You operate a gaming Web site, www .m\udbeast.net, where...

You operate a gaming Web site, www .mudbeast.net, where users must pay a small fee to log on. When you charged $$\$ 2$$ the demand was 280 log-ons per month. When you lowered the price to $$\$ 1.50$$, the demand increased to 560 logons per month. a. Construct a linear demand function for your Web site and hence obtain the monthly revenue $R$ as a function of the log-on fee $x$. b. Your Internet provider charges you a monthly fee of $$\$ 30$$ to maintain your site. Express your monthly profit $P$ as a function of the log-on fee $x$, and hence determine the logon fee you should charge to obtain the largest possible monthly profit. What is the largest possible monthly profit? HINT [See Example 4.]

You operate a gaming Web site, www .mudbeast.net, where users must pay a small fee to log on. When you charged $$\$ 2$$ the demand was 280 log-ons per month. When you lowered the price to $$\$ 1.50$$, the demand increased to 560 logons per month.
a. Construct a linear demand function for your Web site and hence obtain the monthly revenue $R$ as a function of the log-on fee $x$.
b. Your Internet provider charges you a monthly fee of $$\$ 30$$ to maintain your site. Express your monthly profit $P$ as a function of the log-on fee $x$, and hence determine the logon fee you should charge to obtain the largest possible monthly profit. What is the largest possible monthly profit? HINT [See Example 4.]

Key Concepts

Optimization of Profit

Profit optimization involves determining the price point that maximizes the profit function. Typically, this is achieved by using calculus techniques, such as setting the derivative of the profit function equal to zero to find critical points, and then verifying which of these points yields the maximum profit. This process is fundamental in both economics and operations management, driving decisions on pricing and production levels.

Profit Function

The profit function is defined as the revenue function minus the costs associated with producing or offering the service. In this context, it involves subtracting a fixed cost from the revenue. Understanding the profit function is critical for businesses to ascertain the financial viability of pricing strategies by revealing how much profit is made after all expenses are considered.

Linear Demand Function

This concept involves modeling the relationship between the price of a product and the quantity demanded using a straight line. Given two data points for price and demand, one can determine the slope and y-intercept of the linear function, forming an equation that estimates demand for any given price. It is a fundamental tool in economics and helps in predicting consumer behavior based on price changes.

Revenue Function

The revenue function is constructed by multiplying the price of the product by its quantity demanded. It quantifies the total income generated from sales before subtracting any costs. This function is essential in business analysis as it provides insight into how varying prices impact overall sales income.

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