Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 9

Capitolo 4, Problema 9

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as a and b and inversely as the square root of c. y = 12 when a = 3, b = 2, and c = 25. Find y when a = 5, b = 3 and c = 9.

Guida verificata passo dopo passo

Step 1: Identify the type of variation described. Since y varies jointly as a and b and inversely as the square root of c, we can write the variation equation as: \[y = k \cdot a \cdot b \div \sqrt{c}\] where $k$ is the constant of variation.

Step 2: Use the given values $y = 12$, $a = 3$, $b = 2$, and $c = 25$ to find the constant $k$. Substitute these values into the equation: \[12 = k \cdot 3 \cdot 2 \div \sqrt{25}\]

Step 3: Simplify the right side of the equation to solve for $k$. Calculate the square root of 25 and multiply the known values: \[12 = k \cdot 6 \div 5\] Then solve for $k$ by isolating it on one side.

Step 4: Once $k$ is found, write the complete variation equation with $k$ included: \[y = k \cdot a \cdot b \div \sqrt{c}\]

Step 5: Use the new values $a = 5$, $b = 3$, and $c = 9$ to find the new value of $y$. Substitute these values and the constant $k$ into the equation and simplify: \[y = k \cdot 5 \cdot 3 \div \sqrt{9}\]

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Joint and Inverse Variation

Joint variation occurs when a variable depends on the product of two or more variables, while inverse variation means a variable depends on the reciprocal of another. In this problem, y varies jointly as a and b, and inversely as the square root of c, combining both types of variation in one relationship.

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Graphing Logarithmic Functions

Formulating the Variation Equation

To solve variation problems, write an equation expressing the dependent variable in terms of the independent variables and a constant of proportionality. Here, y = k * a * b / √c, where k is found using given values, enabling calculation of y for new inputs.

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Categorizing Linear Equations

Solving for the Constant of Proportionality

Use the provided values of y, a, b, and c to substitute into the variation equation and solve for the constant k. This step is crucial because k allows the equation to model the specific situation accurately before finding y for new variable values.

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Solving Logarithmic Equations

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