Answered

Stefano accidentally dropped his sunglasses off the edge of a canyon as he was looking down. The height, $h(t)$, in meters (as it relates to sea level), of the sunglasses after $t$ seconds, is shown in the table.



Height of Sunglasses over Time




$t$		$h(t)$

0		10

1		5.1

2		-9.6

3		-34.1

4		-68.4

5		-112.5

6		-166.4

7		-230.1






During its descent, the pair of sunglasses passed by a climber in the canyon 6 seconds after Stefano dropped them. To the nearest meter, what is the difference in elevation between Stefano and the climber?


A166 meters
B176 meters
C230 meters
D240 meters

Verified Answer :

  • Identify Stefano's initial height as \(h(0) = 10\) meters and the climber's height as \(h(6) = -166.4\) meters.
  • Calculate the elevation difference by finding the absolute difference: \(|10 - (-166.4)| = 176.4\) meters.
  • Round the resulting elevation difference to the nearest meter to get \(176\) meters.
  • Final answer: \(\boxed{176}\)

Explanation

  1. Problem Analysis

We are given a table representing the height of a pair of sunglasses over time as they fall from a cliff. Stefano dropped the glasses at \(t = 0\) seconds. From the table, we identify that the initial height is \(h(0) = 10\) meters. The sunglasses pass by a climber at \(t = 6\) seconds, and the height at that time is \(h(6) = -166.4\) meters. The objective is to find the elevation difference between Stefano and the climber, which is the difference between these two heights.

  1. Calculating the Elevation Difference

To find the vertical distance (the elevation difference) between Stefano and the climber, we calculate the absolute difference between their respective heights:
\[ \text{Elevation Difference} = |h(0) - h(6)| \]
Substituting the values provided:
\[ \text{Elevation Difference} = |10 - (-166.4)| \]
\[ \text{Elevation Difference} = |10 + 166.4| \]
\[ \text{Elevation Difference} = 176.4 \text{ meters} \]

  1. Rounding to the Nearest Meter and Conclusion

The problem asks for the elevation difference rounded to the nearest meter. Since the calculated value is \(176.4\), and the decimal part \(0.4\) is less than \(0.5\), we round down to the nearest whole number.
\[ 176.4 \approx 176 \]
Therefore, the difference in elevation between Stefano and the climber is approximately \(176\) meters.

Examples

Understanding changes in elevation is crucial in fields like geography, civil engineering, and aviation. For instance, mountaineers use altimeters to track their progress, while engineers designing highways must calculate elevation changes to ensure safe gradients. By calculating differences in height relative to a reference point like sea level, we can effectively measure physical distances in vertical space.

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