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Leya [2.2K]
3 years ago
14

Please help meeeeeeeeeeeeeeee

Mathematics
1 answer:
NikAS [45]3 years ago
4 0
The answer is 81 :)
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Determine an expression for dy<br> dx given that x = sin3<br> (t)andy = cos3<br> (t)
shtirl [24]

Answer:

\frac{dy}{dx} =-\sqrt[3]{\frac{y}{x} }

Step-by-step explanation:

Recall that using the chain rule we can state:

\frac{dy}{dt} =\frac{dy}{dx}*\frac{dx}{dt}

and therefore solve for dy/dx as long as dx/dt is different from zero.

Then we find dy/dt  and dx/dt,

Given that

x=sin^3(t)\\dx/dt = 3 sin^2(t)* cos(t)

And similarly:

y=cos^3(t)\\dy/dt=-3\,cos^2(t)*sin(t)

Therefore, dy/dx can be determined by the quotient of the expressions we just found:

\frac{dy}{dx} =\frac{dy/dt}{dx/dt} =\frac{-3\,cos^2(t)*sin(t)}{3\,sin^2(t)*cos(t)} =-\frac{cos(t)}{sin(t)}

now notice that we can find   cos(t) = \sqrt[3]{y}  from the expression for y,

and   sin(t) = \sqrt[3]{x}  from its expression for x.

Therefore dy/dx can be written in terms of x and y as:

\frac{dy}{dx} =-\frac{cos(t)}{sin(t)}=-\sqrt[3]{\frac{y}{x} }

4 0
3 years ago
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