Find the point on the plane
x − y + z
= 9
that is closest to the point
(3, 4, 6).
(x, y, z)
=?

Answers

Answer 1

The point P on the plane that is closest to Q(3, 4, 6) is (2, 6, 3).

How to find the point on a plane that is closest to a given point in three-dimensional space?

To find the point on the plane that is closest to the given point, we need to find the perpendicular distance from the point to the plane. The vector that is perpendicular to the plane is the normal vector of the plane, which is (1, -1, 1).

Let P be the point on the plane that is closest to the given point Q(3, 4, 6). The vector from Q to P is parallel to the plane's normal vector, so we can write it as a scalar multiple of (1, -1, 1):

P - Q = t(1, -1, 1)

where t is some scalar. We can find t by taking the dot product of both sides with the normal vector of the plane:

(1, -1, 1) ⋅ (P - Q) = 0

Expanding this dot product, we get:

(Px - 3) - (Py - 4) + (Pz - 6) = 0

Simplifying, we get:

Px - Py + Pz = 5

This equation represents all the points on the plane that are equidistant from Q. To find the point that is closest to Q, we need to minimize the distance between Q and P, which is given by ||P - Q||. Substituting the equation for P - Q in terms of t, we get:

\(||P - Q|| = ||t(1, -1, 1)|| = |t|√(1^2 + (-1)^2 + 1^2) =√3|t|\)

To minimize this distance, we need to find the value of t that minimizes |t|. Since we know that Px - Py + Pz = 5, we can substitute Pz = 9 - Px + Py into the expression for |t|:

|t| = |(P - Q) ⋅ (1, -1, 1)| / ||(1, -1, 1)|| = |(Px - Py + Pz - 5) / √3|

= |(Px - Py + (9 - Px + Py) - 5) / √3| = |(4 - Px) / √3|

To minimize |t|, we need to minimize |4 - Px|. Since Px is on the plane, it must satisfy the equation Px - Py + Pz = 5. Substituting Pz = 9 - Px + Py, we get:

Px - Py + (9 - Px + Py) = 5

Simplifying, we get:

Px = 2

Therefore, the point P on the plane that is closest to Q(3, 4, 6) is (2, 6, 3).

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Related Questions

4. Order the set of numbers from least to greatest: 164.8. 814 7 (5 points) 2 o 15 V64 88.14 V64.8 81 2 3. 814 64 o V64 . 3. 814 15 2 V64, 814 8​

Answers

Answer:

For the answer to the question above,

the Order the set of numbers from least to greatest: square root 64, 8 and 1 over 7, 8.14 repeating 14, 15 over 2

15 over 2, square root 64, 8 1 over 7, 8.14 repeating 14.

Step-by-step explanation:

Answer:

15 V64 88.14 V64.8 81 2 3. 814 64 o V64 . 3. 814 15 2 V64, 814 8​

Step-by-step explanation:

help
I need help please help

helpI need help please help

Answers

The difference of the numbers that are given will be:

a. -20

b. -12

c. 11

d. -8

What will the difference be?

It is important to note that the difference simply means the subtraction of the numbers given. In this case, this will be illustrated thus:

a. -8 - 12

= -20

b. 4 - 16

= -12

c. 3 - (-8)

= 3 + 8

= 11

d. -12 - (-4)

= -12 + 4

= -8

It is important to note the following when solving operations:

(minus) × (minus) = plus

(plus) × (minus) = minus

(plus) × (plus) = plus

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helpI need help please help

2. Express the following decimal fractions as a sum of fractions. The denominator should be a power of 10. a)0,32 b)3,003 c)13, 134 d)5,2303​

Answers

Answer:

a) 0.32 = 32/100 = 8/25

b) 3.003 = 3 + 3/1000 = 3000/1000 + 3/1000 = 3003/1000

c) 13.134 = 13 + 134/1000 = 13000/1000 + 134/1000 = 13134/1000

d) 5.2303 = 5 + 2303/10000

Hi Help me with this please :)

Hi Help me with this please :)

Answers

The mean is four

(Brainliest please)

Answer:

4

Step-by-step explanation:

to find the mean you add up all  of the numbers and then divide by the numbr of numbers there are. So to add them all up that will be 3+5+0+4+7+5+5+3=32

so then you divide 32 by 8 which equals 4.

So therefore your answer is 4.

Please give me brainliest.

BRAINLY IF CORRECT!!!! Emmie has $60 to spend on craft supplies. The tax rate is 8.25%. What is the maximum amount she can spend on craft supplies before taxes?

Answers

Answer:

$55.42

Step-by-step explanation:

Price of supplies = 100%

Percentage of tax = 8.25%

Total percentage = 108.25%

Amount allowed to spend = $60

108.25% = 60

60 / 108.25 = .55427

0.55427x100 = 55.427

Round down

$55.42

If my answer is incorrect, pls correct me!

If you like my answer and explanation, mark me as brainliest!

-Chetan K

describe la vida en el barrio que se desarrolla en la pelicula los olvidados de Luis Buñuel teniendo en cuenta sus condiciones economicas y sociales atravez de las distintas actividades desarrolladas por los vecindarios

Answers

Life in the neighborhood depicted in the film "Los Olvidados" is characterized by challenging economic and social conditions.

How do the economic and social conditions shape life in the neighborhood?

The neighborhood in "Los Olvidados" is portrayed as a poverty-stricken area where the residents struggle to make ends meet. The film explores the lives of marginalized individuals most particularly young boys, who are caught in a cycle of poverty, violence, and neglect.

The economic conditions in the neighborhood are harsh, with limited opportunities for employment and a lack of access to basic necessities. As a result, the residents are often forced into criminal activities to survive leading to a high level of violence and crime within the community.

Moreover, the social conditions exacerbate the challenges faced by the neighborhood. There is a sense of abandonment and neglect from the government and wider society with little support or resources available to address the issues faced by the residents.

Full question:

Describes life in the neighborhood that takes place in the film Los Olvidados by Luis Buñuel, taking into account their economic and social conditions through the different activities carried out by the neighborhoods.

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Which equation models the following word problem?The sum of three consecutive integers is 22 more than twice the largest.Find the integers.

Which equation models the following word problem?The sum of three consecutive integers is 22 more than

Answers

The three consecutive integers are

x, (x+1), and (x+2)

Where (x+2) is the largest of them.

22 more than means to add 22.

Twice the largest means 2 times the largest.

So, let us equate the sum of three consecutive integers to 22 plus 2 times (x+2)

\((x)+(x+1)+(x+2)=22+2(x+2)\)

Therefore, the above equation correctly models the given word problem.


Solve the following equations and check your answer.
6x+2=9x+14​

Answers

6x + 2 = 9x + 14

2 - 14 = 9x - 6x

3x = -12

x = -12/3

x = -4

Eric is told by his doctor that he needs to lose 15 pounds. If Eric now weights 285 pounds,then by what percentage should Eric weight decrease?

Answers

Answer:

5.26%

Step-by-step explanation:

15/285=x/100. 15 is the amount of weight he wants to lose. x=5.26%


Which value for x makes the sentence true?
3
4
1+ 4 = 7
0 0 0
521
44
3
co
-3

Which value for x makes the sentence true?341+ 4 = 70 0 0521443co-3

Answers

Answer: 4

Work: 3/4 * 4 + 4 = 7 , 12/4 * 4 + 4 = 7 , 3 + 4 = 7 , 7 = 7

Lisa's age is 4 yrs. older than three times Sammy's age. Lisa is 28 yrs. old. Let's represent Sammy's age.

Which equation can be used to solve for Sammy's age?

a.) 3s = 28
b.) s + 4 = 28
c.) 4s + 3 = 28
d.) 3s + 4 =28

Answers

Answer:

D

Step-by-step explanation:

So it'd be

28=3s+4

28-4=3s

24=3s

24/3=3s/3

8=2

Lisa = 28 yrs

Sammy = Lisa - 4 yrs

Sammy = 28 - 4 yrs

Sammy = 24 yrs

Sammy + 4 = Lisa

S + 4 = 28 ( B )

Can anyone help me solve these equations?
1. 2x + 6 + x -12
2. 1/2 (6x -8) = 3x
3. 5x +1 = -2x -8

Answers

fix the first one,theres no equal sign, and ill solve it

Can anyone help me solve these equations? 1. 2x + 6 + x -12 2. 1/2 (6x -8) = 3x3. 5x +1 = -2x -8
Can anyone help me solve these equations? 1. 2x + 6 + x -12 2. 1/2 (6x -8) = 3x3. 5x +1 = -2x -8

24. walking down jane street, ralph passed four houses in a row, each painted a different color. he passed the orange house before the red house, and he passed the blue house before the yellow house. the blue house was not next to the yellow house. how many orderings of the colored houses are possible?

Answers

The number of orderings of the colored houses is possible in 3 ways

There are 24 possible arrangements for the houses. The number of ways with the blue house next to the yellow house is \($3! \cdot 2!=12$\),

as we can consider the arrangements of B,Y, O, and R.

Thus there are \($24-12$\) arrangements with the blue and yellow houses non-adjacent.

Exactly half of these have the blue house before the yellow house by symmetry, and exactly half of those 6 arrangements have the orange house before the red house (also by symmetry), so our answer is

\(=$12 \cdot \frac{1}{2} \cdot \frac{1}{2}\)

=3

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What is the radius of the circle: x^2+y^2=4
1. -2
2. -4
3. 4
4. 2

Answers

Answer:

2

Step-by-step explanation:

→ The radius of a circle is square root the 4

2

How many pounds does 6.5 dozen cupcakes weigh?
• 1 cupcake = 35.3 grams

Answers

Answer:

35.3g * 78 =2,753.4g

2,753.4g = 6.07lbs

What is the GCF? 3x2-2x

Answers

Answer:

1

Step-by-step explanation:

Everything can be multiplied by 1 in this situation so 1 is the greatest common factor

need help with these 2 questions please

need help with these 2 questions please

Answers

For the first one it’s x=6.6
And the second one is x=-11

Answer:  3. x = 20/3         4. x = -11

Step-by-step explanation:

3.    3x - 5= 15        Add 5 to both sides

            +5   +5

      3x =  20           Now divide both sides by 3

     x = 20/3  

4.   -2x + 7 =29    Subtract 7 from both sides

             -7     -7

      -2x = 22      Divide both sides by  -2

     x = -11

Determine whether each of these compound propositions is satisfiable. (a) (p∨q∨¬r)∧(p∨¬q∨¬s)∧(p∨¬r∨¬s)∧(¬p∨¬q∨¬s)∧(p∨q∨¬s)(b) (¬p∨¬q∨r)∧(¬p∨q∨¬s)∧(p∨¬q∨¬s)∧(¬p∨¬r∨¬s)∧(p∨q∨¬r)∧(p∨¬r∨¬s)

Answers

To determine if a compound proposition is satisfiable, we need to check if there exists an assignment of truth values to the propositional variables that makes the entire proposition true. In (a), we can see that all five clauses have at least one occurrence of variable p. Therefore, p must be true for the entire proposition to be true. If we set p=true, we can then satisfy the first three clauses by setting q=true and r=false. Then we can satisfy the fourth clause by setting s=false. Finally, the fifth clause is already satisfied since p and q are true and r is false. Therefore, (a) is satisfiable. In (b), we can see that each clause has at least one occurrence of either p or ¬p. Therefore, (b) is also satisfiable.

To determine if a compound proposition is satisfiable, we need to check if there exists an assignment of truth values to the propositional variables that makes the entire proposition true. In (a), we can see that all five clauses have at least one occurrence of variable p. Therefore, p must be true for the entire proposition to be true. Then, we can examine the other variables and determine if we can satisfy each clause with a combination of true and false assignments. We can do this by starting with the first clause and finding values that make it true, then moving on to the next clause and so on. In (b), we can use the same method by examining the clauses and finding values that satisfy each one.

In conclusion, both compound propositions (a) and (b) are satisfiable. We can satisfy them by assigning truth values to the propositional variables in a way that makes each clause true. In (a), we need p=true, q=true, r=false, and s=false. In (b), we need p=true, q=false, r=true, and s=false.

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help me please i will give brainlyest

help me please i will give brainlyest

Answers

Answer:

Answer: 1645.

Step-by-step explanation:

Given data:

No. Of days of rental = 11 days.

Solution:

= 325 + 120d

Input the data for (d) into the equation

= 325 + 120(11)

= 325 + 1320

= 1645.

Step-by-step explanation:

Simplify 3x+5m+6m+c please

Answers

Answer:

3x + 11m + c

Step-by-step explanation:

Step 1) Write the Equation: 3x + 5m + 6m + c

Step 2) Add like Terms: 3x + 11m + c

Why is the use of 0 and necessary in a measuring instrument?

Answers

In order for measuring devices to produce correct findings, they must have zero error. Zero error also makes it possible to determine if an instrument is functioning properly or not.

what is measuring instrument?

Instruments used by researchers and practitioners to assist in the assessment or evaluation of subjects, clients, or patients are known as measurement tools. From physical functioning to psychosocial welfare, the tools are used to assess or gather data on a number of aspects.

Electrical instruments are one of three categories for measuring devices. Instrument for electronics. a mechanical instrument.

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Diana usually drives at average rate of 30 mph. If she drives for 2 hours with the speed that is 10 mph more than usual, what distance will she cover?

Answers

Answer:  80 miles

Step-by-step explanation:

10mph + 30mph = 40 mph

Distance = Speed × time

D = 40m/h × 2 h

D = 80 m

find the slope of the line through each pair of points

a. (8,-7) and (5,-3)
b. (-5,9) and (5,11)
c. (-8,-4) and (-4,-9)

Answers

Answer:

I think the answers would be :

a . 4/3

b. 1/5

c . - 5/4

hope it helps u ^^

the denver post reported that a recent audit of los angeles 911 calls showed that 85% were not emergencies. suppose the 911 operators in los angeles have just received five calls. a button hyperlink to the salt program that reads: use salt. (a) what is the probability that all five calls are, in fact, emergencies? (round your answer to five decimal places.) (b) what is the probability that two or more calls are not emergencies? (round your answer to five decimal places.) (c) what is the smallest number of calls that the 911 operators need to answer to be at least 92% (or more) sure that at least one call is, in fact, an emergency? (enter your answer as a whole number.) calls

Answers

(a) The probability that all five calls are emergencies is 0.00008.

(b) The probability that two or more calls are not emergencies is 0.67808.

(c) The smallest number of calls that the 911 operators need to answer to be at least 92% (or more) sure that at least one call is an emergency is 7

(a) If 85% of the calls are not emergencies, then the probability that a call is an emergency is 1 - 0.85 = 0.15. Assuming that the calls are independent, the probability that all five calls are emergencies is

0.15^5 = 0.0000759375

Rounding to five decimal places, the probability is 0.00008.

(b) The probability that a call is not an emergency is 0.85. The probability that two or more calls are not emergencies can be calculated using the complement rule

P(two or more calls are not emergencies) = 1 - P(all five calls are emergencies) - P(one call is not an emergency)

We can calculate P(one call is not an emergency) as

P(one call is not an emergency) = 5C1 * 0.15^1 * 0.85^4 = 0.3218359375

where 5C1 represents "5 choose 1", or the number of ways to choose 1 call out of 5. Using this, we can calculate:

P(two or more calls are not emergencies) = 1 - 0.00008 - 0.3218359375 = 0.6780840625

Rounding to five decimal places, the probability is 0.67808.

(c) To determine the smallest number of calls needed to be at least 92% sure that at least one call is an emergency, we can use the complement rule again

P(at least one call is an emergency) = 1 - P(no calls are emergencies)

We want this probability to be at least 0.92, so we can set up the following inequality

1 - (0.15)^n >= 0.92

where n is the number of calls. Solving for n, we get

n >= log(0.08) / log(0.15)

n >= 6.3979

Since n must be a whole number, we round up to n = 7.

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Du


Find the area of the base and the volume of a cereal box that measures 20. 5 inches high


long, and 3. 25 inches wide.

Answers

The area of the base of the cereal box can be calculated by multiplying its length and width. In this case, the length is 20.5 inches and the width is 3.25 inches. Thus, the area of the base is 66.625 square inches.

The volume of the cereal box can be calculated by multiplying the area of the base by the height. The height of the box is given as 20.5 inches. Therefore, the volume of the cereal box is 66.625 square inches multiplied by 20.5 inches, which equals 1,365.0625 cubic inches.

In summary, the area of the base of the cereal box is 66.625 square inches, and the volume of the box is 1,365.0625 cubic inches.

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Here are two random variables that are uncorrelated but not independent. Let X and Y have the following joint probability mass function:

Answers

The marginal probability mass functions of X and Y are:

P(X = 0) = 0.5, P(X = 1) = 0.5P(Y = 0) = 0.3, P(Y = 1) = 0.4, P(Y = 2) = 0.2

In the given scenario, X and Y are two random variables that are uncorrelated but not independent. The joint probability mass function of X and Y is provided as follows:

P(X = 0, Y = 0) = 0.2

P(X = 0, Y = 1) = 0.1

P(X = 0, Y = 2) = 0.1

P(X = 1, Y = 0) = 0.1

P(X = 1, Y = 1) = 0.2

P(X = 1, Y = 2) = 0.1

To obtain the marginal probability mass functions, we sum the probabilities for each value of X and Y.

For X:

P(X = 0) = 0.2 + 0.1 + 0.1 + 0.1 = 0.5

P(X = 1) = 0.1 + 0.2 + 0.1 + 0.1 = 0.5

Therefore, the marginal probability mass function of X is P(X = 0) = 0.5 and P(X = 1) = 0.5.

For Y:

P(Y = 0) = 0.2 + 0.1 = 0.3

P(Y = 1) = 0.1 + 0.2 + 0.1 = 0.4

P(Y = 2) = 0.1 + 0.1 = 0.2

Hence, the marginal probability mass function of Y is P(Y = 0) = 0.3, P(Y = 1) = 0.4, and P(Y = 2) = 0.2.

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12. Find the perimeter of the isosceles triangle. ​

12. Find the perimeter of the isosceles triangle.

Answers

Answer:

Step-by-step explanation:

First find the length of the two equal sides. Use Pythagoras.

c^2 = a^2 + b^2

c = ?

a = 7

b = 1/2 48 = 24

c^2 = 24^2 + 7^2

c^2 = 576 + 49

c^2 = 625                Take the square root of 625

√c^2 = √625

c = 25

Both of the equal sides = 25

Perimeter and answer = 25 + 25 + 48 = 98

Answer: 25 + 25 + 48 = 98

Step-by-step explanation:

Consider the function p(x) = cos^2x/ sin 2x Which of the following accurately describes the limit as x approaches 0 of the function? (1 point) As x approaches 0, the limit of p(x) approaches a large negative y-value. As x approaches 0, the limit of p(x) does not exist. As x approaches 0, the limit of p(x) approaches 0. As x approaches 0, the limit of p(x) approaches a large positive y-value.

Answers

The function, p(x) = cos²x/ sin 2x  has a limit that is undefined (does not exist) as x approaches 0.

The correct option therefore option b;

b. As x approaches 0, the limit of p(x) does not exist

Here, we have,

We have,

The given function is presented as follows;

p(x) = cos²x/ sin 2x

Required;

The limit of the function as x approaches (zero) 0

we have,

The limit of a function at a given point within the domain of the function is the value of the function as the function's argument approaches a.

Therefore, the limit of the given function at the point x = 0 is given by the function's value as the argument of the function, x approaches 0.

cos²(0) = 1

sin(2×0) = sin(0) = 0

Therefore;

p(x) = cos²0/ sin 2(0) = infinity

Therefore, the limit of the function does not exist as x approaches 0

The correct option is therefore, option b

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Compute the flux of the vector field F = 3i+j - 4k through the oriented rectangular region S shown in 2 S y lying above the square D = {(x, y): 0 < x, y <1} in the zy-plane.

Answers

The flux of the vector field F = 3i+j-4k through the oriented rectangular region S is -3.

To compute the flux of the vector field F = 3i+j-4k through the oriented rectangular region S lying above the square D = {(x, y): 0 < x, y <1} in the zy-plane, we can use the divergence theorem, which states that the flux of a vector field through a closed surface is equal to the triple integral of the divergence of the vector field over the enclosed volume. Since S is not a closed surface, we need to add a surface to close it up.

Let R be the rectangle in the xy-plane with vertices at (0,0), (1,0), (1,1), and (0,1), and let T be the surface consisting of the top and bottom faces of the rectangular box with sides R × [0,2]. Then S is the boundary of T, oriented so that the normal points in the positive z direction.

The divergence of F is div F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z = 0 + 0 - 4 = -4. Therefore, by the divergence theorem,

flux of F through S = ∫∫S F · n dS = ∭T div F dV

= ∭T (-4) dV

= (-4) ∭T dV

= (-4) volume of T

The volume of T is the product of the area of R and the height of the box, which is 2. Therefore, the flux of F through S is

flux of F through S = (-4) volume of T = (-4) × 2 × area of R

= (-4) × 2 × 1 = -8.

Thus, the flux of F through S is -8.

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One angle in an equilateral triangle is 60. What is the measure of another angle?

One angle in an equilateral triangle is 60. What is the measure of another angle?

Answers

Answer:

60

Step-by-step explanation:

all 3 sides are equal length so it has to be 60

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