Final answer:
To find the time (t) when h(t) = -16t² + 8t + 200, we use the quadratic formula after rearranging the terms to zero. After substituting the known values into the quadratic formula, we solve to find t ≈ 3.79 seconds as the physical solution since time cannot be negative.
Step-by-step explanation:
To solve the quadratic equation h(t) = -16t² + 8t + 200 for time (t), we first rearrange the terms to set the equation to zero:
0 = -16t² + 8t + 200
We then use the quadratic formula which is given by:
t = −(b ± √(b² − 4ac)) / (2a)
Where, in our equation, a = -16, b = 8, and c = 200. Substituting these values into the quadratic formula, we get:
t = −(8 ± √(8² − 4(-16)(200))) / (2(-16))
Calculating under the square root:
t = −(8 ± √(64 + 12800)) / (-32)
t = −(8 ± √(12864)) / (-32)
After further simplification:
t = −(8 ± 113.4) / (-32)
We end up with two possible solutions for time:
- t = (8 + 113.4) / 32
- t = (8 - 113.4) / 32
Since time cannot be negative, we discard the solution that gives us a negative time, resulting in:
t = 121.4 / 32 ≈ 3.79 seconds