document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197579>Question 1197579</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The price  -  demand equation and the costfunction for the production of<BR>\n" );
document.write( "honey is given, respectively, by<BR>\n" );
document.write( "x = 5,000 - 100p and C(x) = 2,500 + 4x+ 0.01x2<BR>\n" );
document.write( "where x is the number of bottles that can be sold at a price of $p per<BR>\n" );
document.write( "bottle and C(x) is the total cost (in dollars) of producing x bottles.\r<BR>\n" );
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document.write( "a) Express the price p as a function of the demand x, and find the domain of<BR>\n" );
document.write( "this function. <BR>\n" );
document.write( "b) Find the marginal cost. <BR>\n" );
document.write( "c) Find the revenue function and state its domain. <BR>\n" );
document.write( "d) Find the marginal revenue. <BR>\n" );
document.write( "e) Find R′(2,000) and R′(3,000) and interpret these quantities. <BR>\n" );
document.write( "f) Find the profit function in terms of x. <BR>\n" );
document.write( "g) Find the marginal profit. <BR>\n" );
document.write( "h) Find P′(1,000) and P′(1,500) and interpret these quantities. </SPAN>\n" );
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document.write( "**a) Express the price p as a function of the demand x, and find the domain of this function.**\r<BR>\n" );
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document.write( "* Given: x = 5,000 - 100p<BR>\n" );
document.write( "* Solve for p:<BR>\n" );
document.write( "    * 100p = 5,000 - x<BR>\n" );
document.write( "    * p = (5,000 - x) / 100<BR>\n" );
document.write( "    * p = 50 - 0.01x\r<BR>\n" );
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document.write( "* **Domain:** <BR>\n" );
document.write( "    * The demand (x) must be non-negative (you can't sell a negative number of bottles).<BR>\n" );
document.write( "    * The price (p) must also be non-negative.<BR>\n" );
document.write( "    * <BR>\n" );
document.write( "        * 50 - 0.01x ≥ 0<BR>\n" );
document.write( "        * 0.01x ≤ 50<BR>\n" );
document.write( "        * x ≤ 5,000\r<BR>\n" );
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document.write( "    * **Therefore, the domain of the price function is 0 ≤ x ≤ 5,000**\r<BR>\n" );
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document.write( "**b) Find the marginal cost.**\r<BR>\n" );
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document.write( "* **Marginal Cost:** The derivative of the cost function with respect to x.<BR>\n" );
document.write( "* C(x) = 2,500 + 4x + 0.01x^2<BR>\n" );
document.write( "* C'(x) = 4 + 0.02x\r<BR>\n" );
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document.write( "**c) Find the revenue function and state its domain.**\r<BR>\n" );
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document.write( "* **Revenue (R) = Price (p) * Quantity (x)**<BR>\n" );
document.write( "* R(x) = p * x <BR>\n" );
document.write( "* R(x) = (50 - 0.01x) * x <BR>\n" );
document.write( "* R(x) = 50x - 0.01x^2\r<BR>\n" );
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document.write( "* **Domain:** The domain of the revenue function is the same as the domain of the price function, which is 0 ≤ x ≤ 5,000.\r<BR>\n" );
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document.write( "**d) Find the marginal revenue.**\r<BR>\n" );
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document.write( "* **Marginal Revenue:** The derivative of the revenue function with respect to x.<BR>\n" );
document.write( "* R(x) = 50x - 0.01x^2<BR>\n" );
document.write( "* R'(x) = 50 - 0.02x\r<BR>\n" );
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document.write( "**e) Find R′(2,000) and R′(3,000) and interpret these quantities.**\r<BR>\n" );
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document.write( "* R'(2,000) = 50 - 0.02 * 2,000 = 50 - 40 = 10<BR>\n" );
document.write( "    * When producing 2,000 bottles, the revenue is increasing at a rate of $10 per additional bottle.\r<BR>\n" );
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document.write( "* R'(3,000) = 50 - 0.02 * 3,000 = 50 - 60 = -10<BR>\n" );
document.write( "    * When producing 3,000 bottles, the revenue is decreasing at a rate of $10 per additional bottle.\r<BR>\n" );
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document.write( "**f) Find the profit function in terms of x.**\r<BR>\n" );
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document.write( "* **Profit (P) = Revenue (R) - Cost (C)**<BR>\n" );
document.write( "* P(x) = R(x) - C(x)<BR>\n" );
document.write( "* P(x) = (50x - 0.01x^2) - (2,500 + 4x + 0.01x^2)<BR>\n" );
document.write( "* P(x) = 50x - 0.01x^2 - 2,500 - 4x - 0.01x^2<BR>\n" );
document.write( "* P(x) = 46x - 0.02x^2 - 2,500\r<BR>\n" );
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document.write( "**g) Find the marginal profit.**\r<BR>\n" );
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document.write( "* **Marginal Profit:** The derivative of the profit function with respect to x.<BR>\n" );
document.write( "* P(x) = 46x - 0.02x^2 - 2,500<BR>\n" );
document.write( "* P'(x) = 46 - 0.04x\r<BR>\n" );
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document.write( "**h) Find P′(1,000) and P′(1,500) and interpret these quantities.**\r<BR>\n" );
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document.write( "* P'(1,000) = 46 - 0.04 * 1,000 = 46 - 40 = 6<BR>\n" );
document.write( "    * When producing 1,000 bottles, the profit is increasing at a rate of $6 per additional bottle.\r<BR>\n" );
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document.write( "* P'(1,500) = 46 - 0.04 * 1,500 = 46 - 60 = -14<BR>\n" );
document.write( "    * When producing 1,500 bottles, the profit is decreasing at a rate of $14 per additional bottle.<BR>\n" );
document.write( "**a) Express the price p as a function of the demand x, and find the domain of this function.**\r<BR>\n" );
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document.write( "* **Solve the price-demand equation for p:**<BR>\n" );
document.write( "    * x = 5,000 - 100p<BR>\n" );
document.write( "    * 100p = 5,000 - x<BR>\n" );
document.write( "    * p = (5,000 - x) / 100<BR>\n" );
document.write( "    * p = 50 - 0.01x\r<BR>\n" );
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document.write( "* **Domain of the price function:**<BR>\n" );
document.write( "    * The demand (x) must be non-negative (you cannot sell a negative number of bottles).<BR>\n" );
document.write( "    * 0 ≤ x ≤ 5,000 (The maximum demand occurs when the price is 0)\r<BR>\n" );
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document.write( "**b) Find the marginal cost.**\r<BR>\n" );
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document.write( "* **Marginal Cost (MC):** The derivative of the cost function with respect to x.<BR>\n" );
document.write( "    * MC = C'(x) = d/dx (2,500 + 4x + 0.01x²)<BR>\n" );
document.write( "    * MC = 4 + 0.02x\r<BR>\n" );
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document.write( "**c) Find the revenue function and state its domain.**\r<BR>\n" );
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document.write( "* **Revenue (R):** Price per unit * Number of units sold<BR>\n" );
document.write( "    * R(x) = p * x <BR>\n" );
document.write( "    * R(x) = (50 - 0.01x) * x <BR>\n" );
document.write( "    * R(x) = 50x - 0.01x²\r<BR>\n" );
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document.write( "* **Domain of the revenue function:** <BR>\n" );
document.write( "    * Same as the domain of the price function: 0 ≤ x ≤ 5,000\r<BR>\n" );
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document.write( "**d) Find the marginal revenue.**\r<BR>\n" );
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document.write( "* **Marginal Revenue (MR):** The derivative of the revenue function with respect to x.<BR>\n" );
document.write( "    * MR = R'(x) = d/dx (50x - 0.01x²)<BR>\n" );
document.write( "    * MR = 50 - 0.02x\r<BR>\n" );
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document.write( "**e) Find R′(2,000) and R′(3,000) and interpret these quantities.**\r<BR>\n" );
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document.write( "* **R'(2,000):** <BR>\n" );
document.write( "    * R'(2,000) = 50 - 0.02 * 2,000 = 50 - 40 = 10<BR>\n" );
document.write( "    * Interpretation: When 2,000 bottles are sold, the revenue is increasing at a rate of $10 per additional bottle sold.\r<BR>\n" );
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document.write( "* **R'(3,000):** <BR>\n" );
document.write( "    * R'(3,000) = 50 - 0.02 * 3,000 = 50 - 60 = -10<BR>\n" );
document.write( "    * Interpretation: When 3,000 bottles are sold, the revenue is decreasing at a rate of $10 per additional bottle sold.\r<BR>\n" );
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document.write( "**f) Find the profit function in terms of x.**\r<BR>\n" );
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document.write( "* **Profit (P):** Revenue - Cost<BR>\n" );
document.write( "    * P(x) = R(x) - C(x)<BR>\n" );
document.write( "    * P(x) = (50x - 0.01x²) - (2,500 + 4x + 0.01x²)<BR>\n" );
document.write( "    * P(x) = 46x - 0.02x² - 2,500\r<BR>\n" );
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document.write( "**g) Find the marginal profit.**\r<BR>\n" );
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document.write( "* **Marginal Profit (MP):** The derivative of the profit function with respect to x.<BR>\n" );
document.write( "    * MP = P'(x) = d/dx (46x - 0.02x² - 2,500)<BR>\n" );
document.write( "    * MP = 46 - 0.04x\r<BR>\n" );
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document.write( "**h) Find P′(1,000) and P′(1,500) and interpret these quantities.**\r<BR>\n" );
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document.write( "* **P'(1,000):** <BR>\n" );
document.write( "    * P'(1,000) = 46 - 0.04 * 1,000 = 46 - 40 = 6<BR>\n" );
document.write( "    * Interpretation: When 1,000 bottles are sold, the profit is increasing at a rate of $6 per additional bottle sold.\r<BR>\n" );
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document.write( "* **P'(1,500):** <BR>\n" );
document.write( "    * P'(1,500) = 46 - 0.04 * 1,500 = 46 - 60 = -14<BR>\n" );
document.write( "    * Interpretation: When 1,500 bottles are sold, the profit is decreasing at a rate of $14 per additional bottle sold.<BR>\n" );
document.write( " </SPAN>\n" );
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