a ball is kicked in the air from the top of a cliff. the path the ball travels is given by the equation h(t)=-5t^2+17t+22 where h(t) is the height in metres and t is the time in seconds
a) what is the maximum height that the ball reaches
b)when will the ball hit the ground
h(t)=-5t^2+17t+22
dh/dt=-10t+17 =0 at max height.
t= 10/17
h(max)=-5(10/17)^2 + 17(10/17)+22
when is h(t)=0, that is the ground.
thank you !!!
To find the maximum height that the ball reaches, we need to find the vertex of the quadratic equation h(t) = -5t^2 + 17t + 22. The vertex of a parabolic equation in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the equation.
a) Maximum height:
In this case, a = -5, b = 17, and c = 22. To find the vertex, we substitute these values into the formula: t = -b/(2a).
t = (-17) / [2 * (-5)]
t = 17 / 10
t = 1.7 seconds
Now, to find the maximum height, we substitute this value of t into the equation h(t):
h(1.7) = -5(1.7)^2 + 17(1.7) + 22
h(1.7) = -5(2.89) + 28.9 + 22
h(1.7) = -14.45 + 28.9 + 22
h(1.7) = 36.45 meters
Therefore, the maximum height that the ball reaches is 36.45 meters.
b) When the ball hits the ground:
The ball hits the ground when h(t) equals zero, as this indicates the height is zero.
Setting h(t) = 0, we can solve for t:
-5t^2 + 17t + 22 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. The factorization doesn't seem straightforward in this case, so let's use the quadratic formula:
t = [-b ± sqrt(b^2 - 4ac)] / (2a)
Substituting the values from the equation, we get:
t = [-17 ± sqrt(17^2 - 4(-5)(22))] / (2(-5))
t = [-17 ± sqrt(289 + 440)] / (-10)
t = [-17 ± sqrt(729)] / (-10)
t = [-17 ± 27] / (-10)
This gives us two possible solutions:
1) t = (-17 + 27) / (-10)
t = 10 / (-10)
t = -1 second
2) t = (-17 - 27) / (-10)
t = -44 / (-10)
t = 4.4 seconds
Since time cannot be negative in this context, we discard the negative solution. Therefore, the ball hits the ground after approximately 4.4 seconds.
To summarize:
a) The maximum height that the ball reaches is 36.45 meters.
b) The ball hits the ground after approximately 4.4 seconds.