The height, in metres, of the trajectory of a football is given by h(t)=2+28t-4.9t^2 , where t is the time in flight, in seconds. Determine the maximum height of the football and the time when that height is reached.
Well, determining the maximum height and time at which it is reached is no joke. But I'll give it a try!
To find the maximum height, we need to find the vertex of the quadratic equation h(t) = -4.9t^2 + 28t + 2. And guess what? The vertex can be found with a formula:
t = -b/2a
In this equation, a is the coefficient (-4.9) of t^2 and b is the coefficient (28) of t.
So, plugging in the values, we get:
t = -28/(2 * -4.9) = 2.857
Now that we have the time when the maximum height is reached, let's find that height by plugging it back into the original equation:
h(2.857) = 2 + 28 * 2.857 - 4.9 * (2.857)^2
h(2.857) ≈ 40.42 meters
So, the maximum height of the football is approximately 40.42 meters, and it is reached after approximately 2.857 seconds. Keep it up, football! You're really reaching new heights!
To determine the maximum height of the football, we need to find the vertex of the quadratic function h(t) = 2 + 28t - 4.9t^2.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)).
In this case, a = -4.9, b = 28, and c = 2.
To find the time when the maximum height is reached, we just need to find the value of t at the vertex.
Let's calculate:
t = -b / (2a)
= -28 / (2 * -4.9)
= -28 / -9.8
= 2.857 seconds (rounded to three decimal places)
Now, let's find the maximum height:
h(t) = 2 + 28t - 4.9t^2
Substituting t = 2.857 into the equation:
h(2.857) = 2 + 28(2.857) - 4.9(2.857)^2
= 2 + 80.996 - 36.995
The maximum height of the football is approximately:
h(2.857) ≈ 46.001 meters
Therefore, the maximum height of the football is approximately 46.001 meters and it is reached at approximately 2.857 seconds.
To determine the maximum height of the football and the time when that height is reached, we need to find the vertex of the parabolic function h(t) = 2 + 28t - 4.9t^2.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (h, k), where h = -b/2a and k = f(h), where f(h) is the value of the function at h.
In our case, the quadratic function is h(t) = 2 + 28t - 4.9t^2, which can be rewritten as h(t) = -4.9t^2 + 28t + 2.
Comparing this to the standard form ax^2 + bx + c, we have a = -4.9, b = 28, and c = 2.
To find the time when the maximum height is reached, we can use the formula t = -b/2a.
Plugging in the values, we have t = -28 / (2 * -4.9) = 2.857 seconds.
Now, substitute this value of t back into the function h(t) to find the maximum height.
h(2.857) = -4.9(2.857)^2 + 28(2.857) + 2 = 41.088 meters.
Therefore, the maximum height reached by the football is 41.088 meters, and it is reached at 2.857 seconds.