Find the distance between the point $( 9,9,3 )$ and the plane
$15 x - 3 y - 8 z = 10$. Round your answer to two decimal place.
A) $4.79$
B) $6.79$
C) $4.29$
D) $7.79$
E) $5.79$

Question

Find the distance between the point $( 9,9,3 )$ and the plane 
$15 x - 3 y - 8 z = 10$. Round your answer to two decimal place.
A) $4.79$
B) $6.79$
C) $4.29$
D) $7.79$
E) $5.79$

Quizplus · Accepted Answer

The distance between a point and a plane is given by the formula: 
$$\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$
where $(x_1,y_1,z_1)$ is the point and the equation of the plane is $ax+by+cz+d=0$.

In this case, we have the point $(9,9,3)$ and the plane $15x-3y-8z=10$. To find the distance between them, we first need to write the equation of the plane in the form $ax+by+cz+d=0$. We can do this by adding 10 to both sides and multiplying by -1:
$$15x - 3y - 8z - 10 = 0$$
$$-15x + 3y + 8z + 10 = 0$$
So, we have: 
$$a=-15,\quad b=3,\quad c=8,\quad d=10,\quad x_1=9,\quad y_1=9,\quad z_1=3$$
Using the distance formula, we get:
$$\frac{|(-15)(9)+(3)(9)+(8)(3)+10|}{\sqrt{(-15)^2+3^2+8^2}}$$
$$\frac{|-135+27+24+10|}{\sqrt{338}}$$
$$\frac{146}{\sqrt{338}}\approx 4.29$$
Rounding to two decimal places gives the answer of $\boxed{\textbf{(C)}}$.

Find the Distance Between the Point (9,9,3)( 9,9,3 )(9,9,3) And the Plane

Find the Distance Between the Point $( 9,9,3 )$ And the Plane