the height , h(t) 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 thefootball and the time wehn the height is reached
but how did you get the 28-9.8t?
u dumb bruh
To determine the maximum height of the football and the time when the height is reached, we need to find the vertex of the quadratic equation represented by the given formula for height, h(t).
The equation for height is h(t) = 2 + 28t - 4.9t^2.
The vertex form of a quadratic equation is given by h(t) = a(t - h)^2 + k, where (h, k) represents the coordinates of the vertex.
To find the vertex, we need to rewrite the equation in vertex form. Let's first expand the squared term:
h(t) = -4.9t^2 + 28t + 2
Now, let's factor out the common factor (-4.9):
h(t) = -4.9(t^2 - (28/4.9)t) + 2
Next, we need to complete the square. Take half of the coefficient of the t term (-28/4.9) and square it:
(-28/4.9)/2 = -14/4.9 = -2.857
Now, add and subtract this value from the equation:
h(t) = -4.9(t^2 - (28/4.9)t - 2.857 + 2.857) + 2
= -4.9((t - 2.857)(t - 9.857)) + 2
Now, let's simplify the equation:
h(t) = -4.9(t - 2.857)(t - 9.857) + 2
The vertex form of the equation is h(t) = -4.9(t - 2.857)(t - 9.857) + 2.
From this equation, we can see that the maximum height occurs at the vertex (2.857, k), where k is the value of h(t) at t = 2.857.
To find the maximum height, we substitute t = 2.857 into the h(t) equation:
h(2.857) = -4.9(2.857 - 2.857)(2.857 - 9.857) + 2
= -4.9(0)(-7) + 2
= 0 + 2
= 2
Therefore, the maximum height of the football is 2 meters.
To find the time when the height is reached, we already know that the x-coordinate of the vertex is t = 2.857.
Therefore, the time when the height is reached is 2.857 seconds.
to get this, we get the derivative of h(t) with respect to t, and equate it to zero:
h(t) = 2 + 28t - 4.9t^2
0 = 28 - 9.8t
9.8t = 28
t = 2.86 s
*note that the derivative of a function is the slope of the tangent line at the given point (in this problem, the point referred is the time, t, that we're solving). since the given equation is a parabola (concave downward), it has a maximum point, and at this point, the slope is zero (that's why we equate to zero)
hope this helps~ :)