✏️MATH VARIATIONS
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#1: Let us say y varies jointly as x, z, and w, When x=3, z=4, and w=10, the value of y=24.
• Identify the constant of the variation referred as k.
- [tex] 24 = k(3)(4)(10) [/tex]
- [tex] \frac{24}{120} = \frac{k(\cancel{120})}{\cancel{120}} \\ [/tex]
- [tex] \frac{1}{5} = k \:\: or \:\: k = \frac{1}{5} \\ [/tex]
a) What is y if x=4, z=2, and w=15
- [tex] y = \frac{1}{5}xzw \\ [/tex]
- [tex] y = \frac{1}{5}(4)(2)(5) \\ [/tex]
- [tex] y = \frac{1}{5}(40) \\[/tex]
- [tex] y = \frac{40}{5} \\ [/tex]
[tex] \large \therefore \underline{\boxed{\tt \purple{y = 8}}} [/tex]
b) What is w if y=16, x=5, and z=8
- [tex] y = \frac{1}{5}xzw \\ [/tex]
- [tex] 16 = \frac{1}{5}(5)(8)w \\ [/tex]
- [tex] 16 = \frac{1}{5}(40)w \\ [/tex]
- [tex] 16 = \frac{40}{5}w \\ [/tex]
- [tex] \frac{16}{8} = \frac{\cancel8w}{\cancel8} \\ [/tex]
- [tex] 2 = w \:\: or \:\: w = 2 [/tex]
[tex] \large \therefore \underline{\boxed{\tt \purple{w = 2}}} [/tex]
[tex] \: [/tex]
#2: Consider that a=12, b=6 and c=2. When b=3 and c=3
a) What is a if a varies directly as b but inversely as c.
• Identify the constant of the variation referred as k.
- [tex] a = \frac{kb}{c} \\ [/tex]
- [tex] 12 = \frac{k(6)}{2} \\ [/tex]
- [tex] \frac{12}{3} = \frac{k(\cancel3)}{\cancel3} \\ [/tex]
- [tex] 4 = k \:\: or \:\: k = 4 [/tex]
• Identify the value of a if b=3 and c=3.
- [tex] a = \frac{4b}{c} \\ [/tex]
- [tex] a = \frac{4(3)}{3} \\ [/tex]
- [tex] a = \frac{12}{3} \\ [/tex]
[tex] \large \therefore \underline{\boxed{\tt \purple{a = 4}}} [/tex]
b) What is a if a varies directly as b and as the square of c.
• Identify the constant of the variation referred as k.
- [tex] 12 = k(6)(2)^2 [/tex]
- [tex] 12 = k(6)(4) [/tex]
- [tex] \frac{12}{24} = \frac{k(\cancel{24})}{\cancel{24}} \\ [/tex]
- [tex] \frac{1}{2} = k \:\: or \:\: k = \frac{1}{2} \\ [/tex]
• Identify the value of a if b=3 and c=3.
- [tex] a = \frac{1}{2}bc^2 \\ [/tex]
- [tex] a = \frac{1}{2}(3)(3)^2 \\ [/tex]
- [tex] a = \frac{1}{2}(3)(9) \\[/tex]
- [tex] a = \frac{1}{2}(27) \\ [/tex]
- [tex] a = \frac{27}{2} \\ [/tex]
[tex] \large \therefore \underline{\boxed{\tt \purple{a = \frac{27}{2}}}} [/tex]
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