Jess is driving to Pittsburgh. Suppose that the distance to her destination (in miles) is a linear function of her
total driving time (in minutes). Jess is 49 miles away from her destination after 45 minutes of driving, and
she is 37 miles away from her destination after 61 minutes of driving. Find the function and determine how
many miles away from her destination she will be after 73 minutes of driving?

To solve this problem, we need to find the linear function that represents Jess's distance from her destination as a function of time, and then use that function to determine how far she will be from her destination after 73 minutes.

### Step 1: Define the linear function.
Let \( d(t) \) be the distance (in miles) from her destination as a function of driving time \( t \) (in minutes). Since it's a linear function, we can express it as:

\[
d(t) = mt + b
\]

where \( m \) is the slope (rate of change in distance per minute) and \( b \) is the y-intercept (initial distance from her destination).

### Step 2: Use the given points to create equations.
We know the following:
- After 45 minutes of driving, Jess is 49 miles away. So, \( d(45) = 49 \).
- After 61 minutes of driving, Jess is 37 miles away. So, \( d(61) = 37 \).

These two points allow us to set up the following system of equations:

1. \( 49 = 45m + b \)
2. \( 37 = 61m + b \)

### Step 3: Solve for \( m \) and \( b \).
We start by subtracting the first equation from the second to eliminate \( b \):

\[
(37 - 49) = (61m + b) - (45m + b)
\]

Simplifying:

\[
-12 = 16m
\]

Solving for \( m \):

\[
m = \frac{-12}{16} = -\frac{3}{4}
\]

Now, substitute \( m = -\frac{3}{4} \) into one of the original equations to find \( b \). Using the first equation:

\[
49 = 45\left(-\frac{3}{4}\right) + b
\]

Simplifying:

\[
49 = -\frac{135}{4} + b
\]

Convert 49 to a fraction with denominator 4:

\[
49 = \frac{196}{4}
\]

Now solve for \( b \):

\[
\frac{196}{4} = \frac{-135}{4} + b
\]

\[
b = \frac{196}{4} + \frac{135}{4} = \frac{331}{4}
\]

So, \( b = \frac{331}{4} = 82.75 \).

### Step 4: Write the linear function.
The linear function for the distance is:

\[
d(t) = -\frac{3}{4}t + 82.75
\]

### Step 5: Determine the distance after 73 minutes.
To find how many miles Jess is away from her destination after 73 minutes, substitute \( t = 73 \) into the function:

\[
d(73) = -\frac{3}{4}(73) + 82.75
\]

Simplifying:

\[
d(73) = -\frac{219}{4} + 82.75 = -54.75 + 82.75 = 28
\]

So, after 73 minutes of driving, Jess will be 28 miles away from her destination.

### Final Answer:
Jess will be 28 miles away from her destination after 73 minutes of driving.

---

Would you like more details on any step or have any questions?

Here are 5 related questions to explore:

1. How do we calculate the slope of a linear function from two points?
2. What does the slope represent in this driving problem?
3. How would the equation change if Jess started driving closer to her destination?
4. Can you find out how long it would take for Jess to reach her destination (i.e., when \( d(t) = 0 \))?
5. How would you describe the relationship between distance and time if Jess were driving faster?

**Tip:** When working with linear functions, remember that the slope \( m \) represents the rate of change in the dependent variable per unit of the independent variable.

Jess is driving to Pittsburgh. Suppose that the distance to her destination (in miles) is a linear function of her total driving time (in minutes). Jess is 49 miles away from her destination after 45 minutes of driving, and she is 37 miles away from her destination after 61 minutes of driving. Find the function and determine how many miles away from her destination she will be after 73 minutes of driving.

Discover how to find the linear function representing Jess's distance from Pittsburgh over time. Learn to calculate the slope and intercept to determine how far Jess will be from her destination after 73 minutes of driving. Suitable for Grades 8-10, this problem focuses on algebraic concepts of linear functions and slope-intercept form.

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