A school is straveling to calgary. they can leave tonight byb bus or save 2 hours by leaving tomorrow morninng and using the express train which travels 20km/h fastere than the bus . if the distance between the school and calgary is s800km determine how long it will take to travel by bus

Let's solve the problem step by step.

### Given:
- The distance between the school and Calgary is \( 800 \) km.
- The express train travels \( 20 \) km/h faster than the bus.
- The school can save \( 2 \) hours by taking the express train instead of the bus.

### To Find:
The time it will take to travel by bus.

### Let:
- \( v \) be the speed of the bus in km/h.
- \( v + 20 \) be the speed of the express train in km/h.

### Step 1: Express the time taken for each mode of transportation
- Time taken by bus: \( \text{Time}_\text{bus} = \frac{800}{v} \)
- Time taken by train: \( \text{Time}_\text{train} = \frac{800}{v + 20} \)

### Step 2: Set up the equation based on the time saved
Since the train saves 2 hours compared to the bus, we can write the equation as:
\[
\frac{800}{v} - \frac{800}{v + 20} = 2
\]

### Step 3: Solve the equation

Let's solve this equation for \( v \).

\[
\frac{800(v + 20) - 800v}{v(v + 20)} = 2
\]
\[
\frac{800v + 16000 - 800v}{v(v + 20)} = 2
\]
\[
\frac{16000}{v(v + 20)} = 2
\]
\[
16000 = 2v(v + 20)
\]
\[
8000 = v^2 + 20v
\]
\[
v^2 + 20v - 8000 = 0
\]

This is a quadratic equation. We can solve for \( v \) using the quadratic formula:

\[
v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 20 \), and \( c = -8000 \).

Let's calculate the value of \( v \).The two solutions for the speed \( v \) of the bus are \( 80 \) km/h and \( -100 \) km/h. Since speed cannot be negative, we take \( v = 80 \) km/h as the speed of the bus.

### Step 4: Calculate the time taken by the bus
The time taken by the bus to travel 800 km is:
\[
\text{Time}_\text{bus} = \frac{800}{80} = 10 \text{ hours}
\]

So, it will take 10 hours to travel by bus.

Would you like further details or have any questions? 

Here are some related questions:
1. How would the time change if the distance were 900 km instead of 800 km?
2. What would be the time saved if the bus traveled at 100 km/h instead of 80 km/h?
3. How long would it take if the train traveled 30 km/h faster than the bus?
4. What is the total time spent traveling if the bus makes a 1-hour stop halfway?
5. How does the speed difference impact the overall travel time?

**Tip:** When solving quadratic equations, always consider the physical context to choose the correct root.

Learn how to calculate travel time between a school and Calgary using a bus and an express train. Solve the problem step-by-step, determining the time taken by each mode of transportation and understanding the impact of speed differences on travel time. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation