Word Problem Please need help on how to solve this

Question

A manufacturer sells a product for $18 per unit, selling all that is produced. Total costs are made up of fixed costs of $1100 and variable costs of $2.50 per unit.

1. Find an expression for profit (𝜋).
2. Find the break-even level of production (to the nearest unit).
3. Find the break-even level of production if variable costs increase by 20%.
4. Suppose the manufacturer can only sell at a price of $18 per unit when selling less than 500 units. For quantities above 500 units the price drops to $15 per unit. Write out a split revenue function (𝑟) that reflects this.

Answer 1

Profit is the difference between cost and revenue from sales of the product. The revenue in this case (R) is $18 x the no. of units sold (18n). The costs are a combination of variable and fixed costs: C = 2.5n + 1100.

So the equation for profit is π = R - C or 18n - (2.5n + 1100). Simplifies to π = 15.5n - 1100.

Breakeven occurs when R = C ( no profit), so set 18n = 2.5n + 1100 and solve for n. n = 70.96 or ≅ 71 units. If you plug 71 units into the profit equation you'll see the the answer is approximately 0 (due to rounding). If you graph the R and C functions individually (straight lines), they'll meet at n = 71 where R and C is ≅ $1,278.

If variable costs increase by 20%, C = 3n +1100 and B/E occurs with approximately 73 units.

I'll leave (4) to another tutor.