To find a linear model for this situation, we assume a linear relationship between the price of the widget and time. Let's define the following variables:

• Let $P(t)$ be the price of the widget at time $t$.
• Let $t$ represent the number of years since the initial price of $610.

We know:

• At $t = 0$, $P(0) = 610$.
• At $t = 5$, $P(5) = 505$.

Step 1: Find the slope $m$

The slope $m$ of a linear function is calculated using the formula: $m = \frac{P(5) - P(0)}{5 - 0}$ Substitute the values for $P(5)$ and $P(0)$: $m = \frac{505 - 610}{5} = \frac{-105}{5} = -21$

Step 2: Write the equation of the line

The general form of a linear equation is: $P(t) = mt + b$ where:

• $m = -21$ (slope), and
• $b$ is the y-intercept, or initial price.

We already know $P(0) = 610$, so we can find $b$ directly: $P(0) = -21(0) + b \Rightarrow b = 610$

Thus, the linear model is: $P(t) = -21t + 610$

Final Answer:

The linear model for the price $P(t)$ at time $t$ is: $\boxed{P(t) = -21t + 610}$

This model describes the price of the widget as it decreases over time. For each year that passes, the price decreases by $21.

Would you like further details on how this formula is used? Here are some related questions to explore:

1. What would the price be after 10 years using this linear model?
2. How long would it take for the price to drop to $400?
3. What is the interpretation of the slope in this context?
4. If the trend continued, when would the price be zero?
5. Can we assume this model indefinitely, or might there be limitations to this linear trend?

Tip: Linear models are useful for estimating trends but are generally best for short-term predictions, especially if real-world factors may alter the long-term behavior.