Show your work in solving problems. Business Math

$1.$ A manufacturer plans to introduce a new type of shirt based on the following information.

The selling price is $$35\mathrm{.}00$; variable cost per unit is $$15\mathrm{.}00$; fixed costs are $$8200\mathrm{.}00$; and capacity per period is $740$ units.

a$)$ Calculate the break$-$even point

$($i$)$ in units

$($ii$)$ in dollars

$($iii$)$ as a percent of capacity

b$)$ Draw a detailed break$-$even chart.

c$)$ Calculate the break$-$even point $($in units$)$ if fixed costs are reduced to $$7000\mathrm{.}00$

d$)$ Calculate the break$-$even point $($in dollars$)$ if the selling price is increased to $$40\mathrm{.}00$

$2.$ Olfert Greenhouses has compiled the following estimates for operations.

Sales $$150000$

Fixed cost $$45200$

Variable costs $67500112700$

Net income $$37300$

Capacity is a sales volume of $$175000.$

Perform a break$-$even analysis showing

a$)$ an algebraic statement of

$($i$)$ the revenue function;

$($ii$)$ the cost function;

b$)$ a detailed break$-$even chart.

$3.$ The Excellent DVD Company sells DVDs for $$62$ each. Manufacturing cost is $$22\mathrm{.}70$ per DVD; marketing costs are $$7\mathrm{.}75$ per DVD; and royalty payments are $15\%$ of the selling price. The fixed cost of preparing the DVDs is $$227300.$ Capacity is $20000$ DVDs$.$

a$)$ Draw a detailed break$-$even chart.

b$)$ Compute the break$-$even point

$($i$)$ in units;

$($ii$)$ in dollars;

$($iii$)$ as a percent of capacity.

c$)$ Determine the break$-$even point in units if fixed costs are increased by $$3300$ while manufacturing cost is reduced $$1\mathrm{.}65$ per DVD$.$

d$)$ Determine the break$-$even point in units if the selling price is increased by $10\%$ while fixed costs are increased by $$2900.$

$4.$ Sana launched a perfume business through Amazon and achieved sales of $$17,380$ in her first year. She had a total of $$8,342\mathrm{.}40$ in variable costs and $$2,100$ in fixed costs. Her capacity for sales was $$30,000$ this year. Provide an algebraic statement representing her revenue and cost functions, and then compute her break$-$even point in terms of dollars and as a percentage of capacity.