Questions: A manufacturer produces a product at a cost of 25.80 per unit. The manufacturer has a fixed cost of 900.00 per day. Each unit retails for 38.00. Let x represent the number of units produced in a 5-day period. (a) Write the total cost C as a function of x. C(x) = (b) Write the revenue R as a function of x. R(x) = (c) Write the profit P as a function of x. (Hint: The profit function is given by P(x) = R(x) - C(x).) P(x) =

A manufacturer produces a product at a cost of 25.80 per unit. The manufacturer has a fixed cost of 900.00 per day. Each unit retails for 38.00. Let x represent the number of units produced in a 5-day period.

(a) Write the total cost C as a function of x.

C(x) = 

(b) Write the revenue R as a function of x.

R(x) = 

(c) Write the profit P as a function of x. (Hint: The profit function is given by P(x) = R(x) - C(x).)

P(x) =
Transcript text: A manufacturer produces a product at a cost of $25.80 per unit. The manufacturer has a fixed cost of $900.00 per day. Each unit retails for $38.00. Let x represent the number of units produced in a 5-day period. (a) Write the total cost C as a function of x. C(x) = (b) Write the revenue R as a function of x. R(x) = (c) Write the profit P as a function of x. (Hint: The profit function is given by P(x) = R(x) - C(x).) P(x) =
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Solution

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Solution Steps

Solution Approach

(a) The total cost \( C(x) \) includes the fixed cost and the variable cost per unit. The fixed cost is $900.00 per day, and since we are considering a 5-day period, the fixed cost for 5 days is \( 900 \times 5 \). The variable cost per unit is $25.80, so the total cost function \( C(x) \) is the sum of the fixed cost for 5 days and the variable cost multiplied by the number of units \( x \).

(b) The revenue \( R(x) \) is the total income from selling \( x \) units. Each unit retails for $38.00, so the revenue function \( R(x) \) is the retail price per unit multiplied by the number of units \( x \).

(c) The profit \( P(x) \) is the difference between the revenue and the total cost. Therefore, the profit function \( P(x) \) is given by \( P(x) = R(x) - C(x) \).

Step 1: Total Cost Calculation

The total cost \( C(x) \) for producing \( x \) units over a 5-day period is calculated as follows:

\[ C(x) = \text{Fixed Cost for 5 Days} + \text{Variable Cost per Unit} \times x \]

Given that the fixed cost per day is \( 900 \) and the variable cost per unit is \( 25.80 \):

\[ C(x) = 900 \times 5 + 25.80 \times x = 4500 + 25.80x \]

For \( x = 100 \):

\[ C(100) = 4500 + 25.80 \times 100 = 4500 + 2580 = 7080.0 \]

Step 2: Revenue Calculation

The revenue \( R(x) \) from selling \( x \) units is given by:

\[ R(x) = \text{Retail Price per Unit} \times x \]

With a retail price of \( 38.00 \):

\[ R(x) = 38.00 \times x \]

For \( x = 100 \):

\[ R(100) = 38.00 \times 100 = 3800.0 \]

Step 3: Profit Calculation

The profit \( P(x) \) is determined by the difference between revenue and total cost:

\[ P(x) = R(x) - C(x) \]

Substituting the values for \( x = 100 \):

\[ P(100) = 3800.0 - 7080.0 = -3280.0 \]

Final Answer

The calculations yield the following results:

  • Total Cost \( C(100) = 7080.0 \)
  • Revenue \( R(100) = 3800.0 \)
  • Profit \( P(100) = -3280.0 \)

Thus, the final answers are: \[ \boxed{C(100) = 7080.0} \] \[ \boxed{R(100) = 3800.0} \] \[ \boxed{P(100) = -3280.0} \]

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