On Mars, gravity is less than it is on Earth. If you were to kick a Mars rock at an initial velocity of 56 feet per second from the top of 30-foot hill, the rock's height h would be modeled by the equation h=- 1.9t^2+56t+30 where t is time in seconds.

Question

A: How high would the rock be after 20 seconds?
B: How high would the rock get?
C: How long would the rock be in the air?

Answer 1

A:
To find out how high the rock would be after 20 seconds, we can plug t=20 into the equation h=-1.9t^2+56t+30:

h = -1.9(20)^2 + 56(20) + 30
h = -1.9(400) + 1120 + 30
h = -760 + 1120 + 30
h = 390 feet

Therefore, the rock would be 390 feet high after 20 seconds.

B:
To find out how high the rock would get, we need to find the maximum height of the rock.
The maximum height occurs at the vertex of the parabolic function.
The vertex of a parabola is given by the formula t = -b/2a
In this case, a=-1.9 and b=56.

t = -56 / (2 * -1.9)
t = 14.74 seconds

Now we can plug this value of t back into the equation h=-1.9t^2+56t+30 to find the maximum height:

h = -1.9(14.74)^2 + 56(14.74) + 30
h = -1.9(217.5876) + 832.24 + 30
h = -413.4168 + 832.24 + 30
h = 448.8232 feet

Therefore, the rock would reach a maximum height of 448.8232 feet.

C:
To find out how long the rock would be in the air, we need to find the total time it takes for the rock to hit the ground. This occurs when h=0:

0=-1.9t^2+56t+30

Using the quadratic formula, we have:
t = (-56 ± √(56^2 - 4*(-1.9)*(30))) / (2*(-1.9))

Calculating the values gives t=0.523 seconds and t=29.696 seconds.

Therefore, the rock would be in the air for approximately 29.7 seconds.