Arc Length and Area of Sector│Please Help!

Question

1. If you rotate the red gear 130 degrees, how many degrees will the blue gear rotate?
2. If you rotate the green gear 38 degrees, how many degrees will the red gear rotate? Round your answer to the nearest hundredth.

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Answer 1

These questions explore the topic of arc length. The arc length that passes through the point where the gears touch when one gear rotates with a second gear will be the same for both the gears. With this insight we just need the formula for arc length (s) in terms of the radius of each gear (r) and the angle through which each gear rotates (θ):

s = r · θ

For the first question, we are asked to find the angle the blue gear will rotate if the red gear rotates through an angle of 130°. We know that the radius of the red gear is 1.6 units, so we can find the arc length by multiplying the two numbers together. Before we do, though, we need to convert the angle from degrees to radians by multiplying by the factor π/180°. Our angle is then found to be:

θ = 130° · π / 180° ≈ 2.269

Now we can plug this in to find the arc length:

s = 1.6 · 2.269 ≈ 3.630

Now that we know the arc length that is shared between the gears, we can use the same formula for the arc length to find out how much angle the blue gear rotates through since we already know the radius. We start by rearranging the formula to give an expression for the angle in terms of the arc length and radius:

θ = s / r

So we divide the arc length from above by the radius of the blue gear (r = 3.6 units) to find the answer in radians:

θ = 3.630 / 3.6 ≈ 1.008

Now we must convert this to degrees by multiplying by our conversion factor again to get the final answer, making sure to round to two decimal places at the end:

θ = 1.008 * 180° / π = 57.78°

The second question can be answered using exactly the same process. I would be glad to guide you through this problem or answer any follow up questions you may have. Best of luck!