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The weight (W) of a cylindrical metal rod varies jointly as its length (L) and the square of its diameter (d). A rod that is 8 cm long with a diameter of 2 cm weighs 240 grams. What is the weight of a rod that is 12 cm long with a diameter of 3 cm?
This article serves as a complete study guide. We will break down exactly what joint and combined variation mean, how to set up the equations, where to find the best Kuta worksheets, and how to solve common problem types step by step. Before you download a worksheet, you need a rock-solid conceptual foundation. Direct Variation (Review) [ y = kx ] Meaning: As (x) increases, (y) increases at a constant rate (k is the constant of variation). Inverse Variation (Review) [ y = \frackx ] Meaning: As (x) increases, (y) decreases proportionally. Joint Variation (The New Concept) Definition: A quantity varies jointly as two (or more) other quantities if it is directly proportional to their product.
[ y = \frackxz ] or [ y = \frack \cdot (product\ of\ direct\ variables)product\ of\ inverse\ variables ]
(y) varies jointly as (x) and (z). (y=24) when (x=2, z=3). [ 24 = k \cdot 2 \cdot 3 ] [ 24 = 6k ] [ k = 4 ] Step 3: Rewrite the Equation with (k) Now that you know (k=4), rewrite the equation: (y = 4xz). Step 4: Solve for the Unknown Use the second set of conditions (e.g., "Find (y) when (x=5, z=10)"). [ y = 4 \cdot 5 \cdot 10 ] [ y = 200 ]
"varies directly as (x) and inversely as (z)".
Introduction In the world of Algebra 2 and Precalculus, few topics bridge the gap between abstract equations and real-world physical laws quite like variation. While direct and inverse variation are the building blocks, joint and combined variation represent the next level of complexity—and the level where many students begin to struggle.
