The problem describes the following:

• In 1920, the record for a certain race was 45.3 seconds.
• In 1980, the record was 43.5 seconds.
• The function $R(t)$ represents the record in seconds, where $t$ is the number of years since 1920.

Now, let's break this down step by step.

Part (a) Find a linear function that fits the data.

We need to find a linear function in the form:

$R(t) = mt + b$

where:

• $t$ is the number of years since 1920, so $t = 0$ in 1920.
• $R(0) = 45.3$ (the record in 1920).
• $R(60) = 43.5$ (since 1980 is 60 years after 1920).

We can use these two points to calculate the slope $m$ and then determine $b$.

The slope $m$ is given by:

$m = \frac{R(60) - R(0)}{60 - 0} = \frac{43.5 - 45.3}{60} = \frac{-1.8}{60} = -0.03$

Thus, the linear equation so far is:

$R(t) = -0.03t + b$

Using the point $R(0) = 45.3$ to find $b$:

$45.3 = -0.03(0) + b \implies b = 45.3$

So, the linear function is:

$R(t) = -0.03t + 45.3$

Part (b) Use the function to predict the record in 2003 and 2006.

For 2003:

• $t = 2003 - 1920 = 83$

$R(83) = -0.03(83) + 45.3 = -2.49 + 45.3 = 42.81 \, \text{seconds}$

For 2006:

• $t = 2006 - 1920 = 86$

$R(86) = -0.03(86) + 45.3 = -2.58 + 45.3 = 42.72 \, \text{seconds}$

Part (c) Find the year when the record will be 42.57 seconds.

We need to find $t$ such that $R(t) = 42.57$. Using the linear equation:

$42.57 = -0.03t + 45.3$

Solving for $t$:

$42.57 - 45.3 = -0.03t$ $-2.73 = -0.03t$ $t = \frac{-2.73}{-0.03} = 91$

Thus, $t = 91$, which corresponds to the year:

$1920 + 91 = 2011$

Final Answers:

• (a) The linear function is $R(t) = -0.03t + 45.3$.
• (b) The predicted record is 42.81 seconds in 2003, and 42.72 seconds in 2006.
• (c) The record will be 42.57 seconds in the year 2011.

Would you like further details or explanations on any of these parts?

Here are 5 related questions to extend your understanding:

1. What happens to the record as time progresses according to this model?
2. How would the model change if there was a nonlinear improvement in the record?
3. Can you think of real-world factors that might invalidate this linear model in the long run?
4. How would you adjust the function if the record improved faster in the years closer to 1920?
5. What other types of models could be used to predict performance improvements over time?

Tip: Always check the units of time and ensure the variable definitions (e.g., $t =$ years since a specific date) match consistently in your equation.