A car goes one kilometre at 30km/h and then goes another kilometre to 40 km/h. The average speed (in km/h) of the car for 2 km is

Options

A. `33(3)/(7)`
B. `33(5)/(7)`
C. 35
D. `34(2)/(7)`

Answer

To find the average speed of the car for the total distance of 2 kilometers, we will follow these steps:

### Step 1: Understand the formula for average speed
The average speed is calculated using the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]

### Step 2: Calculate the total distance
In this problem, the car travels a total distance of:
\[ \text{Total Distance} = 1 \text{ km} + 1 \text{ km} = 2 \text{ km} \]

### Step 3: Calculate the time taken for each segment of the journey
1. For the first kilometer at a speed of 30 km/h:
\[ \text{Time}_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{1 \text{ km}}{30 \text{ km/h}} = \frac{1}{30} \text{ hours} \]

2. For the second kilometer at a speed of 40 km/h:
\[ \text{Time}_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{1 \text{ km}}{40 \text{ km/h}} = \frac{1}{40} \text{ hours} \]

### Step 4: Calculate the total time taken
Now, we need to add the time taken for both segments:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = \frac{1}{30} + \frac{1}{40} \]

To add these fractions, we need a common denominator, which is 120:
\[ \text{Total Time} = \frac{4}{120} + \frac{3}{120} = \frac{7}{120} \text{ hours} \]

### Step 5: Calculate the average speed
Now we can substitute the total distance and total time into the average speed formula:
\[ \text{Average Speed} = \frac{2 \text{ km}}{\frac{7}{120} \text{ hours}} = 2 \times \frac{120}{7} = \frac{240}{7} \text{ km/h} \]

### Step 6: Convert the fraction to a mixed number
To convert \(\frac{240}{7}\) into a mixed number:
\[ 240 \div 7 = 34 \quad \text{(with a remainder of 2)} \]
Thus, \(\frac{240}{7} = 34 \frac{2}{7} \text{ km/h}\)

### Final Answer
The average speed of the car for the 2 km journey is:
\[ \text{Average Speed} = 34 \frac{2}{7} \text{ km/h} \]

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