Problem 70 - Contemporary Precalculus

Question

Jack and Jill are salespersons in the suit department of a clothing store. Jack is paid \(\$ 200\) per week plus \(\$ 5\) for each suit he sells, whereas Jill is paid \(\$ 10\) for every suit she sells. (a) Let \(f(x)\) denote Jack's weekly income, and let \(g(x)\) denote Jill's weekly income from selling \(x\) suits. Find the rules of the functions \(f\) and \(g\). (b) Use algebra or a table to find \(f(20)\) and \(g(20), f(35)\) and \(g(35), f(50)\) and \(g(50)\) (c) If Jack sells 50 suits a week, how many must Jill sell to have the same income as Jack?

Step-by-Step Solution

Answer

Answer: Jill needs to sell 45 suits.

Step 1: Find the rules of functions f(x) and g(x) for Jack and Jill's weekly income

Jack is paid \(200 per week plus \)5 for each suit he sells, so we can represent his weekly income, f(x), as: f(x) = 200 + 5x Jill is paid $10 for each suit she sells, so we can represent her weekly income, g(x), as: g(x) = 10x

Step 2: Calculate f(20), g(20), f(35), g(35), f(50), and g(50)

We'll use the functions f(x) and g(x) to calculate the weekly income for Jack and Jill, respectively, when they sell 20, 35, and 50 suits: f(20) = 200 + 5(20) = 200 + 100 = 300 g(20) = 10(20) = 200 f(35) = 200 + 5(35) = 200 + 175 = 375 g(35) = 10(35) = 350 f(50) = 200 + 5(50) = 200 + 250 = 450 g(50) = 10(50) = 500

Step 3: Determine the number of suits Jill must sell to match Jack's income for selling 50 suits

To find the number of suits Jill needs to sell to have the same income as Jack, we must find the value of x for which g(x) equals Jack's income for selling 50 suits (which we calculated as f(50) = 450): g(x) = 10x 450 = 10x Now, we'll solve for x: x = 450/10 x = 45 Jill needs to sell 45 suits to have the same income as Jack when he sells 50 suits.

Key Concepts

Piecewise Functions

In mathematics, piecewise functions are a type of function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Though Jack and Jill's income functions aren't technically piecewise since they don't have different expressions for separate intervals, understanding piecewise concepts can still be useful. Piecewise functions are essential when different conditions lead to different outcomes for sections of a domain. Some classic examples include tax brackets or shipping rates that change after certain thresholds.

If Jack and Jill’s pay structures included changing pay per suit at different sales milestones (for example, a bonus for every 10 suits sold), their pay would be based on piecewise functions:

• Jack's pay would change when he hits different sales tiers.
• Jill might earn differently after selling over a certain number of suits.

Although this exercise doesn’t provide such a structure, understanding piecewise functions is critical as they often appear in real-world scenarios, such as in salary calculations where bonuses or commissions depend on reaching specific targets.

Algebra Application

Algebra is a powerful tool that allows us to express real-life problems in mathematical terms and solve them systematically. In the given exercise, algebra helps us calculate Jack and Jill's weekly income based on the number of suits they sell. By establishing the function rules:

• Jack's income: \(f(x) = 200 + 5x\), which accounts for his base salary and commission per suit.
• Jill's income: \(g(x) = 10x\), where her income is purely commission based.

To use algebra efficiently:

• Set up the equations based on the problem description.
• Substitute specific values to find the number of suits needed or test different sales outcomes.

If Jack sells a certain number of suits, you can quickly compute his income by plugging that number into his function. Similarly, calculating Jill's income is straightforward. Understanding these algebraic expressions and solving for unknowns help analyze potential earnings and make strategic decisions, such as determining break-even sales needed for equal incomes.

Function Evaluation

Function evaluation involves finding the value of a function for a given input, essentially plugging numbers into a function to compute the output. Here, Jack and Jill's income over a week is analyzed by evaluating their income functions for various suit numbers. This step-by-step method is pivotal in assessing their earnings.

• Evaluating \(f(20)\) shows Jack's income when selling 20 suits. Inputting 20 into Jack's function \(f(x) = 200 + 5x\) gives \(f(20) = 300\).
• Similarly, evaluate Jill's income for 20 suits by using \(g(20) = 10(20) = 200\).

Through function evaluation, you can compare Jack's base pay advantage against Jill's pure commission structure across different sales volumes. For specific calculations like \(f(50)\) and \(g(50)\), you systematically provide exact numeric results that depict varying sales scenarios.

By practicing function evaluation, students can better understand the relationship between input values and the resulting output, connecting abstract math concepts directly to practical, real-world applications.