Question

an inventor is designing a machine that will have two gears. the first gear will complete one revolution every 8x squared y seconds. the second gear will complete one revolution every 6x to the third power y squared seconds. how often will both of the gears be in their starting positions at the same time​

Answer 1

every 24x^3y^2 seconds

Explanation:

To determine how often both gears will be in their starting positions at the same time, we need to find a common multiple of the two revolution times.

The first gear completes one revolution every 8x^2y seconds, and the second gear completes one revolution every 6x^3y^2 seconds.

To find a common multiple, we need to factorize the revolution times:

8x^2y = (2^3)(x^2)(y)

6x^3y^2 = (2)(3)(x^3)(y^2)

The common multiple should include all the prime factors with their highest powers. Therefore, the common multiple would be:

(2^3)(3)(x^3)(y^2) = 24x^3y^2 seconds.

So, both gears will be in their starting positions at the same time every 24x^3y^2 seconds.