Final answer:
To solve the equation y = -16t^2 + 32t + 46, we apply the quadratic formula with coefficients a = -16, b = 32, and c = 46, calculating the discriminant and finding two possible time values, t ≈ -0.97 and t ≈ 2.97.
Step-by-step explanation:
To solve the quadratic equation y = -16t^2 + 32t + 46 using the quadratic formula, we start by identifying the coefficients a, b, and c in the standard form of a quadratic equation, which is at^2 + bt + c = 0. In this case, a = -16, b = 32, and c = 46. The quadratic formula is given by t = (-b ± √(b^2 - 4ac)) / (2a).
Applying the formula, we calculate the discriminant (the part under the square root) first: b^2 - 4ac = (32)^2 - 4(-16)(46). Simplifying, we get 1024 + 2944 = 3968. The square root of the discriminant, √3968, is approximately 63.
Now, we can find both possible values for t:
-
- t = (-32 + 63) / (2 * -16) = 31 / -32 = -0.96875
-
- t = (-32 - 63) / (2 * -16) = -95 / -32 = 2.96875
Therefore, the solutions to the equation are t ≈ -0.97 and t ≈ 2.97, taking into account the real values where time is typically measured.