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Ksju [112]
4 years ago
12

Theodore has 6 gallons of milk. he uses 1/2 of it to make hot chocolate. then, he uses 2/3 of the milk he has left to make cooki

es. how many gallons of milk does theodore have left after making hot chocolate and cookies?
Mathematics
1 answer:
Anna11 [10]4 years ago
8 0
Theodore firs uses 1/2 of milk to make hot chocolate: 1 - 1/2 = 1/2. So 1/2 of milk is left. After that, he uses 3/2 of the milk he has left to make cookies. It means:
1/2 - ( 2/3 * 1/2 ) = 1/2 - 1/3 = 3/6 - 2/6 = 1/6
At the end 1/6 of the milk Theodore have left.
1/6 * 6 = 1 gallon
Answer: After making hot chocolate and the cookies 1 gallon of milk Theodore have left.
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I'll use subscript notation for brevity, i.e. \frac{\partial f}{\partial x}=f_x.

By the chain rule,

z_u=z_xx_u+z_yy_u

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We have

z=x^4+xy^3\implies\begin{cases}z_x=4x^3+y^3\\z_y=3xy^2\end{cases}

and

\begin{cases}x=uv^4+w^3\\y=u+ve^w\end{cases}\implies\begin{cases}x_u=v^4\\x_v=4uv^3\\x_w=3w^2\\y_u=1\\y_v=e^w\\y_w=ve^w\end{cases}

When u=1,v=1,w=0, we have

\begin{cases}x(1,1,0)=1\\y(1,1,0)=2\end{cases}\implies\begin{cases}z_x(1,2)=12\\z_y(1,2)=12\end{cases}

and the partial derivatives take on values of

\begin{cases}x_u(1,1,0)=1\\x_v(1,1,0)=4\\x_w(1,1,0)=0\\y_u(1,1,0)=1\\y_v(1,1,0)=1\\y_w(1,1,0)=1\end{cases}

So we end up with

\boxed{\begin{cases}z_u(1,1,0)=24\\z_v(1,1,0)=60\\z_w(1,1,0)=12\end{cases}}

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