A man drove a car at an average rate of speed of 45 miles per hour for the first 3 hours of a 7-hour car trip. If the average rate speed for the entire trip was 53 miles per hour, what was the average rate of speed in miles per hour for the remaining part of the trip?

Options

A. `50`
B. `55`
C. `57`
D. `59`

Answer

To find the average rate of speed for the remaining part of the trip, we can follow these steps:

### Step 1: Calculate the distance traveled in the first part of the trip
The man drove for 3 hours at a speed of 45 miles per hour.

\[ \text{Distance} = \text{Speed} \times \text{Time} = 45 \text{ miles/hour} \times 3 \text{ hours} = 135 \text{ miles} \]

### Step 2: Calculate the total distance for the entire trip
The average speed for the entire trip is given as 53 miles per hour, and the total time for the trip is 7 hours.

\[ \text{Total Distance} = \text{Average Speed} \times \text{Total Time} = 53 \text{ miles/hour} \times 7 \text{ hours} = 371 \text{ miles} \]

### Step 3: Calculate the distance remaining for the second part of the trip
To find the distance remaining after the first part of the trip, we subtract the distance traveled in the first part from the total distance.

\[ \text{Remaining Distance} = \text{Total Distance} - \text{Distance Traveled} = 371 \text{ miles} - 135 \text{ miles} = 236 \text{ miles} \]

### Step 4: Calculate the time remaining for the second part of the trip
The remaining time for the trip is the total time minus the time spent in the first part.

\[ \text{Remaining Time} = \text{Total Time} - \text{Time Spent} = 7 \text{ hours} - 3 \text{ hours} = 4 \text{ hours} \]

### Step 5: Calculate the average speed for the remaining part of the trip
Now, we can find the average speed for the remaining part of the trip using the remaining distance and remaining time.

\[ \text{Average Speed for Remaining Trip} = \frac{\text{Remaining Distance}}{\text{Remaining Time}} = \frac{236 \text{ miles}}{4 \text{ hours}} = 59 \text{ miles/hour} \]

Thus, the average rate of speed for the remaining part of the trip is **59 miles per hour**.