assuming that the petrol burnt in the motor boat varies as the cube of its velocity,the most economical speed of the boat which going against a speed of 8kmph is

To determine the most economical speed of a motorboat that burns petrol in relation to the cube of its velocity, especially when going against a current of 8 km/h, we can apply some principles of physics and calculus. The relationship between speed and fuel consumption is crucial for optimizing performance and efficiency.

Understanding the Relationship

When we say that the petrol burnt varies as the cube of the velocity, we can express this mathematically. Let’s denote:

• v = speed of the boat (in km/h)
• c = constant of proportionality
• F = fuel consumption

We can write the fuel consumption as:

F = c * v³

Considering the Current

When the boat is moving against a current of 8 km/h, the effective speed of the boat relative to the water is:

v_eff = v - 8

Thus, the fuel consumption while moving against the current becomes:

F = c * (v - 8)³

Finding the Minimum Fuel Consumption

To find the most economical speed, we need to minimize the fuel consumption function. This involves taking the derivative of the fuel consumption function with respect to the effective speed and setting it to zero.

Calculating the Derivative

First, we can expand the cubic term:

F = c * (v³ - 24v² + 192v - 512)

Next, we differentiate F with respect to v:

dF/dv = c * (3v² - 48v + 192)

Setting this derivative equal to zero to find critical points:

3v² - 48v + 192 = 0

Solving the Quadratic Equation

We can simplify this equation:

v² - 16v + 64 = 0

Using the quadratic formula, v = [16 ± √(16² - 4 * 1 * 64)] / (2 * 1), we find:

v = [16 ± √(256 - 256)] / 2

This simplifies to:

v = 16 / 2 = 8 km/h

Interpreting the Result

Since the boat is moving against a current of 8 km/h, the effective speed of the boat becomes:

v_eff = 8 - 8 = 0 km/h

This indicates that the most economical speed of the boat, when considering the current, is actually at the point where it is not moving against the current at all. In practical terms, this means that to achieve the best fuel efficiency, the boat should ideally not attempt to move against the current at all.

Conclusion

In summary, when navigating against a current of 8 km/h, the most economical speed for the motorboat is effectively zero relative to the water. This highlights the importance of considering environmental factors like currents when planning travel to optimize fuel consumption.