Difference between revisions of "Math 22 Business and Economics Applications"

From Math Wiki
Jump to navigation Jump to search
Line 7: Line 7:
 
|-
 
|-
 
|Notice: <math>\overline{C}=\frac{C}{x}=\frac{2x^2+348x+7200}{x}=2x+348+\frac{7200}{x}</math>
 
|Notice: <math>\overline{C}=\frac{C}{x}=\frac{2x^2+348x+7200}{x}=2x+348+\frac{7200}{x}</math>
 +
|-
 +
|Then, <math>\overline{C} '=2-\frac{7200}{x^2}=0</math>, so <math>x^2=3600</math>, so <math>x=\pm\sqrt{3600}=\pm 60=60</math> since <math>x</math> is positive.
 +
|}
 +
 +
'''2)''' Find the price that will maximize profit for the demand and cost functions, where <math>p</math> is the price, <math>x</math> is the number of units, and <math>C</math> is the cost. Given the demand function <math>p(x)=90-x</math> and the cost function <math>C(x)=100+30x</math>.
 +
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|Notice: The revenue function <math>R(x)=x\cdot p(x)=x(90-x)=90x-x^2</math>
 +
|-
 +
|The Profit function is <math>P(x)=R(x)-C(x)=90x-x^2-(100+30x)=90x-x^2-100-30x=-x^2+60x-100</math>
 +
|-
 +
|Then, <math>P'(x)=-2x+60=0</math>, so <math>x=30</math>
 +
|-
 +
|So, <math>p(30)=90-30=60</math>
 +
|-
 +
|Therefore, the price is <math>\$ 60</math> a unit will maximize the profit.
 
|-
 
|-
 
|Then, <math>\overline{C} '=2-\frac{7200}{x^2}=0</math>, so <math>x^2=3600</math>, so <math>x=\pm\sqrt{3600}=\pm 60=60</math> since <math>x</math> is positive.
 
|Then, <math>\overline{C} '=2-\frac{7200}{x^2}=0</math>, so <math>x^2=3600</math>, so <math>x=\pm\sqrt{3600}=\pm 60=60</math> since <math>x</math> is positive.

Revision as of 06:51, 2 August 2020

Optimization in Business and Economics

1) Find the number of 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 x} that minimizes the average cost 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 \overline{C}} when 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=2x^2+348x+7200}

Solution:  
Notice: 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 \overline{C}=\frac{C}{x}=\frac{2x^2+348x+7200}{x}=2x+348+\frac{7200}{x}}
Then, 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 \overline{C} '=2-\frac{7200}{x^2}=0} , so 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^2=3600} , so 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=\pm\sqrt{3600}=\pm 60=60} since 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} is positive.

2) Find the price that will maximize profit for the demand and cost functions, where 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} is the price, 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} is the number of units, 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 C} is the cost. 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(x)=90-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+30x} .

Solution:  
Notice: The revenue 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 R(x)=x\cdot p(x)=x(90-x)=90x-x^2}
The Profit function 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)=90x-x^2-(100+30x)=90x-x^2-100-30x=-x^2+60x-100}
Then, 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)=-2x+60=0} , so 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=30}
So, 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(30)=90-30=60}
Therefore, the price 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 \$ 60} a unit will maximize the profit.
Then, 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 \overline{C} '=2-\frac{7200}{x^2}=0} , so 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^2=3600} , so 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=\pm\sqrt{3600}=\pm 60=60} since 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} is positive.




Return to Topics Page

This page were made by Tri Phan

Retrieved from "https://wiki.math.ucr.edu/index.php?title=Math_22_Business_and_Economics_Applications&oldid=2374"