Question

emma is mountain climbing and starts the day at 182.5 meters above sea level. each hour she descends 36.5 meters. the table shows the function relating emmas height to the number of hours she is mountain climbing. find the intercepts and describe what they mean in the context of the situation.
the x - intercept is __ and it means __.
the y - intercept is __ and it means __.

Explanation:

Step1: Find the y - intercept

The y - intercept is the value of \(y\) when \(x = 0\). From the table, when \(x=0\), \(y = 182.5\). In the context of the problem, this means Emma's initial height (at \(0\) hours) is \(182.5\) meters above sea level.

Step2: Find the x - intercept

We know the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((0,182.5)\) and \((5,0)\), \(m=\frac{0 - 182.5}{5-0}=- 36.5\).
The equation of the line is \(y=mx + b\), where \(b = 182.5\) and \(m=-36.5\), so \(y=-36.5x + 182.5\).
To find the x - intercept, set \(y = 0\):

$$
LATEXBLOCK0
$$

In the context of the problem, this means it takes Emma \(5\) hours to reach sea level (\(y = 0\) meters).

Answer:

The y - intercept is \(182.5\) and it means Emma's initial height is \(182.5\) meters. The x - intercept is \(5\) and it means it takes Emma \(5\) hours to reach sea level.