Answer:
[tex]\LARGE\text{$\tt{1.)\:y=10}$}[/tex]
[tex]\LARGE\text{$\tt{2.)\:x=16}$}[/tex]
[tex]\LARGE\text{$\tt{3.)\:y=\frac{968}{9}\:\:or\:\:107\frac{5}{9}}$}[/tex]
[tex]\LARGE\text{$\tt{4.)\:f=405}$}[/tex]
[tex]\LARGE\text{$\tt{5.)\:a=18}$}[/tex]
Step-by-step explanation:
1. If y varies directly as x, and x = 9 when y = 15, find y when x = 33.
Mathematical Equation:
[tex]\LARGE\text{$y=kx$}[/tex]
For the constant of variation (k):
[tex]x=9[/tex], [tex]y=15[/tex]
[tex]y=kx\\15=9k\\\frac{15}{9}=\frac{9k}{9}\\\boldsymbol{\frac{5}{3}=k}[/tex]
For the value of y:
[tex]k=\frac{5}{3}[/tex], [tex]x=33[/tex]
[tex]y=kx\\y=(\frac{5}{3})(6)\\y=(5)(2)\\\boxed{\boldsymbol{y=10}}[/tex]
2. If y varies inversely as x, and y = 32 when x = 3, find x when y = 6.
Mathematical Equation:
[tex]\LARGE\text{$y=\frac{k}{x}$}[/tex]
For the constant of variation (k):
[tex]y=32[/tex], [tex]x=3[/tex]
[tex]y=\frac{k}{x}\\32=\frac{k}{3}\\k=(32)(3)\\\boldsymbol{k=96}[/tex]
For the value of x:
[tex]k=96[/tex], [tex]y=6[/tex]
[tex]y=\frac{k}{x}\\6=\frac{96}{x}\\6x=96\\\frac{6x}{6}=\frac{96}{6}\\\boxed{\boldsymbol{x=16}}[/tex]
3. If y varies jointly as x and z, and y = 33 when x = 9 and z = 12, find y when x = 16 and z = 22.
Mathematical Equation:
[tex]\LARGE\text{$y=kxz$}[/tex]
For the constant of variation (k):
[tex]y=33[/tex], [tex]x=9[/tex], [tex]z=12[/tex]
[tex]y=kxz\\33=(9)(12)z\\33=108k\\\frac{33}{108}=\frac{108k}{108}\\\boldsymbol{\frac{11}{36}=k}[/tex]
For the value of y:
[tex]k=\frac{11}{36}[/tex], [tex]x=16[/tex], [tex]z=22[/tex]
[tex]y=kxz\\y=(\frac{11}{36})(16)(22)\\y=(\frac{11}{18})(8)(22)\\y=(\frac{11}{9})(8)(11)\\y=\frac{(11)(8)(11)}{9}\\\boxed{\boldsymbol{y=\frac{968}{9}\:\:or\:\:107\frac{5}{9}}}[/tex]
4. If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f when g = 3 and h = 6.
Mathematical Equation:
[tex]\LARGE\text{$f=kgh^3$}[/tex]
For the constant of variation (k):
[tex]f=200[/tex], [tex]g=5[/tex], [tex]h=4[/tex]
[tex]f=kgh^3\\200=(5)(4)^3k\\200=(5)(64)k\\200=320k\\\frac{200}{320}=\frac{320k}{320}\\\boldsymbol{\frac{5}{8}=k}[/tex]
For the value of f:
[tex]k=\frac{5}{8}[/tex], [tex]g=3[/tex], [tex]h=6[/tex]
[tex]f=kgh^3\\f=(\frac{5}{8})(3)(6)^3\\f=(\frac{5}{8})(3)(216)\\f=(5)(3)(27)\\\boxed{\boldsymbol{f=405}}[/tex]
5. A varies directly as b and inversely as c. If a = 15 when y = 20 and z =40, find a when b = 12 and c = 20.
Mathematical Equation:
[tex]\LARGE\text{$a=\frac{kb}{c}$}[/tex]
For the constant of variation (k):
[tex]a=15[/tex], [tex]y=20[/tex], [tex]z=40[/tex]
[tex]a=\frac{kb}{c}\\15=\frac{20k}{40}\\15=\frac{k}{2}\\(15)(2)=k\\\boldsymbol{30=k}[/tex]
For the value of y:
[tex]k=30[/tex], [tex]b=12[/tex], [tex]c=20[/tex]
[tex]a=\frac{kb}{c}\\a=\frac{(30)(12)}{20}\\a=\frac{(3)(12)}{2}\\a=(3)(6)\\\boxed{\boldsymbol{18}}[/tex]
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