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
Order the following measures from least to greatest: 0.5 m, 30 in., 1.25 ft, 17 cm
Answer:
30 in, .5 m , 1.25 ft, 17 cm
Step-by-step explanation:
Answer:
17cm, 1.25 ft, 0.5 m, 30 inches
Step-by-step explanation:
I converted all of them to meters before deciding which one was least and which one was greatest.
The Current Population Survey conducted by the Census Bureau records the incomes of a large sample of U.S. households each month. What will be the relationship between the mean and median of the collected data
Explanation:
There are 3 possible cases that describe the relationship between the mean and median of the collected data.
First case:
If the distribution is symmetric then, mean = median.
Second case:
For the positively skewed frequency distribution, the mean is greater than the median.
Third case:
For the negatively skewed frequency distribution mean is less than the median.
Which of the following represents a function? OA X -5-1 9 8 -1 y 1 7 23 17 1 B. y NS x -4 NO 4 C. {(0,1),(3,2), (-8,3), (-7,2), (3,4)} D. -4 5 9 3 13
Given:
The different relations in the options.
To find:
The relation which represents a function.
Solution:
A relation is called a function if there exist a unique output (y-value) for each input (x-value).
In option A, there exist two outputs y=7 and y=1 for a single input x=-1. So, this is not a function.
In option B, there exist two outputs y=-5 and y=2 for a single input x=-1. So, this is not a function.
In option C, there exist two outputs y=2 and y=4 for a single input x=3. So, this is not a function.
In option D, there exist only one value of y for each x. So, this is a function.
Therefore, the correct option is D.
Answer:
D
Step-by-step explanation:
you probalby just want the answer
Maria orders a salad and a drink for lunch at a resturant. A salad costs $4.75 and a drink costs $1.65. How much tip should maria leave the server if she wants to tip her 18%?
A. $5.24
B. $7.55
C. $6.40
D. $1.15
help now plsssssssss!!!
Answer:
5
Step-by-step explanation:
Descefdsvxerdazdxsazd4r
out of 50 flags, 35 are triangular and of the remaining 15 some are rectangular and some are pentagonal. Satish chooses one flag randomly from all 50. Estimate the highest probability is that he picks which flag
Answer:
triangular
Step-by-step explanation:
triangular flags have a probability of 35/50, so it's the highest.
a number has the same digit in its hundreds place and its hundredths place. how many times greater is the value of the digit in the hundreds place then the value of the digit in the hundredths place?
Answer:
The answer would be 1000.
Step-by-step explanation:
Im not completely sure if this is correct. but i hope it helps!
Have an amazing day!
Answer: it should be 1000
Translate this verbal expression to an algebraic expression.
Two times the sum of x and 10.
A 2(x - 10)
B 2x + 10
C 2(x + 10)
D 2 + x + 10
Answer:
C.
Step-by-step explanation:
It's c because you need the sum of x and 10 but both multiplied by 2. So it wouldn't be A nor B because they are both subtracting. That would leave you with C and D and d doesn't have anything to do with multiplying. I dunno made sense to me.
. What is the y-intercept of the graph of y = -2x + 3?
A. (0, -2)
B. (0, 3)
C. (-2, 0)
D. (3, 0)
Answer:
B (0, 3)
when a graph is in slope-intercept form (y =mx +b) then the last number is the y-intercept. in the equation m=the slope and b=the y-intercept. then you plug it into a point (x, y)
Step-by-step explanation:
I hope this helped :)
Answer:
B(0,3)
when a graph in slope intercept from ( y= mx+b) then the last number is the y - intercept in the equation m= the slope and b = the y - intercept then you you plug it into a point ( x, y)
I need help with these two question help me ASAP........
It’s 1:00 pm and I haven’t finish my homework plz helpp
Answer:
Concept: Vectors
The line demonstrates a 90 degree angle Hence they are Perpendicular by geometric qualitiesGive brainlistAnswer:
Omg...so complete fast......
Help I don’t get this
A ladybug and ant move at constant speeds. The diagrams with tick marks show their positions at different times. Each tick mark represents 1 centimeter.
In your textbook pg. 111, scale the vertical and horizontal axes by labeling each grid line with a number. You will need to use the time and distance information in the tick-mark diagrams.
Which line shows the ladybugs movement? The ant?
Unit rate for the Ladybug? The ant?
How long does it take for the ladybug to travel 12 cm? The ant?
Answer:
From the graph we can say that ant is moving faster than the ladybug
Slope of the ant's movement will be steeper than the ladybug
Hence, line "v" denotes the speed of ant and line "u" denotes the speed of ladybug.Unit rate/slope of the graph is (distance traveled/elapsed time)
Unit rate of ant is 3/2 cm/sec and that of ladybug is 1 cm/sec.Ant takes (12×2/3) = 8 sec and ladybug takes (12×1) = 12 sec to travel 12 cm.Remember ant will take less time to travel a certain distance than ladybug.
Write an expression to explain the following word problem: Jeremy elimbed
25 feet up the mountain while Jessica olimbed 32 and a half feet up the
mountain higher than Jeremy
Answer:
Jessica = J(Jeremy)+32.5
Step-by-step explanation:
Jessica equals Jeremy + 32.5 feet.
Hope this helps, have a great day! :D
PLEASE HELP I DONT WANT TO FAIL
Answer:
1
Step-by-step explanation:
12/8=1.5
15/10=1.5
1.5/1.5=1
Find the area of the parallelogram shown below.
7
5
6
square units
Answer:
35
Step-by-step explanation:
A=bh=7*5=35
base= 7
height= 5
the formula is bh so you just multiply them and that's the area
Answer:
35 Sq. Units
Step-by-step explanation:
The Intuition for why the area of a parallelogram is A=bh (area = base x height)
The formula for the area of a parallelogram is base times height, just like the formula for the area of a rectangle.
Have a great day!
You had $25 to spend on school lunches. After buying five of them, you have $15 left. How much did each lunch cost?
Answer:
2 each
Step-by-step explanation:
simple math
6/9 of 27 students in a class play soccer some of the students play offense and the rest play defense the ratio of the students playing offense to the total number of students playing soccer is 1 to3 how many students play defense
12 students play defense. 1st step. 6/9 of 27 is 18. 2nd. the ratio of the students playing offense to the total number of students playing soccer is 1 to 3 which means the total amount of students is '3', in the ratio it's 3 because it's simplified (i like to think of it as 'groups'). since we know that there are a total of 18 students and there's 3 total 'groups' we divide 18 into 3 which is 6. so there's 6 students in each group. we subtract 6 (1 group) from 18 ( the total number of students) which is 12, so there are 12 students playing defense. if you wrote the ratio again without simplifying it would be 6:18
The histogram gives information about a fitness club
Answer:
a)
b) 99
Step-by-step explanation:
i'm not sure about a
but for b i got 99 members
Mrs. Rodriguez has a blueprint of her new house. The actual living room has a width of 20 feet and a length of 24 feet. If the scale used to make the blueprint is 12 inch = 4 feet, what is the length of the living room on the blueprint?
What is 7.44444444444 in fraction form
Answer:
186/25 i am pretty sure
Step-by-step explanation:
The population standard deviation for the height of college basketball players is 2.9 inches. If we want to estimate a 99% confidence interval for the population mean height of these players with a 0.45 margin of error, how many randomly selected players must be surveyed
Answer:
276
Step-by-step explanation:
Given :
Standard deviation (σ) = 2.9 inches
Error margin (E) = 0.45
Confidence interval = 99%
Using the relation to obtain the sample size (n)
n = [(Zcrit * σ) / E]²
Zcritical at 99% confidence interval = 2.576
n = [(2.576 * 2.9) / 0.45]²
n = [7.4704 / 0.45]²
n = 16.600888^2
n = 275.58951
n = 276 samples
Sample size
what is equivalent to 2/3 divided by 5/6
Answer:
the answer is 4/5
Step-by-step explanation:
2/3 times the reciprocal of 6/5 and then that will give you 4/5
–2z − 7 need this done now
Answer:
the value of the z is -3.5
Step-by-step explanation:
According to the question, the following information is given
-2z - 7
Based on the above information
We assume that we have to determine the z value
so
-2z - 7 = 0
-2z = 7
z = 7 ÷ -2
= -3.5
Hence, the value of the z is -3.5
The same is to be considered and relevant too
" If $6000 is borrowed at 7.5% simple interest for 2 yr, then the amount of interest is "
Answer:
u
Step-by-step explanation:
What is the value of x
Group of answer choices
60 degrees
30 degrees
15 degrees
45 degrees
Answer:
x = 15 degrees
Step-by-step explanation:
This is a right angle, since we know that a right angle contains 90 degrees, we can form an equation.
[tex]30+4x=90[/tex]
Now, we solve for x.
[tex]-30+30+ 4x = -30+90[/tex]
[tex]4x=60[/tex]
[tex]\frac{4x}{4}=\frac{90}{4}[/tex]
[tex]x=15[/tex]
The variable s varies directly as the square of t. When s = 4, t = 12. Nick’s work finding the value of t when s = 48 is shown:
What is the first error, if any, in Nick’s work?
See in pic:
Answer:
C - He substituted incorrectly when calculating the constant of variation.
The first error, in Nick’s work is given inn option C, i.e. He substituted incorrectly when calculating the constant of variation.
What is constant of variation?Constant of variation is the ratio between two variables in a direct variation or the product of two variables in an inverse variation.
We have,
s = 4, and t = 12
Now according to the question,
The variable s varies directly as the square of t,
i.e.
s = kt²
So,
Substituting given values of s and t,
s = kt²
i.e.
4 = k × 12²
⇒
[tex]k = \frac{4}{144}=\frac{1}{36}[/tex],
Now,
for 48,
i.e.
s = kt²,
We get,
[tex]48 = \frac{1}{36} t^2[/tex],
i.e.
[tex]t^2=36 *48=1728[/tex]
[tex]t=\sqrt{1728} = \sqrt{3*576} =24\sqrt{3}[/tex]
So, this is the correct value for t.
Hence, we can say that the first error, in Nick’s work is given inn option C, i.e. He substituted incorrectly when calculating the constant of variation.
To know more about constant of variation click here
https://brainly.com/question/14254277
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please help :) worth 10 points
Answer:
a. (0,3)
b. (-4,-3)
c. (-2,0)
d. (-1 1/3, 1)
e. (4,9)
f. (-3 1/3, -2)
Step-by-step explanation:
Find missing Heights
Answer:
15 feet. 10 plus 15=25 s0 therefore it is 15
which pair of of expressions are equivalent? A. 10(x+4) and 10x+14 B. 3(x+5) and 3x + 5 C. x+x+x+1+1+1 and 3(x+1) D. x+x+x+x+1+1+1+1 and 8x
Answer:
C. x+x+x+1+1+1 and 3(x+1)
Step-by-step explanation:
Algebraic Expressions
Some algebraic expressions are equivalent to others if the correct operations are done on them.
A. 10(x+4) must be operated by multiplying 10 by x and 10 by 4, thus:
10(x+4) = 10x + 40.
This choice is not correct
B. 3(x+5) must be operated by multiplying 3 by x and 3 by 5, thus:
3(x+5) = 3x + 15.
This choice is not correct
C. x+x+x+1+1+1 is equivalent to an expression where like terms are collected, that is. x+x+x=3x and 1+1+1=3, thus:
x+x+x+1+1+1=3x+3=3(x+1)
This choice is correct
D. x+x+x+x+1+1+1+1 in a similar way as explained above, like terms must be correctly collected: D. x+x+x+x=4x and 1+1+1+1=4, thus
D. x+x+x+x+1+1+1+1 = 4x+4 = 4(x+1)
This choice is not correct