Question 1197717
### 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.