1. Domain:
The domain of the function represents all possible values of time (t) for which the function is defined. In this case, time cannot be negative as it is not physically possible to have a negative time value. Additionally, we can consider the time at which the ball hits the ground as the end point of the domain.
Since the ball is dropped from a certain height and hits the ground, the domain will be from the initial time the ball is dropped (t=0) to the time it hits the ground. The time at which the ball hits the ground can be found by setting h(t) = 0:
-4.9t^2 + 10t + h0 = 0
-4.9t^2 + 10t + 4 = 0
To find the time, you can solve the above quadratic equation for t. The solution(s) will give you the time at which the ball hits the ground, which will be the upper bound of the domain.
2. Range:
The range of the function represents all possible heights (h) that the ball can have above the ground. Since the function is a quadratic function with a downward-opening parabola (-4.9t^2 term), the maximum height occurs at the vertex of the parabola. The vertex of the parabola is given by the formula:
t_vertex = -b / (2a)
Substitute the values of a = -4.9 and b = 10 into the formula to find the time at which the maximum height is reached. Then, substitute this time back into the function to find the maximum height above the ground.
The range would be from the initial height of the ball (h0) to the maximum height reached above the ground.
A ball is dropped from different heights, and its height above the ground is measured at various time intervals. The data can be modeled by the quadratic function h(t)=-4.9t^2+10t+h0, where h0 is the initial height in meters. Determine the domain and range of this situation given the initial height is 4 meters
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