Isaiah and andrew are college students who decided to rent an apartment that costs $755 a month. isaiah receives $250 a month from his parents and andrew's parents give him $225 a month to them pay their rent. they both work at ringaling telemarketing company, making $10.00 an hour.

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kaleahlove13

16.03.2023

Mathematics

By the chain rule, if z=f(x,y)=f(x(t),y(t)) , then

$\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial f}{\partial x}\dfrac{\mathrm dx}{\mathrm dt} + \dfrac{\partial f}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}$

We have f(x,y)=(x+y)e^y , for which

• $\dfrac{\partial f}{\partial x} = e^y$

• $\dfrac{\partial f}{\partial y} = e^y + (x+y)e^y = (x+y+1)e^y$

and x(t)=5t and y(t)=1-t^2 , so that

• $\dfrac{\mathrm dx}{\mathrm dt} = 5$

• $\dfrac{\mathrm dy}{\mathrm dt} = -2t$

Putting everything together, we get

$\dfrac{\mathrm dz}{\mathrm dt} = 5e^y - 2t(x+y+1)e^y \\\\ \dfrac{\mathrm dz}{\mathrm dt} = 5e^{1-t^2} - 2t(5t-t^2+2)e^{1-t^2} = \boxed{\left(2t^3-10t^2-4t+5\right)e^{1-t^2}}$

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