Question

A motor boat went downstream motor boat for \(120 \mathrm{~km}\) and immediately returned. It took the boat 15 hrs to complete the road trip. If the speed of the river were twice as high the trip downstream and back would take 24 hours. What is the speed of the stream? (a) \(3.5 \mathrm{~km} / \mathrm{h}\) (b) \(4 \mathrm{~km} / \mathrm{h}\) (c) \(6 \mathrm{~km} / \mathrm{h}\) (d) \(8 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: The speed of the stream is 6 km/h.

Step-by-step solution

Define the variables

Let x be the speed of the motorboat in still water, and y be the speed of the stream. The total distance downstream and upstream is the same (120 km), so the downstream distance equals the upstream distance.

Write the equations

Since distance equals speed multiplied by time, we can write the equation for the time taken for the downstream and upstream trip as follows: Downstream time: \(\dfrac{120}{x+y}\) Upstream time: \(\dfrac{120}{x-y}\) The problem states that the total time taken for the downstream and upstream trip is 15 hours, so we can write the first equation: \(\dfrac{120}{x+y} + \dfrac{120}{x-y} = 15\) Now, we need to represent the given information about the river speed being twice as high. If the river speed doubles, the downstream trip will be faster, and the upstream trip will be slower. We can write the equation for this scenario as follows: \(\dfrac{120}{x+(2y)} + \dfrac{120}{x-(2y)} = 24\) This gives us two equations.

Solve the system of equations

Now we have the following system of equations: 1) \(\dfrac{120}{x+y} + \dfrac{120}{x-y} = 15\) 2) \(\dfrac{120}{x+2y} + \dfrac{120}{x-2y} = 24\) We can multiply each equation by its respective denominator to eliminate fractions. Doing so gives us: 1) \(120(x-y) + 120(x+y) = 15(x^2-y^2)\) 2) \(120(x-2y) + 120(x+2y) = 24(x^2-4y^2)\) Now, we can simplify the two equations: 1) \(240x = 15(x^2-y^2)\) 2) \(240x = 24(x^2-4y^2)\) Now let's solve the first equation for \(x\): \(240 = 15(x^2 - y^2) / x\) \(x = 15 (x^2 - y^2) / 240\) \(x = (x^2 - y^2) / 16\) We can substitute this into Equation (2): \(\dfrac{x^2 - y^2}{16} = (x^2 - 4y^2) / 10\) Multiplying both sides by 160 to eliminate fractions, we get: \(10(x^2 - y^2) = 16(x^2 - 4y^2)\) Simplifying: \(10x^2 - 10y^2 = 16x^2 - 64y^2\) Rearranging and simplifying further, we get: \(6x^2 = 54y^2\) Thus, \(x^2 = 9y^2\), which means \(x = 3y\). Now, plug in the value of \(x\) in terms of \(y\) into the first equation: \(x = \dfrac{x^2 - y^2}{16}\) \(3y = \dfrac{(3y)^2 - y^2}{16}\) \(48y = 8y^2\) \(y = 6\) Since we found the value of \(y\), which represents the speed of the stream, the speed of the stream is \(6 \mathrm{~km} / \mathrm{h}\). So the correct answer is (c) \(6 \mathrm{~km} / \mathrm{h}\).

Key concepts

Distance and Time Calculation

Understanding how distance and time interact is key in solving problems involving moving objects. The basic formula that connects these concepts is \[ \text{Distance} = \text{Speed} \times \text{Time} \]This formula can be rearranged to find speed or time if you know the other two variables:- To find speed: \( \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} \)- To find time: \( \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \)These calculations become crucial when dealing with practical scenarios such as traveling in a boat or a car.
In our context of upstream and downstream, understanding distance and time helps us grasp why the total travel time changes when the stream's speed alters.

Upstream and Downstream Problems

Navigating the waters involves understanding the effects of the current on travel. When moving downstream, the current aids the travel, effectively increasing the boat's speed, while going upstream, the current slows down the travel. **Key Ideas:** - **Downstream speed**: The speed of the boat plus the speed of the stream. - **Upstream speed**: The speed of the boat minus the speed of the stream. The challenge comes from noting how these speeds affect time. When a boat covers a set distance in both downstream and upstream conditions, the time taken is calculated separately and then summed up for total time.
In problems like ours, by setting up equations for each part of the trip (upstream and downstream), we can solve for unknown variables like the speed of the stream.

Solving Systems of Equations

When you encounter a practical problem with multiple conditions, like the speed of a boat affected differently downstream and upstream, using systems of equations comes in handy. **What Are Systems of Equations?** A system of equations is a set of two or more equations with the same variables. Solving them together can reveal those variables. - In our problem, each equation represents a different scenario: normal stream speed and doubled stream speed. **Solving Techniques:** - **Substitution:** Solve one equation for one variable and substitute into the other. - **Elimination:** Add or subtract equations to eliminate a variable. In our exercise, expressing the scenarios as separate equations and then using algebraic techniques allows us to simplify and find the solution. For instance, by setting equations for the travel times and distances, we could determine the stream's speed by eliminating boat speed from the equations.
Ultimately, solving systems of equations is a powerful tool in breaking down complex problems into manageable calculations.