Cost, Revenue, and Profit The cost C (in dollars) of...

Cost, Revenue, and Profit The cost $C$ (in dollars) of producing $x$ dome tents is $C=200+45 x$. The revenue $R$ (in dollars) for selling $x$ dome tents is $R=120 x-x^{2}$, where $0 \leq x \leq 60$. The profit $P$ is the difference between revenue and cost. (a) Perform the subtraction required to find the polynomial representing profit $P$. (b) Determine the profit when 40 tents are produced and sold. (c) Find the change in profit when the number of tents produced and sold increases from 40 to 50 .

Cost, Revenue, and Profit The cost $C$ (in dollars) of producing $x$ dome tents is $C=200+45 x$. The revenue $R$ (in dollars) for selling $x$ dome tents is $R=120 x-x^{2}$, where $0 \leq x \leq 60$. The profit $P$ is the difference between revenue and cost.
(a) Perform the subtraction required to find the polynomial representing profit $P$.
(b) Determine the profit when 40 tents are produced and sold.
(c) Find the change in profit when the number of tents produced and sold increases from 40 to 50 .

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Key Concepts

Change in Function Values

Analyzing the change in function values involves calculating the difference in outputs when the input changes by a certain amount. This concept is important in understanding incremental changes, such as determining how profit varies when production levels increase or decrease.

Evaluating Functions at Specific Values

Evaluating a function means substituting a specific numerical value into the function to compute its output. This technique is widely used in mathematical applications to determine real-world quantities, such as computing the profit for a given level of production or sales.

Profit Function

A profit function is used to determine the net gain or loss in a scenario by subtracting total costs from total revenues. This function serves as a key tool in business and economics to assess the performance of an operation, helping decision-makers determine the outcome of production and sales activities.

Polynomial Subtraction and Simplification

Polynomial subtraction involves the operation of subtracting one polynomial from another and then combining like terms to simplify the resulting expression. This process is crucial in algebra when modifying and combining functions, such as when determining profit by reducing the revenue polynomial by the cost polynomial.

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