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Chapter 18: Problem 45

Solve the given applied problems involving variation. The power \(P\) required to propel a ship varies directly as the cube of the speed \(s\) of the ship. If 5200 hp will propel a ship at \(12.0 \mathrm{mi} / \mathrm{h}\), what power is required to propel it at \(15.0 \mathrm{mi} / \mathrm{h} ?\)

Short Answer

Expert verified
The power required is approximately 10152 horsepower.

Step by step solution

01

Understand Direct Variation

In direct variation, one quantity is directly proportional to some power of another quantity. This can be expressed as \( P = k \cdot s^3 \), where \( k \) is the constant of proportionality and \( s \) is the speed.
02

Find the Constant of Proportionality

Use the given information that 5200 hp is required at a speed of 12 mi/h to find \( k \). Substitute the values into the equation: \( 5200 = k \cdot (12)^3 \). This simplifies to \( 5200 = k \cdot 1728 \). Solve for \( k \): \( k = \frac{5200}{1728} \).
03

Simplify the Constant

Calculate \( k \) by dividing: \( k = \frac{5200}{1728} = 3.00926 \). This is an approximate value for the constant of proportionality.
04

Calculate Power for New Speed

Now that we have the constant \( k = 3.00926 \), substitute the new speed of 15 mi/h into the direct variation equation to find the new power: \( P = 3.00926 \cdot (15)^3 \).
05

Calculate New Power

Calculate \( (15)^3 = 3375 \). Then \( P = 3.00926 \cdot 3375 = 10152 \) hp. So, 10152 horsepower is required to propel the ship at 15 mi/h.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportionality Constant
In the context of direct variation, a proportionality constant is a crucial element that helps link two directly varying quantities. When we say two quantities vary directly, it signifies that as one quantity increases, the other increases in a constant ratio.
For instance, in our exercise, the power \( P \) necessary to move a ship is directly proportional to the cube of its speed \( s \). This is mathematically represented as \( P = k \cdot s^3 \), where \( k \) is our proportionality constant. A proportionality constant remains consistent across all conditions unless the underlying system changes.
  • In our example, when the speed of the ship is 12 mi/h and requires 5200 hp, we find \( k \) by rewriting the equation as \( 5200 = k \cdot (12)^3 \).
  • After calculating, we discover \( k \approx 3.00926 \), reflecting the sensitivity of power needs to speed changes for this particular ship.
Speed and Power Relationship
The relationship between speed and power in this exercise is grounded in the concept of direct variation. Here, power is not just directly proportional to speed; instead, it is proportional to the cube of the speed.
This implies that even a small increase in speed results in a much larger increase in power requirement. Such a cubic relationship illustrates how power needs can escalate quickly.
  • The equation \( P = k \cdot s^3 \) helps visualize this connection. For a ship, going from 12 mi/h to 15 mi/h significantly increases power from 5200 hp to 10152 hp.
  • Knowing this relationship allows engineers to predict how ships will perform at various speeds and design engines that adapt to these demands.
Mathematical Modeling
Mathematical modeling involves creating equations to represent real-world situations. In our exercise, the relationship between the ship's speed and the required power is captured through a direct variation model.
Models like \( P = k \cdot s^3 \) allow predictions and strategic decisions in engineering and design. By using known data (like needing 5200 hp at 12 mi/h), we establish the proportionality constant. This constant then aids in modeling scenarios like determining the power needed at 15 mi/h.
  • With a clear mathematical model, complex predictions are possible with simple calculations.
  • Such models are pivotal not only in marine engineering but also in various fields, including aerodynamics and mechanical systems.
  • It's essential to recognize when direct or more complex variations describe a situation accurately, impacting how models are formulated and used.

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Most popular questions from this chapter

Find the required ratios. The specific gravity of a substance is the ratio of its density to the density of water. If the density of steel is 487 lb/ft \(^{3}\) and that of water is \(62.4 \mathrm{lb} / \mathrm{ft}^{3},\) what is the specific gravity of steel?

Express the meaning of the given equation in a verbal statement, using the language of variation. ( \(k\) and \(\pi\) are constants.) $$V=\pi r^{2} h$$

Solve the given applied problems involving variation. The \(x\) -component of the acceleration of an object moving around a circle with constant angular velocity \(\omega\) varies jointly as \(\cos \omega t\) and the square of \(\omega .\) If the \(x\) -component of the acceleration is \(-11.4 \mathrm{ft} / \mathrm{s}^{2}\) when \(t=1.00 \mathrm{s}\) for \(\omega=0.524 \mathrm{rad} / \mathrm{s},\) find the \(x\) -component of the acceleration when \(t=2.00 \mathrm{s}\).

Solve the given applied problems involving variation. The blade tip speed of a wind turbine is directly proportional to the rotation rate \(\omega .\) For a certain wind turbine, the blade tip speed is \(125 \mathrm{mi} / \mathrm{h}\) when \(\omega=12.0 \mathrm{r} / \mathrm{min}\). Find the blade tip speed when \(\omega=17.0 \mathrm{r} / \mathrm{min}\)

Solve the given applied problems involving variation. The force \(F\) between two parallel wires carrying electric currents is inversely proportional to the distance \(d\) between the wires. If a force of \(0.750 \mathrm{N}\) exists between wires that are \(1.25 \mathrm{cm}\) apart, what is the force between them if they are separated by \(1.75 \mathrm{cm} ?\)
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