To determine the correct equation representing the height \( h(t) \) of the elevator as a function of the time \( t \), we can analyze the provided data in the table.
From the table:
- At \( t = 0 \) seconds, \( h(0) = 500 \) feet
- At \( t = 5 \) seconds, \( h(5) = 475 \) feet
- At \( t = 10 \) seconds, \( h(10) = 450 \) feet
- At \( t = 15 \) seconds, \( h(15) = 425 \) feet
We observe that the elevator descends by 25 feet every 5 seconds. Therefore, the rate of descent can be calculated as follows:
\[ \text{Rate of descent} = \frac{25 \text{ feet}}{5 \text{ seconds}} = 5 \text{ feet per second} \]
Since the elevator is descending, the height decreases over time. This means that for every second that passes, the height decreases by 5 feet.
Thus, we can express this relationship mathematically. The height \( h(t) \) as a function of time \( t \) can be modeled with the following equation:
\[ h(t) = -5t + 500 \]
This equation tells us that at \( t = 0 \) seconds, the height is 500 feet, and for every additional second \( t \), the height decreases by 5 feet.
Now, we can check which of the options given matches this equation:
- \( h(t) = 5t + 500 \) ⇒ Incorrect, as this would imply the height increases.
- \( h(t) = 5t - 500 \) ⇒ Incorrect, as this does not match our calculated relationship.
- \( h(t) = -5t + 500 \) ⇒ Correct.
- \( h(t) = -5t - 500 \) ⇒ Incorrect, as this does not correctly account for the starting height.
Therefore, the correct equation is:
\[ \boxed{h(t) = -5t + 500} \]
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