Find the shortest distance from the point $( 3,9,8 )$ to the plane $3 x + 9 y + 4 z = 16$
Select the correct answer.
A) $D = 61$
B) $D = 90$
C) $D = 10.29$
D) $D = \sqrt { 61 }$
E) $D = 106$

Question

Quizplus · Accepted Answer

We can use the formula for distance from a point to a plane which is:

$$\frac{\lvert Ax_0 + By_0 + Cz_0 + D \rvert}{\sqrt{A^2+B^2+C^2}}$$

where $(x_0,y_0,z_0)$ is the point and $Ax+By+Cz+D=0$ is the equation of the plane.

In this case, $A=3$, $B=9$, $C=4$, $D=-16$, and $(x_0,y_0,z_0)=(3,9,8)$, so the distance is:

$$\frac{\lvert 3(3) + 9(9) + 4(8) - 16 \rvert}{\sqrt{3^2+9^2+4^2}}=\frac{77}{\sqrt{106}}\approx 10.29$$

Therefore, the correct answer is C.

Find the Shortest Distance from the Point (3,9,8)( 3,9,8 )(3,9,8) To the Plane

Find the Shortest Distance from the Point $( 3,9,8 )$ To the Plane