Problem 34 - Algebra and Trigonometry

Question

Power Needed to Propel a Boat The power \(P\) (measured in horsepower, hp) needed to propel a boat is directly proportional to the cube of the speed \(s\) . An \(80-\) hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots.

Step-by-Step Solution

Answer

The power needed is 270 hp.

Step 1: Understand the Relationship

The problem states that the power \(P\) is directly proportional to the cube of the speed \(s\). This relationship can be written as \(P = k s^3\), where \(k\) is the constant of proportionality.

Step 2: Find the Constant of Proportionality

We are given that an 80-hp engine is needed for a speed of 10 knots. Use this information to find \(k\):\[80 = k (10)^3\]Solving for \(k\):\[k = \frac{80}{1000} = 0.08\]

Step 3: Use the Constant to Find New Power Requirement

Now that we have \(k = 0.08\), we can find the power needed for 15 knots using the formula \(P = k s^3\):\[P = 0.08 \cdot (15)^3\]Calculate \((15)^3 = 3375\), thus:\[P = 0.08 \cdot 3375 = 270\]

Step 4: Conclusion

The power needed to propel the boat at 15 knots is 270 hp.

Key Concepts

Power and Speed Relationship

The relationship between power needed and speed is significant in understanding mechanical systems, especially when it comes to boats. In this context, power is how much energy output is required from an engine to maintain a certain speed. When we say that power is directly proportional to the cube of the speed, this means as the speed of the boat increases, the power needed increases as well but at a faster rate.This particular relationship is represented by the formula:

• \( P = k s^3 \)

Here, \( P \) is the power, \( s \) is the speed, and \( k \) is a constant. This tells us that if we know the constant, we can calculate the power required for any given speed. Understanding this relationship is crucial as it helps in designing engines that are efficient and meet specific speed requirements. It emphasizes that a small increase in speed requires a significantly larger increase in power, hence understanding how power scales with speed is essential for practical applications in engineering.

Cube Function

A cube function is a type of mathematical function where the variable is raised to the third power. In the formula \( P = k s^3 \), the speed \( s \) is cubed. This cube gives an indication of how changes in speed will dramatically affect power.When you compute the cube of a speed, it grows exponentially:

• If speed doubles, the power requirement increases by eight times.
• If speed triples, the power requirement increases by twenty-seven times.

This is because the cube of a number is that number time itself three times, expressed as \( s^3 = s \times s \times s \).This nonlinear scaling is particularly important to recognize in mechanical and engineering calculations. It means that improvements in engine efficiency or design might be necessary when planning for higher speeds.

Constant of Proportionality

The constant of proportionality \( k \) plays a crucial role in the relationship between power and speed. This constant allows the formula \( P = k s^3 \) to be tailored to specific scenarios, such as different boat designs or engine types.To determine \( k \), you need:

• One set of known values for power and speed.

In our exercise, we started with an 80-hp engine at 10 knots. By substituting these known values into the equation, \( 80 = k (10)^3 \), we solve for \( k \):\[ k = \frac{80}{1000} = 0.08 \]This constant remains applicable as long as the external conditions (like water current and boat design) do not change.Once \( k \) is known, it can be used within the direct proportionality formula to determine the power needed for different speeds, making it a vital component to understand and calculate correctly.