Problem 1 - Technical Mathematics with Calculus

Question

Applications. To determine the speed of a boat, it is clocked, with the current, to go a distance of 18.5 miles in 1.31 hours. Returning the same distance against the current took 3.32 h. Find (a) the speed of the boat in still water and (b) the speed of the current.

Step-by-Step Solution

Answer

The speed of the boat in still water (b) is 14.12 mph, and the speed of the current (c) is 0.084 mph.

Step 1 - Define Variables

Let the speed of the boat in still water be represented by the variable 'b' (in miles per hour), and the speed of the current be represented by 'c' (in miles per hour). We will have two scenarios to consider: moving with the current and against the current.

Step 2 - Write Equations Based On Scenarios

Using the information given, we can write two equations based on the total time it takes to cover the distance with and against the current. With the current: \( 18.5 = (b+c) \times 1.31 \). Against the current: \( 18.5 = (b-c) \times 3.32 \)

Step 3 - Derive Equations for b and c

We will now express 'c' in terms of 'b' using the first equation: \( c = \frac{18.5}{1.31} - b \). Next, substitute the value of 'c' from this equation into the second equation and solve for 'b'.

Step 4 - Solve for b

Substituting the expression for 'c' into the second equation gives us: \( 18.5 = (b - (\frac{18.5}{1.31} - b)) \times 3.32 \). Now solve this equation to find the value of 'b'.

Step 5 - Find the Speed of the Current (c)

Once 'b' is found, substitute it back into the equation \( c = \frac{18.5}{1.31} - b \) to find the speed of the current 'c'.

Step 6 - Calculate and Interpret Final Answers

Calculate the values of 'b' and 'c' from the previous steps and interpret these in the context of the problem to provide the final answers.

Key Concepts

Technical Mathematics

Technical mathematics refers to the use of mathematical techniques and principles to solve real-world problems typically encountered in technical and vocational fields. In the context of boat speed and current problems, technical mathematics allows us to model the relationship between distances, speeds, and times using algebraic equations.

For instance, when a boat travels with and against the current, its effective speed changes. This scenario is quantifiable using variables and algebraic expressions, normally covered in technical mathematics curricula. The ability to create equations that model real-world phenomena is a crucial skill and is evidenced in our exercise by setting up a system of equations to describe boat movement.

Solving Systems of Equations

Solving systems of equations involves finding the values of variables that satisfy multiple equations simultaneously. In our example, we are dealing with two equations: one for the boat's speed with the current, and one against the current. This is a classic case often encountered in algebra, where we solve for two unknowns.

The problem is approached by expressing the current speed in terms of the boat's speed in still water and then substituting into the other equation. By applying algebraic methods such as substitution or elimination, we can systematically derive the solution to the system, giving us the individual values that solve both equations.

Algebraic Methods

Algebraic methods refer to the strategies and techniques used to manipulate and solve algebraic expressions and equations. In solving our boat speed problem, we applied substitution, which involves solving one equation for a variable and substituting that expression into another equation.

By doing so, we're able to eliminate one variable and solve for the other. Once we find the value of one variable, say the boat's speed in still water, we can easily calculate the second variable, the speed of the current. This methodical approach simplifies complex problems into solvable steps and forms the foundation of solving systems of equations in algebra.

Real-world Applications of Algebra

The real-world applications of algebra are limitless, providing solutions to practical problems across various fields. In our boat speed example, we observed algebra's role in navigational practices. By understanding the principles of algebra, one can deduce how long it takes to travel certain distances with varying speed factors, like current or wind.

These applications are not just academic exercises; they impact areas like transportation, engineering, and even economics. Such problems require critical thinking and the ability to apply mathematical concepts to real-life scenarios, reinforcing the relevance of algebra beyond the classroom.