1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
IrinaK [193]
3 years ago
15

Is it true that the planes x + 2y − 2z = 7 and x + 2y − 2z = −5 are two units away from the plane x + 2y − 2z = 1?

Mathematics
1 answer:
zhuklara [117]3 years ago
8 0

Lets Find It Out..

First we'll find the equation of ALL planes parallel to the original one.

As a model consider this lesson:

Equation of a plane parallel to other

The normal vector is:
<span><span>→n</span>=<1,2−2></span>

The equation of the plane parallel to the original one passing through <span>P<span>(<span>x0</span>,<span>y0</span>,<span>z0</span>)</span></span>is:

<span><span>→n</span>⋅< x−<span>x0</span>,y−<span>y0</span>,z−<span>z0</span>>=0</span>
<span><1,2,−2>⋅<x−<span>x0</span>,y−<span>y0</span>,z−<span>z0</span>>=0</span>
<span>x−<span>x0</span>+2y−2<span>y0</span>−2z+2<span>z0</span>=0</span>
<span>x+2y−2z−<span>x0</span>−2<span>y0</span>+2<span>z0</span>=0</span>

Or

<span>x+2y−2z+d=0</span> [1]
where <span>a=1</span>, <span>b=2</span>, <span>c=−2</span> and <span>d=−<span>x0</span>−2<span>y0</span>+2<span>z0</span></span>

Now we'll find planes that obey the previous formula and at a distance of 2 units from a point in the original plane. (We should expect 2 results, one for each half-space delimited by the original plane.)
As a model consider this lesson:

Distance between 2 parallel planes

In the original plane let's choose a point.
For instance, when <span>x=0</span> and <span>y=0</span>:
<span>x+2y−2z=1</span> => <span>0+2⋅0−2z=1</span> => <span>z=−<span>12</span></span>
<span>→<span>P1</span><span>(0,0,−<span>12</span>)</span></span>

In the formula of the distance between a point and a plane (not any plane but a plane parallel to the original one, equation [1] ), keeping <span>D=2</span>, and d as d itself, we get:

<span><span>D=<span><span>|a<span>x1</span>+b<span>y1</span>+c<span>z1</span>+d|</span><span>√<span><span>a2</span>+<span>b2</span>+<span>c2</span></span></span></span></span>
<span>2=<span><span><span>∣∣</span>1⋅0+2⋅0+<span>(−2)</span>⋅<span>(−<span>12</span>)</span>+d<span>∣∣</span></span><span>√<span>1+4+4</span></span></span></span>
<span><span>|d+1|</span>=2⋅3</span> => <span><span>|d+1|</span>=6</span>First solution:
<span>d+1=6</span> => <span>d=5</span>
<span>→x+2y−2z+5=0</span>Second solution:
<span>d+1=−6</span> => <span>d=−7</span>
<span>→x+2y−2z−7=<span>0</span></span></span>
You might be interested in
Andrea has 18 yards of fabric. How many 4 inch pieces can she cut from her original place?
Galina-37 [17]
4inches = 1/3 of a foot (4inches x 3 = 12 inches) - so she can cut 3 out of each foot

1 yard = 3 feet - so she can cut 3 x 3 = 9 out of each yard

18 x 9 = 162 pieces
8 0
3 years ago
Read 2 more answers
Kendra bought a magazine for $3 and four paperback books for $5 each the expression 3 + 4 x 5 represents the total cost in dolla
Elodia [21]
I´m pretty sure that the terms would just be 3, 4, and 5. It depends what kind of problem it is. 
3 0
3 years ago
How I can solve this
gavmur [86]

Answer:

\boxed{\textrm{B)}\qquad y>4}.

Step-by-step explanation:

We start by subtracting 1 from both sides of the second inequality:

2x > 5 \iff 2x - 1 > 5 - 1 \iff 2x - 1 > 4.

Therefore, we conclude that:

y > 2x - 1 > 4 \implies y > 4.

Therefore, every point that satisfies the system has a y-coordinate greater than 4.

7 0
3 years ago
The average wage for an electrician in CA is $25.47/hour. That is 11% above the national average. What is the national average?​
yan [13]

Answer:

$22.95/hour.

Step-by-step explanation:

It is given that the average wage for an electrician in CA is $25.47/hour. That is 11% above the national average.

Let as consider x be the national average.

11% above the national average is

x+\dfrac{11}{100}x=x+0.11x=1.11x

The average wage for an electrician in CA is 11% above the national average. So,

1.11x=25.47

Divide both sides by 1.11.

x=\dfrac{25.47}{1.11}

x=22.945945...

x=22.95

Therefore, the national average is $22.95/hour.

3 0
3 years ago
Do you need to restrict the domain of hte inverse of a cubic function
aalyn [17]
I dont believe so but im not 100% sure i hoped i helped
3 0
3 years ago
Other questions:
  • Ebert used to make $22 an hour, but got a 10% raise. How much more will he make in a 40 work week with the raise? Give steps!
    9·2 answers
  • I need your help plz say fast this answer​
    9·1 answer
  • What does it mean for rows to be linearly independent?
    11·1 answer
  • —BRAINLIEST—A person and a tree cast a shadow at the same time of day. Round your answer to the nearest hundredth.
    7·2 answers
  • Jim’s family is choosing whether to go to a restaurant for lunch or for dinner. The cost of a salad is $7 at lunch and $9 at din
    14·2 answers
  • Pls help:( this is due by the end of the day!! 16 points!
    6·1 answer
  • There are
    6·2 answers
  • Simplify each expression.<br> (3m^1/2•27n^1/4)^4
    10·1 answer
  • Parker offers to pay for 5 of his friends to play laser tag,
    11·2 answers
  • A clock hand rotated from 12 to 6 this is​
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!