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The image contains a set of questions regarding a function that models the distance \( d(t) \) in miles based on the number of hours \( t \) spent driving from Los Angeles, California, to Gilbert, Arizona. The function provided is:

\[
d(t) = 400 - 66t
\]

### Let's break down and address the questions:

**a. Evaluate \( d(2.3) \) and explain its meaning in the problem context.**

To evaluate \( d(2.3) \), substitute \( t = 2.3 \) into the function:

\[
d(2.3) = 400 - 66(2.3)
\]
\[
d(2.3) = 400 - 151.8
\]
\[
d(2.3) = 248.2
\]

**Meaning:** After 2.3 hours of driving, the distance remaining to reach Gilbert, Arizona, is 248.2 miles.

---

**b. Solve \( d(t) = 100 \) and explain its meaning in the problem context.**

Set \( d(t) = 100 \) and solve for \( t \):

\[
400 - 66t = 100
\]
\[
66t = 400 - 100
\]
\[
66t = 300
\]
\[
t = \frac{300}{66} \approx 4.545
\]

**Meaning:** It takes approximately 4.55 hours of driving to reduce the remaining distance to 100 miles from Gilbert, Arizona.

---

**c. Determine the vertical intercept(s) of the function and explain the meaning in the problem context.**

The vertical intercept occurs when \( t = 0 \):

\[
d(0) = 400 - 66(0) = 400
\]

**Meaning:** When \( t = 0 \), the initial distance from Gilbert, Arizona, is 400 miles. This is the starting point of the trip.

---

**d. Determine the horizontal intercept(s) of the function and explain the meaning in the problem context.**

The horizontal intercept occurs when \( d(t) = 0 \). Set the equation to 0 and solve for \( t \):

\[
400 - 66t = 0
\]
\[
66t = 400
\]
\[
t = \frac{400}{66} \approx 6.06
\]

**Meaning:** It takes approximately 6.06 hours to complete the journey, at which point the distance to Gilbert, Arizona, will be 0 miles.

---

Would you like more details or have any questions about the solutions? Here are some follow-up questions to expand on this topic:

1. How would you interpret the rate of change in the function \( d(t) = 400 - 66t \)?
2. What would happen to the function if the speed of the car were to change? How would that reflect in the equation?
3. Could you use this function to determine the average speed of the vehicle? How?
4. How does the concept of intercepts help in understanding the journey's progress?
5. What other factors (e.g., traffic, stops) could affect the real-world accuracy of this model?

**Tip:** Always check the units when solving word problems to ensure consistency, especially when interpreting rates and distances.

As you drive from Los Angeles, California, back to Gilbert, Arizona, your distance from home d(t) in miles is a function of the number of hours t you have been driving. This function is d(t) = 400 - 66t. Show all work.

Learn how to evaluate and solve the linear distance-time function d(t) = 400 - 66t. This problem focuses on key algebraic concepts, including function evaluation, solving for variables, and finding intercepts. Suitable for students in Grades 9-11.

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