The function $z(x,y)$ is given by $z(x,y)=5x^2y^4$, $(x,y\geq 0)$. Determine the point $(x,y)$ where the line with slope $-4$ is tangent to the level curve of the function $z(x,y)$ with $z$-value 83886080.
Antwoord 1 correct
Correct
Antwoord 2 optie
$(x,y)=(1,64)$
Antwoord 2 correct
Fout
Antwoord 3 optie
$(x,y)=(2,16)$
Antwoord 3 correct
Fout
Antwoord 4 optie
$(x,y)=(4,1)$
Antwoord 4 correct
Fout
Antwoord 1 optie
$(x,y)=(4,32)$
Antwoord 1 feedback
Correct: $$\begin{align*}
\dfrac{z'_x(x,y)}{z'_y(x,y)}&= \dfrac{10xy^4}{20x^2y^3}\\
& = \dfrac{y}{2x}
\end{align*}$$
$$\begin{align*}
-4&=- \dfrac{y}{2x},
\end{align*}$$
and hence, $y=8x$.
Plugging in $z(x,y)$:
$$\begin{align*}
5x^2(8x)^4 & = 83886080\\
20480x^6&=83886080\\
x^6 & = 4096\\
x & = 4.
\end{align*}$$
Hence, $y=8\cdot 4=32$.
Go on.
\dfrac{z'_x(x,y)}{z'_y(x,y)}&= \dfrac{10xy^4}{20x^2y^3}\\
& = \dfrac{y}{2x}
\end{align*}$$
$$\begin{align*}
-4&=- \dfrac{y}{2x},
\end{align*}$$
and hence, $y=8x$.
Plugging in $z(x,y)$:
$$\begin{align*}
5x^2(8x)^4 & = 83886080\\
20480x^6&=83886080\\
x^6 & = 4096\\
x & = 4.
\end{align*}$$
Hence, $y=8\cdot 4=32$.
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Antwoord 4 feedback