Problem 8 - Precalculus Essentials

Question

You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of \(\$ 300\) and you pay \(70 \%\) of the manufacturer's recommended list price. Plan \(\mathbf{B}\) offers an annual membership fee of \(\$ 40\) and you pay \(90 \%\) of the manufacturer's recommended list price. a. Express the total yearly amount paid to the warehouse under plan \(\mathrm{A}, f,\) as a function of the dollars of merchandise purchased during the year, \(x\). E. Express the total yearly amount paid to the warehouse under plan \(\mathrm{B}, g,\) as a function of the dollars of merchandise purchased during the year, \(x\). c. How many dollars of merchandise would you have to purchase in a year to pay the same amount under both plans? What will be the total yearly amount paid to the warehouse for each plan?

Step-by-Step Solution

Answer

Plan A is represented as \(f(x) = 300 + 0.70x\) and Plan B as \(g(x) = 40 + 0.90x\). If the total dollar amount of merchandise is $1300, both plans will cost the same, amounting to a total of $1110 per year.

Step 1: Express Plan A as Function

Plan A offers an annual membership fee of $300 and additional fees of 70% of the manufacturer's recommended list price. Therefore, the total yearly amount can be expressed as a function \(f(x) = 300 + 0.70x\), where \(x\) represents the amount of dollars of merchandise purchased during the year.

Step 2: Express Plan B as Function

Similar to Plan A, Plan B can be expressed as a function of dollars spent on merchandise. It has an annual membership fee of $40 and the members pay 90% of the recommended list price. Thus, the function \(g(x) = 40 + 0.90x\) is the total yearly amount paid to the warehouse under Plan B.

Step 3: Compare the Two Plans

To find the dollar amount at which both plans cost the same, we set both functions equal to each other and solve for \(x\), the merchandise's dollar amount. This gives the equality \(300 + 0.70x = 40 + 0.90x\).

Step 4: Solve the Equality

Subtracting \(0.70x\) and \(40\) from both sides of the equation results in \(260 = 0.20x\). Finally, dividing both sides by \(0.20\) gives \(x = 1300\). This implies that if the merchandise's total dollar amount is $1300, both plans will cost the same.

Step 5: Calculate the Total Yearly Amount

Substituting \(x = 1300\) into either \(f(x)\) or \(g(x)\) gives the total yearly amount paid to the warehouse under both plans, which is \(f(1300) = g(1300) = 300 + 0.70*1300 = 40 + 0.90*1300 = 1110\).

Key Concepts

Annual Membership Fee

An annual membership fee is a fixed cost that you pay once per year to access certain services or benefits. In this exercise, both Plan A and Plan B require an upfront annual membership fee.

• Plan A charges a membership fee of \(\\(300\).
• Plan B charges a membership fee of \(\\)40\).

This fee is paid regardless of how much you spend. It’s important to recognize that the membership fee provides access to lower prices, which saves money on future purchases. The total cost for the year will be influenced by not just the items you buy, but also this fixed fee.

Break-Even Point

The break-even point is the amount where two different plans cost the same. In this problem, the break-even point tells us how much merchandise you need to purchase for Plan A and Plan B to have the same total yearly cost. The calculation involves setting the total costs of both plans equal to each other and solving for the dollar amount spent.
When you solve for the break-even point:

• First, you consider the equation for the total costs: \(300 + 0.70x = 40 + 0.90x\).
• The goal is to find the value of \(x\) where the two expressions yield the same result, which is the same total cost.
• Solving this equation, we find that \(x = 1300\) dollars of merchandise is the break-even point.

Understanding the break-even point is crucial as it helps you decide which plan might be more advantageous based on your shopping habits.

Equations

Equations are mathematical statements that show the equality of two expressions. In this task, we use them to model each plan's total cost relative to the merchandise purchased.
Plan A's total yearly cost equation is:

• \(f(x) = 300 + 0.70x\)

This shows that you pay a fixed \(\\(300\) plus 70% of the total merchandise cost.
Plan B's total cost equation is:

• \(g(x) = 40 + 0.90x\)

This shows a lower fixed fee of \(\\)40\), but you pay 90% of the total merchandise cost.
By setting these equations equal, \(300 + 0.70x = 40 + 0.90x\), we found the break-even point. Solving equations like these helps you to predict how changes in one variable affect another and is foundational for solving many kinds of real-world problems.