A ball is thrown into the air from a height of 7ft. The height, h, of the ball after t seconds, is given by the equation h= −4.9t^2 + 34t + 7 .
What is the maximum height the ball reaches?
Select one:
a. between 45ft and 50ft
b. between 50ft and 55ft
c. between 55ft and 60ft
d. over 60ft (is this one correct?)
well I will cheat and use calculus
at to dh/dt = 0
0= -9.8 t + 34
t = 34/9.8 = 3.47 s
so
h= −4.9(3.47)^2 + 34(3.47) + 7
= -59 + 118 +7
= 66
so yes
h= −4.9t^2 + 34t + 7
parabola look for vertex
4.9 t^2 -34 t = -h + 7
t^2 - 6.94 t = -h/4.9 + 1.43
t^2 - 6.94 t + 12.0 = -h/4.9+ 13.43
(t-3.47)^2 = -h/4.9+ 65.8/4.9 = -(1/4.9)(h-65.8)
so t = 3.47 and h = 65.8 at vertex
−4.9t^2 + 34t + 7
the vertex is at t = -b/2a = 34/9.8
the height is h = c - b^2/4a = 7 + 34^2/19.6
To find the maximum height the ball reaches, we need to determine the vertex of the parabolic equation h = -4.9t^2 + 34t + 7.
The t-coordinate of the vertex can be found using the formula t = -b/(2a), where a, b, and c are the coefficients in the equation in the form ax^2 + bx + c. In this case, a = -4.9 and b = 34.
t = -34/(2 * -4.9)
t ≈ 3.47 seconds
To find the maximum height, substitute this value of t back into the equation:
h = -4.9(3.47)^2 + 34(3.47) + 7
h ≈ 60.67 feet
Therefore, the maximum height the ball reaches is over 60 feet. The correct answer is d. over 60ft.