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.