Problem 56
Question
A cruise ship can sail 28 mph in calm water. Sailing with the Gulf Stream, the ship can sail \(170 \mathrm{mi}\) in the same amount of time it takes to sail \(110 \mathrm{mi}\) against the Gulf Stream. Find the rate of the Gulf Stream.
Step-by-Step Solution
Verified Answer
The rate of the Gulf Stream is 6 mph.
1Step 1: Setting up the Equations
Let’s denote the speed of the Gulf Stream as \(x\). When the ship is moving with the stream, its speed is (\(28 + x\)) mph whereas against the stream, it is (\(28 - x\)) mph. We should note that the time it takes to sail a certain distance is given by the formula \(Time = Distance / Speed \). Therefore, the time taken to sail 170 miles with the stream is \(170 / (28 + x)\), and the time taken to sail 110 miles against the stream is \(110 / (28 - x)\). We know that these two times are equal.
2Step 2: Solving the Equation
Given our understanding from step 1, our equation is \(170 / (28 + x) = 110 / (28 - x)\). Solve this equation by cross- multiplying and simplify to get \(170 * (28 - x) = 110 * (28 + x)\). Simplifying gives \(4760 - 170x = 3080 + 110x\). This gives a linear equation \(280x = 1680\). Solve for \(x\).
3Step 3: Getting the rate of the Gulf Stream
To solve for \(x\), divide both sides by 280 to get \(x = 1680 / 280 = 6\).
Key Concepts
Algebraic EquationsDistance/Speed/Time RelationshipLinear Equations
Algebraic Equations
An algebraic equation is a statement that two expressions, which can contain both numbers and variables, are equal. In the context of the Gulf Stream problem, we are using an algebraic equation to represent the relationship between the cruise ship's speed, the Gulf Stream's speed, and the distances traveled both with and against the current.
When we set up our initial equations, we are effectively translating the problem statement into a mathematical language which allows us to manipulate the variables to find a solution.
For example, by designating the ship's speed as a constant 28 mph and the variable speed of the Gulf Stream as 'x', we can describe complex scenarios in a simple, solvable manner. Students often struggle with setting up these equations from word problems, so it's important to identify key pieces of information that lead to the construction of the equation, such as distances traveled and the ship's differing speeds with and against the Gulf Stream.
When we set up our initial equations, we are effectively translating the problem statement into a mathematical language which allows us to manipulate the variables to find a solution.
For example, by designating the ship's speed as a constant 28 mph and the variable speed of the Gulf Stream as 'x', we can describe complex scenarios in a simple, solvable manner. Students often struggle with setting up these equations from word problems, so it's important to identify key pieces of information that lead to the construction of the equation, such as distances traveled and the ship's differing speeds with and against the Gulf Stream.
Distance/Speed/Time Relationship
The distance/speed/time relationship is a fundamental concept in physics and mathematics, often encapsulated in the simple formula:
\[\begin{equation} Time = \frac{Distance}{Speed} \end{equation}\]
In our Gulf Stream scenario, we apply this relationship twice: once for the ship traveling with the stream and again for it traveling against the stream. This relationship tells us how long it takes for an object to cover a certain distance at a constant speed.
A common stumbling block for students is not recognizing that the time remains consistent for both parts of the journey, regardless of the speed changes due to the Gulf Stream. Understanding this principle is crucial for setting up the algebraic equations correctly. It allows us to equate the time expressions for both parts of the journey, paving the way to solve for the unknown variable, which is the speed of the Gulf Stream.
\[\begin{equation} Time = \frac{Distance}{Speed} \end{equation}\]
In our Gulf Stream scenario, we apply this relationship twice: once for the ship traveling with the stream and again for it traveling against the stream. This relationship tells us how long it takes for an object to cover a certain distance at a constant speed.
A common stumbling block for students is not recognizing that the time remains consistent for both parts of the journey, regardless of the speed changes due to the Gulf Stream. Understanding this principle is crucial for setting up the algebraic equations correctly. It allows us to equate the time expressions for both parts of the journey, paving the way to solve for the unknown variable, which is the speed of the Gulf Stream.
Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be graphed as straight lines, hence the name 'linear'. Simplifying complex scenarios and solving for an unknown can often be achieved by understanding and manipulating linear equations.
In our Gulf Stream exercise, once the equation
\[\begin{equation} 170 / (28 + x) = 110 / (28 - x) \end{equation}\]
is established, cross-multiplication transforms it into a linear equation where we can solve for 'x'. Students may find it insightful to note that most real-life problems involving rates, ratios, and proportions can be broken down into linear equations. The clarity and simplicity of linear equations make them powerful tools for solving various problems, particularly in physics and engineering contexts.
In our Gulf Stream exercise, once the equation
\[\begin{equation} 170 / (28 + x) = 110 / (28 - x) \end{equation}\]
is established, cross-multiplication transforms it into a linear equation where we can solve for 'x'. Students may find it insightful to note that most real-life problems involving rates, ratios, and proportions can be broken down into linear equations. The clarity and simplicity of linear equations make them powerful tools for solving various problems, particularly in physics and engineering contexts.
Other exercises in this chapter
Problem 55
Simplify. $$\frac{5 x+7}{6 x y^{2}}-\frac{4 x-3}{8 x^{2} y}$$
View solution Problem 55
Multiply. $$\frac{2 x^{2}+5 x+2}{2 x^{2}+7 x+3} \cdot \frac{x^{2}-7 x-30}{x^{2}-6 x-40}$$
View solution Problem 56
Simplify. $$\frac{x-2}{8 x^{2}}-\frac{x+7}{12 x y}$$
View solution Problem 56
Multiply. $$\frac{x^{2}-4 x-32}{x^{2}-8 x-48} \cdot \frac{3 x^{2}+17 x+10}{3 x^{2}-22 x-16}$$
View solution