Q. 67 - Calculus

Question

Leila has been gathering data on the population density of caribou in a valley of the Selkirk Range in British Columbia, Canada. In winter, the caribou stay close to the bottom of the valley. Leila models the population for February with the function 

$$p_{F}(x,y) = 13 e^{-(y+0.5x-2)^2} e^{-0.2(x-1)^2}$$, where $$x$$ and $$y$$ are measured in miles from the center of the valley.

(a) Where are the caribou most likely to be found? 

(b) During the summer, there is a stream in the valley. What is a vector in the direction of the stream? 

Step-by-Step Solution

Answer

(a). Caribou is most likely to be found in the region $$y= \dfrac{-x}{2} +2$$

(b). The vector in the direction of the stream in the valley is $$4i-2j$$

Step 1. Given Information

The model created by Leila of population density of caribou in a valley of the Selkirk Range in British Columbia, Canada for february month is given by $$p_{F}(x,y) = 13 e^{-(y+0.5x-2)^2} e^{-0.2(x-1)^2}$$

Step 2. Solution

To determine where caribou is most likely to be found, we need to maximize the given function $$p_{F}(x,y) = 13 e^{-(y+0.5x-2)^2} e^{-0.2(x-1)^2}$$ . Since, the powers of exponential is negative the function becomes

\begin{equation}p_{F}(x,y) = 13 e^{-(y+0.5x-2)^2} e^{-0.2(x-1)^2}  = 13 e^{-(y+0.5x-2)^2-0.2(x-1)^2}\end{equation}

Also, looking at the function, we know that it is obviously a decreasing function. Plotting the function in Mathematica, we have

Now, we can see from the plot that the caribou are distributed up and down the valley along the line

$$y= \dfrac{-x}{2} +2$$

Step 3. Given information

From the plot, we have that the 

caribou are distributed up and down the valley along the line $$y= \dfrac{-x}{2} +2$$

Step 4. Solution

Also, plotting the function in 3D for part (b), we get the bottom of the valley in the direction of $$4i-2j$$.

So, the vector in the direction of the stream in the valley is $$4i-2j$$.