{"id":11163,"date":"2015-07-14T18:38:40","date_gmt":"2015-07-14T18:38:40","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11163"},"modified":"2015-09-10T18:15:08","modified_gmt":"2015-09-10T18:15:08","slug":"solve-problems-involving-joint-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/solve-problems-involving-joint-variation\/","title":{"raw":"Solve problems involving joint variation","rendered":"Solve problems involving joint variation"},"content":{"raw":"

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation<\/strong>. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c<\/em>, cost, varies jointly with the number of students, n<\/em>, and the distance, d<\/em>.<\/p>\r\n\r\n

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A General Note: Joint Variation<\/h3>\r\n

Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\r\n

For instance, if x<\/em>\u00a0varies directly with both y<\/em>\u00a0and z<\/em>, we have x\u00a0<\/em>= kyz<\/em>. If x<\/em>\u00a0varies directly with y<\/em>\u00a0and inversely with z<\/em>, we have [latex]x=\\frac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.<\/p>\r\n\r\n<\/div>\r\n

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Example 4: Solving Problems Involving Joint Variation<\/h3>\r\n

A quantity x<\/em>\u00a0varies directly with the square of y<\/em>\u00a0and inversely with the cube root of z<\/em>. If x\u00a0<\/em>= 6 when y\u00a0<\/em>= 2 and z\u00a0<\/em>= 8, find x<\/em>\u00a0when y\u00a0<\/em>= 1 and z\u00a0<\/em>= 27.<\/p>\r\n\r\n<\/div>\r\n

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Solution<\/h3>\r\n

Begin by writing an equation to show the relationship between the variables.<\/p>\r\n\r\n

[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\r\n

Substitute x\u00a0<\/em>= 6, y\u00a0<\/em>= 2, and z\u00a0<\/em>= 8 to find the value of the constant k<\/em>.<\/p>\r\n\r\n

[latex]\\begin{cases}6=\\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ 6=\\frac{4k}{2}\\hfill \\\\ 3=k\\hfill \\end{cases}[\/latex]<\/div>\r\n

Now we can substitute the value of the constant into the equation for the relationship.<\/p>\r\n\r\n

[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\r\n

To find x<\/em>\u00a0when y\u00a0<\/em>= 1 and z\u00a0<\/em>= 27, we will substitute values for y<\/em>\u00a0and z<\/em>\u00a0into our equation.<\/p>\r\n\r\n

[latex]\\begin{cases}x=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\hfill \\\\ \\text{ }=1\\hfill \\end{cases}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Try It 3<\/h3>\r\n

x<\/em>\u00a0varies directly with the square of y<\/em>\u00a0and inversely with z<\/em>. If x\u00a0<\/em>= 40 when y\u00a0<\/em>= 4 and z\u00a0<\/em>= 2, find x<\/em>\u00a0when y\u00a0<\/em>= 10 and z\u00a0<\/em>= 25.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation<\/strong>. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c<\/em>, cost, varies jointly with the number of students, n<\/em>, and the distance, d<\/em>.<\/p>\n

\n

A General Note: Joint Variation<\/h3>\n

Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n

For instance, if x<\/em>\u00a0varies directly with both y<\/em>\u00a0and z<\/em>, we have x\u00a0<\/em>= kyz<\/em>. If x<\/em>\u00a0varies directly with y<\/em>\u00a0and inversely with z<\/em>, we have [latex]x=\\frac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.<\/p>\n<\/div>\n

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Example 4: Solving Problems Involving Joint Variation<\/h3>\n

A quantity x<\/em>\u00a0varies directly with the square of y<\/em>\u00a0and inversely with the cube root of z<\/em>. If x\u00a0<\/em>= 6 when y\u00a0<\/em>= 2 and z\u00a0<\/em>= 8, find x<\/em>\u00a0when y\u00a0<\/em>= 1 and z\u00a0<\/em>= 27.<\/p>\n<\/div>\n

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Solution<\/h3>\n

Begin by writing an equation to show the relationship between the variables.<\/p>\n

[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\n

Substitute x\u00a0<\/em>= 6, y\u00a0<\/em>= 2, and z\u00a0<\/em>= 8 to find the value of the constant k<\/em>.<\/p>\n

[latex]\\begin{cases}6=\\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ 6=\\frac{4k}{2}\\hfill \\\\ 3=k\\hfill \\end{cases}[\/latex]<\/div>\n

Now we can substitute the value of the constant into the equation for the relationship.<\/p>\n

[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\n

To find x<\/em>\u00a0when y\u00a0<\/em>= 1 and z\u00a0<\/em>= 27, we will substitute values for y<\/em>\u00a0and z<\/em>\u00a0into our equation.<\/p>\n

[latex]\\begin{cases}x=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\hfill \\\\ \\text{ }=1\\hfill \\end{cases}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Try It 3<\/h3>\n

x<\/em>\u00a0varies directly with the square of y<\/em>\u00a0and inversely with z<\/em>. If x\u00a0<\/em>= 40 when y\u00a0<\/em>= 4 and z\u00a0<\/em>= 2, find x<\/em>\u00a0when y\u00a0<\/em>= 10 and z\u00a0<\/em>= 25.<\/p>\n

Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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