An object is projected directly upward from the ground. After t seconds its distance in feet above the ground is s = 144t -16t2
How long does it take for the object to reach its maximum height?
What is the maximum height the object reaches? When does it hit the ground?
To find the maximum height, we need to find the vertex of the parabolic function s = 144t -16t^2. The vertex has a t value of -b/2a, where a = -16 and b = 144. So:
t = -b/2a = -144/(2(-16)) = 4.5
To find the maximum height, we substitute 4.5 for t in the equation:
s = 144t -16t^2 = 144(4.5) -16(4.5)^2 = 324 feet
So the object reaches a maximum height of 324 feet.
To find when the object hits the ground, we set s = 0 and solve for t:
0 = 144t -16t^2
0 = t(144 -16t)
t = 0 or t = 9
t = 0 corresponds to when the object is first launched and t = 9 corresponds to when it hits the ground. So the object hits the ground after 9 seconds.
Therefore, the object takes 4.5 seconds to reach its maximum height, reaches a maximum height of 324 feet, and hits the ground after 9 seconds.
To find the time it takes for the object to reach its maximum height, we need to find the vertex of the equation. The vertex can be found using the formula t = -b / (2a), where a = -16 and b = 144.
t = -144 / (2 * -16)
t = 144 / 32
t = 4.5
Therefore, it takes the object 4.5 seconds to reach its maximum height.
To find the maximum height, we substitute the time obtained into the equation s = 144t - 16t^2.
s = 144 * 4.5 - 16 * (4.5^2)
s = 648 - 16 * 20.25
s = 648 - 324
s = 324
The maximum height the object reaches is 324 feet.
To find when the object hits the ground, we set s (the distance above the ground) equal to 0 and solve for t.
0 = 144t - 16t^2
16t^2 = 144t
16t^2 - 144t = 0
16t(t - 9) = 0
So, t = 0 or t = 9.
Since the object was projected directly upward from the ground, t = 0 represents the time at the start, not when it hits the ground. Therefore, the object hits the ground after 9 seconds.
To summarize:
- The object takes 4.5 seconds to reach its maximum height.
- The maximum height the object reaches is 324 feet.
- The object hits the ground after 9 seconds.
To find the time it takes for the object to reach its maximum height, we can differentiate the equation with respect to time and set it equal to zero.
The equation given for the distance of the object is s = 144t - 16t^2.
Differentiating with respect to time (t) gives us the velocity function v(t) = 144 - 32t.
Setting v(t) = 0, we get:
144 - 32t = 0
32t = 144
t = 4.5 seconds.
Therefore, it takes the object 4.5 seconds to reach its maximum height.
To find the maximum height the object reaches, we can substitute the value of t = 4.5 seconds into the equation for distance.
s = 144t - 16t^2
s = 144(4.5) - 16(4.5)^2
s = 648 - 324
s = 324 feet.
Therefore, the maximum height the object reaches is 324 feet.
To find when the object hits the ground, we set the distance s equal to zero and solve for t.
0 = 144t - 16t^2
16t^2 - 144t = 0
t(16t - 144) = 0
From this equation, we can see that t = 0 or 16t - 144 = 0.
Solving 16t - 144 = 0, we get:
16t = 144
t = 9 seconds.
Therefore, the object hits the ground after 9 seconds.