Mathematics
Grade 11
15 min
Find the constant of variation
Find the constant of variation
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1
Introduction & Learning Objectives
Learning Objectives
Define the constant of variation (k) and its role in linking variables.
Identify the type of variation (direct, inverse, joint, combined) from a verbal description or equation.
Write the correct general formula for each of the four types of variation.
Algebraically isolate the constant of variation, k, in any variation formula.
Calculate the specific value of k when given a set of corresponding variable values.
Interpret the meaning of the constant of variation within the context of a real-world problem.
Ever wonder why a 10-minute phone charge gives you 20% battery, but a 20-minute charge gives you 40%? 🔋 That predictable link is controlled by a 'constant of variation' we're about to master.
This tutorial focuses exclusively on finding t...
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Key Concepts & Vocabulary
TermDefinitionExample
VariationA mathematical relationship describing how one variable changes in a predictable way in response to the change in one or more other variables.The total cost of apples varies with the number of apples you buy.
Constant of Variation (k)A non-zero constant that quantifies the relationship between variables in a variation problem. It is the specific numerical value that links the variables together.If you earn $15 per hour, the constant of variation between your earnings (E) and hours worked (h) is 15. The equation is E = 15h.
Direct VariationA relationship where two variables increase or decrease together. If one variable doubles, the other variable also doubles.The distance you travel (d) varies directly with time (t) if your speed is constant. d = kt.
Inverse...
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Core Formulas
Direct Variation Formula
y = kx => k = y/x
Use this when a quantity 'y' is said to vary directly with 'x'. To find the constant k, divide the output variable (y) by the input variable (x).
Inverse Variation Formula
y = k/x => k = xy
Use this when a quantity 'y' varies inversely with 'x'. To find the constant k, multiply the two variables together.
Joint Variation Formula
z = kxy => k = z/(xy)
Use this when a quantity 'z' varies jointly with 'x' and 'y'. To find k, divide 'z' by the product of the other variables.
Combined Variation Formula
z = (kx)/y => k = (zy)/x
Use this for mixed relationships, like 'z varies directly with x and inversely with y'....
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Challenging
The volume V of a cone is given by the formula V = (1/3)πr²h. If the volume is said to vary jointly with the height (h) and the square of the radius (r), what is the constant of variation, k?
A.π
B.3π
C.π/3
D.3
Challenging
The variable P varies directly as Q and inversely as R, with a constant of variation k. In an experiment, Q is found to vary directly with the square of T, with a constant c = 3. If P = 18 when T = 2 and R = 4, what is the constant of variation, k, for the original relationship between P, Q, and R?
A.3
B.6
C.9
D.12
Challenging
Newton's Law of Universal Gravitation states F = G(m₁m₂)/r², where F is the gravitational force, m₁ and m₂ are masses, r is the distance between them, and G is a constant. If F is said to vary jointly with the masses and inversely with the square of the distance, what is the constant of variation?
A.G, the universal gravitational constant
B.The product of the masses, m₁m₂
C.The square of the distance, r²
D.The value depends on the specific masses and distance
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