Q 62. - Calculus

Question

Emmy is tracking down another source of contamination at Hanford, this time in a warehouse containing numerous $55$-gallon drums of waste. Detectors placed throughout the facility have measured a certain amount of radiation at different points. From these measurements, Emmy has constructed a polynomial approximation to the radiation given by

$f(x,y)=-0.0156x^2-0.0392y^2+0.0268xy+1.7x+20$

where the radiation is measured in microrems per hour and the distances $x$ and $y$ are measured in feet from a corner of the warehouse. At what location in the ware-

house should Emmy start her search for the contaminated drum?

Step-by-Step Solution

Answer

The required direction is 

Step 1: Given Information

It is given that function is

$f(x,y)=1.2-0.2x^2-0.3y^2+0.1xy-0.25x$

Step 2: To find the direction in which the person descend most steeply at the given point

Points are $(0.5,-0.5)$

The gradient of $f(x, y)$ is given by

$\nabla f(x,y)=f_x(x,y)\mathrm{i}+f_y(x,y)\mathrm{j}+f_z(x,y)\mathrm{k}$

$\nabla f(x,y)=f_x(x,y)\mathrm{i}+f_y(x,y)\mathrm{j}$

$=\frac{\partial }{\partial x}\left(1.2-0.2x^2-0.3y^2+0.1xy-0.25x\right)\mathrm{i}+\frac{\partial }{\partial y}\left(1.2-0.2x^2-0.3y^2\right.+0.1xy-0.25x)\mathrm{j}$

$=(-0.4x+0.1y-0.25)\mathrm{i}+(-0.6y+0.1x)\mathrm{j}$

At $(0.5,-0.5)$

$\nabla f(0.5,-0.5)=⟨-0.5,0.35⟩$ is increasing function

Thus function decreases in direction $-\nabla f(0.5,-0.5)=-⟨-0.5,0.35⟩$

Hence, required points are

$⟨0.5,-0.35⟩$

Step 3: The direction derivative at a point x 0 , y 0 using gradient

It is given by  $D_{\mathbf{u}}f\left(x_0,y_0\right)=\nabla f\left(x_0,y_0\right)·\mathbf{u}$ 

The direction derivative at given direction is zero since it neither ascend not descend

$D_{\mathbf{u}}f\left(x_0,y_0\right)=\nabla f\left(x_0,y_0\right)·\mathbf{u}$

$=0$

Hence, gradient of given function at $(0.5,-0.5)$ is

$\nabla f(0.5,-0.5)=⟨-0.5,0.35⟩$

Assuming  be unit vector

$\nabla f\left(x_0,y_0\right)·\mathbf{u}=0$

$-u_1+0.7u_2=0$

$u_1{}^2+u_2{}^2=1$

Using $u_1=0.7u_2$ in above equation

${\left(0.7u_2\right)}^2+u_2^2=1$

$u_2=\pm \frac{1}{\sqrt{1.49}}$

Using $u_2=\pm \frac{1}{\sqrt{1.49}}\text{ in }{u}_1=0.7u_2$

$u_1=\pm \frac{0.7}{\sqrt{1.49}}$

The required direction is