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 boat in still water? (a) \(20 \mathrm{~km} / \mathrm{h}\) (b) \(18 \mathrm{~km} / \mathrm{h}\) (c) \(15 \mathrm{~km} / \mathrm{h}\) (d) \(16 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: (a) 20 km/h

Step-by-step solution

Assign variables and equations for each scenario

Let the speed of the boat in still water be b km/h and the speed of the river be r km/h. Then, the total time for each scenario is given by the downstream trip time added to the upstream trip time. For the first scenario (river speed r and total trip time 15 hours): Downstream time = (120) / (b + r) Upstream time = (120) / (b - r) Total trip time = Downstream time + Upstream time = 15 For the second scenario (river speed 2r and total trip time 24 hours): Downstream time = (120) / (b + 2r) Upstream time = (120) / (b - 2r) Total trip time = Downstream time + Upstream time = 24

Write equations for each scenario

Now we can write the equations: (1) (120) / (b + r) + (120) / (b - r) = 15 (2) (120) / (b + 2r) + (120) / (b - 2r) = 24

Solve equations simultaneously

We first solve equation (1) for r: r = [(120 / (15 - (120 / (b - r))) - b] / 2 Now, substitute this expression for r into equation (2): (120) / (b + 2[(120 / (15 - (120 / (b - r))) - b] / 2) + (120) / (b - 2[(120 / (15 - (120 / (b - r))) - b] / 2) = 24 This is a nonlinear equation involving b only. You can use numerical methods (e.g., Newton-Raphson method, bisection method) or graphing to solve for b.

Solve for b (speed of the boat in still water)

After numerically solving the nonlinear equation, we find the solution: b ≈ 20 km/h

Check the answer options

Now, check the available answer options to see which one matches the calculated value: (a) \(20 \mathrm{~km} / \mathrm{h}\) (b) \(18 \mathrm{~km} / \mathrm{h}\) (c) \(15 \mathrm{~km} / \mathrm{h}\) (d) \(16 \mathrm{~km} / \mathrm{h}\) The correct answer is (a) \(20 \mathrm{~km} / \mathrm{h}\).

Key concepts

Boat Speed in Still Water

Understanding the boat speed in still water is a fundamental concept in solving river speed problems. In mathematical terms, the speed of the boat in still water is the velocity of the boat when it is not affected by the current of a river or stream. To understand the boat's speed in a flowing river, you need to differentiate between upstream and downstream movements.

Let's consider a theoretical speed, where the boat is not influenced by the current. This speed is what we refer to as the 'boat speed in still water'. In the given exercise, we denote this speed as 'b'. It's crucial to realize that this speed is constant and does not change whether the boat is going downstream or upstream; instead, it's the river's speed that adds to or subtracts from this base speed to give the effective speed of the boat in the river.

When calculating problems, setting the boat speed in still water as a variable allows us to create equations that account for the speed of the river, helping us to solve for unknowns using algebraic methods.

Downstream and Upstream Concepts

When a boat navigates a river, it encounters two types of movements - downstream and upstream. These terms are used to describe the direction of movement relative to the river's current.

• Downstream: This is when the boat is moving in the same direction as the river's current. In this scenario, the speed of the current aids the boat's movement. Mathematically, we calculate downstream speed by adding the speed of the river (denoted as 'r') to the boat's speed in still water.
• Upstream: Opposite to downstream, in upstream movement, the boat is going against the current. This makes the travel slower as the current's speed subtracts from the boat's speed in still water. The upstream speed is found by subtracting the river's speed from the boat's speed in still water.

In the context of the given problem, the downstream and upstream concepts directly influence the time it takes for the boat to cover the same distance. The actual time spent in each direction will vary depending on the strength of the river's current.

Speed, Distance, and Time Calculations

The relationship between speed, distance, and time is a cornerstone of many physics and mathematics problems, including river speed problems. The formula connecting these three quantities is expressed as: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]To apply this formula, you have to be familiar with the concepts of distance being the length of the path travelled by the boat, speed being the rate at which the boat travels, and time being the duration of the trip.

In the exercise, you're given the total time for a round trip and a change in time if the speed of the river is doubled. By using the speed equation twice - once for downstream and once for upstream - you can set up two equations that help you solve for the unknown boat speed in still water. The distance remained constant at 120 km; it's the varying speed that affects the total time. This application shows how intimately speed, distance, and time are related and how altering one variable impacts the others.

When solving real-life river problems, students must piece these calculations together to determine missing variables like the boat's speed in still water or the speed of the river. Ensuring a solid grasp of these formulas and their manipulation is vital for accurately solving such exercises.