Bot GPT 4-o mini To determine the domain of the volleyball's height in terms of time when it reaches its maximum height, we can follow these steps:
-
Identify the Type of Equation: The given equation \( h(t) = -16t^2 + 33t + 6 \) is a quadratic function, and it opens downwards (since the coefficient of \( t^2 \) is negative). This means it has a maximum point (the vertex).
-
Find the Vertex: The formula to find the time \( t \) at which the maximum height occurs for a quadratic equation \( at^2 + bt + c \) is given by: \[ t = -\frac{b}{2a} \] In this case, \( a = -16 \) and \( b = 33 \).
Plugging in these values: \[ t = -\frac{33}{2 \cdot -16} = \frac{33}{32} \] This value of \( t \) represents the time at which the ball reaches its maximum height.
-
Define the Domain: The domain of the height function can also be established by considering the physical context of the problem—specifically, when the volleyball is in the air.
- The ball is served at \( t = 0 \) when it leaves the player's hand.
- It will reach the ground when its height \( h(t) \) becomes 0. To find this, we need to solve \( -16t^2 + 33t + 6 = 0 \).
-
Finding When the Volleyball Hits the Ground: Use the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our function, \( a = -16\), \( b = 33 \), and \( c = 6 \): \[ t = \frac{-33 \pm \sqrt{33^2 - 4 \cdot (-16) \cdot 6}}{2 \cdot -16} \] Calculate the discriminant: \[ 33^2 = 1089 \] \[ 4 \cdot (-16) \cdot 6 = -384 \] Hence: \[ b^2 - 4ac = 1089 + 384 = 1473 \] Now, substituting into the quadratic formula: \[ t = \frac{-33 \pm \sqrt{1473}}{-32} \] We will only consider the positive root because time cannot be negative: \[ t \approx \frac{-33 + 38.4}{-32} \quad (\text{using } \sqrt{1473} \approx 38.4) \] \[ t \approx \frac{5.4}{-32} \quad (which will give a positive time solution) \]
Therefore, the domain will be \( t \) from \( 0 \) to the positive solution of \( t \) when \( h(t) = 0 \).
-
Calculating the Exact Time when it Hits the Ground: The specific roots would give the total flight time for the volleyball before it hits the ground.
Final Domain Conclusion: The volleyball is in the air for \( 0 \leq t \leq T \) where \( T \) is the positive root found. You can substitute the square root back to find the approximate positive time it will land on the ground.
In conclusion, the domain of the function describing the volleyball’s height is: \[ [0, t_{ground}] \] Where \( t_{ground} \) is the time at which the ball lands, which can be approximated as above.
1 answer