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 30
days, is given by g(x)=150,000csc(π30x)
What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?
To find the minimum distance between the comet and Earth, we need to find the minimum value of the function g(x) over the interval 0 ≤ x ≤ 30 .
First, note that csc(x) is always greater than or equal to 1, so the minimum value of the function occurs when csc(π/2) = 1 . This happens when x = 15 days.
Plugging x = 15 into the function g(x) , we get:
g(15) = 150,000csc(π/2) = 150,000
Therefore, the minimum distance between the comet and Earth is 150,000 kilometers and it occurs at x = 15 days.
This corresponds to the constant 150,000 in the equation.