Contribution Margin, Break-Even Sales, Cost-Volume-Profit Chart, Margin of Safety, and Operating Leverage Youngsville Industries Inc. expects to maintain the same inventories at the end of 20Y5 as at the beginning of the year. The total of all production costs for the year is therefore assumed to be equal to the cost of goods sold. With this in mind, the various department heads were asked to submit estimates of the costs for their departments during the year. A summary report of these estimates is as follows

Estimated Fixed Cost

Estimated Variable Cost (per unit sold)

Production Costs:

Direct Materials

$48

Direct Labor

$38

Factory Overhead

$150,000

$25

Selling Expenses:

Sales salaries and commissions

$125,000

$10

Advertising

$50,000

Travel

$15,000

Miscellaneous selling expense

$10,000

$2

Administrative Expenses:

Office and officers' salaries

$140,000

Supplies

$10,000

$5

Miscellaneous administrative expense

$10,000

$2

Total

$510,000

$130

It is expected that 22,175 units will be sold at a price of $170 a unit. Maximum sales within the relevant range are 27,500 units.

Giving the following information:

Units sold= 450,000

Total Sales= $12,150,000

Total variable cost= $6,925,500

Total fixed cost= $3,242,673

First, we need to calculate the unitary selling price and variable cost. Then, we calculate the contribution margin.

Selling price= 12,150,000/450,000= $27

Unitary variable cost= 6,925,500/450,000= $15.39

Contribution margin per unit= 27 - 15.39= $11.61

To calculate the contribution margin ratio, we need to use the following formula:

Contribution margin ratio= contribution margin/ selling price

Contribution margin ratio= 11.61/27

Contribution margin ratio= 0.43

To calculate the break-even point both in units and dollars, we need to use the following formulas:

Break-even point in units= fixed costs/ contribution margin per unit

Break-even point in units= 3,242,673 / 11.61= 279,300 units

Break-even point (dollars)= fixed costs/ contribution margin ratio

Break-even point (dollars)= 3,242,673/0.43= $7,541,100

Finally, we need to incorporate the desired profit to the break-even point formula:

Break-even point in units= (3,242,673 + 426,087) / 11.61

Break-even point in units= 316,000 units

Step-by-step explanation:

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).