a person drives witha constant speed of 89.5 km/hour except for a 22 minute rest stop. if the person's average speed is 77.8 km/hour, how far is the trip?

To solve this problem, we'll use the relationship between distance, speed, and time, which is given by the formula:

\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

### Step 1: Express the average speed formula
The average speed \( \text{v}_{\text{avg}} \) is given by:

\[
\text{v}_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}}
\]

Let's denote:
- \( D \) as the total distance of the trip.
- \( T \) as the driving time without the rest stop.
- \( T_{\text{rest}} \) as the duration of the rest stop in hours.
- \( T_{\text{total}} \) as the total time, which is \( T + T_{\text{rest}} \).

### Step 2: Convert the rest stop time to hours
The rest stop is 22 minutes, which needs to be converted to hours:

\[
T_{\text{rest}} = \frac{22}{60} \, \text{hours} = \frac{11}{30} \, \text{hours}
\]

### Step 3: Set up the equations
We know:
- The driving speed is \( 89.5 \, \text{km/h} \).
- The average speed is \( 77.8 \, \text{km/h} \).

The total distance \( D \) is covered by driving for \( T \) hours at 89.5 km/h:

\[
D = 89.5 \times T
\]

The average speed is:

\[
77.8 = \frac{D}{T + \frac{11}{30}}
\]

### Step 4: Solve for the total distance \( D \)
Substitute \( D = 89.5 \times T \) into the average speed formula:

\[
77.8 = \frac{89.5 \times T}{T + \frac{11}{30}}
\]

Now, solve for \( T \) first:

\[
77.8 \times \left(T + \frac{11}{30}\right) = 89.5 \times T
\]

\[
77.8T + \frac{77.8 \times 11}{30} = 89.5T
\]

\[
\frac{77.8 \times 11}{30} = 89.5T - 77.8T
\]

\[
\frac{77.8 \times 11}{30} = 11.7T
\]

\[
T = \frac{77.8 \times 11}{11.7 \times 30}
\]

### Step 5: Calculate \( T \) and \( D \)
Now compute \( T \) and then \( D \):

\[
T \approx \frac{855.8}{351} \approx 2.44 \, \text{hours}
\]

Finally, calculate the total distance:

\[
D = 89.5 \times 2.44 \approx 218.28 \, \text{km}
\]

So, the total distance of the trip is approximately **218.3 km**.

Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. What is the effect of increasing the rest stop duration on the total distance?
2. How would the total distance change if the driving speed was 10 km/h slower?
3. Can the average speed ever be higher than the driving speed?
4. What if there were two rest stops of different durations?
5. How would the calculations change if the speed varied during the trip?

### Tip:
When solving problems involving average speed, remember that it accounts for the entire trip, including any stops or changes in speed, which is why it's always lower than the driving speed if there are any stops.

Learn how to calculate the total distance of a trip using concepts of speed, distance, and time. This problem involves determining distance covered at constant and average speeds, including a rest stop duration. Suitable for high school students studying mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation