Question
A hang glider pilot's goggles fall off when he is about 2,104 feet above the ground. The height above the ground of the goggles can be modeled by the equation h(t) = -16t^2 + 2,104, where t is the number of seconds after the goggles are dropped and h (t) is the height in feet.
What is the meaning of in this context?
What is the meaning of in this context?
Answers
GPT-4o mini
In the context of the given problem, the equation \( h(t) = -16t^2 + 2,104 \) models the height \( h(t) \) of the goggles above the ground as a function of time \( t \) in seconds after they are dropped.
Here’s the meaning of the components of the equation:
1. **\( h(t) \)**: This represents the height in feet of the goggles above the ground at time \( t \).
2. **\( -16t^2 \)**: This term represents the effect of gravity on the goggles as they fall. The coefficient -16 is derived from the acceleration due to gravity, which is approximately \( 32 \, \text{ft/s}^2 \) (the negative sign indicates that the height decreases over time). The \( t^2 \) term indicates that the height decreases at a rate that is dependent on the square of the time elapsed, which is characteristic of objects in free fall under the influence of gravity.
3. **\( 2,104 \)**: This is the initial height (in feet) from which the goggles are dropped. It indicates that at \( t = 0 \), when the goggles are just released, they are at 2,104 feet above the ground.
Overall, the equation describes the trajectory of the goggles as they fall from the initial height of 2,104 feet to the ground, with the height decreasing over time due to the acceleration of gravity.
Here’s the meaning of the components of the equation:
1. **\( h(t) \)**: This represents the height in feet of the goggles above the ground at time \( t \).
2. **\( -16t^2 \)**: This term represents the effect of gravity on the goggles as they fall. The coefficient -16 is derived from the acceleration due to gravity, which is approximately \( 32 \, \text{ft/s}^2 \) (the negative sign indicates that the height decreases over time). The \( t^2 \) term indicates that the height decreases at a rate that is dependent on the square of the time elapsed, which is characteristic of objects in free fall under the influence of gravity.
3. **\( 2,104 \)**: This is the initial height (in feet) from which the goggles are dropped. It indicates that at \( t = 0 \), when the goggles are just released, they are at 2,104 feet above the ground.
Overall, the equation describes the trajectory of the goggles as they fall from the initial height of 2,104 feet to the ground, with the height decreasing over time due to the acceleration of gravity.