Let's analyze the problem together.

Problem Details:

• Rafael is driving to Seattle, and the distance to his destination is a linear function of his driving time.
• At 34 minutes, Rafael is 68 miles away from his destination.
• At 52 minutes, he is 55.4 miles away.
• We need to find how many miles he will have left after 62 minutes of driving.

Approach:

1. Find the rate of change (slope): Since the distance decreases linearly with time, we can compute the slope using the two points: $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{55.4 - 68}{52 - 34}$ This will give us the rate at which Rafael is reducing his distance per minute.

2. Equation of the line: The equation follows the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept (distance at 0 minutes).

   Use one of the points (e.g., $(34, 68)$) to solve for $b$.

3. Predict the distance at 62 minutes: Once we have the equation of the line, plug in $x = 62$ (minutes) to find the remaining distance.

Let's calculate step by step.

1. Calculate the slope:

$\text{slope} = \frac{55.4 - 68}{52 - 34} = \frac{-12.6}{18} = -0.7$ So, Rafael is decreasing his distance by 0.7 miles per minute.

2. Find the equation of the line:

Using the point $(34, 68)$ in the slope-intercept form $y = mx + b$: $68 = -0.7(34) + b$ $68 = -23.8 + b$ $b = 68 + 23.8 = 91.8$ Thus, the equation of the line is: $y = -0.7x + 91.8$

3. Find the distance after 62 minutes:

Substitute $x = 62$ into the equation: $y = -0.7(62) + 91.8$ $y = -43.4 + 91.8 = 48.4$ So, after 62 minutes of driving, Rafael will have 48.4 miles left to his destination.

Would you like a deeper explanation on any step?

Related Questions:

1. How would the equation change if Rafael's initial distance was different?
2. What if the rate of decrease in distance was nonlinear? How would that affect the model?
3. How can you interpret the y-intercept in real-world terms?
4. How would the time needed to reach the destination be determined using this equation?
5. What happens if the slope is positive? How does that impact the distance over time?

Tip:

Always check if the problem involves a linear relationship when given two points. This can guide your approach to find the equation of the line and make future predictions.