Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Camping You are choosing between camping outfitters. Outfitter A charges a reservation fee of $$\$ 150$$ plus a daily guide fee of $$\$ 70 .$$Outfitter \(\mathrm{B}\) charges a reservation fee of $$\$ 75$$ plus a daily guide fee of $$\$ 90 .$$ Estimate when the cost of Outfitter A equals the cost of Outfitter \(\mathrm{B}\).

Step-by-Step Solution

Answer

The costs of using Outfitter A and Outfitter B will be equal after approximately 4 days. Since you can't hire an outfitter for a fraction of a day, you may need to consider other factors to decide between them if you plan to hire for around 4 days.

Step 1: Setting up the equations

To approach this problem, formulate two equations, one for each outfitter. Outfitter A charges a reservation fee of \(\$150\) plus a daily guide fee of \(\$70\). This can be represented by the equation \(A = 150 + 70d\), where \(A\) is the total cost and \(d\) is the number of days. On the other hand, Outfitter B charges a reservation fee of \(\$75\) plus a daily guide fee of \(\$90\). This can be represented by the equation \(B = 75 + 90d\), where \(B\) is the total cost and \(d\) is the number of days.

Step 2: Equating the two equations

The next step is to find when the cost of Outfitter A is equal to the cost of Outfitter B. This involves setting the two equations equal to each other and solving for \(d\). Therefore: \(150 + 70d = 75 + 90d\).

Step 3: Solve for \(d\)

Now, solve for \(d\). Subtract \(70d\) from both sides, then subtract \(75\) from both sides to simplify the equation. \(20d = 75\). Then divide the sides by \(20\). This will give you the number of days they are equal: \(d = 3.75\).

Key Concepts

System of Linear Equations

Understanding the concept of a system of linear equations is fundamental in algebra. In real-world terms, a linear equation is a straight-line relationship between two variables. When we have two such relationships that we're looking to compare or find a point of intersection—such as comparing costs between two service providers—we are dealing with a system of these equations.

In the camping outfitter example, each outfitter's cost structure creates a separate linear equation, which depends on the number of camping days, represented by 'd'. We translated the cost for Outfitter A and Outfitter B into two equations that can be analyzed together. This system of equations can be visualized on a graph as two lines, and the point of intersection represents the number of days where their costs are equivalent. The key to solving these problems is establishing each equation correctly and then finding the values that satisfy both equations at the same time.

Algebraic Problem-Solving

Algebraic problem-solving requires a step-by-step approach to manipulate and solve equations. In the context of the system of equations word problems, we apply a specific strategy to find the solution. The first step, as seen in the camping scenario, was to translate the word problem into algebraic equations that represent the cost for each day.

After the equations are set up, we look for values that make both equations true. This often involves rearranging the equations—via addition, subtraction, multiplication, or division—to isolate the variable we're solving for. In our case, we isolated 'd', which represents the number of days. By maintaining such systematic approaches, algebraic problem-solving becomes a matter of following logical steps to arrive at a solution.

Equating Coefficients

Equating coefficients is a method used to solve systems of equations where you make the coefficients of the variable terms equal so as to find the solution. In the Outfitter A and B problem, we have 'd' representing days associated with different coefficients in each equation.

When we set the equations equal to each other, we are essentially equating the total costs, which gives us an equation that solely involves the variable 'd'. By manipulating this new equation—subtracting the lesser coefficient from the greater one—we're left with an equation that can be solved for 'd'. Applying this method of equating coefficients streamlines the process of finding at which point two scenarios—like the costs of two outfitters—become equal.