Find an equation of the sphere containing all surface points P = (x, y, z) such that the distance from P to A(−3, 6, 3) is "twice the distance from P to" B(6, 2, −3).
Answer:
Equation of Sphere = [tex]x^{2}[/tex] + [tex]y^{2}[/tex] + [tex]z^{2}[/tex] - 18x - 4/3y +10z + 142/3 = 0
Step-by-step explanation:
Data Given:
P = (x,y,z)
Distance from P to A (-3,6,3) = Twice the distance from P to B(6,2,-3)
Solution:
Find the equation of the sphere:
It is given that:
PA = 2PB
Squaring both sides:
[tex](PA)^{2} = (2PB)^{2}[/tex]
[tex](PA)^{2} = 4 (PB)^{2}[/tex]
[tex](x - (-3))^{2}[/tex] + [tex](y - 6)^{2}[/tex] + [tex](z-3)^{2}[/tex] = 4 x [tex](x-6)^{2}[/tex] + [tex](y-2)^{2}[/tex] + [tex](z-(-3))^{2}[/tex]
Solving the above equation:
[tex](x + 3))^{2}[/tex] + [tex](y - 6)^{2}[/tex] + [tex](z-3)^{2}[/tex] = 4 x {[tex](x-6)^{2}[/tex] + [tex](y-2)^{2}[/tex] + [tex](z + 3))^{2}[/tex]}
[tex]x^{2}[/tex] + 9 + 6x + [tex]y^{2}[/tex] + 36 - 12y + [tex]z^{2}[/tex] + 9 - 6z = 4 { [tex]x^{2}[/tex] + 36 - 12x + [tex]y^{2}[/tex] + 4 - 4y + [tex]z^{2}[/tex] + 9 + 6z}
[tex]x^{2}[/tex] + 9 + 6x + [tex]y^{2}[/tex] + 36 - 12y + [tex]z^{2}[/tex] + 9 - 6z = 4[tex]x^{2}[/tex] + 144 - 48x +4[tex]y^{2}[/tex] + 16 - 16y + 4[tex]z^{2}[/tex] + 36 + 24z
Putting the right hand side = 0 and solving the equation:
[tex]x^{2}[/tex] - 4[tex]x^{2}[/tex] + [tex]y^{2}[/tex] - 4[tex]y^{2}[/tex] + [tex]z^{2}[/tex] - 4[tex]z^{2}[/tex] + 6x + 48x - 12y +16y -6z - 24z + 9 + 36 + 9 -144 - 16 - 36 = 0
-3[tex]x^{2}[/tex] - 3[tex]y^{2}[/tex] -3[tex]z^{2}[/tex] + 54x + 4y -30z -142 = 0
Taking (-) common
- ( 3[tex]x^{2}[/tex] + 3[tex]y^{2}[/tex] + 3[tex]z^{2}[/tex] - 54x - 4y +30z + 142) = 0
3[tex]x^{2}[/tex] + 3[tex]y^{2}[/tex] + 3[tex]z^{2}[/tex] - 54x - 4y +30z + 142 = 0
dividing the whole equation by 3
[tex]x^{2}[/tex] + [tex]y^{2}[/tex] + [tex]z^{2}[/tex] - 54/3x - 4/3y +30/3z + 142/3 = 0
[tex]x^{2}[/tex] + [tex]y^{2}[/tex] + [tex]z^{2}[/tex] - 54/3x - 4/3y +30/3z + 142/3 = 0
[tex]x^{2}[/tex] + [tex]y^{2}[/tex] + [tex]z^{2}[/tex] - 18x - 4/3y +10z + 142/3 = 0
The equation of the sphere for all surface point P such that [tex]AP = 2\cdot BP[/tex] is [tex]x^{2}+y^{2}+z^{2}-18\cdot x -\frac{4}{3}\cdot y + 10\cdot z +\frac{142}{3} = 0[/tex].
Procedure - Determination of the equation of a sphereGeometric locus and Pythagorean expansionMathematically speaking, we have the following geometric locus:
[tex]AP = 2\cdot BP[/tex] (1)
Then, we expand the expression presented above by Pythagorean theorem, used to determine the length of a line segment in the field of analytical geometry:
[tex]\sqrt{(x +3)^{2} + (y-6)^{2} + (z-3)^{2}} = 2\cdot \sqrt{(x-6)^{2}+(y-2)^{2}+(z+3)^{2}}[/tex]
[tex](x+3)^{2}+(y-6)^{2}+(z-3)^{2} = 4\cdot (x-6)^{2}+4\cdot (y-2)^{2}+4\cdot (z+3)^{2}[/tex]
ResolutionNow we expand and simplify the resulting expression:
[tex]x^{2}+6\cdot x + 9 + y^{2}-12\cdot y + 36 + z^{2}-6\cdot z + 9 = 4\cdot (x^{2}-12\cdot x +36) + 4\cdot (y^{2}-4\cdot y +4) + 4\cdot (z^{2}+6\cdot z + 9)[/tex]
[tex]x^{2} + y^{2} + z^{2} + 6\cdot x -12\cdot y - 6\cdot z +54 = 4\cdot x^{2} +4\cdot y^{2} + 4\cdot z^{2} -48\cdot x -16\cdot y +24\cdot z +196[/tex]
[tex]3\cdot x^{2} + 3\cdot y^{2} + 3\cdot z^{2} -54\cdot x -4\cdot y + 30\cdot z + 142 = 0[/tex]
[tex]x^{2}+y^{2}+z^{2}-18\cdot x -\frac{4}{3}\cdot y + 10\cdot z +\frac{142}{3} = 0[/tex] (2)
This expression corresponds to the general form of the equation of the sphere for all surface point P such that [tex]AP = 2\cdot BP[/tex]. [tex]\blacksquare[/tex]
To learn more on geometric loci, we kindly invite to check this verified question: https://brainly.com/question/521553
HELP PLSSSS ASAP ILL MARK AS BRAINLIEST AND 30 POINTS
Answer:
A- 4.50
B- unit price is 2 dollar per a hat. Price would be 24 dollars
C- unit price is .40
D- 69
Step-by-step explanation:
If pencils sell at 6 for 59¢, how many pencils can be bought for $2.95?
30
First you divide 2.95 by .59, which is 5.
Then you multiply 6 by 5, and get 30!
The perimeter of the rectangle shown is 96 feet. Determine the length and width of the rectangle.
Answer:
45
Step-by-step explanation:
detrain of the value they was given
Answer:
Length = 37 feet
width = 11 feet
Step-by-step explanation:
First, let us find the value of x.
They've already given that the perimeter of this rectangle is 96 feet.
So,
4x - 7 + 4x - 7 + x + x = 96
10x - 14 = 96
10x = 96 + 14
10x = 110
Divide both sides by x.
x = 11 feet
And now we can find the value of the width and length.
Given that,
Length = ( 4x - 7 ) ⇒ ( 4 × 11 - 7 )
44 - 7
37 feet
Width = x ⇒ 11 feet
Let me know if you have any other questions :-)
The number of leaves that fall off a tree n days after day 1 is given by the function f (n) = 3(2)".
What does the value 3 mean in this function
Answer:
the amount of leaves that fall of the tree in one day
Step-by-step explanation:
n is the amount of days
f is the amount of leaves that fall in a day
I GIVEEEEE BRASINLILST
Answer:
Answer: Choice G.
Step-by-step explanation:
Dilations
Suppose a point P(x,y) is dilated across the origin, that is, the origin is the center of dilation, by a scale factor k.
The image of P is P' whose coordinates are P'(kx,ky).
If an image is formed by several points and each one is dilated by the same factor, then:
If |k| > 1, the image maps to a stretched or larger image.
If |k| < 1, the image maps to a shrunk or shorter image.
We have a hexagon graphed on a coordinate grid. The hexagon is dilated to create a larger hexagon. This means k must be greater than 1 and it should multiply each original coordinate (x,y).
Options F and H don't multiply by the coordinates but add to them, thus they are not valid answers.
Option J actually multiplies by the factor 0.2 but since it's less than 1, it does not correspond to an enlargement.
Only option G multiplies by a factor 4/3=1.3 greater than 1.
Answer: Choice G.
Solve the proportion. Please show me step by step how to solve this. Thanks
4.6/6 = w/3
9514 1404 393
Answer:
w = 2.3
Step-by-step explanation:
It looks like you have ...
4.6/6 = w/3
To solve this, you need to clear the 3 from the denominator of the w term. Do that by multiplying both sides of the equation by 3.
3(4.6/6) = 3(w/3)
4.6(3/6) = w . . . . simplify (rearrange the product to make mental arithmetic easier: 3/6 = 1/2)
w = 2.3 . . . . . . . . do the arithmetic
Answer:
2.3
Step-by-step explanation:
There are three types of sweet in a jar and this table gives
some information about them.
Type of sweet Eclair Humbug Mint
Number of sweets (see picture 4 3x + 6 2x)
A sweet is chosen at random.
The probability it is an eclair is
15
Work out the probability it is a humbug.
Answer:
Answer:
2/3
Step-by-step explanation:
Given:
Number of eclairs = 3
Number of humbugs =2x+6=2(x+3)
Number of mints = x
Therefore, total number of sweets in the jar is,
s=3+2x+6+x=3x+9=3(x+3)
Now, probability of selecting a sweet to be humbug is given as:
p(humbug)= number of humbugs/number of sweets
p(humbug)=2(x+3)/3(x+3)=2/3
Therefore, the probability that it is a humbug is 2/3
Step-by-step explanation:
-5(2x - 7) + 24 = 89
Answer:
x=-3
Step-by-step explanation:
First you must distribute -5 to 2x and -7
so you get
-10x+35+24=89
then you will subtract 35 from both sides and get
-10x+24=54 then you will subtract 24 from each side and get
-10x=30 then you will divide -10 and get
x=-3
A jet travels 4648 miles against the wind in 8 hours and 5368 miles with the wind in the same amount of time. What is the rathe of the jet in still air and what is the rate of the wind?
Answer:
The rate of the jet in still air is 738 mph and the rate of the wind is 90 mph
Step-by-step explanation:
Check: (738+90)*6 = 648*6 = 3888 mike's against the wind
(738-90)*6 = 828*6 = 4968 miles with the wind
jacks cow is tied to a beanstalk with a piece of rope that is 15ft long. what is the exact area of the circle in which a cow can graze?
Answer:
705.5 ft^2
Step-by-step explanation:
a=πr²
a=π15²
a=π225
a=(3.14)(225)
a=705.5
Express square root of 169^5 using rational exponents
Answer:
The square root is 371293.
Step-by-step explanation:
We solve this question using properties of exponents.
The square root of 169 is 13, so the exponent given can be written as:
[tex]169^{5} = (13^{2})^5 = 13^{10}[/tex]
The square root of [tex]13^{10}[/tex] is given by:
[tex]\sqrt{13^{10}} = 13^{\frac{10}{2}} = 13^5 = 371293[/tex]
The square root is 371293.
Triangle V P M. Angle V is 52 degrees, angle P is 78 degrees, angle M is x degrees. What is the measure of AngleM?
Answer:
The answer is 50
Step-by-step explanation:
Got it right on edge
Answer:
50 the guy under me got it right on edge 2021
Step-by-step explanation:
Hopes this helps!!! ^^
I need youre helpppp
Answer:
B , C, and D
Step-by-step explanation:
x + y and x+ z both make a straight line which is 180 degrees, and Z and Y are congruent angles.
Given the diagram to the right,determine m
Answer:
The measure of Angle S is 75 (degrees).
Step-by-step explanation:
Key skills needed to solve this are: Sums of a triangle, one variable equation, operations
1) All the angles in a triangle add up to equal 180 degrees.
2) We can use this to say that Angle S + Angle R + Angle T is 180 degrees.
3) Subsitute in the expression for each angle: 7x+40+8x+25+5x+15 = 180
4) Combine like terms
7x + 8x + 5x is 20x & 40 + 25 + 15 is 80
This means that 20x + 80 = 180
5) Move 80 to the right side: 20x = 100
6) Divide by 20 andbget x equal to 5
7) To find Angle S, you need to plug in 5 into the expression.
7x + 40 = 7(5) + 40 = 35 + 40 = 75 degrees
The measure of Angle S is 75
Pls help best will get Brainliest
Missing addend: 1/2 + 1/4
Missing Sums: 5/8 and 3/4
Step-by-step explanation:
The pattern of the addend is that the first term increases by 1/8 and the second term is always 1/4. If only the first term increases by 1/8, that means the total sum will increase by 1/8 each time also.
what is the answers to 3 1/4 *1 1/3 this is a 15 point offer!!!
Answer:
3 1/4*1 1/3 = 4 1/3
Step-by-step explanation:
I have a confusion is it 3 1/4 *1 1/3 does 3 and 1 have space or 31/4 *11/3 fully joined!?
Based on the graph of the line below, which describes the relationship between the value of the y-intercept and the
value of the slope?
10
oc
6
4
2
2
-10 -3 -4 -22
4
B
х
8 10
6
28
10
Answer:
y intercept = 3
[tex]m = \frac{1}{3}[/tex] --- slope
Step-by-step explanation:
Given
The attached graph
Required
Determine the slope and the y intercept
The [tex]y\ intercept[/tex] is at the point where [tex]x = 0[/tex].
From the graph:
[tex]y = 3;\ when\ x = 0[/tex]
Hence,
y intercept = 3
Next, we calculate the slope using:
[tex]m = \frac{y_1 - y_2}{x_1 - x_2}[/tex]
Where:
[tex](x_1,y_1) = (0,3)[/tex]
[tex](x_2,y_2) = (9,6)[/tex]
So, we have:
[tex]m = \frac{3 - 6}{0 - 9}[/tex]
[tex]m = \frac{-3}{- 9}[/tex]
[tex]m = \frac{3}{9}[/tex]
[tex]m = \frac{1}{3}[/tex]
Which number is irrational?
1) -4
2) 2/9
3) [tex]\sqrt{11}[/tex]
4) 8.25
Which expression is equivalent to
14x + 35 - 21y
Expression 1: 2x + 5 - 3y
Expression 2: 7(2x + 5 - 3y)
Expression 3: 7xy(2 + 5 - 3)
Expression 4: 7x(2+5+3y)
Answer:
Expression 1
Step-by-step explanation:
14x + 35 - 21y
[tex]14x + 35 - 21y \\ \frac{14x}{7} + \frac{35}{7} - \frac{21y}{7} \\ 2x + 5 - 3y[/tex]
Answer:
Its Expression 2: 7(2x + 5 - 3y)
Step-by-step explanation:
just got it right
Which graph represents a proportional relationship?
Answer:
we need a picture
Step-by-step explanation:
The function of f is defined as follows
Answer:
f(-121) = -5
f(31) = 3
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Function NotationStep-by-step explanation:
Step 1: Define
f(x) = ∛(x - 4)
f(-121) is x = -121
f(31) is x = 31
Step 2: Evaluate
f(-121)
Substitute in x: f(-121) = ∛(-121 - 4)[Radical] Subtract: f(-121) = ∛(-125)[Radical] Evaluate: f(-121) = -5f(31)
Substitute in x: f(31) = ∛(31 - 4)[Radical] Subtract: f(31) = ∛27[Radical] Evaluate: f(31) = 3What is the answer to 3x^2+2x^2+5x^2 ?
Answer:
10x²
Step-by-step explanation:
all have the same exponential value so you can add them directly
A population has a standard elevation of 25.29 what is the approximate standard error of the mean for a sample size of 175
Answer:
D. 1.91
(got it right on edmentum)
for your assurance please look at the picture below↓Step-by-step explanation:
The standard error is 1.91.
How is Standard Error Calculated ?For a data set the standard error is calculated by the ratio of standard deviation and square root of the sample size.
[tex]\rm Standard \; Error= \dfrac{ \sigma}{\sqrt{n}}[/tex]
Substituting the values
Standard Deviation = 25.29
Sample size = 175
Standard Error = 25.29 / √175
Standard Error = 1.91
Therefore the standard error is 1.91.
To know more about Standard Error
https://brainly.com/question/13179711
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Which situation cannot be represented by the equation x + 10 = 45?
A Marissa spent $45 on a hat and a shirt. The hat cost $10. What is x, the cost of the shirt in
dollars?
B Nicholas rode his bike 45 miles last week. He rode 10 miles on Tuesday and the rest of the
miles on Wednesday. What is x, the number of miles Nicholas rode his bike on
Wednesday?
C Two players scored a total of 45 points in a game. One player scored 10 points. What is x,
the number of points scored by the other player?
D There are 45 students in a group. There are also 10 adults in the group. What is x, the
total number of students and adults in the group?
Answer: D
because the other options actually make sense ;)
Answer:
D
Step-by-step explanation:
Match each mixed number to it's improper fraction.
Answer:
9/2
5/3
13/4
3/2
Step-by-step explanation:
HELP ME QUICK!!!!.!.!..
PLS HELP GIVING 25 POINTS PLS PLS
Answer A is 28
Step-by-step explanation:
Answer:
oop
Step-by-step explanation:
Sammy solved 68 math problems in 17 minutes. At what rate did Sammy solve the math problems?
Answer:
4 math problems per minute
Step-by-step explanation:
68 ÷ 17 = 4
4 math problem per minute
Answer:
9.71428571429 or 9.7
Step-by- step explanation:
ok so for this we need to divide 68 into 17 , we get the large number above.
Please help!!
a. What is the x intercept of the line graphed above?
b. What is the Y intercept of the line?
c. What is the slope of the line?