Okay, let's tackle this problem step by step. So, Charlie is riding a unicycle, and we need to find the speed in inches per second. The wheel has a radius of 9 inches and makes 2 revolutions every 3 seconds. Hmm, speed... I think speed here refers to the linear speed of the unicycle as it moves forward. 

First, I remember that the linear speed of a rotating object is related to its angular speed. The formula for linear speed (v) is v = rω, where r is the radius and ω is the angular speed in radians per second. But wait, the problem gives the number of revolutions, not radians. So maybe I need to convert revolutions to radians first. 

Another approach could be to find the distance traveled in a certain time. Since each revolution covers the circumference of the wheel, the distance per revolution is 2πr. If the wheel makes 2 revolutions every 3 seconds, then the total distance covered in 3 seconds is 2 times the circumference. Then, dividing that distance by the time (3 seconds) will give the speed in inches per second. That might be a simpler way without dealing with angular velocity. Let me check both methods to make sure.

Method 1: Using angular speed. The wheel makes 2 revolutions every 3 seconds. So the angular speed ω is 2 revolutions per 3 seconds. Since one revolution is 2π radians, ω = 2 * 2π / 3 = 4π/3 radians per second. Then, linear speed v = rω = 9 * (4π/3) = 12π inches per second. Wait, that seems high. Let me verify.

Wait, 9 inches radius, so circumference is 2π*9 = 18π inches per revolution. If it does 2 revolutions, that's 36π inches in 3 seconds. So 36π / 3 = 12π inches per second. Oh, same result. So both methods give the same answer. So that's 12π inches per second. But the question asks for inches per second, so maybe we need to compute it numerically or leave it in terms of π? The problem doesn't specify, but since π is a common factor, perhaps leaving it as 12π is acceptable. But let me check the problem statement again. It says "what is the unicycle's speed in inches per second?" It doesn't specify to approximate π, so probably 12π is the exact answer. 

Wait, but let me make sure I didn't make a mistake in the first method. Let's go through it again. Angular speed ω is 2 revolutions per 3 seconds. Each revolution is 2π radians, so ω = 2*(2π)/3 = 4π/3 rad/s. Then v = rω = 9*(4π/3) = 12π in/s. Yep, that's correct. 

Alternatively, calculating distance: 2 revolutions each 3 seconds. Each revolution is 2π*9 = 18π inches. So 2 revolutions would be 36π inches. Then speed is 36π inches / 3 seconds = 12π inches per second. Same result. So both methods confirm the answer is 12π in/s. 

Wait, but maybe the problem expects a numerical value? Let me check again. The problem says "inches per second," but doesn't specify to approximate π. So in mathematical problems, unless told otherwise, we usually leave π as is. So 12π inches per second is the answer. 

Alternatively, if they had wanted a decimal, they might have specified. So I think 12π is the correct answer here. Let me just make sure there's no unit conversion issues. The radius is in inches, and the time is in seconds, so the units should work out. Inches per second is correct. 

Another sanity check: 1 revolution per second would be 18π inches per second. If it's 2 revolutions every 3 seconds, that's 2/3 revolutions per second, so 2/3 * 18π = 12π. Yep, same thing. So all methods lead to 12π. 

Therefore, the unicycle's speed is 12π inches per second.
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