antoninapride

08.04.2021 •

Mathematics

Ty earns $14 an hour working on weekdays and $21 an hour working on weekends. If he wants to
make at least $600 by working a total of 36 hours
in a week, to the nearest hour, at least how many
hours does he need work on the weekends?

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Ответ:

martinezlittleman

02.07.2022

Mathematics

$\mathrm dx+(x\cot y+\sin y)\,\mathrm dy=0$

Multiply both sides by $\sin y$ :

$\sin y\,\mathrm dx+(x\cos y+\sin^2y)\,\mathrm dy=0$

The ODE is now exact, since

$\dfrac{\partial(\sin y)}{\partial y}=\cos y$

$\dfrac{\partial(x\cos y+\sin^2y)}{\partial x}=\cos y$

so there exists a solution of the form $\Psi(x,y)=C$ . This solution satisfies

$\dfrac{\partial\Psi}{\partial x}=\sin y$

$\dfrac{\partial\Psi}{\partial y}=x\cos y+\sin^2y$

Integrating both sides of the first PDE wrt gives

$\Psi(x,y)=x\sin y+f(y)$

and differentiating wrt gives

$\dfrac{\partial\Psi}{\partial y}=x\cos y+\sin^2y=x\cos y+\dfrac{\mathrm df}{\mathrm dy}$

$\implies\dfrac{\mathrm df}{\mathrm dy}=\sin^2y=\dfrac{1-\cos2y}2$

$\implies f(y)=\dfrac y2-\dfrac{\sin2y}4+C$

So the ODE has solution

$\Psi(x,y)=x\sin y+\dfrac y2-\dfrac{\sin2y}4=C$

which can be rewritten and simplified as

$\Psi(x,y)=\boxed{\sin y(2x-\cos y)+y=C}$

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Ty earns $14 an hour working on weekdays and $21 an hour working on weekends. If he wants to make at least $600 by working a total of 36 hours in a week, to the nearest hour, at least how many hours does he need work on the weekends?