Question

4.5.7 check your understanding
jacob is hiking in wyoming and wants to determine the temperature change based on his elevation. the temperature, t, in degrees fahrenheit for jacobs altitude, h, in meters above sea level is given by $h(t) = -dfrac{2}{3}t + 50$ as shown in the graph.
graph: altitude (meters) on y - axis from 0 to 50, temperature (°f) on x - axis from 0 to 80, line starting at (0,50) and ending at (75,0)
what is the domain of the function in the context?
$\square \leq t \leq \square$

Explanation:

Step1: Identify the graph's t - intercepts

The function is \( h(t)=-\frac{2}{3}t + 50 \). From the graph, we can see the temperature \( t \) (the independent variable) starts at the y - intercept of the graph (when \( h = 0 \)) and goes to the initial temperature when \( h = 50 \) (the y - intercept of the function \( h(t) \)). First, find when \( h(t)=0 \):
Set \( h(t) = 0 \), so \( 0=-\frac{2}{3}t + 50 \).
Add \( \frac{2}{3}t \) to both sides: \( \frac{2}{3}t=50 \).
Multiply both sides by \( \frac{3}{2} \): \( t = 50\times\frac{3}{2}=75 \).

Step2: Determine the domain bounds

From the graph, when \( h = 50 \) (the initial altitude), \( t = 0 \) (by looking at the y - intercept of the graph where the line starts at \( (0,50) \) in terms of \( (t,h) \) with \( t \) on the x - axis and \( h \) on the y - axis). And when \( h = 0 \), \( t = 75 \) (from the x - intercept we calculated). Since \( t \) represents temperature and in the context of the hike, the temperature ranges from \( t = 0 \) (when altitude is 50 meters) to \( t = 75 \) (when altitude is 0 meters). So the domain of \( t \) in the context is from \( 0 \) to \( 75 \).

Answer:

\( 0\leq t\leq75 \)