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Chapter 3: Problem 89

A paddle-steamer takes five days to travel from St Louis to New Orleans, and takes seven days for the return journey. Assuming that the rate of flow of the current is constant, calculate how long it takes for a raft to drift from St Louis to New Orleans.

Short Answer

Expert verified
The raft takes 35 days to drift from St Louis to New Orleans.

Step by step solution

01

Understand the problem

The paddle-steamer takes different amounts of time to travel from St Louis to New Orleans and back, due to the influence of the current's flow. We are asked to find out how long a raft, which is only influenced by the current, would take to drift from St Louis to New Orleans.
02

Define Variables

Let the speed of the paddle-steamer in still water be \( v_p \) and the speed of the current be \( v_c \). The distance between St Louis and New Orleans is \( d \).
03

Set up the equations for the paddle-steamer

For the trip from St Louis to New Orleans: \[ d = (v_p + v_c) \times 5. \]For the return trip from New Orleans to St Louis:\[ d = (v_p - v_c) \times 7. \]
04

Combine the equations to eliminate \(v_p\)

From the two equations, you can equate the expressions for \(d\):\[ 5(v_p + v_c) = 7(v_p - v_c). \]
05

Simplify the equation

Expand both sides and simplify:\[ 5v_p + 5v_c = 7v_p - 7v_c. \]Rearranging terms gives:\[ 12v_c = 2v_p. \]
06

Solve for the current's speed \(v_c\)

Divide both sides by 12:\[ v_c = \frac{v_p}{6}. \]
07

Calculate the time for the raft to drift

The speed of the raft is equal to the speed of the current \(v_c\). Thus, the time \(t\) for the raft to drift the distance \(d\) is:\[ d = v_c \times t, \]\[ t = \frac{d}{v_c}, \]\[ t = \frac{d}{\frac{v_p}{6}} = 6 \times \frac{d}{v_p}. \] We know from the problem that:\[ d = (v_p + v_c) \times 5, \ d = (\frac{7v_p}{6}) \times 5, \ d = \frac{35v_p}{6}. \] Substituting in, we find\[ t = 6 \times \frac{\frac{35v_p}{6}}{v_p} = 35. \]
08

Conclude the solution

The raft takes 35 days to drift from St Louis to New Orleans.

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Key Concepts

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

Speed and Distance Problems
When dealing with speed and distance problems, it's crucial to understand how the relationship between speed, distance, and time interacts. The basic formula used here is:
  • Speed = Distance / Time
  • Time = Distance / Speed
  • Distance = Speed 脳 Time
If you know any two of these variables, you can solve for the third. This exercise features a paddle-steamer traveling two different times over the same distance because the river current affects its speed. So, when analyzing problems like this, account for external factors affecting speed, such as water currents or wind.
Using equations to express the relations helps solve for unknowns. Here, the problem is complicated by the two speeds the paddle-steamer experiences: with the current and against it.
Algebraic Equations
Algebraic equations are crucial in word problems as they allow us to express relationships mathematically. In this case, we set up two equations to represent the paddle-steamer's journeys.We used the following equations:
  • For the trip from St Louis to New Orleans: \( d = (v_p + v_c) \times 5 \)
  • For the return trip: \( d = (v_p - v_c) \times 7 \)
These equations help us understand how the river current affects the boat's travel time and allows us to eliminate unknowns. By equating and simplifying these equations, we can find relationships between variables that aren't immediately obvious. It's vital to follow the steps of simplifying and rearranging equations systematically to isolate and solve for unknowns, like the speed of the river current in this instance.
River Current Problems
River current problems often involve understanding how the flow of water affects movement through it. In these cases, current adds to or subtracts from the speed of an object like a boat.
  • When moving with the current, the effective speed is the sum of the object's speed in still water and the current's speed.
  • Against the current, the speed is reduced by the current's speed.
  • For objects like rafts, which drift, the only speed at play is the current's speed itself.
Understanding these principles is essential when setting up equations. We resolved the exercise by recognizing that the raft's journey time depended solely on the river current speed. Given the raft's passive drift, the current's speed determined how fast it traveled the distance from St Louis to New Orleans. This required us to solve the equation for the current's speed in relation to the paddle-steamer's movement.

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Most popular questions from this chapter

A team of mowers had to mow two fields, one twice as large as the other. The team spent half-a-day mowing the larger field. After that the team split: one half continued working on the big field and finished it by evening; the other half worked on the smaller field, and did not finish it that day - but the remaining part was mowed by one mower in one day. How many mowers were there?

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