Key: *** = my answer
8. A catapult launches a boulder with an upward velocity of 148 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h = –16t2 + 148t + 30. How long does it take the boulder to reach its maximum height? What is the boulder’s maximum height? Round to the nearest hundredth, if necessary.
a. Reaches a maximum height of 30 feet in 9.25 seconds.
b. Reaches a maximum height of 640.5 feet in 4.63 seconds.
c. Reaches a maximum height of 1,056.75 feet in 4.63 seconds.
d. Reaches a maximum height of 372.25 feet in 4.63 seconds.
I think its C.
dh/dt = -32t + 148
= 0 for a max of h
32t = 148
t = 148/32 = 4.625
when t = 4.625
h = -16(4.625)^2 + 148(4.625) + 3 0 = 372.25
looks like d
look for vertex of the parabola in algebra. (You would be able to do it much faster with calculus or physics)
t^2 - 9.25 t - 1.875 = -h/16
t^2 - 9.25 t = -h/16 + 1.875
t^2-9.25t+4.265^2 = -h/16 +1.875+21.39
( t-4.265)^2 = -1/16( h - 372.25)
So, I think it is d and at 4.265 seconds
To find the time it takes for the boulder to reach its maximum height, we need to find the vertex of the parabolic function. The formula for the vertex of a parabola in the form of h = at^2 + bt + c is given by t = -b / (2a).
In the given function h = -16t^2 + 148t + 30, we can identify a = -16, b = 148, and c = 30.
Using the formula t = -b / (2a), we can substitute the values to find the time it takes for the boulder to reach its maximum height:
t = -148 / (2 * -16)
t = 148 / 32
t = 4.625
Therefore, it takes the boulder approximately 4.63 seconds to reach its maximum height.
To find the maximum height of the boulder, we substitute this time value into the function h = -16t^2 + 148t + 30:
h = -16(4.63)^2 + 148(4.63) + 30
h = -16(21.4369) + 684.44 + 30
h = -342.99 + 684.44 + 30
h = 371.45
Therefore, the boulder reaches a maximum height of approximately 371.45 feet.
*** The correct answer is d. Reaches a maximum height of 372.25 feet in 4.63 seconds.