Sylvia launches a bottle rocket into the air. The function h(t)=115t−16t2+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.
(1 point)
Responses
-10 seconds
-10 seconds
-8 seconds
-8 seconds
10 seconds
10 seconds
8 seconds
To find out how many seconds it took for the bottle rocket to reach a height of 200 feet, we can set the function h(t) equal to 200 and solve for t:
115t - 16t^2 + 300 = 200
16t^2 - 115t + 100 = 0
Using the quadratic formula, t = (-(-115) ± sqrt((-115)^2 - 4(16)(100)))/(2(16))
t = (115 ± sqrt(115^2 - 6400))/32
t = (115 ± sqrt(13225 - 6400))/32
t = (115 ± sqrt(6825))/32
Since the square root of 6825 ≈ 82.60, the two possible values for t are:
t = (115 + 82.60)/32 ≈ 5.42 seconds
t = (115 - 82.60)/32 ≈ 1.04 seconds
Rounding to the nearest second, we get:
t ≈ 5 seconds