Difference between revisions of "022 Exam 1 Sample A, Problem 5"

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(Created page with "Find the marginal revenue and marginal profit at <math style="vertical-align: -3%">x=4</math>, given the demand function ::<math>p=\frac{200}{\sqrt{x}}</math> <span class=...")
 
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Find the marginal revenue and marginal profit at <math style="vertical-align: -3%">x=4</math>, given the demand function  
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<span class="exam">Find the marginal revenue and marginal profit at <math style="vertical-align: -3%">x=4</math>, given the demand function  
  
 
::<math>p=\frac{200}{\sqrt{x}}</math>  
 
::<math>p=\frac{200}{\sqrt{x}}</math>  
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! Foundations: &nbsp;  
 
! Foundations: &nbsp;  
 
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|-
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|Recall that the demand function, <math style="vertical-align: -25%">p(x)</math>, relates the price per unit <math style="vertical-align: -20%">p</math> to the number of units sold, <math style="vertical-align: 0%">x</math>.
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|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:
 
Moreover, we have several important important functions:
 
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::<math>R(x)=x\cdot p(x).</math>
 
::<math>R(x)=x\cdot p(x).</math>
 
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|Finally, marginal profit at <math style="vertical-align: -20%">x_0</math> units is defined to be the effective cost of the next unit produced, and is precisely <math style="vertical-align: -25%">P'(x_0)</math>.  Similarly, marginal revenue or cost would be <math style="vertical-align: -25%">R'(x_0)</math> or <math style="vertical-align: -25%">C'(x_0)</math>, respectively.
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|Finally, marginal profit at <math style="vertical-align: -20%">x_0</math> units is defined to be the effective cost of the next unit produced, and is precisely <math style="vertical-align: -22%">P'(x_0)</math>.  Similarly, marginal revenue or cost would be <math style="vertical-align: -22%">R'(x_0)</math> or <math style="vertical-align: -22%">C'(x_0)</math>, respectively.
  
 
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
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|'''Write the Basic Equation:'''  
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|'''Find the Important Functions:''' We have
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|
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::<math>R(x)\,\,=\,\,x\cdot p(x)\,\,=\,\,x\cdot \frac{200}{\sqrt {x}}\,\,=\,\,200 \sqrt{x}.</math>
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|-
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|From this,
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|-
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|
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::<math>P(x)\,\,=\,\,R(x)-C(x)\,\,=\,\,200 \sqrt{x}- \left( 100+15x+3x^{2} \right) .</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;
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|-
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|'''Find the Marginal Revenue and Profit:''' The marginal revenue is
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|-
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|
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::<math>R'(x)\,\,=\,\,\left(200 \sqrt{x}\right) '\,\,=\,\,200\cdot \frac{1}{2}\cdot\frac{1}{\sqrt{x}}\,\,=\,\,\frac{100}{\sqrt{x}}, </math>
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|-
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|while the marginal profit is
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|-
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|
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::<math>P'(x)\,\,=\,\,R'(x)-C'(x)\,\,=\,\,\frac{100}{\sqrt{x}}-(30+6x).</math>
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|-
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|At <math style="vertical-align: -3%">x=4</math>, we find
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|
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::<math>R'(4)\,\,=\,\,\frac{100}{\sqrt{4}}\,\,=\,\,50. </math>
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|-
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|On the other hand, marginal profit is
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|-
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|
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::<math>P'(4)\,\,=\,\,\frac{100}{\sqrt{4}}-(30+6x(4))\,\,=\,\,50-54\,\,=\,\,-4.</math>
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|Thus, it is <u>''not''</u> profitable to produce another item.
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;
 
!Final Answer: &nbsp;
 
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|With units, we have that the ladder is sliding down the wall at <math style="vertical-align: -25%">-3/2</math>&thinsp; feet per second.
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|
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::<math>R'(4)\,\,=\,\,50;\qquad P'(4)\,\,=\,\,-4. </math>
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|-
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|Thus, it is <u>''not''</u> profitable to produce another item.
 
|}
 
|}
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[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]

Revision as of 19:12, 13 April 2015

Find the marginal revenue and marginal profit 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=4} , given 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=\frac{200}{\sqrt{x}}}

and the 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(x)=100+15x+3x^{2}.}

Should the firm produce one more item under these conditions? Justify your answer.

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).}
Finally, marginal profit 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_0} units is defined to be the effective cost of the next unit produced, and is precisely 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_0)} . Similarly, marginal revenue or cost would be 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_0)} or 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_0)} , respectively.

 Solution:

Step 1:  
Find the Important Functions: 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 \frac{200}{\sqrt {x}}\,\,=\,\,200 \sqrt{x}.}
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)\,\,=\,\,200 \sqrt{x}- \left( 100+15x+3x^{2} \right) .}
Step 1:  
Find the Marginal Revenue and Profit: The marginal revenue is
while the marginal profit 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)\,\,=\,\,R'(x)-C'(x)\,\,=\,\,\frac{100}{\sqrt{x}}-(30+6x).}
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=4} , 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 R'(4)\,\,=\,\,\frac{100}{\sqrt{4}}\,\,=\,\,50. }
On the other hand, marginal profit 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'(4)\,\,=\,\,\frac{100}{\sqrt{4}}-(30+6x(4))\,\,=\,\,50-54\,\,=\,\,-4.}
Thus, it is not profitable to produce another item.
Final Answer:  
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'(4)\,\,=\,\,50;\qquad P'(4)\,\,=\,\,-4. }
Thus, it is not profitable to produce another item.


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