Question

Cost of a Commodity The price \(p,\) in dollars, of a certain commodity and the quantity \(x\) sold follow the demand equation $$ p=-\frac{1}{5} x+200 \quad 0 \leq x \leq 1000 $$ Suppose that the cost \(C,\) in dollars, of producing \(x\) units is $$ C=\frac{\sqrt{x}}{10}+400 $$ Assuming that all items produced are sold, find the cost \(C\) as a function of the price \(p\)

Short answer

The cost function is \(C = \frac{\sqrt{1000 - 5p}}{10} + 400\)

Step-by-step solution

- Understand the Given Demand Equation

The given demand equation is \[ p = -\frac{1}{5}x + 200 \] where \( p \) is the price in dollars and \( x \) is the quantity sold.

- Solve for Quantity \( x \)

Rearrange the demand equation to solve for \( x \) in terms of \( p \): \[ p = -\frac{1}{5}x + 200 \] Subtract 200 from both sides: \[ p - 200 = -\frac{1}{5}x \] Multiply both sides by -5 to isolate \( x \): \[ x = 1000 - 5p \]

- Substitute \( x \) in the Cost Function

The cost equation is given by \[ C = \frac{\sqrt{x}}{10} + 400 \] We substitute \( x = 1000 - 5p \) into this cost equation: \[ C = \frac{\sqrt{1000 - 5p}}{10} + 400 \]

- Simplify the Resulting Function

The final function can be written as: \[ C = \frac{\sqrt{1000 - 5p}}{10} + 400 \] This describes the cost \( C \) as a function of the price \( p \).

Key concepts

Cost Function

In economics, the cost function represents the total cost of producing a specific quantity of goods. In our problem, the cost function is given by \[ C = \frac{\frac{\text{sqrt{x}}}{10} + 400} \]. Here, the variable \( C \) represents the total cost, and \( x \) represents the number of units produced.
To create a more flexible understanding:

• The term \( \frac{\text{sqrt{x}}}{10} \) represents the variable costs. These change depending on how many units are produced.
• The number 400 represents the fixed costs. These remain constant regardless of the number of units produced.

It’s essential to break down the cost function to understand how each part contributes to the total cost.
In our context, we need to find the relationship between the cost and the price given—all items produced are sold. This involves taking further steps to express \( C \) in terms of \( p \) (price).

Price and Quantity Relationship

The relationship between price and quantity is typically expressed through a demand equation. In this exercise, the demand equation is given by \[ p = -\frac{1}{5}x + 200 \ 0 \ x \ 1000 \]. This means the price \( p \) decreases as the quantity \( x \) sold increases.

• If you rearrange the equation, you can solve for \( x \) in terms of \( p \). This can be represented as \( x = 1000 - 5p \).
• This relationship is essential; it shows how changing the price impacts the quantity sold.

Understanding this relationship helps in identifying how price adjustments can influence demand, which is crucial for optimizing production and pricing strategies.

In our specific case, this relationship allows us to reframe the cost equation into one that factors in price and not just quantity. This step-by-step progression is critical to ensure thorough comprehension.

Solving for Variables

Solving for a variable implies isolating that variable on one side of an equation. In our exercise, we need to rearrange the demand equation to solve for the quantity \( x \). Let’s break it down:

• We start with the equation \( p = -\frac{1}{5}x + 200 \).
• Subtracting 200 from both sides results in \( p - 200 = -\frac{1}{5}x \).
• Multiplying both sides by -5 isolates \( x \) giving \( x = 1000 - 5p \).

Once we have \( x \) expressed in terms of \( p \), this new expression can be substituted back into the cost function to find \( C \) as a function of \( p \).
The final substitution in the cost function \[ C = \frac{\text{sqrt{1000 - 5p}}}{10} + 400 \]
showcases how solving for one variable can redefine equations to express relationships in different formats. This exercise is pivotal for understanding how changing one variable affects others in mathematical and economic models.