A bus is moving with a uniform speed travelling a certain distance in a certain time. The speed of the bus is directly proportional to the distance travelled and inversely proportional to the square root of time. It travels is 60 km in 4 hours at a speed of 40 km/h. Then find how much distance will travel in 9 hours at a speed of 44 km/h?

Answer

To solve the problem step by step, we will use the relationship given in the question about speed, distance, and time.

### Step 1: Understand the relationship
The speed \( S \) of the bus is directly proportional to the distance \( D \) travelled and inversely proportional to the square root of time \( T \). This can be expressed as:

\[ S = k \cdot \frac{D}{\sqrt{T}} \]

where \( k \) is a constant.

### Step 2: Set up the equation for the first scenario
From the problem, we know that:
- Distance \( D_1 = 60 \) km
- Time \( T_1 = 4 \) hours
- Speed \( S_1 = 40 \) km/h

Substituting these values into the equation gives:

\[ 40 = k \cdot \frac{60}{\sqrt{4}} \]

### Step 3: Calculate \( k \)
Now, we can simplify the equation:

\[ 40 = k \cdot \frac{60}{2} \quad (\text{since } \sqrt{4} = 2) \]

This simplifies to:

\[ 40 = k \cdot 30 \]

Now, solve for \( k \):

\[ k = \frac{40}{30} = \frac{4}{3} \]

### Step 4: Set up the equation for the second scenario
Now we need to find the distance \( D_2 \) when:
- Speed \( S_2 = 44 \) km/h
- Time \( T_2 = 9 \) hours

Using the same relationship, we have:

\[ 44 = \frac{4}{3} \cdot \frac{D_2}{\sqrt{9}} \]

### Step 5: Simplify the equation
Since \( \sqrt{9} = 3 \), we can substitute that into the equation:

\[ 44 = \frac{4}{3} \cdot \frac{D_2}{3} \]

This simplifies to:

\[ 44 = \frac{4D_2}{9} \]

### Step 6: Solve for \( D_2 \)
Now, multiply both sides by 9 to eliminate the fraction:

\[ 44 \cdot 9 = 4D_2 \]

Calculating \( 44 \cdot 9 \):

\[ 396 = 4D_2 \]

Now, divide both sides by 4:

\[ D_2 = \frac{396}{4} = 99 \text{ km} \]

### Final Answer
The distance the bus will travel in 9 hours at a speed of 44 km/h is **99 km**.

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