Sue launches a bottle rocket into the air. The function h(t) = 115t - 16t^2 + 300 gives the height of the bottle rocket, in feet, t seconds after it is launched. Using a table or graph, about how many seconds did it take for the bottle rocket to reach a height of 200 feet? Round your answer to the nearest second. Select the correct answer from the following. Show your work.
a) - 10 seconds
b) 8 seconds
c) -8 seconds
d) 10 seconds
To find the number of seconds it took for the bottle rocket to reach a height of 200 feet, we need to set the height function h(t) equal to 200 and solve for t:
115t - 16t^2 + 300 = 200
-16t^2 + 115t + 100 = 0
To find the solutions for t, we can use the quadratic formula:
t = (-115 ± √(115^2 - 4(-16)(100)) / 2(-16)
t = (-115 ± √(13225 + 6400)) / -32
t = (-115 ± √19625) / -32
t = (-115 ± 140) / -32
This gives us two possible solutions: t = (-115 + 140) / -32 = 25 / -32 or t = (-115 - 140) / -32 = -255 / -32 = 8.
Since time cannot be negative, the correct answer is 8 seconds. Therefore, the bottle rocket took about 8 seconds to reach a height of 200 feet.
So, the correct answer is:
b) 8 seconds