# Example Of Essay On Math 202 Category W Writing Assignment

A general requirement for an emerging company is to determine the factors that would affect the economy of their goods and services. As part of developing a company, it has been decided that the company that I am supposed to design would specialize in manufacturing liquor. To be more specific, the company would specialize in manufacturing bottles of wine.

The production cost of wine was determined by conducting informal research. The informal research was performed primarily through the internet. According to Wine-Searcher (2014), farmers on average sell their bottles of wine to retailers for about $19. Thus, to proceed with this report, let us assume that the production of a bottle of wine costs $19. Note that this detail on the production of a bottle of wine was based on the accounts of the owners of Chateau Montelena (Wine-Searcher, 2014).

If we assume that there exists a fixed cost on miscellaneous fees including rent and other utilities, it would affect the total cost of manufacturing the product. In this case, let us assume that the said cost is fixed at $7,500 per month. With this, the linear cost function, denoted as C(x), would be the sum of the total cost of production per month, which is determined by the production cost per unit and the number of units produced, and the fixed cost, which is at $7,500 per month.

Cx=19x+7500

(Equation 1)

In Equation 1, the x denotes the quantity of units produced. Note that there exists an upper limit in the amount of units that can be produced, and this upper limit of the quantities of product depends on the principal capital, which is the allowed amount of money allotted for the manufacturing. Let us assume that the said amount of money sums up to $150,000 per month. In this case, we can determine the number of units, x, that can be produced.

150000=19x+7500

142500=19x

x=7500

Therefore, with $150,000 of money allotted for production, there will be 7,500 bottles of wine that can be produced per month while considering a fixed cost of $7,500 per month and the production cost of $19 per unit.

One other player in the market equilibrium of a certain product is how much it is demanded within its consumers. Moreover, the demand for a product could affect its price. Let us assume that a price-demand equation exists.

x=9600-30p

(Equation 2)

Equation 2 refers to the said price-demand equation, where x represents the number of bottles of wine demanded per month while p represents the price per bottle of wine sold in dollars. In this case, there would be a revenue function, R(x), which will be determined by the quantities of product demanded per month and the price per unit sold.

Rx=x(9600-x30)

(Equation 3)

Equation 3 refers to the revenue function which is the product of the number of units demanded and the price per unit. The relevance of the revenue function is that it helps in determining the feasible range of units that can be demanded per month. That means that the revenue function must always be positive. To solve for the values that would make the revenue value positive, let us solve for the values of x that would make the revenue zero.

0=x(9600-x30)

x=0, 9600

Then, the revenue function will be evaluated considering the points where x=0 and x=9600 as critical points.

Because the revenue function, R(x), and the linear cost function, C(x), are determined, the profit function, P(x), can be inferred.

Px=Rx-Cx=x9600-x30-(19x+7500)

(Equation 4)

0=x9600-x30-(19x+7500)

x=4515-1589601, x=4515+1589601

Since x must always be an integer, the breakeven points are when x is 24 and 9005. If we assume that the company is currently producing 1000 units per month, then the marginal cost and marginal revenue can be inferred.

marginal cost=C'x=19

marginal revenue=R'x=320-x15

When x is at 1000 units, then the marginal cost would be $19 while the marginal revenue would be $253.33. According to this analysis, the revenue is still much higher than the cost, therefore, the production must still proceed after 1000 units.

## For the optimal production level, there is a necessity to interpret P’(x).

P'x=320-x15-19=301-x15

(Equation 5)

## The optimal production level would be when P’(x) is zero.

0=301-x15

x=4515

## Therefore, the maximum profit would be when the number of produced units is 4515 bottles of wine.

With this report, it must be transparent that evaluation of marginal analysis to business operations. The marginal cost determines the change in the cost of production per unit while the marginal revenue represents the change in revenue per unit change. Thus, relating these two terms across business operations is necessary because it determines the profit per unit change.

## Work Cited

Wine-Searcher. (2014, June 10). Gross Margins: Breaking Down the Price of a Bottle of Wine. Retrieved from http://www.wine-searcher.com/m/2014/06/gross-margins-breaking-down-wine-prices

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