Math CentralQuandaries & Queries


Question from Ashutosh, a student:

Jose can remember that the length of an arc is 440cm, but he cannot remember the radius of the arc or the angle at the center. He does know that the angle was a whole number of degrees and the radius was less than 100cm. Find three possible angles and write down the size of each of the possible radii.

Hi Ashutosh.

The length of an arc is related to the radius and the central angle in the formula

s = rθ

where s is the arc length, r is the radius and θ is the angle in radians. Remember that to convert from radians to degrees, 2π radians = 360 degrees, so if φ = the angle in degrees, then the equation becomes this:

s = φ r π / 180

You know s = 440 cm and r < 100 cm. Thus,

s = 440 = φ r π / 180 < φ 100 π / 180

This simplifies:

φ 100 π / 180 > 440

φ > (440)(180) / (100π) = 252.1

φ ≥ 253 (since the angle is a whole number of degrees).

So you know the angle must be at least 253 degrees.

To complete the problem, pick any three integer angles larger than 253 degrees (and less than 360 degrees obviously) and compute the corresponding radius using the second equation I showed you.

Stephen La Rocque.

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