The following planes are parallel. $$ \begin{array}{l}4 x+3 y+5 z+10=0 ext { and } 8 x+6 y+10 z=0 \ ext { Find the distance between the planes. } \ d= ext { Ex: } 5.67\end{array} $$

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The following planes are parallel. $$ \begin{array}{l}4 x+3 y+5 z+10=0 	ext { and } 8 x+6 y+10 z=0 \ 	ext { Find the distance between the planes. } \ d=	ext { Ex: } 5.67\end{array} $$

UpStudy AI · Accepted Answer

To find the distance between two parallel planes, we can use the formula:

$$ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2 + c^2}} $$

where:
- $$ d $$ is the distance between the planes,
- $$ c_1 $$ and $$ c_2 $$ are the constants on the right side of the equations of the planes,
- $$ a $$, $$ b $$, and $$ c $$ are the coefficients of $$ x $$, $$ y $$, and $$ z $$ in the equations of the planes.

Given the equations of the planes:
$$ 4x + 3y + 5z + 10 = 0 $$
$$ 8x + 6y + 10z = 0 $$

We can rewrite the second equation as:
$$ 8x + 6y + 10z - 0 = 0 $$

Now, we can identify the coefficients and constants:
- For the first plane: $$ a = 4 $$, $$ b = 3 $$, $$ c = 5 $$, and $$ c_1 = 10 $$
- For the second plane: $$ a = 8 $$, $$ b = 6 $$, $$ c = 10 $$, and $$ c_2 = 0 $$

Substitute these values into the formula to find the distance between the planes.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left|10-0\right|}{\sqrt{4^{2}+3^{2}+5^{2}}}$$
- step1: Remove 0:
  $$\frac{\left|10\right|}{\sqrt{4^{2}+3^{2}+5^{2}}}$$
- step2: Add the numbers:
  $$\frac{\left|10\right|}{\sqrt{50}}$$
- step3: Calculate the absolute value:
  $$\frac{10}{\sqrt{50}}$$
- step4: Simplify the root:
  $$\frac{10}{5\sqrt{2}}$$
- step5: Reduce the fraction:
  $$\frac{2}{\sqrt{2}}$$
- step6: Multiply by the Conjugate:
  $$\frac{2\sqrt{2}}{\sqrt{2}\times \sqrt{2}}$$
- step7: Multiply the numbers:
  $$\frac{2\sqrt{2}}{2}$$
- step8: Reduce the fraction:
  $$\sqrt{2}$$
The distance between the two parallel planes is approximately 1.414214 units.
The distance between the two parallel planes is approximately 1.414214 units.

The following planes are parallel.

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