You are setting up a part-time business with an initial investment of $27,000. the unit cost of the product is $11.30, and the selling price is $18.80. (a) find equations for the total cost c (in dollars) and total revenue r (in dollars) for x units. c(x) = r(x) = (b) find the break-even point by finding the point of intersection of the cost and revenue equations. units (c) how many units would yield a profit of $1000? units

A. The function to express the cost is:

     C(X) = $11.3*X + $27,000

    The function to express the revenue is:

     R(X) = $18.8*X

B.  The break-even point is obtained for 3600 units

C. It is needed 3734 units to yield a profit of $1,000.

Explanation:

The expression for the cost of bussines is obtained by adding the initial investment to the cost of production of X units:

C(X) = $11.3*X + $27,000

Meanwhile the revenue for selling X is units is represented as:

R(X) = $18.8*X

Then the break-even point is found by equalling C(X) with R(X):

R(X) = C(X)

$18.8*X = $11.3*X + $27,000

$18.8*X  - $11.3*X = $27,000

$ 7.5*X = $27,000

X = $27,000  / $7.5

X = 3,600 units

The profit of the bussines P(X) can be expressed sustracting the cost C(X) to the revenue R(X):

P(X) = R(X) - C(X)

P(X) = $18.8 * X - ($11.3*X + $27,000)

Equalling the profit to $1000 we will find the amount of units for selling:

$1,000 = $18.8*X - ($11.3*X + $27,000)

$1,000 = $18.8*X - $11.3*X - $27,000

$1,000 = $7.5*X - $27,000

$1,000 + $27,000 = $7.5*X  

$28,000 = $7.5*X  

$28,000 / $7.5 = X  

X =  3734 units

the investment was $14,174.196

explanation:

a lump sum means that the father of the example made one deposit and leave it be.

we need to know the value of a capital that generated at 4.05% per year during 28 years the sum of $48,613.24

now clearing the formula to solve for capital we got:

and we reach the answer investment = $14,174.20