Difference between revisions of "022 Exam 2 Sample B, Problem 8"

Revision as of 17:46, 15 May 2015

Find the quantity that produces maximum profit, given demand function $p=70-3x$ and cost function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = 120 - 30x + 2x^2.}

Foundations:
Recall that the demand function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} , relates the price per unit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to the number of units sold, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Moreover, we have several important important functions:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x)} , the total cost to produce Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)} , the total revenue (or gross receipts) from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)} , the total profit from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units.
In particular, we have the relations
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)=R(x)-C(x),}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)=x\cdot p(x).}
Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero.

Solution:

Step 1:
Find the Profit Function: We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)\,=\,x\cdot p(x)\,=\,x\cdot (70-3x)\,=\,70x-3x^2.}
From this,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)\,=\,R(x)-C(x)\,=\,70x-3x^2- \left(120 - 30x + 2x^2 \right)\,=\,-120 + 100 x - 5 x^2 .}

Step 2:
Find the Maximum: The equation for marginal revenue is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)\,=\,-120 + 100 x - 5 x^2 .}
Applying our power rule to each term, we find
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(x)\,=\,100-10x\,=\,8(15-x).}
The only root of this occurs at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=10} , and this is our production level to achieve maximum profit.

Final Answer:
Maximum profit occurs when we produce 10 items.

Return to Sample Exam

@@ 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>''']]

Difference between revisions of "022 Exam 2 Sample B, Problem 8"

Revision as of 17:46, 15 May 2015

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