The airline cost for x thousand passengers to travel round...

The airline cost for $x$ thousand passengers to travel round trip from New York to Atlanta is given by $$C(x)=\sqrt{0.3 x+1}$$ where $C(x)$ is measured in millions of dollars and $x \geq 0 .$ a. Find the airline's cost for $10,000$ passengers $(x=10)$ to travel from New York to Atlanta. b. If the airline charges $\$ 320$ per passenger, find the profit made by the airline for flying $10,000$ passengers from New York to Atlanta. c. Approximate the number of passengers who traveled from New York to Atlanta if the total cost for the airline was $\$ 4$ million.

The airline cost for $x$ thousand passengers to travel round trip from New York to Atlanta is given by
$$C(x)=\sqrt{0.3 x+1}$$
where $C(x)$ is measured in millions of dollars and $x \geq 0 .$
a. Find the airline's cost for $10,000$ passengers $(x=10)$ to travel from New York to Atlanta.
b. If the airline charges $\$ 320$ per passenger, find the profit made by the airline for flying $10,000$ passengers from New York to Atlanta.
c. Approximate the number of passengers who traveled from New York to Atlanta if the total cost for the airline was $\$ 4$ million.

Key Concepts

Function Evaluation

This concept involves substituting a specific input value into a function to determine the corresponding output. It is a fundamental technique used to interpret and analyze the behavior of functions, such as determining the cost or revenue when given a certain number of units or passengers.

Interpreting Cost Functions

A cost function mathematically represents how the cost incurred by a company varies with the level of activity or output—in this case, the number of passengers. Understanding such functions is crucial for assessing expenses and making strategic decisions in economics and business operations.

Revenue, Cost, and Profit Relationships

This concept details the relationship between revenue, cost, and profit. Revenue is generated by sales (or in this instance, charging a fee per passenger), while cost is the expense of providing the service. Profit is calculated as the difference between revenue and cost. Analyzing these relationships helps companies identify financial performance and make pricing decisions.

Solving Equations Involving Square Roots

Equations that include square root expressions require special algebraic techniques to isolate the square root term and then eliminate it (often by squaring both sides of the equation). This is important for determining unknown variables when the function has a square root component, as seen in many applied contexts such as physics, economics, and engineering.

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