1. A projectile is fired straight up from a height of 6 feet. Its height (h) in feet after t seconds is given by h = 6 + 192t -16t^2.
Answer:
h = -16t^2 + 192t + 6
h = -16( t^2 -12t +36) + 6 +576
h = -16 (t-6)^2 +582
The maximum height will be 582 feet.
2. The path of a basketball on the moon can be modelled by the equation h(d) -9d^2+90d+2 where h(d) is the height of the basketball in meters and d is the horizontal distance of the ball from the player. Dtermine
a) height of ball before it is show
h(0) = 2
The height of the ball is 2 m
b) The maximum height
h(d) = -9 ( d^2 - 10d + 25) + 2 + 225
h(d) = -9 ( d-5)^2 + 227
The maximum height is 227 m
c) How long will it take the ball to reach the maximum height?
Since the vertex is (5,227), it will take 5 seconds
all appears correct.
Thank you
a) height of ball before it is shot: The height of the ball before it is shot is 2 meters.
b) The maximum height: The maximum height of the ball can be found by evaluating the equation h(d) = -9 (d^2 - 10d + 25) + 2 + 225. Simplifying, we get h(d) = -9(d-5)^2 + 227. Therefore, the maximum height is 227 meters.
c) How long will it take the ball to reach the maximum height: Since the vertex of the parabolic equation is at (5,227), it will take 5 seconds for the ball to reach the maximum height.
To answer Question 1:
The equation h = 6 + 192t - 16t^2 describes the height of the projectile as a function of time. To find the maximum height, we need to find the vertex of the parabolic equation.
The vertex of a parabola in the form ax^2 + bx + c can be found using the formula x = -b / (2a).
In this case, a = -16 and b = 192. Plugging these values into the formula, we get:
t = -192 / (2 * -16)
t = -192 / -32
t = 6
So, the maximum height is reached after 6 seconds.
To find the actual maximum height, we substitute the value of t = 6 back into the equation:
h = 6 + 192(6) - 16(6^2)
h = 6 + 1152 - 576
h = 6 + 576
h = 582
Therefore, the maximum height is 582 feet.
To answer Question 2:
a) To find the height of the ball before it is shot, we need to evaluate the equation h(d) = -9d^2 + 90d + 2 when d = 0.
h(0) = -9(0)^2 + 90(0) + 2
h(0) = 0 + 0 + 2
h(0) = 2
Therefore, the height of the ball before it is shot is 2 meters.
b) To find the maximum height of the basketball, we need to find the vertex of the parabolic equation h(d) = -9d^2 + 90d + 2.
The vertex of a parabola can be found using the formula d = -b / (2a), where a = -9 and b = 90.
d = -90 / (2 * -9)
d = -90 / -18
d = 5
So, the maximum height is reached when the basketball is at a horizontal distance of 5 meters.
To find the actual maximum height, we substitute the value of d = 5 back into the equation:
h(d) = -9(5)^2 + 90(5) + 2
h(d) = -9(25) + 450 + 2
h(d) = -225 + 450 + 2
h(d) = 227
Therefore, the maximum height of the basketball is 227 meters.
c) To find how long it takes for the ball to reach the maximum height, we can use the fact that the vertex of the parabola occurs at the halfway point of the ball's flight time.
In this case, the maximum height is reached at d = 5 meters, which means the halfway point is at d = 2.5 meters.
Given that the ball is shot with an initial velocity (assuming it is shot horizontally), we can use the formula d = v0t + (1/2)at^2, where v0 is the initial velocity, a is the acceleration (in this case, due to gravity), and t is time.
Since, in this case, we are not given specific values for these variables, we cannot determine the time it takes for the ball to reach the maximum height.