The cost of fuel running the engine of an army tank is proportional to the square of the speed and Rs. 64 per hour for a Speed of 16 kmph. Other costs amount to Rs. 400 per hour. The tank has to make a journey of 400 km at a constant speed. The total cost for the journey at this most economical speed is :

Options

A. Rs 6000
B. Rs 8000
C. Rs 10000
D. Rs 11000

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the relationship between cost and speed
The cost of fuel is proportional to the square of the speed. We can express this relationship mathematically as:
\[ C = k \cdot x^2 \]
where \( C \) is the cost, \( x \) is the speed, and \( k \) is the constant of proportionality.

### Step 2: Find the value of \( k \)
We know that at a speed of 16 km/h, the cost is Rs. 64 per hour. We can substitute these values into the equation to find \( k \):
\[ 64 = k \cdot (16)^2 \]
\[ 64 = k \cdot 256 \]
To find \( k \), we divide both sides by 256:
\[ k = \frac{64}{256} = \frac{1}{4} \]

### Step 3: Write the total cost function
Now that we have the value of \( k \), we can write the total cost function for the tank:
\[ C = \frac{1}{4} x^2 + 400 \]
Here, Rs. 400 is the fixed cost per hour.

### Step 4: Determine the time taken for the journey
The tank has to make a journey of 400 km at a constant speed \( x \). The time taken \( t \) to complete the journey can be expressed as:
\[ t = \frac{400}{x} \]

### Step 5: Express the total cost in terms of speed
The total cost for the journey can be expressed as the cost per hour multiplied by the time taken:
\[ \text{Total Cost} = \left( \frac{1}{4} x^2 + 400 \right) \cdot t \]
Substituting \( t \):
\[ \text{Total Cost} = \left( \frac{1}{4} x^2 + 400 \right) \cdot \frac{400}{x} \]

### Step 6: Simplify the total cost expression
Now we simplify the total cost expression:
\[ \text{Total Cost} = \left( \frac{1}{4} x^2 + 400 \right) \cdot \frac{400}{x} \]
\[ = \frac{400}{x} \cdot \frac{1}{4} x^2 + \frac{400}{x} \cdot 400 \]
\[ = 100 x + \frac{160000}{x} \]

### Step 7: Find the most economical speed
To minimize the total cost, we can differentiate the total cost function with respect to \( x \) and set the derivative to zero:
Let \( C(x) = 100x + \frac{160000}{x} \)
Differentiating:
\[ C'(x) = 100 - \frac{160000}{x^2} \]
Setting \( C'(x) = 0 \):
\[ 100 = \frac{160000}{x^2} \]
\[ 100x^2 = 160000 \]
\[ x^2 = 1600 \]
\[ x = 40 \text{ km/h} \]

### Step 8: Calculate the total cost at the most economical speed
Now, we substitute \( x = 40 \) back into the total cost function:
\[ \text{Total Cost} = 100(40) + \frac{160000}{40} \]
\[ = 4000 + 4000 \]
\[ = 8000 \]

### Final Answer
The total cost for the journey at the most economical speed is Rs. 8000.

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