Worksheet covering direct, inverse, and joint variation concepts, including equations, problem-solving steps, and practice questions.

Let's go through each problem step by step, using the definitions and steps provided in the worksheet.

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Part 1: Identify the type of variation and name the constant of variation

1) $ y = 2x $
- This is direct variation because $ y $ is proportional to $ x $.
- General form: $ y = kx $ → here $ k = 2 $
✔ Answer: Direct variation; constant $ k = 2 $

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2) $ \frac{x}{5} = y $
- Rewrite: $ y = \frac{1}{5}x $
- This is direct variation with $ k = \frac{1}{5} $
✔ Answer: Direct variation; constant $ k = \frac{1}{5} $

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3) $ xy = 12 $
- Solve for $ y $: $ y = \frac{12}{x} $
- This is inverse variation: $ y = \frac{k}{x} $, where $ k = 12 $
✔ Answer: Inverse variation; constant $ k = 12 $

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4) $ D = \frac{3}{4}gh $
- $ D $ varies jointly with $ g $ and $ h $ (both variables are multiplied).
- General form: $ D = kgh $ → here $ k = \frac{3}{4} $
✔ Answer: Joint variation; constant $ k = \frac{3}{4} $

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Part 2: Translate statements into formulas

5) E varies jointly as M and the square of V.
- "Jointly" means product of variables.
- Square of V = $ V^2 $
- So: $ E = kM V^2 $
✔ Answer: $ E = kM V^2 $

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6) The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure P.
- Directly with $ T $: $ V \propto T $
- Inversely with $ P $: $ V \propto \frac{1}{P} $
- Combined: $ V = k \cdot \frac{T}{P} $
✔ Answer: $ V = k \frac{T}{P} $

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7) The mass, M, of a cement block varies jointly as the length, L, width, W, and thickness, T.
- All three variables are multiplied together.
- So: $ M = kLWT $
✔ Answer: $ M = kLWT $

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8) P varies directly as the square of V and inversely as R.
- Directly with $ V^2 $, inversely with $ R $
- So: $ P = k \cdot \frac{V^2}{R} $
✔ Answer: $ P = k \frac{V^2}{R} $

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Part 3: Write an equation and solve

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9) If $ y $ varies inversely as $ x $ and $ y = 2 $ when $ x = 8 $, find $ x $ when $ y = 14 $.

- Inverse variation: $ y = \frac{k}{x} $
- Step 1: Find $ k $ using given values:
$$
2 = \frac{k}{8} \Rightarrow k = 2 \times 8 = 16
$$
- So formula: $ y = \frac{16}{x} $
- Now find $ x $ when $ y = 14 $:
$$
14 = \frac{16}{x} \Rightarrow x = \frac{16}{14} = \frac{8}{7}
$$
✔ Answer: $ x = \frac{8}{7} $

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10) Suppose $ y $ varies jointly with $ x $ and $ z $. If $ y = 20 $ when $ x = 2 $ and $ z = 5 $, find $ y $ when $ x = 14 $ and $ z = 8 $.

- Joint variation: $ y = kxz $
- Use given values to find $ k $:
$$
20 = k \cdot 2 \cdot 5 = 10k \Rightarrow k = \frac{20}{10} = 2
$$
- So: $ y = 2xz $
- Now plug in $ x = 14 $, $ z = 8 $:
$$
y = 2 \cdot 14 \cdot 8 = 224
$$
✔ Answer: $ y = 224 $

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11) If $ y $ varies inversely as $ x $ and $ x = 7 $ when $ y = 21 $, find $ y $ when $ x = 42 $.

- Inverse variation: $ y = \frac{k}{x} $
- Use $ x = 7 $, $ y = 21 $ to find $ k $:
$$
21 = \frac{k}{7} \Rightarrow k = 21 \times 7 = 147
$$
- So: $ y = \frac{147}{x} $
- When $ x = 42 $:
$$
y = \frac{147}{42} = 3.5 = \frac{7}{2}
$$
✔ Answer: $ y = \frac{7}{2} $ or $ 3.5 $

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12) Find $ y $ when $ x = 1.5 $, if $ y $ varies directly as $ x $ and $ y = -16 $ when $ x = 6 $.

- Direct variation: $ y = kx $
- Use $ y = -16 $, $ x = 6 $ to find $ k $:
$$
-16 = k \cdot 6 \Rightarrow k = \frac{-16}{6} = -\frac{8}{3}
$$
- So: $ y = -\frac{8}{3}x $
- Now plug in $ x = 1.5 = \frac{3}{2} $:
$$
y = -\frac{8}{3} \cdot \frac{3}{2} = -\frac{24}{6} = -4
$$
✔ Answer: $ y = -4 $

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✔ Final Answers Summary:

1. Direct; $ k = 2 $
2. Direct; $ k = \frac{1}{5} $
3. Inverse; $ k = 12 $
4. Joint; $ k = \frac{3}{4} $
5. $ E = kM V^2 $
6. $ V = k \frac{T}{P} $
7. $ M = kLWT $
8. $ P = k \frac{V^2}{R} $
9. $ x = \frac{8}{7} $
10. $ y = 224 $
11. $ y = \frac{7}{2} $
12. $ y = -4 $

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