Skip to main content
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 15
Chapter 4, Problem 15

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.

Verified step by step guidance
1
Identify the type of variation described: "x varies jointly as y and z" means x is proportional to the product of y and z, so we write \(x = k \cdot y \cdot z\) where \(k\) is a constant of proportionality.
Since x also varies inversely as the square root of w, include this inverse relationship by dividing by \(\sqrt{w}\). The equation becomes \(x = \frac{k \cdot y \cdot z}{\sqrt{w}}\).
Write the full equation expressing the relationship: \(x = \frac{k \cdot y \cdot z}{\sqrt{w}}\).
To solve for y, multiply both sides of the equation by \(\sqrt{w}\) to get rid of the denominator: \(x \cdot \sqrt{w} = k \cdot y \cdot z\).
Finally, isolate y by dividing both sides by \(k \cdot z\): \(y = \frac{x \cdot \sqrt{w}}{k \cdot z}\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m

Key Concepts

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

Joint Variation

Joint variation describes a relationship where one variable varies directly as the product of two or more other variables. In this case, x varies jointly as y and z means x is proportional to the product y * z, expressed as x = k * y * z, where k is a constant.

Inverse Variation

Inverse variation means one variable varies inversely as another, so as one increases, the other decreases proportionally. Here, x varies inversely as the square root of w, meaning x is proportional to 1 divided by √w, or x = k / √w.
Recommended video:
4:30
Graphing Logarithmic Functions

Solving for a Variable in an Equation

Solving for y involves isolating y on one side of the equation. After writing the joint and inverse variation equation, algebraic manipulation such as multiplication, division, and taking square roots is used to express y explicitly in terms of x, z, w, and the constant k.
Recommended video:
05:28
Equations with Two Variables
Related Practice
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

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→1+, f(x)→x\(\to\)1^{+},\(\text{ }\)f(x)\(\to\) __

Textbook Question

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

5
views
Textbook Question

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\)