, in case you forgot, there are 5,280 feet in a mile. So we're gonna get one mile is equal to 5,280 feet. So then we see that the foot measurement cancels out. And so summarizing what we've discovered so far, we're gonna have 13 over 12 times 5,280 miles in this calculation here. Well, what about the angular velocity? I'm gonna put that here. So we're working with revolutions per minute, but we wanna be in miles per hour, which converted the distance here into miles. In terms of angle measure, the angle does need to be in radians so that the conversion works properly. That was required when using these formulas. Like the arc length, the area sector formula, this angular velocity, you do have to be in radians here. So to convert the 75 revolutions into radians, we're just gonna get 75. Well, one revolution is two pi radians, like so. But really radians is like a unitless quantity there. So we don't even have to write the radians in there. We're gonna get 75 times two pi. We have to convert the one minute into hours, for which case there are 60 minutes in an hour. So 60 minutes in one hour, like so. And oftentimes people are confused about what should it be? Should it be 60 minutes over one hour or one hour over 60 minutes? So by times you by 60 or dividing by six, that's a big deal. Well, this is why I write the units in here. We need to have the units cancel out. So if I'm dividing by minutes, because it's per minute, like we saw up here, and I need to have a minute on top to cancel out, and then one hour to go on the bottom, like so. So putting this together here, we then have 75 times two pi times 60, and this is then gonna be over hours. And so in terms of units, we have the correct unit measurement, miles per hour, like so. And so now it just comes down to simplifying this number, of course. And we could try to find an exact fraction. You'll have things like 12 goes into 65 times, that cancels out right there. 5,280 does have some common factors of two and five. And so we can continue to simplify this thing. I'm gonna kind of skip over that step if that's all right. We end up with 325 pi over 176 miles per hour, if you want the exact answer. But an approximate answer would also be appropriate here, in which case you get 5.80 miles per hour. So if you use your calculator to get past some of this fraction simplification process, that is completely acceptable.