Questions: Marginal Cost, Revenue, and Profit for Producing LCD TV wholesale unit price (p) by the following equation. [ p=-0.005 x+170 ] The weekly total cost (in dollars) incurred by Pulsar for producing (x) [ C(x)=0.000003 x^3-0.02 x^2+110 x+75,000 ] (a) Find the revenue function (R). [ R(x)= ] Find the profit function (P). [ P(x)= ] (b) Find the marginal cost function (C'). [ C'(x)= ] Find the marginal revenue function (R'). [ R'(x)= ] Find the marginal profit function (P'). [ P'(x)= ] (c) Compute the following values. (Round your answers to two decimal places) [ C'(1,500)= ] [ R'(1,500)= ] [ P'(1,500)= ] (d) Sketch (C(x)).

Marginal Cost, Revenue, and Profit for Producing LCD TV wholesale unit price (p) by the following equation. [ p=-0.005 x+170 ] The weekly total cost (in dollars) incurred by Pulsar for producing (x) [ C(x)=0.000003 x^3-0.02 x^2+110 x+75,000 ] (a) Find the revenue function (R). [ R(x)= ] Find the profit function (P). [ P(x)= ] (b) Find the marginal cost function (C'). [ C'(x)= ] Find the marginal revenue function (R'). [ R'(x)= ] Find the marginal profit function (P'). [ P'(x)= ] (c) Compute the following values. (Round your answers to two decimal places) [ C'(1,500)= ] [ R'(1,500)= ] [ P'(1,500)= ] (d) Sketch (C(x)).
Transcript text: Marginal Cost, Revenus, and Profit for Producing LCD TV wholesale unit price $p$ by the following equation. \[ p=-0.005 x+170 \] The weekly total cost (in dollars) incurred by Pulsar for producing $x$ \[ C(x)=0.000003 x^{3}-0.02 x^{2}+110 x+75,000 \] (a) Find the revenue function $R$. \[ R(x)= \] Find the profit function $P$. \[ P(x)= \] (b) Find the marginal cost function $C^{\prime}$. \[ C^{\prime}(x)= \] Find the marginal revenue function $R^{\prime}$. \[ R^{\prime}(x)= \] Find the marginal profit function $P^{\prime}$. \[ P^{\prime}(x)= \] (c) Compute the following values. (Round your answers to two decimal places) \[ \begin{array}{l} C^{\prime}(1,500)=\square \\ R^{\prime}(1,500)=\square \\ P^{\prime}(1,500)=\square \end{array} \] (d) Sketch $C(x)$.
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Solution

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Solution Steps

Step 1: Find the revenue function \( R(x) \)

The revenue function \( R(x) \) is given by the product of the price per unit \( p \) and the number of units \( x \). The price per unit is given by \( p = -0.005x + 170 \).

\[ R(x) = p \cdot x = (-0.005x + 170) \cdot x = -0.005x^2 + 170x \]

Step 2: Find the profit function \( P(x) \)

The profit function \( P(x) \) is the difference between the revenue function \( R(x) \) and the cost function \( C(x) \).

\[ P(x) = R(x) - C(x) = (-0.005x^2 + 170x) - (0.000003x^3 - 0.02x^2 + 110x + 75000) \]

Simplifying, we get:

\[ P(x) = -0.000003x^3 + 0.015x^2 + 60x - 75000 \]

Step 3: Find the marginal cost function \( C'(x) \)

The marginal cost function \( C'(x) \) is the derivative of the cost function \( C(x) \).

\[ C(x) = 0.000003x^3 - 0.02x^2 + 110x + 75000 \]

Taking the derivative:

\[ C'(x) = 0.000009x^2 - 0.04x + 110 \]

Final Answer

  • Revenue function: \( R(x) = -0.005x^2 + 170x \)
  • Profit function: \( P(x) = -0.000003x^3 + 0.015x^2 + 60x - 75000 \)
  • Marginal cost function: \( C'(x) = 0.000009x^2 - 0.04x + 110 \)

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