A boy runs for 10 minutes at a uniform speed of 9 km `h^(-1)`. At what speed should he run for the next 20 minutes so that the average speed becomes 12 km `h^(-1)` ?

Options

A. `13.5 Km h^(-1)`
B. `10.2 km h^(-1)`
C. `8.2 km h^(-1)`
D. `7.72 km h^(-1)`

Explanation

To solve the problem step by step, we need to find the speed at which the boy should run for the next 20 minutes to achieve an average speed of 12 km/h over the total time of 30 minutes.

### Step 1: Convert the time from minutes to hours
The boy runs for 10 minutes initially and then for 20 minutes. We need to convert these times into hours for our calculations.

- 10 minutes = \( \frac{10}{60} = \frac{1}{6} \) hours
- 20 minutes = \( \frac{20}{60} = \frac{1}{3} \) hours

### Step 2: Calculate the distance covered in the first part
Using the formula for distance, \( \text{Distance} = \text{Speed} \times \text{Time} \), we can find the distance covered in the first 10 minutes at a speed of 9 km/h.

- Distance = Speed × Time
- Distance = \( 9 \, \text{km/h} \times \frac{1}{6} \, \text{h} = \frac{9}{6} \, \text{km} = 1.5 \, \text{km} \)

### Step 3: Set up the equation for average speed
The average speed is defined as the total distance divided by the total time. We want the average speed to be 12 km/h over the total time of 30 minutes (or \( \frac{1}{2} \) hours).

Let \( V \) be the speed for the next 20 minutes (or \( \frac{1}{3} \) hours). The total distance covered will be the distance from the first part plus the distance from the second part.

- Total Distance = Distance from first part + Distance from second part
- Total Distance = \( 1.5 \, \text{km} + V \times \frac{1}{3} \)

The total time is \( \frac{1}{2} \) hours.

Using the average speed formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = 12 \, \text{km/h} \]

Substituting the values:
\[ 12 = \frac{1.5 + V \times \frac{1}{3}}{\frac{1}{2}} \]

### Step 4: Solve for V
To eliminate the fraction, multiply both sides by \( \frac{1}{2} \):
\[ 12 \times \frac{1}{2} = 1.5 + V \times \frac{1}{3} \]
\[ 6 = 1.5 + \frac{V}{3} \]

Now, isolate \( \frac{V}{3} \):
\[ 6 - 1.5 = \frac{V}{3} \]
\[ 4.5 = \frac{V}{3} \]

Multiply both sides by 3 to solve for \( V \):
\[ V = 4.5 \times 3 = 13.5 \, \text{km/h} \]

### Final Answer
The speed at which the boy should run for the next 20 minutes is **13.5 km/h**.

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