An object is fired vertically upward with an initial speed of 68.6 meters/second. After t seconds, the height is shown by

y = -4.9 t^2 + 68.6 t meters. Given that this object has an initial speed of 68.6 m/s, what is the maximum height it will reach? How long did it take the rocket to reach the maximum height?

dy/dx = -9.8t + 68.6

= 0 for a max of y
9.8t = 68.6
t = 7

after 7 sec it will have a max of
-4.9(49) + 68.6 or 240.1 ft

or

complete the square on
y = -4.9 t^2 + 68.6 t
= -4.9(t^2 - 14t + 49 - 49)
= -4.9(t-7)^2 + 240.1

vertex is (7, 240.1) , and the parabola is downward

Max is 240.1 , when t = 7

To find the maximum height the object will reach, we need to determine the vertex of the equation y = -4.9t^2 + 68.6t.

The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula:
h = -b/2a, where h represents the x-coordinate of the vertex.

In this case, the equation is y = -4.9t^2 + 68.6t, which means a = -4.9 and b = 68.6.

Using the formula, we can calculate the x-coordinate of the vertex:
t_vertex = -68.6 / (2 * -4.9)
= 68.6 / 9.8
≈ 7 seconds (rounded to 1 decimal place)

To find the maximum height, we substitute t = 7 into the given equation:
y_max = -4.9(7)^2 + 68.6(7)
= -4.9(49) + 480.2
= -240.1 + 480.2
= 240.1 meters

Therefore, the maximum height the object will reach is approximately 240.1 meters.

The object takes 7 seconds to reach the maximum height.

To find the maximum height the object will reach, we need to determine the vertex of the quadratic equation y = -4.9t^2 + 68.6t. The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula:

t = -b / 2a

In our case, a = -4.9 and b = 68.6. Plug in these values into the formula to find the time at which the object reaches its maximum height:

t = -68.6 / (2 * -4.9)
t = 7 seconds

To find the maximum height, substitute the value of t back into the equation y = -4.9t^2 + 68.6t:

y = -4.9(7^2) + 68.6(7)
y = -4.9(49) + 480.2
y = -240.1 + 480.2
y = 240.1 meters

So, the object will reach a maximum height of 240.1 meters, and it will take 7 seconds to reach that height.