In Exercises 11–20, 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.

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1. Identify the type of variation described in the problem. Here, x varies jointly as y and z, which means x is directly proportional to the product of y and z. Additionally, x varies inversely as the square root of w, meaning x is inversely proportional to the square root of w.
2. Write the initial equation for the variation. Since x varies jointly as y and z, and inversely as the square root of w, the equation can be expressed as: x = k \cdot \frac{y \cdot z}{\sqrt{w}}, where k is the constant of proportionality.
3. Isolate y in the equation to solve for y in terms of x, z, w, and k. Multiply both sides of the equation by \sqrt{w} to eliminate the denominator: x \cdot \sqrt{w} = k \cdot y \cdot z.
4. Divide both sides of the equation by k \cdot z to solve for y: y = \frac{x \cdot \sqrt{w}}{k \cdot z}.
5. The equation expressing y in terms of x, z, w, and k is y = \frac{x \cdot \sqrt{w}}{k \cdot z}. This equation shows how y is determined based on the values of x, z, w, and the constant k.

Key Concepts

Joint Variation

Joint variation occurs when a variable is directly proportional to the product of two or more other variables. In this case, if x varies jointly as y and z, it means that x = k(yz) for some constant k. Understanding this concept is crucial for setting up the initial equation based on the relationships described in the problem.

Inverse Variation

Inverse variation describes a relationship where one variable increases as another decreases. Specifically, if x varies inversely as the square root of w, it can be expressed as x = k' / √w, where k' is another constant. This concept is essential for incorporating the inverse relationship into the equation that needs to be solved.

Solving for a Variable

Solving for a variable involves rearranging an equation to isolate the variable of interest, in this case, y. This process often requires using algebraic techniques such as multiplication, division, and factoring. Mastery of this concept is necessary to manipulate the equation derived from the relationships and find the value of y.