Answer :
The formula in terms of x and y for dy/dx is -3y/x
The value of dy/dx at the point (81,16) is -0.5926
The question is incomplete, the full question is given below:
A manufacturer can produce 2970 cell phones when x dollars is spent on labor and y dollars is spent on capital. the equation that relates x and y is [tex]55x^{\frac{3}{4} } y^{\frac{1}{4} } =2970[/tex]
a. Find a formula in terms of x and y for [tex]\frac{dy}{dx}[/tex]
b. Find the value of dy/dx at the point (81,16). Round your answer to 4 decimal places
Part a
A formula in terms of x and y for [tex]\frac{dy}{dx}[/tex]
We differentiate equation that relates x and y with respect to x by implicit differentiation:
=[tex]\frac{d}{dx} (55x^{\frac{3}{4} } y^{\frac{1}{4} } )=\frac{d}{dx} (2970)[/tex]
=[tex]55(x^{\frac{3}{4} } \frac{d}{dx} (y^{\frac{1}{4} } )+y^{\frac{1}{4} } ((x^{\frac{3}{4} } ))=0[/tex]
=[tex]x^{\frac{3}{4} } y^{\frac{-3}{4} } \frac{dy}{dx} =3y^{\frac{1}{4} } x^{\frac{-1}{4} }[/tex]
= [tex]\frac{dy}{dx} =\frac{-3y^{0.5}x^{-0.5} }{x^{0.75}y^{-0.75} }[/tex]
[tex]=\frac{-3y}{x}[/tex]
=-3y/x
Part b
Value of dy/dx at the point (81,16)
dy/dx at (81,16), x=81, y=16
dy/dx=-3y/x
dy/dx=-3*16/81
dy/dx=-48/81
dy/dx=-16/27
=-0.5926
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