Toto throws a ball up in the air. The height of the ball can be modelled using the equation h=-16.1t^2+73.5t+5.5 where t is time in seconds. At what time will the ball reach its maximum height? What is the maximum height? How long does it take for the ball to hit the ground?
as with any parabola ax^2+bx+c, the vertex is at
(-b/2a, (4ac-b^2)/4a)
it hits the ground when
-16.1t^2+73.5t+5.5 = 0
so plug in your quadratic formula.
for time to maximum height and that maximum height you need the vertex of this parabola.
Use the method you learned finding that vertex
as to hitting the ground, set
-16.1t^2+73.5t+5.5 = 0 and solve the quadratic.
reject any negative value of t.
To find the time at which the ball reaches its maximum height, you need to determine the vertex of the parabolic equation. The vertex of a parabola in the form of y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)), where a, b, and c are coefficients of the equation.
In this case, the equation is h = -16.1t^2 + 73.5t + 5.5. Comparing this to the general form y = ax^2 + bx + c, we have a = -16.1, b = 73.5, and c = 5.5.
To find the time at the maximum height, we substitute -b/2a for t in the equation and solve for t.
t = -b/2a
t = - 73.5 / (2 * -16.1)
t = - 73.5 / -32.2
t ≈ 2.28 seconds
So, the ball will reach its maximum height at approximately 2.28 seconds.
To find the maximum height, substitute the value of t (2.28 seconds) into the equation h = -16.1t^2 + 73.5t + 5.5 and solve for h.
h = -16.1(2.28)^2 + 73.5(2.28) + 5.5
h ≈ 76.83 feet
Thus, the maximum height of the ball is approximately 76.83 feet.
Finally, to find how long it takes for the ball to hit the ground, we set the equation h = -16.1t^2 + 73.5t + 5.5 equal to 0 and solve for t.
-16.1t^2 + 73.5t + 5.5 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
t = (-73.5 ± √(73.5^2 - 4*(-16.1)*5.5)) / (2*(-16.1))
Calculating this, we find two solutions for t:
t ≈ 0.42 seconds and t ≈ 4.51 seconds
Since time cannot be negative, we discard the negative solution.
Therefore, it takes approximately 4.51 seconds for the ball to hit the ground.