a. Using technology to sketch the graph of the function g(t) = -16t^2 + 140t, we get a downward-opening parabolic curve. The x-axis represents time (t) in seconds, and the y-axis represents the height (g(t)) in feet.
b. To find the height of the model rocket after 5 seconds, we substitute t = 5 into the function:
g(5) = -16(5)^2 + 140(5) = -16(25) + 700 = -400 + 700 = 300 feet
Therefore, the height of the model rocket after 5 seconds is 300 feet.
c. To determine the approximate time at which the model rocket is at a height of 200 feet, we set the function g(t) equal to 200 and solve for t:
-16t^2 + 140t = 200
This equation can be solved using technology or factoring techniques, which gives us approximately t = 1.61 seconds.
Therefore, after approximately 1.61 seconds, the model rocket is at a height of 200 feet.
d. The maximum height the model rocket reaches corresponds to the vertex of the parabolic curve described by the function. The vertex of a parabola with equation g(t) = -16t^2 + 140t can be found using the formula t = -b / (2a), where a = -16 and b = 140.
t = -140 / (2 * (-16)) = -140 / (-32) = 4.375
To find the maximum height, substitute this value back into the function:
g(4.375) = -16(4.375)^2 + 140(4.375) = -16(19.140625) + 612.5 = -306.25 + 612.5 = 306.25 feet
Therefore, the maximum height the model rocket reaches is 306.25 feet, and it occurs at approximately 4.375 seconds.
e. The domain of the function g(t) = -16t^2 + 140t is all real numbers, as there are no restrictions on time in the equation. However, in the context of the problem, the domain of the situation may be limited to a specific range of time based on practical constraints. For example, if the rocket is launched at t = 0 and observed until it lands, the domain of the problem situation may be limited to t ≥ 0. This means we are considering time starting from the moment of launch and continuing until the rocket lands or until a specified time limit is reached.