MATH VARIATIONS
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Solve:
x varies jointly as w, y, and z. If x = 18, when w = 2, y = 6 and z = 5, find x when w = 5, y = 12 and z = 3
[tex] \implies \sf \large x = kwyz[/tex]
[tex]\large \tt \red{given} \begin{cases} \sf \: x = 18 \\ \sf \: w = 2 \\ \sf \: y = 6 \\ \sf \: z = 5\end{cases} \\ \\ [/tex]
[tex]\implies \sf \large 18= k(2)(6)(5)[/tex]
[tex]\implies \sf \large 18= k(12)(5)[/tex]
[tex]\implies \sf \large 18= k(60)[/tex]
[tex]\implies \sf \large \frac{18}{60} = \frac{k( \cancel{60})}{ \cancel{60}} \\ [/tex]
[tex]\implies \sf \large \frac{18 \div 6}{60 \div 6} = k \\ [/tex]
[tex]\implies \sf \large \therefore \: k = \frac{3}{10} \\ [/tex]
[tex] \: [/tex]
[tex]\large \tt \red{given(2)} \begin{cases} \sf \: k = \frac{3}{10} \\ \sf \: w = 5 \\ \sf \: y = 12 \\ \sf \: z = 3\end{cases} \\ \\ [/tex]
[tex]\implies \sf \large x = \frac{3}{10} (5)(12)(3) \\ [/tex]
[tex]\implies \sf \large x = \frac{3}{10} (60)(3) \\ [/tex]
[tex]\implies \sf \large x = \frac{3}{10} (180) \\ [/tex]
[tex]\implies \sf \large x = \frac{540}{10} \\ [/tex]
[tex] \large \implies \huge \sf \orange{x = 54}[/tex]
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Solve:
If z varies inversely as x and y. z = 60 when x =
12 and y = 4. Find y when z = 80 and x = 3
[tex]\implies \sf \large z = \frac{k}{xy} \\ [/tex]
[tex]\large \tt \red{given} \begin{cases} \sf \: z = 60 \\ \sf \: x = 12 \\ \sf \: y = 4\end{cases} \\ \\ [/tex]
[tex]\implies \sf \large 60 = \frac{k}{12(4)} \\[/tex]
[tex]\implies \sf \large 60 = \frac{k}{48} \\[/tex]
[tex]\implies \sf \large 60 \times 48= \frac{k}{ \cancel{48}} \times \cancel{48} \\[/tex]
[tex]\implies \sf \large \therefore \: k = 2880[/tex]
[tex] \: [/tex]
[tex]\large \tt \red{given(2)} \begin{cases} \sf \: z = 80 \\ \sf \: x = 3 \\ \sf \: k = 2880\end{cases} \\ \\ [/tex]
[tex]\implies \sf \large 80 \times 3= \frac{2880}{ \cancel3y} \times \cancel3 \\[/tex]
[tex]\implies \sf \large 240 = \frac{2880}{y} \\[/tex]
[tex]\implies \sf \large 240y= 2880 [/tex]
[tex]\implies \sf \large \frac{ \cancel{240}y}{ \cancel{240}} = \frac{2880}{240} \\ [/tex]
[tex] \large\implies \sf \huge \orange{y = 12}[/tex]
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