To solve the problem, let's determine the rate of change and the peak elevation step-by-step.
1. Identify the given data points:
We are given two points:
[tex]\[
(x_1, y_1) = (2, 1400) \quad \text{and} \quad (x_2, y_2) = (4, 1000)
\][/tex]
2. Calculate the rate of change (slope):
The rate of change (slope) [tex]\( m \)[/tex] can be computed using the formula for the slope of a line between two points:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values:
[tex]\[
m = \frac{1000 - 1400}{4 - 2} = \frac{-400}{2} = -200 \text{ feet per hour}
\][/tex]
3. Determine the linear equation:
The general form of the linear equation for this situation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept (elevation at the peak when hours hiked is zero).
4. Find the y-intercept [tex]\( b \)[/tex]:
We will use one of the provided points and the slope to calculate [tex]\( b \)[/tex]. Using the point [tex]\( (2, 1400) \)[/tex], we substitute into the linear equation [tex]\( y = mx + b \)[/tex]:
[tex]\[
1400 = (-200)(2) + b
\][/tex]
[tex]\[
1400 = -400 + b
\][/tex]
[tex]\[
b = 1400 + 400 = 1800
\][/tex]
5. Conclusion:
The rate of change is [tex]\( -200 \)[/tex] feet per hour, and the peak is at an elevation of [tex]\( 1800 \)[/tex] feet.
Hence, in a function for this situation, the rate of change is [tex]\(-200\)[/tex] feet per hour and the peak is at an elevation of [tex]\( 1800 \)[/tex] feet.
