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
Define joint variation and combined variation.
Translate a verbal statement of joint variation into a mathematical equation.
Translate a verbal statement of combined variation into a mathematical equation.
Identify and correctly place the constant of variation, k, in an equation.
Differentiate between joint, combined, direct, and inverse variation within a single problem statement.
Correctly incorporate powers and roots of variables into variation equations.
Ever wonder how the force of gravity depends on both the masses of planets and their distance apart? 🪐 That complex relationship is a perfect example of combined variation!
This tutorial introduces joint and combined variation, which are powerful tools for modeling real-world scenarios where one qua...
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Key Concepts & Vocabulary
TermDefinitionExample
Constant of Variation (k)A non-zero constant that acts as a scaling factor in a variation relationship. It links the independent and dependent variables.In the direct variation equation y = 5x, the constant of variation 'k' is 5.
Joint VariationA relationship where a variable varies directly as 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 variables are in the denominator of the equation.The statement 'y varies inversely as x' is written as the equation y = k/x.
Combined VariationA relationship that involves a combination of direct or joint variation and inverse varia...
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Core Formulas
The Joint Variation Equation
y = kx_1x_2...x_n
Use this structure when a quantity 'y' is said to 'vary jointly as' or 'be jointly proportional to' two or more other quantities (x₁, x₂, etc.). All varying quantities appear in the numerator with the constant k.
The Combined Variation Equation
y = \frac{kx_1x_2...}{z_1z_2...}
Use this structure for problems that mix direct/joint and inverse variation. Variables that 'y' varies 'directly' or 'jointly' with (x₁, x₂, etc.) go in the numerator. Variables that 'y' varies 'inversely' with (z₁, z₂, etc.) go in the denominator.
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Challenging
A variable, Q, varies jointly as the square of A and the cube root of B, and inversely as the product of the square root of C and the fourth power of D. Which equation correctly represents this relationship?
A.Q = (kA² ³√B) / (√C D⁴)
B.Q = (k√C D⁴) / (A² ³√B)
C.Q = kA² ³√B √C D⁴
D.Q = k(A² + ³√B) / (√C + D⁴)
Challenging
In a chemical reaction, the rate, R, is observed to triple when the concentration of reactant X is tripled. The rate increases by a factor of nine when the concentration of reactant Y is tripled. The rate is halved when the volume, V, is doubled. Which equation best models the reaction rate, R?
A.R = kXY / V
B.R = kX²Y / V
C.R = kXY² / V
D.R = kX³Y / V²
Challenging
The Stefan-Boltzmann law describes the power radiated from a body. This power, P, is directly proportional to the surface area, A, of the object. It is also found to be directly proportional to the fourth power of the body's temperature, T. Which single variation equation correctly combines these two facts?
A.P = k(A + T⁴)
B.P = kA / T⁴
C.P = kAT⁴
D.P = k(A / T)⁴
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