The price p, in dollars, of a certain product and the...

The price $p,$ in dollars, of a certain product and the quantity $x$ sold obey the demand equation $$p=-\frac{1}{4} x+100 \quad 0 \leq x \leq 400$$ Suppose that the cost $C$, in dollars, of producing $x$ units is $$C=\frac{\sqrt{x}}{25}+600$$ Assuming that all items produced are sold, find the cost $C$ as a function of the price $p$ [Hint: Solve for $x$ in the demand equation and then form the composite. $]$

The price $p,$ in dollars, of a certain product and the quantity $x$ sold obey the demand equation
$$p=-\frac{1}{4} x+100 \quad 0 \leq x \leq 400$$
Suppose that the cost $C$, in dollars, of producing $x$ units is
$$C=\frac{\sqrt{x}}{25}+600$$
Assuming that all items produced are sold, find the cost $C$ as a function of the price $p$ [Hint: Solve for $x$ in the demand equation and then form the composite. $]$

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Key Concepts

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations and solving for variables using operations such as addition, subtraction, multiplication, division, and the extraction of roots. This is an essential skill for isolating variables and composing functions effectively.

Domain Considerations

Domain considerations involve understanding and applying the restrictions on the values that variables can take in a function. This ensures that when functions are composed, the resulting expression is valid and respects the original constraints laid out in the problem.

Composite Functions

Composite functions involve combining two functions where the output of one function becomes the input of another. This concept is critical when you need to express a function in terms of a different variable by substituting one function into another.

Inverse Functions

Inverse functions are used to reverse the roles of dependent and independent variables. When you solve for a variable in one equation with respect to another (for example, expressing quantity in terms of price), you are essentially finding an inverse relationship that allows you to substitute into another function.

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