To find the slope of Francis's hike and the equation representing his elevation as a function of time, we can use the information provided:
- After 1 hour, the elevation is 100 feet.
- After 5 hours, the elevation is 360 feet.
Step 1: Calculate the slope
The slope \( m \) can be calculated using the formula:
\[ m = \frac{{\text{change in elevation}}}{{\text{change in time}}} \]
Let's find the change in elevation and change in time:
- Change in elevation: \( 360 \text{ feet} - 100 \text{ feet} = 260 \text{ feet} \)
- Change in time: \( 5 \text{ hours} - 1 \text{ hour} = 4 \text{ hours} \)
Now, we can plug these values into the slope formula:
\[ m = \frac{{260 \text{ feet}}}{{4 \text{ hours}}} = 65 \text{ feet per hour} \]
Step 2: Write the equation
We can use the point-slope form of the equation for a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( m \) is the slope,
- \( (x_1, y_1) \) is a point on the line.
We can use the point \( (1, 100) \):
\[ y - 100 = 65(x - 1) \]
Step 3: Simplify the equation
Now, let's simplify the equation:
\[ y - 100 = 65x - 65 \] \[ y = 65x + 35 \]
Conclusion
The slope of Francis's hike is \( 65 \text{ feet per hour} \), and the equation representing his elevation in feet as a function of time in hours is:
\[ y = 65x + 35 \]
where \( y \) is the elevation and \( x \) is the time in hours.