022 Exam 2 Sample B, Problem 8 - Revision history

MathAdmin at 00:32, 18 May 2015

2015-05-18T00:32:29Z

MathAdmin at 15:00, 17 May 2015

2015-05-17T15:00:04Z

MathAdmin at 14:59, 17 May 2015

2015-05-17T14:59:42Z

MathAdmin at 01:48, 16 May 2015

2015-05-16T01:48:04Z

MathAdmin at 01:46, 16 May 2015

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MathAdmin at 01:41, 16 May 2015

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MathAdmin: Created page with " Find the quantity that produces maximum profit, given demand function $p = 70 - 3x$ and cost function
2015-05-16T01:40:03Z

Created page with "<span class="exam"> Find the quantity that produces maximum profit, given demand function <math style="vertical-align: -15%">p = 70 - 3x</math> and cost function <math..."

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<span class="exam">
Find the quantity that produces maximum profit, given demand function <math style="vertical-align: -15%">p = 70 - 3x</math> and cost function  <math style="vertical-align: -8%">C = 120 - 30x + 2x^2.</math>

@@ Line 45: / Line 45: @@
 !Step 2: &nbsp;
 |-
-|'''Find the Maximum:''' The equation for marginal revenue is
+|'''Find the Maximum:''' The equation for profit at a given production level is
 |-

@@ Line 56: / Line 56: @@
 ::<math>P'(x)\,=\,100-10x\,=\,10(10-x).</math>
 |-
-|The only root of this occurs at <math style="vertical-align: 0%">x=10</math>, and this is our production level to achieve maximum profit.
+|The only root of this occurs at <math style="vertical-align: -3%">x=10</math>, and this is our production level to achieve maximum profit.
 |-
+|

@@ Line 56: / Line 56: @@
 ::<math>P'(x)\,=\,100-10x\,=\,10(10-x).</math>
 |-
-|The only root of this occurs at <math style="vertical-align: -5%">x=10</math>, and this is our production level to achieve maximum profit.
+|The only root of this occurs at <math style="vertical-align: 0%">x=10</math>, and this is our production level to achieve maximum profit.
 |-
+|

@@ Line 54: / Line 54: @@
 |-
+|
-::<math>P'(x)\,=\,100-10x\,=\,8(15-x).</math>
+::<math>P'(x)\,=\,100-10x\,=\,10(10-x).</math>
 |-
 |The only root of this occurs at <math style="vertical-align: -5%">x=10</math>, and this is our production level to achieve maximum profit.

@@ Line 34: / Line 34: @@
 |-
+|
-::<math>R(x)\,=\,x\cdot p(x)\,=\,x\cdot (90-3x)\,=\,90x-3x^2.</math>
+::<math>R(x)\,=\,x\cdot p(x)\,=\,x\cdot (70-3x)\,=\,70x-3x^2.</math>
 |-
 |From this,
 |-
+|
-::<math>P(x)\,=\,R(x)-C(x)\,=\,90x-3x^2- \left(200-30x+x^2 \right)\,=\,120x-4x^2-200 .</math>
+::<math>P(x)\,=\,R(x)-C(x)\,=\,70x-3x^2- \left(120 - 30x + 2x^2 \right)\,=\,-120 + 100 x - 5 x^2 .</math>
 |}
@@ Line 49: / Line 49: @@
 |-
+|
-::<math>P(x)\,=\,120x-4x^2-200 .</math>
+::<math>P(x)\,=\,-120 + 100 x - 5 x^2 .</math>
 |-
 |Applying our power rule to each term, we find
 |-
+|
-::<math>P'(x)\,=\,120-8x\,=\,8(15-x).</math>
+::<math>P'(x)\,=\,100-10x\,=\,8(15-x).</math>
 |-
-|The only root of this occurs at <math style="vertical-align: -5%">x=15</math>, and this is our production level to achieve maximum profit.
+|The only root of this occurs at <math style="vertical-align: -5%">x=10</math>, and this is our production level to achieve maximum profit.
 |-
+|
@@ Line 64: / Line 64: @@
 !Final Answer: &nbsp;
 |-
-|Maximum profit occurs when we produce 15 items.
+|Maximum profit occurs when we produce 10 items.
 |}
 [[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:41, 16 May 2015
Line 1:		Line 1:
	<span class="exam">		<span class="exam">
	Find the quantity that produces maximum profit, given demand function <math style="vertical-align: -15%">p = 70 - 3x</math> and cost function  <math style="vertical-align: -8%">C = 120 - 30x + 2x^2.</math>		Find the quantity that produces maximum profit, given demand function <math style="vertical-align: -15%">p = 70 - 3x</math> and cost function  <math style="vertical-align: -8%">C = 120 - 30x + 2x^2.</math>
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	! Foundations:
		+	\|-
		+	\|Recall that the '''demand function''', <math style="vertical-align: -25%">p(x)</math>, relates the price per unit <math style="vertical-align: -17%">p</math> to the number of units sold, <math style="vertical-align: 0%">x</math>.
		+	Moreover, we have several important important functions:
		+	\|-
		+	\|
		+	*<math style="vertical-align: -20%">C(x)</math>, the '''total cost''' to produce <math style="vertical-align: 0%">x</math> units;<br>
		+	*<math style="vertical-align: -20%">R(x)</math>, the '''total revenue''' (or gross receipts) from producing <math style="vertical-align: 0%">x</math> units;<br>
		+	*<math style="vertical-align: -20%">P(x)</math>, the '''total profit''' from producing <math style="vertical-align: 0%">x</math> units.<br>
		+	\|-
		+	\|In particular, we have the relations
		+	\|-
		+	\|
		+	::<math>P(x)=R(x)-C(x),</math>
		+	\|-
		+	\|and
		+	\|-
		+	\|
		+	::<math>R(x)=x\cdot p(x).</math>
		+	\|-
		+	\|Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero.
		+
		+	\|}
		+	'''Solution:'''
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Step 1:
		+	\|-
		+	\|'''Find the Profit Function:''' We have
		+	\|-
		+	\|
		+	::<math>R(x)\,=\,x\cdot p(x)\,=\,x\cdot (90-3x)\,=\,90x-3x^2.</math>
		+	\|-
		+	\|From this,
		+	\|-
		+	\|
		+	::<math>P(x)\,=\,R(x)-C(x)\,=\,90x-3x^2- \left(200-30x+x^2 \right)\,=\,120x-4x^2-200 .</math>
		+	\|}
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Step 2:
		+	\|-
		+	\|'''Find the Maximum:''' The equation for marginal revenue is
		+
		+	\|-
		+	\|
		+	::<math>P(x)\,=\,120x-4x^2-200 .</math>
		+	\|-
		+	\|Applying our power rule to each term, we find
		+	\|-
		+	\|
		+	::<math>P'(x)\,=\,120-8x\,=\,8(15-x).</math>
		+	\|-
		+	\|The only root of this occurs at <math style="vertical-align: -5%">x=15</math>, and this is our production level to achieve maximum profit.
		+	\|-
		+	\|
		+	\|}
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Final Answer:
		+	\|-
		+	\|Maximum profit occurs when we produce 15 items.
		+	\|}
		+
		+
		+	[[022_Exam_2_Sample_B\|'''<u>Return to Sample Exam</u>''']]