Mathematics
Grade 11
15 min
Solve variation equations: Set 2
Solve variation equations: Set 2
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1
Introduction & Learning Objectives
Learning Objectives
Formulate equations for joint variation.
Formulate equations for combined variation.
Solve for the constant of variation (k) in multi-variable scenarios.
Use derived variation equations to find unknown values in complex problems.
Solve problems where a variable varies with the square, cube, or square root of another variable.
Translate complex verbal descriptions of variation into precise mathematical models.
Ever wonder how engineers calculate the stress on a beam based on its length, width, and depth? 🏗️ That's the power of joint and combined variation in action!
In this tutorial, we move beyond basic direct and inverse relationships to tackle more complex scenarios: joint and combined variation. Mastering these concepts allows you to model real-w...
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Key Concepts & Vocabulary
TermDefinitionExample
Joint VariationA relationship where a variable varies directly as the product of two or more other variables. If y varies jointly with x and z, it means y is directly proportional to the product xz.The area of a triangle (A) varies jointly with its base (b) and height (h). The equation is A = (1/2)bh, where k = 1/2.
Combined VariationA relationship that involves a combination of direct (or joint) and inverse variation in a single statement.The time (t) it takes to build a house varies directly with the size of the house (s) and inversely with the number of workers (w). The equation is t = ks/w.
Constant of Variation (k)The non-zero constant that acts as a multiplier in a variation equation. It is the specific value that links the variables in a particular model.In th...
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Core Formulas
Joint Variation Formula
y = kx_1x_2...x_n
Use this when a quantity 'y' varies directly with several other quantities (x_1, x_2, etc.) simultaneously. All variables that vary directly are multiplied together in the numerator with k.
Combined Variation Formula
y = (kx_1x_2...) / (z_1z_2...)
Use this for problems involving a mix of direct and inverse variation. Variables with a direct relationship go in the numerator with k, and variables with an inverse relationship go in the denominator.
General Power Variation Formula
y = kx^n or y = k / x^n
This is the fundamental structure when a problem states that a variable varies with the 'square', 'cube', 'square root' (n=1/2), or any other power of another variable.
4 more steps in this tutorial
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Challenging
If y varies jointly with x and z, what happens to y if x is doubled and z is halved?
A.y is quadrupled
B.y is doubled
C.y remains unchanged
D.y is halved
Challenging
The 'safe load' (S) for a horizontal beam varies jointly with its breadth (b) and the square of its depth (d), and inversely with its length (L). Two beams are made of the same material. Beam A is 4m long, 0.1m broad, and 0.2m deep. Beam B is 8m long, 0.12m broad, and 0.3m deep. What is the ratio of the safe load of Beam B to Beam A (S_B / S_A)?
A.27/20
B.27/10
C.9/10
D.9/5
Challenging
The time (t) required to complete a project varies directly with the number of tasks (n) and inversely with the product of the number of workers (w) and their average skill level (s). A project with 100 tasks takes 5 days for 10 workers with a skill level of 2. If the number of tasks is increased to 150, and the number of workers is reduced to 8, what skill level would be required to complete the project in 6 days?
A.2.50
B.2.75
C.3.00
D.3.125
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