SOLUTION: PART 1 A University has a printer system designed for profit. The printer has a flat cost of $50 every month to run, and it costs the university two cents for every page printed.

Question 1197717: PART 1
A University has a printer system designed for profit. The printer has a flat cost of $50 every month to run, and it costs the university two cents for every page printed. The university sells each printed page for 10 cents.
a. Write an equation to describe the relationship between:
i) the costs (C) and the number of pages (p)
ii) income (I) and the number of pages (p)
b. Draw a graph to represent the costs and income monthly.
c. How many pages need to be sold to break even every month?
d. Check your answer to part C algebraically.
PART 2
A company makes and sells fidget spinners. They are sold on eBay for $8 each.
Using spreadsheet or Desmos, make a table of values showing the income (I) for different quantities sold (n).
The fixed over-head expenses or Desmos, make a table of values for the number sold against the cost to produce this number of fidget spinners.
a. What is the break-even point?
b. What does this mean in terms of the number of fidget spinners and the cost/income?
c. What is the slope of the income line?
d. What is the Income equation (I) if n is the number of fidget spinners sold?
e. What is the slope of the cost line?
f. What is the Cost equation (C) if n is the number of fidget spinners sold (remember to add in the fixed costs)?
g. If the fixed costs of $515 were reduced, with everything else remaining equal, describe how the break-even point would change (in relation to the graph)?

Answer by onyulee(41) (Show Source): You can put this solution on YOUR website!
### Part 1: University Printer System
#### a. Write equations
- **i) Cost equation $C(p)$:**
The cost includes a flat fee of $50 and 2 cents per page:
\[
C(p) = 50 + 0.02p
\]
- **ii) Income equation $I(p)$:**
The income comes from selling pages at 10 cents per page:
\[
I(p) = 0.10p
\]
#### b. Graph of Costs and Income
- The cost graph $C(p)$ is a straight line starting at $50 with a slope of $0.02$.
- The income graph $I(p)$ is a straight line starting at $0 with a slope of $0.10$.

Let me know if you'd like a plotted graph.
#### c. Break-Even Point
To break even, the income equals the cost:
\[
C(p) = I(p)
\]
\[
50 + 0.02p = 0.10p
\]
#### d. Solve Algebraically
\[
50 = 0.10p - 0.02p
\]
\[
50 = 0.08p
\]
\[
p = \frac{50}{0.08} = 625
\]
Thus, **625 pages need to be sold to break even.**
---
### Part 2: Fidget Spinner Company
#### Create Tables
- **Income $I(n)$:** Income per fidget spinner is $8.
\[
I(n) = 8n
\]
- **Cost $C(n)$:** Cost includes fixed overhead of $515 and variable costs:
Let me know the variable cost per spinner for precise calculations.
#### a. Break-Even Point
To find the break-even point:
\[
I(n) = C(n)
\]
Substitute the income and cost equations and solve for $n$.
#### b. Interpretation of Break-Even
The break-even point represents the number of fidget spinners that must be sold for the income to cover the costs. At this point:
\[
\text{Income = Costs}
\]
#### c. Slope of Income Line
The slope of the income line represents the rate at which income increases per unit sold:
\[
\text{Slope of Income} = 8
\]
#### d. Income Equation
\[
I(n) = 8n
\]
#### e. Slope of Cost Line
The slope of the cost line represents the rate at which cost increases per unit produced:
\[
\text{Slope of Cost} = \text{Variable Cost (value needed)}
\]
#### f. Cost Equation
If the fixed cost is $515 and the variable cost per spinner is $c$:
\[
C(n) = 515 + cn
\]
#### g. Effect of Reduced Fixed Costs
Reducing the fixed costs would lower the $y$-intercept of the cost line, shifting it downward. The break-even point would decrease, meaning fewer spinners need to be sold to cover the costs.
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