Problem 2 - Intermediate Algebra

Question

Fill in the blanks. When a boat travels ____ , the speed of the boat is increased by the current. When a boat travels____, the speed of the boat is decreased by the current.

Step-by-Step Solution

Answer

Downstream; Upstream.

Step 1: Understanding Boat and Current Interaction

A boat moving in a water body is influenced by the speed of the current. When a boat is moving along the direction of the water flow, the water current adds to the speed of the boat. Conversely, when it moves against the water flow, the current reduces the boat's effective speed.

Step 2: Filling the First Blank

For the first blank, we need to identify the scenario in which the speed of the boat is increased due to the current. This happens when the boat is traveling in the same direction as the current, termed as 'downstream'. Hence, we fill the first blank with 'downstream'.

Step 3: Filling the Second Blank

For the second blank, we need to identify the scenario where the speed of the boat is decreased by the current. This occurs when the boat is moving in the opposite direction to the current, known as 'upstream'. Therefore, we complete the second blank with 'upstream'.

Key Concepts

Downstream

When talking about downstream movement, it's essential to understand how the current affects the boat's speed. Imagine you're on a river, and you're moving in the same direction as the water flow. In this case, the boat benefits from an additional push from the current.
This extra force effectively increases the boat's speed compared to its speed in still water. - Consider the speed of the boat in still water as \(S_b\) and the speed of the current as \(S_c\). - The resultant speed when moving downstream will be the sum of these speeds, given by the equation: \[ S_{ ext{downstream}} = S_b + S_c \] Watching these principles in action helps visualize why traveling downstream is generally easier and faster. The current aids in propelling the boat forward, reducing the effort needed to maintain a desired speed.

Upstream

On the other hand, moving upstream presents a distinct challenge. Here, the boat faces resistance from the current, as it moves against the flow of water. This resistance leads to a decrease in the effective speed of the boat.- Imagine the boat's speed in still water is \(S_b\) and the current's speed is \(S_c\).- When moving upstream, the actual speed of the boat is reduced by the speed of the current, illustrated by the equation: \[ S_{ ext{upstream}} = S_b - S_c \] This concept explains why traveling upstream requires more effort and time. The current works against the direction of the boat, slowing down its overall pace. Understanding this principle can help in planning for additional energy and time when navigating against the current.

Boat Speed Calculations

Calculating the effective speed of a boat involves considering both the boat's speed in still water and the influence of the current. These calculations help you understand how the boat's speed changes based on direction, whether it’s downstream or upstream.
A clear grasp of these principles is particularly useful for navigation and logistical planning.- **Downstream Calculation**: When moving downstream, use the formula: \[ S_{ ext{downstream}} = S_b + S_c \] This calculation shows that the boat's speed increases with the addition of current speed.- **Upstream Calculation**: When moving upstream, apply the formula: \[ S_{ ext{upstream}} = S_b - S_c \] Here, the current's speed is subtracted, reflecting the decrease in the boat's effective speed.These formulas are instrumental when planning trips and estimating travel time across bodies of water with flowing currents. Recognizing the effects of the current on boat speed allows for more accurate and efficient travel management.