The Squeaky Clean Window Cleaning Company from Exercise 35...

The Squeaky Clean Window Cleaning Company from Exercise 35 charges companies $\$ 9$ per window cleaned plus a travel charge of $\$ 50$. a. Write an equation for the total revenue for cleaning windows at a business depending on the number of windows cleaned. b. How much will the Squeaky Clean Window Cleaning Company charge a business to clean 50 windows? c. Write an equation for the profit made by the Squeaky Clean Window Cleaning Company depending on the number of windows cleaned. d. How much profit will the company make from cleaning 80 windows for a business? e. How many windows must the company clean in a day to break even? (Breaking even means that profit $=0 .$

The Squeaky Clean Window Cleaning Company from Exercise 35 charges companies $\$ 9$ per window cleaned plus a travel charge of $\$ 50$.
a. Write an equation for the total revenue for cleaning windows at a business depending on the number of windows cleaned.
b. How much will the Squeaky Clean Window Cleaning Company charge a business to clean 50 windows?
c. Write an equation for the profit made by the Squeaky Clean Window Cleaning Company depending on the number of windows cleaned.
d. How much profit will the company make from cleaning 80 windows for a business?
e. How many windows must the company clean in a day to break even? (Breaking even means that profit $=0 .$

Key Concepts

Linear Functions

Linear functions are mathematical models written in the form y = mx + b, where m represents a constant rate of change and b is the initial fixed value. They are used to represent relationships with constant increments, such as the addition of a flat fee plus a per-unit charge.

Fixed and Variable Components

Many real-world pricing structures, including revenue models, can be broken into a fixed component (a constant fee that does not vary with quantity) and a variable component (a fee that scales with the number of items or services provided). Understanding these components is essential when setting up and analyzing equations for revenue, cost, and profit.

Function Evaluation

Function evaluation involves substituting specific values into an equation to determine an outcome. This is important in calculating exact totals, such as finding the total charge for a given number of windows by substituting that value into the revenue equation.

Profit Function

The profit function models net earnings by ideally subtracting costs from revenue. Even when given a revenue model, defining a corresponding profit function may involve accounting for fixed costs or additional expenses, which is critical for understanding the financial viability of a service.

Break-even Analysis

Break-even analysis determines the point at which total revenue equals total costs, resulting in zero profit. Finding this point helps businesses understand the minimum quantity of service or product they must sell to cover all expenses and avoid losses.

Transcript

00:02 So for problem 36, the first question asked about writing the revenue equation.

00:05 So we'll say r is per revenue and x will be for number of windows cleaned.

00:20 So write it, r is a total, so we'll say r equals.

00:23 And it's $9 per window.

00:25 So it would be non -times however many windows there are, which we don't know yet.

00:29 That's why we represent it with x.

00:31 And then we would add the $50 that they're going to charge.

00:38 So this is how much revenue would be.

00:41 Then part b says, okay, so how much revenue would they bring in for 50 windows? if they clean 50 windows.

00:48 So we'd say r equals 9 times 50 plus 50, which 9 times 50 is 450 plus another 50 is $500.

01:02 And then part c ask us to write the profit.

01:05 Equation.

01:06 So we'll say p is for profit and x would still be for the number of windows cleaned.

01:17 To write it, we'd have to take p, the total profit.

01:21 To get that, we would take what they're going to charge and take away or subtract their costs are.

01:33 So they charge nine dollars per window, but it also cost them $1 .50 per window for the materials as it says in.

01:44 The question above.

01:48 Then they charge the $50, but it cost them $530 on the same day to pay people and for insurance.

02:01 So here's the equation, but we can simplify that by combining like terms.

02:05 9x minus 150x is 750x...

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