Problem 49 - Intermediate Algebra

Question

Jet Skis. A Jet Ski rider can travel 10 miles against the current of the lower Mississippi River in \(\frac{1}{2}\) hour and make the return trip with the current in \(\frac{1}{3}\) hour. Find the speed of the Jet Ski in still water and the speed of the current.

Step-by-Step Solution

Answer

The speed of the Jet Ski in still water is 25 mph and the speed of the current is 5 mph.

Step 1: Define variables

Let the speed of the Jet Ski in still water be \( x \) miles per hour, and the speed of the current be \( y \) miles per hour. When traveling against the current, the effective speed is \( x - y \), and when traveling with the current, it is \( x + y \).

Step 2: Write equations for each trip

For the trip against the current, the speed is \( x - y \) and it takes \( \frac{1}{2} \) hour to travel 10 miles. Hence, the equation is: \[ x - y = \frac{10}{\frac{1}{2}} = 20 \]For the return trip with the current, the speed is \( x + y \) and it takes \( \frac{1}{3} \) hour to travel 10 miles. Hence, the equation is:\[ x + y = \frac{10}{\frac{1}{3}} = 30 \]

Step 3: Solve the system of equations

We have two equations:1. \( x - y = 20 \)2. \( x + y = 30 \)Add the two equations to eliminate \( y \) :\[ (x - y) + (x + y) = 20 + 30 \]\[ 2x = 50 \]\[ x = 25 \]Now, substitute \( x = 25 \) into the first equation to find \( y \):\[ 25 - y = 20 \]\[ y = 5 \]

Step 4: Verify the solution

Check the values of \( x \) and \( y \) by substituting back into both equations:For \( x - y = 20 \):\[ 25 - 5 = 20 \] which is correct.For \( x + y = 30 \):\[ 25 + 5 = 30 \] which is also correct.Both equations are satisfied, confirming that the solution is correct.

Key Concepts

System of Equations

When solving word problems involving speeds, especially with different conditions like currents or wind, it's often helpful to employ a system of equations.
By using algebra, we set up equations based on the relationships described in the problem. In our example with the Jet Ski, the speed is affected by the current's flow.

• The speed of the Jet Ski in still water is represented by a variable, let's say \( x \), and the speed of the current by \( y \).
• We form equations based on the given information, such as distances covered and the time taken.
• Our goal is to find the unknowns \( x \) and \( y \).

By setting up and solving the equations simultaneously, we can determine both the speed of the Jet Ski in still water and the speed of the current.
This approach allows us to find precise solutions when dealing with two or more interdependent quantities.

Word Problems

Word problems can often seem challenging because they require translating a story into a mathematical framework.
However, with careful reading and a methodical approach, you can tackle these problems with success.

• First, identify the quantities involved and what you're being asked to find; in this case, the Jet Ski’s speed in still water and the current’s speed.
• Assign variables to these unknowns to simplify the relationships described in the problem.
• Then, translate the situation into mathematical equations using known formulas for speed, distance, and time.

By approaching word problems with a clear plan, and by understanding what each part of the problem represents, the math becomes more accessible and logical.
Additionally, checking your solutions in the context of the original problem helps ensure that you've translated the scenario accurately.

Speed and Distance Calculations

Speed, distance, and time are interrelated concepts that are commonly encountered in algebra problems.
The formula \( \,\text{Speed} = \frac{\text{Distance}}{\text{Time}} \,\) is key to solving these problems.

• Speed - This is how fast an object is moving, typically expressed in units such as miles per hour (mph).
• Distance - The length of the path an object travels, expressed in units like miles or kilometers.
• Time - The duration over which the movement occurs, often measured in hours or seconds.

In problems like the Jet Ski scenario, different sets of conditions (with and against the current) affect the effective speed. Therefore, different equations for each trip arise, helping us isolate variables of interest.
Breaking down the problem into these key elements allows for a systematic analysis and clear mathematical formulation to find the solution.