(4 pts) If a rock is thrown vertically upward from the surface of Mars with an initial velocity of 15m/sec then the height of the rock after t seconds is h=15t-1.86t^2 (h in meters and t in seconds). The
rock will reach its maximum height when the velocity=0 m/sec. How long does it take for the rock to
reach its maximum height, and what is the maximum height?

Step-by-step explanation:

I see you're in college math, so we'll solve this with calculus, since it's the easiest way anyway.

The position equation is

 That equation will give us the height of the rock at ANY TIME during its travels. I could find the height at 2 seconds by plugging in a 2 for t; I could find the height at 12 seconds by plugging in a 12 for t, etc.

The first derivative of position is velocity:

v(t) = -3.72t + 15 and you stated that the rock will be at its max height when the velocity is 0, so we plug in a 0 for v(t):

0 = -3.72t + 15 and solve for t:\

-15 = -3.72t so

t = 4.03 seconds. This is how long it takes to get to its max height. Knowing that, we can plug 4.03 seconds into the position equation to find the height at 4.03 seconds:

s(4.03) = -1.86(4.03)² + 15(4.03) so

s(4.03) = 30.2 meters.

Calculus is amazing. Much easier than most methods to solve problems like this.

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