Mathematics
Grade 11
15 min
Write joint and combined variation equations Set 1
Write joint and combined variation equations Set 1
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1
Introduction & Learning Objectives
Learning Objectives
Translate a verbal statement of joint variation into a mathematical equation.
Translate a verbal statement of combined variation into a mathematical equation.
Identify the constant of variation (k) and its role in variation equations.
Differentiate between joint, combined, direct, and inverse relationships within a single problem statement.
Correctly place variables in the numerator or denominator based on the type of variation.
Incorporate powers and roots into variation equations as described in a problem.
Ever wondered why a large pizza seems like a much better deal than a small one? 🍕 The area of the pizza varies with the square of its radius, a relationship we can model with variation equations!
This tutorial focuses on translating complex relation...
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Key Concepts & Vocabulary
TermDefinitionExample
Constant of Variation (k)A non-zero constant that acts as a multiplier in a variation equation. It defines the specific relationship between the variables.In the equation for the circumference of a circle, C = 2πr, the term '2π' can be considered the constant of variation, k. C varies directly with r.
Joint VariationA relationship where one variable depends on the product of two or more other variables.The statement 'y varies jointly as x and z' is written as the equation y = kxz.
Inverse VariationA relationship where one variable increases as another variable decreases. The variable is placed in the denominator of the equation.The statement 'y varies inversely as x' is written as the equation y = k/x.
Combined VariationA relationship th...
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Core Formulas
Joint Variation Formula
y = k * x * z * ...
Use this when a variable varies directly with the product of two or more other variables. The dependent variable equals k times the product of all independent variables.
Combined Variation Formula
y = (k * [product of direct variables]) / [product of inverse variables]
Use this for statements involving both direct and inverse variation. Variables with a direct relationship go in the numerator with k; variables with an inverse relationship go in the denominator.
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Challenging
The variable Z varies jointly as the square of A and the cube root of B, and inversely as the product of C and the square of D. Which equation correctly models this complex relationship?
A.Z = (k * A^2 * ³√B) / (C * D^2)
B.Z = (k * A^2 * B^3) / (C * D^2)
C.Z = (k * C * D^2) / (A^2 * ³√B)
D.Z = k * A^2 * ³√B * C * D^2
Challenging
Given the equation P = (k * a^2 * √b) / (c * d^3), which of the following is the most accurate verbal description?
A.P varies jointly as a squared and the square root of b, and inversely as c and d cubed.
B.P varies jointly as the square of a and the square root of b, and inversely as the product of c and the cube of d.
C.P varies combined as a squared and b, and inversely as c and d cubed.
D.P varies directly as a squared and inversely as the product of c, d cubed, and the square root of b.
Challenging
The stress, S, in a beam varies jointly as the bending moment, M, and the distance, c, from the neutral axis, and inversely as the moment of inertia, I. Which equation correctly models this and avoids all common pitfalls?
A.S = Mc/I
B.S = k(M+c)/I
C.S = kMc/I
D.S = kI/Mc
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