Which of the following functions corresponds to the level curve in the following graph?
Antwoord 1 correct
Correct
Antwoord 2 optie
$z(x,y)=6x + 2y$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$y(x) = 2 - \tfrac{1}{3}x$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$z(x,y) = \min\{6x,2y\}$.
Antwoord 4 correct
Fout
Antwoord 1 optie
$z(x,y=2x + 6y$.
Antwoord 1 feedback
Correct: The shape of the level curves indicates that the general form of the function of which these are level curves is equal to $z(x,y) = ax + by + c$ (see Example 1).
On the level curve with value $k=12$ are the points $(x,y)=(0,2)$ and $(x,y)=(6,0)$; hence, we know that
$$ (1)~z(0,2) = 2b + c = 12 \iff c = 12-2b \quad \text{and}\quad (2)~z(6,0) = 6a + c = 12\iff c = 12-6a .$$
On the level curve with value $k=18$ are the points $(x,y)=(0,3)$ and $(x,y)=(9,0)$; hence, we know that
$$ (3)~z(0,3) = 3b + c = 18\iff c = 18-3b \quad \text{and}\quad (4)~z(9,0) = 9a + c = 18\iff c = 18-9a .$$
Combining $(1)$ and $(3)$ gives
$$\begin{align*}
12-2b &= 18 - 3b\\
b &= 6\\
c &= 12 - 2\cdot6 = 18 - 3\cdot 6 = 0.
\end{align*}$$
Combining $(2)$ and $(4)$ gives
$$\begin{align*}
12-6a &= 18 - 9a\\
3a &= 6\\
a &= 2\\
c &= 12 - 6\cdot2 = 18 - 9\cdot2 = 0.
\end{align*}$$
Hence, the function of which the level curves are given is $z(x,y) = 2x + 6y$.
Go on.
On the level curve with value $k=12$ are the points $(x,y)=(0,2)$ and $(x,y)=(6,0)$; hence, we know that
$$ (1)~z(0,2) = 2b + c = 12 \iff c = 12-2b \quad \text{and}\quad (2)~z(6,0) = 6a + c = 12\iff c = 12-6a .$$
On the level curve with value $k=18$ are the points $(x,y)=(0,3)$ and $(x,y)=(9,0)$; hence, we know that
$$ (3)~z(0,3) = 3b + c = 18\iff c = 18-3b \quad \text{and}\quad (4)~z(9,0) = 9a + c = 18\iff c = 18-9a .$$
Combining $(1)$ and $(3)$ gives
$$\begin{align*}
12-2b &= 18 - 3b\\
b &= 6\\
c &= 12 - 2\cdot6 = 18 - 3\cdot 6 = 0.
\end{align*}$$
Combining $(2)$ and $(4)$ gives
$$\begin{align*}
12-6a &= 18 - 9a\\
3a &= 6\\
a &= 2\\
c &= 12 - 6\cdot2 = 18 - 9\cdot2 = 0.
\end{align*}$$
Hence, the function of which the level curves are given is $z(x,y) = 2x + 6y$.
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: This is the equation of the level curve, not of the equation of the function of which the level curve is shown..
See Example 1 and try again.
See Example 1 and try again.