Difference between revisions of "Math 22 Optimization Problems"
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| − | ==Solving Optimization Problems= | + | ==Solving Optimization Problems== |
| − | '''Find Maximum Area''': Find the length and width of a rectangle that has 80 meters perimeter. | + | '''Find Maximum Area''': Find the length and width of a rectangle that has 80 meters perimeter and a maximum area. |
| + | |||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |Let <math>l</math> be the length of the rectangle in meter. | ||
| + | |- | ||
| + | |and <math>w</math> be the width of the rectangle in meter. | ||
| + | |- | ||
| + | |Then, the perimeter <math>P=2(l+w)=80</math>, so <math>l+w=40</math>, then <math>l=40-w</math> | ||
| + | |- | ||
| + | |Area <math>A=l.w=(40-w)w=40w-w^2</math> | ||
| + | |- | ||
| + | |<math>A'=40-2w=0</math>, then <math>w=20</math>, so <math>l=40-w=40-20=20</math> | ||
| + | |- | ||
| + | |Therefore, <math>l=w=20</math> | ||
| + | |} | ||
| − | |||
Revision as of 07:24, 1 August 2020
Solving Optimization Problems
Find Maximum Area: Find the length and width of a rectangle that has 80 meters perimeter and a maximum area.
| Solution: |
|---|
| Let be the length of the rectangle in meter. |
| and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle w} be the width of the rectangle in meter. |
| Then, the perimeter Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P=2(l+w)=80} , 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 l+w=40} , 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 l=40-w} |
| Area 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 A=l.w=(40-w)w=40w-w^2} |
| 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 A'=40-2w=0} , 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 w=20} , 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 l=40-w=40-20=20} |
| Therefore, 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 l=w=20} |
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