Problem 28 - Algebra and Trigonometry

Question

Power Needed to Propel a Boat The power P (measured in horse power, 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 to drive the boat at 15 knots is 270 hp.

Step 1: Understand the Proportionality

The power \( P \) needed is directly proportional to the cube of the speed \( s \). This can be expressed as \( P = k s^3 \) where \( k \) is the proportionality constant. Your task is to find this constant using the information given.

Step 2: Calculate the Proportionality Constant

We know that when the speed \( s = 10 \) knots, the power \( P = 80 \) hp. Using \( P = k s^3 \), substitute the known values: \( 80 = k \cdot 10^3 \). Solve for \( k \): \( k = \frac{80}{1000} = 0.08 \).

Step 3: Use the Constant to Find New Power

Now that we have \( k = 0.08 \), we want to find the power \( P \) when the speed is \( 15 \) knots. Plug in the values into the expression: \( P = 0.08 \cdot 15^3 \).

Step 4: Solve the Equation for New Speed

Calculate \( 15^3 = 3375 \). Then multiply by the constant: \( P = 0.08 \cdot 3375 = 270 \). Thus, the power needed is 270 hp.

Key Concepts

Power and Speed Relationship

The relationship between power and speed in this scenario is a great example of direct proportion, more specifically, a cubic relationship.
Power is directly proportional to the cube of the speed, which means that as speed increases, the power required to propel the boat increases dramatically. So, if the speed of the boat doubles, the power doesn't just double; instead, it increases eight times—since doubling the speed means cubing 2 (i.e., 2^3=8). This type of relationship is called a cubic relationship.

• Directly proportional means there's a constant multiplier in play.
• The relationship is represented mathematically as:
  \( P = k \, s^3 \)
  where \( P \) is power, \( s \) is speed, and \( k \) is the proportionality constant.

This cubic relationship shows just how much more power is needed when speed increases. It's important to keep this in mind when working with engines and speeds, as it might require much more energy than initially expected.

Proportionality Constant

The proportionality constant, often represented by \( k \), plays a vital role in understanding the relationship between two directly proportional variables.
In our boat example, the constant is essential to determine how much power relates to a given speed.

• At a known speed of 10 knots with an 80-hp engine, you determine \( k \) by plugging known values into the equation \( P = k s^3 \).
• So, \( 80 = k \cdot 10^3 \) simplifies to the constant \( k = \frac{80}{1000} = 0.08 \).

Once you have this constant, it can be used to find power requirements at different speeds. The constant connects these two measurements smoothly, revealing how power changes with speed.

Solving Equations

In this exercise, solving equations is about deriving the power needed at a new speed, given the determined constant.
Once you have the constant \( k \), you can find the power for any other speed using the formula \( P = k s^3 \). For instance:

• With \( k = 0.08 \) and a speed \( s = 15 \) knots, substitute into \( P = 0.08 \cdot 15^3 \).
• Calculate \( 15^3 \) which equals 3375.
• Then multiply by 0.08 to find the new power: \( P = 0.08 \cdot 3375 = 270 \) hp.

This methodical approach is crucial for accurately determining power requirements and is representative of how equations can be solved in practical applications. Remember, solving such equations requires both understanding the relationship and doing the math accurately.