How many inches would a point on the outer edge of the large gear travel in a 150º rotation? (the gear has a radius of 4 inches) explain how you would solve this.

In a full rotation, 360°, a point on the edge of a gear would travel a distance equal to its circumference.  The circumference of a circle is:

C=2πr, and since we are only traveling 150°, we need to set up an appropriate ratio for circumference to the distance the point travels:

d/C=150/360

d=5C/12 and since C=2πr

d=10πr/12

d=5πr/12, and since r=4in

d=5*4π/12 in

d=20π/12 in

d=5π/3 in

d≈5.24 in (to nearest hundredth of an inch)

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