Question

a laser rangefinder is locked on a comet approaching earth. the distance g(x)in kilometers, of the comet after x days, for x in the interval 0 to 36 days, is given by g(x) = 150,000 csc((pi/36)x). What is the period of the comet's approach cycle?

Answer 1

The period of the comet's approach cycle, as described by the function g(x) = 150,000 csc((pi/36)x), is 72 days. This is calculated by solving for x in the equation (pi/36)x = 2pi, taking into account the natural period of the co-secant function.

Step-by-step explanation:

The period of a comet's approach cycle described by the function g(x) = 150,000 csc((pi/36)x) is determined by the co-secant (csc) trigonometric function. The trigonometric functions sine and co-secant have a natural period of 2π, which means they repeat their values every 2π radians. However, since there is a coefficient (π/36) multiplying the variable x, the function's period is changed.

To find the period of the function, we set the inside of the csc function equal to 2π and solve for one cycle:
(π/36)x = 2π which simplifies to x = 2π × (36/π) = 72 days. Therefore, the period of the comet's movement as described by the function is 72 days. This tells us that the comet completes one full approach cycle with respect to distance from the Earth every 72 days according to this laser rangefinder model.