Vanessa starts driving to her office at 7:48am; she must arrive by 9:00am. She starts her trip driving at an average speed of 32 miles per hour. With half of the time remaining to get to her office, she increases her average speed by 25% for the rest of the trip. She arrives at her office 9 minutes early. How many miles did she travel?

To solve this problem, we need to determine the total distance Vanessa traveled based on her driving speeds and the time she took. Let's break it down step by step.

### Step 1: Time and Speed Details
1. **Time window:** Vanessa starts at 7:48 AM and must arrive by 9:00 AM. Therefore, she has 72 minutes (1 hour and 12 minutes) to reach her office.
2. **Early arrival:** She arrives 9 minutes early, so she actually took \( 72 - 9 = 63 \) minutes to reach her office.

### Step 2: Setup the problem
Let the total distance be \( D \) miles. Let's denote the time she traveled at 32 miles per hour as \( t_1 \) (in hours), and the remaining time as \( t_2 \) (in hours).

- During the first part of the trip:
  \[
  t_1 = t_2
  \]
  She travels at 32 miles per hour.
  
- During the second part of the trip, she increases her speed by 25%, so she travels at:
  \[
  32 \times 1.25 = 40 \text{ miles per hour}
  \]

Since she spent half of her remaining time driving at 40 mph, we have:
\[
t_1 + t_2 = \frac{63}{60} \text{ hours}
\]

### Step 3: Distance Calculation

1. The distance covered in the first half of the journey:
   \[
   \text{Distance}_1 = 32 \times t_1
   \]

2. The distance covered in the second half of the journey:
   \[
   \text{Distance}_2 = 40 \times t_2
   \]

Thus, the total distance \( D \) is:
\[
D = 32 \times t_1 + 40 \times t_2
\]

### Step 4: Time Relationship
Since \( t_2 = t_1 \), we can substitute \( t_2 \) in the distance equation:
\[
D = 32 \times t_1 + 40 \times t_1 = 72 \times t_1
\]

The total time is:
\[
t_1 + t_2 = t_1 + t_1 = 2t_1
\]

Given that:
\[
2t_1 = \frac{63}{60} \text{ hours}
\]
So:
\[
t_1 = \frac{63}{120} = \frac{21}{40} \text{ hours}
\]

### Step 5: Final Distance Calculation
Substituting \( t_1 \) into the distance equation:
\[
D = 72 \times \frac{21}{40} = \frac{72 \times 21}{40} = \frac{1512}{40} = 37.8 \text{ miles}
\]

### Answer
Vanessa traveled a total distance of **37.8 miles**.

Would you like to delve deeper into any of the steps or have any questions? Here are some related questions:

1. How would the result change if she didn't increase her speed?
2. How much time did she spend at 40 mph?
3. What would be the impact on the total distance if she increased her speed by 30% instead?
4. If Vanessa drove the entire distance at 32 mph, what time would she have arrived?
5. What if Vanessa left 10 minutes later but drove faster?

**Tip:** Always double-check units when calculating time, speed, and distance to avoid errors.

Explore how Vanessa's commute problem is solved step-by-step. Find out how she managed to arrive 9 minutes early at her office, traveling at two different speeds. Vanessa drove a total distance of 37.8 miles.

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