Emmy is trying to get information about the water table below the Hanford reservation. She has drilled wells that show that the water table can be found at (0, 0,−35) and (300, 0,−38). She drills one more well and finds the water

table at (0, 300,−37).

(a) Find a plane that approximates the water table.

(b) If she drills another hole at x = 300, y = 300, how deep does she expect to find the water table?

Consider the plane that contains the points $P=(0,0,-35),Q=(300,0,-38)$ and $R=(0,300,-37)$

The goal is to determine the planar equation that approximates the water table.

To find the equation of the plane, first show that the points are non-collinear.

The vector $\overrightarrow{PQ}$is:

$$\begin{gathered} \overrightarrow{PQ}=⟨300-0,0-0,-38+35⟩ \\ =⟨300,0,-3⟩ \end{gathered}$$

The vector $\overrightarrow{QR}$is:

$$\begin{gathered} \overrightarrow{QR}=⟨0-300,300-0,-37+38⟩ \\ =⟨-300,300,1⟩ \end{gathered}$$

The vector $\overrightarrow{PR}$is:

$$\begin{gathered} \overrightarrow{PR}=⟨0-0,0-300,-35+37⟩ \\ =⟨0,-300,2⟩ \end{gathered}$$

Because neither of the vectors is a scalar multiple of the other, they are non-collinear. Calculate the normal vector by using the cross-product of any two vectors to determine the plane's equation.

The normal vector $\mathbf{N}=\overrightarrow{PQ}\times \overrightarrow{PR}$is given by:

$$\begin{gathered} =-900\mathbf{i}-600\mathbf{j}-900\mathbf{k} \\ =⟨-900,-600,-900⟩ \\ =\{3,2,3⟩ \end{gathered}$$

The equation of the plane is:

$$\begin{gathered} 3(x-0)+2(y-0)+3(z+35)=0 \\ 3x+2y+3z+105=0\left(\text{Simplify}\right) \end{gathered}$$

The equation of the plane that approximates water table is $3x+2y+3z+105=0$

The objective is to find the depth of the water table if the point is drilled at $x=300$ and $y=300$

To find the depth of the water table, substitute the value of $x=300$ and $y=300$ in the equation of the plane $3x+2y+3z+105=0$

The depth of the water table is:

$3x+2y+3z+105=0$(Equation of the plane)

$3\left(300\right)+2\left(300\right)+3z+105=0$(Substitution)

$$\begin{gathered} 900+600+3z+105=0 \\ 3z=-1605\text{(Simplify)} \\ z=-535 \end{gathered}$$

The negative sign denotes that the depth is below the level of the surface.

Therefore, the depth of the water table is $535$