A bottle rocket launches into the air. The equation h=-16t^2+64t+9 gives the height of the ball, h in feet, as a function of t, number of seconds after launch. What is the maximum height the ball reaches? How many seconds does it take the ball to reach that height?
you are basically looking for the vertex of this downwards opening parabola
the t of the vertex is -b/2a
= -64/-32 = 2 seconds
it will take 2 seconds to reach the maximum height
when t = 2
h = -16(4) + 64(2) + 9
= 73 ft
THANK YOU SO MUCH
To find the maximum height the ball reaches, we need to determine the vertex of the parabolic function given by the equation h = -16t^2 + 64t + 9.
The formula for the x-coordinate of the vertex is given by x = -b / (2a), where the coefficients of the quadratic equation are a = -16 and b = 64.
Substituting the values into the formula, we have:
t = -64 / (2*(-16))
t = -64 / (-32)
t = 2
So, the ball takes 2 seconds to reach its maximum height.
To find the maximum height, we need to substitute this value back into the original equation h = -16t^2 + 64t + 9:
h = -16(2)^2 + 64(2) + 9
h = -16(4) + 128 + 9
h = -64 + 128 + 9
h = 73
Therefore, the maximum height the ball reaches is 73 feet and it takes 2 seconds to reach that height.
To find the maximum height the ball reaches, we need to determine the vertex of the parabolic function represented by the equation h = -16t^2 + 64t + 9. The vertex represents the highest point of the parabola, which in this case corresponds to the maximum height of the ball.
The vertex of a parabola in the form ax^2 + bx + c can be found using the formula t = -b / (2a), where t represents the time, a is the coefficient of the t^2 term (-16 in this case), and b is the coefficient of the t term (64 in this case).
In our equation, a = -16 and b = 64. Substituting these values into the formula, we can calculate the time at which the maximum height is reached:
t = -b / (2a)
t = -64 / (2*(-16))
t = -64 / (-32)
t = 2 seconds
So, it takes the ball 2 seconds to reach its maximum height.
To find the actual maximum height, substitute the value of t (2 seconds) back into the equation h = -16t^2 + 64t + 9:
h = -16(2)^2 + 64(2) + 9
h = -16(4) + 128 + 9
h = -64 + 128 + 9
h = 73 feet
Therefore, the maximum height the ball reaches is 73 feet and it takes 2 seconds for the ball to reach that height.