The value of z is 58 when z is directly proportional to [tex]\sqrt{x}[/tex] and indirectly proportional to [tex]y^{3}[/tex]
We have :
z is directly proportional to [tex]\sqrt{x}[/tex]
z is inversely proportional to [tex]y^{3}[/tex]
Now, we introduce a constant of proportionality, then
[tex]z = k\frac{\sqrt{x} }{y^{3} }[/tex]
It is given that z = 248 when x is 9 and y is 4.
Then, Applying these values in the equation,
[tex]z = k\frac{\sqrt{x} }{y^{3} }[/tex]
[tex]248 = k\frac{\sqrt{9} }{\ 4^{3} } \\\\248 = k \frac{3}{64} \\\\k = \frac{248*64}{3} =\frac{15872}{3} =5290.67[/tex]
Now, we need to calculate the value of z when x = 64 and y =9.
[tex]z = k\frac{\sqrt{x} }{y^{3} }[/tex]
[tex]z =\frac{15872}{3} *\frac{\sqrt{64} }{9^{3} } = \frac{15872*8}{3*729} =\frac{126976}{2187} \\z = 58.05[/tex]
Hence, the value of z = 58.05
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