{"id":226,"date":"2015-09-18T20:09:29","date_gmt":"2015-09-18T20:09:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=226"},"modified":"2016-11-08T00:22:21","modified_gmt":"2016-11-08T00:22:21","slug":"formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/formulas\/","title":{"raw":"Formulas","rendered":"Formulas"},"content":{"raw":"An equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].\r\n\r\nA formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].\r\n
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Example 11: Using a Formula<\/h3>\r\nA right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 3.<\/b> Right circular cylinder[\/caption]\r\n\r\n<\/div>\r\n
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Solution<\/h3>\r\nEvaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].\r\n
[latex]\\begin{array}\\text{ }S\\hfill&=2\\pi r\\left(r+h\\right) \\\\ \\hfill& =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ \\hfill& =2\\pi\\left(6\\right)\\left(15\\right) \\\\ \\hfill& =180\\pi\\end{array}[\/latex]<\/div>\r\nThe surface area is [latex]180\\pi [\/latex] square inches.\r\n\r\n<\/div>\r\n
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Try It 11<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\/ Figure 4<\/b>[\/caption]\r\n\r\nA photograph with length L<\/em> and width W<\/em> is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a matte for a photograph with length 32 cm and width 24 cm.\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

An equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\n

A formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].<\/p>\n

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Example 11: Using a Formula<\/h3>\n

A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi[\/latex].<\/p>\n

\"A<\/p>\n

Figure 3.<\/b> Right circular cylinder<\/p>\n<\/div>\n<\/div>\n

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Solution<\/h3>\n

Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].<\/p>\n

[latex]\\begin{array}\\text{ }S\\hfill&=2\\pi r\\left(r+h\\right) \\\\ \\hfill& =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ \\hfill& =2\\pi\\left(6\\right)\\left(15\\right) \\\\ \\hfill& =180\\pi\\end{array}[\/latex]<\/div>\n

The surface area is [latex]180\\pi[\/latex] square inches.<\/p>\n<\/div>\n

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Try It 11<\/h3>\n
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Figure 4<\/b><\/p>\n<\/div>\n

A photograph with length L<\/em> and width W<\/em> is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a matte for a photograph with length 32 cm and width 24 cm.<\/p>\n

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

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