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

Quizplus · Accepted Answer

The distance between a point and a plane can be found using the formula:

$$\frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$$

where $(x_0,y_0,z_0)$ is a point on the plane and $(a,b,c)$ is the normal vector to the plane.

We need to find the equation of the plane in the form $(ax+by+cz+d)=0$. So we first rearrange the given equation:

$$15x - 3y - 8z = 10 \implies 15x - 3y - 8z - 10 = 0$$

Now we can identify the coefficients and the constant:

$$a=15, b=-3, c=-8, d=-10$$

Next, we need to find a point on the plane. To do this, we can set one of the variables to any value and solve for the other two variables. For example, if we let $z=0$, we get:

$$15x - 3y = 10 \implies y = 5x - \frac{10}{3}$$

So one point on the plane is $(1,5,-\frac{10}{3})$, which means:

$$x_0=1, y_0=5, z_0=-\frac{10}{3}$$

Now we can use the formula to find the distance:

$$\frac{|15(9) - 3(9) - 8(3) - 10|}{\sqrt{15^2 + (-3)^2 + (-8)^2}} \approx 4.29$$

Therefore, the answer is 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