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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 14
Chapter 4, Problem 14

Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube root of z and inversely as y.

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Identify the given variation relationships: x varies directly as the cube root of z, and inversely as y. This means we can write the equation as \(x = k \frac{\sqrt[3]{z}}{y}\), where \(k\) is the constant of proportionality.
Write the equation explicitly: \(x = k \frac{z^{\frac{1}{3}}}{y}\).
To solve for \(y\), multiply both sides of the equation by \(y\) to get rid of the denominator: \(xy = k z^{\frac{1}{3}}\).
Next, isolate \(y\) by dividing both sides by \(x\): \(y = \frac{k z^{\frac{1}{3}}}{x}\).
The equation is now solved for \(y\): \(y = \frac{k z^{\frac{1}{3}}}{x}\). To find the specific value of \(k\), you would need additional information such as a set of values for \(x\), \(y\), and \(z\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Direct Variation

Direct variation describes a relationship where one variable is proportional to another. If x varies directly as the cube root of z, it means x = k * ∛z for some constant k. This shows that as ∛z increases, x increases proportionally.
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Inverse Variation

Inverse variation means one variable changes in the opposite way to another, expressed as x = k / y. Here, x varies inversely as y, so as y increases, x decreases proportionally, and vice versa, with k being a constant.
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Solving for a Variable in an Equation

Solving for y involves manipulating the equation algebraically to isolate y on one side. This requires understanding operations like multiplication, division, and roots to rewrite the equation in terms of y, making it the subject of the formula.
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Related Practice
Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 6x2+x>1

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Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=−x4+x2f(x)=−x^4+x^2

Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x3+x2−3x+1

Textbook Question

Divide using long division. State the quotient, and the remainder, r(x). (x4+2x3−4x2−5x−6)/(x2+x−2)

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Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3+4x2−3x−6f(x)=x^3+4x^2−3x−6

Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.