a meteorite slams into the earths crust and penetrates d=(1200t-36000t^2)m in t seconds after impact. What is the maximum depth of penetration?
dd/dt = 1200 - 72000t
= 0 for a max
72000t = 1200
t = 1/60
d = 1200(1/60) - 36000(1/3600)
= 10 m
Thanks Alot I was really puzzled on this one.
To find the maximum depth of penetration, we need to determine the point at which the meteorite stops penetrating further into the Earth's crust. This occurs when the velocity of the meteorite becomes zero, indicating that it has reached its maximum depth.
Given the equation of the depth of penetration, d = (1200t - 36000t^2)m, we need to find the time at which the velocity is zero. The velocity can be obtained by taking the derivative of the equation with respect to time. Let's do that:
First, differentiate d with respect to t:
v = d/dt (1200t - 36000t^2)
v = 1200 - 72000t
To find the time when the velocity is zero, we set v = 0 and solve for t:
0 = 1200 - 72000t
Solving this equation, we get:
72000t = 1200
t = 1200 / 72000
t = 0.0167 s
Now that we have the time at which the velocity is zero, we can substitute this value back into the equation for d to find the maximum depth of penetration.
Substituting t = 0.0167 into the equation d = (1200t - 36000t^2), we get:
d = (1200 * 0.0167 - 36000 * 0.0167^2)m
d ≈ 19.56m
Therefore, the maximum depth of penetration is approximately 19.56 meters.