Difference between revisions of "Math 22 Business and Economics Applications"
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==Optimization in Business and Economics== | ==Optimization in Business and Economics== | ||
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| + | '''1)''' Find the number of units <math>x</math> that minimizes the average cost per unit <math>\overline{C}</math> when <math>C=2x^2+348x+7200</math> | ||
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| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
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| + | |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. | ||
| + | |} | ||
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| + | '''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>. | ||
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| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
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| + | |Notice: The revenue function <math>R(x)=x\cdot p(x)=x(90-x)=90x-x^2</math> | ||
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| + | |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> | ||
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| + | |Then, <math>P'(x)=-2x+60=0</math>, so <math>x=30</math> | ||
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| + | |So, <math>p(30)=90-30=60</math> | ||
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| + | |Therefore, the price is <math>\$ 60</math> a unit will maximize the profit. | ||
| + | |} | ||
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[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Latest revision as of 06:51, 2 August 2020
Optimization in Business and Economics
1) Find the number of units 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: |
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| 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. |
This page were made by Tri Phan