We can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.
Given the equation of the surface is xy + 15x + z^2 = 209.
Let's determine the shortest distance from the surface to the origin.
The shortest distance between the surface and the origin is given by the perpendicular distance, which can be calculated as follows:
Firstly, we need to determine the gradient of the surface, which is the vector normal to the surface.
For this purpose, we need to write the surface equation in the standard form, which is:xy + 15x + z^2 = 209 xy + 15x + (0)z^2 - 209 = 0 (the coefficients of x, y, and z are a, b, and c).
The gradient of the surface is given by the vector: ∇f = (a, b, c) = (y + 15, x, 2z) at the point P(x, y, z), which is a point on the surface.
Here, the normal vector is ∇f = (y + 15, x, 2z).
Now, let's consider a point A on the surface which is closest to the origin.
Let the coordinates of A be (a, b, c).
Therefore, the position vector of A is given by: OA = ai + bj + ck.
The direction of the position vector is in the direction of the normal vector, and therefore: OA is parallel to ∇f.
Thus, we can write: OA = λ∇f = λ(y + 15)i + λxj + 2λzkWhere λ is a scalar.
Since A lies on the surface, we have: a*b + 15a + c^2 = 209.
We also know that OA passes through the origin.
Therefore, the position vector of A is perpendicular to the direction vector OA.
This gives us: OA·OA = 0⟹ (ai + bj + ck)·(λ(y + 15)i + λxj + 2λzk) = 0
Simplifying this equation gives us:aλ(y + 15) + bλx + c(2λz) = 0
Also, we know that OA passes through the origin.
Therefore, the magnitude of OA is equal to the distance of A from the origin.
Hence, we can write: |OA| = √(a^2 + b^2 + c^2)
The value of λ can be obtained from the equation: aλ(y + 15) + bλx + c(2λz) = 0orλ = -2cz / (b + a(y + 15))
Substituting this value of λ in OA, we get: OA = λ(y + 15)i + λxj + 2λzk= -2cz/(b + a(y + 15)) (y + 15)i - 2cz/(b + a(y + 15)) xj - 4cz^2/(b + a(y + 15))k
Substituting this value of λ in |OA|, we get: |OA| = √[(2cz/(b + a(y + 15)))^2 + (2cz/(b + a(y + 15)))^2 + (4cz^2/(b + a(y + 15)))^2] = 2cz√[(y + 15)^2 + x^2 + 4z^2] / |b + a(y + 15)|
The distance of the point A from the origin is |OA|, which is minimized when the denominator is maximized. The denominator is given by |b + a(y + 15)|.
Thus, we have to maximize the denominator with respect to a and b. The condition for maximum value of the denominator is obtained by differentiating the denominator with respect to a and b separately and equating it to zero. The values of a and b obtained from these equations are substituted in the equation a*b + 15a + c^2 = 209 to obtain the coordinates of the point A, which is closest to the origin.
Hence, we can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.
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Select the correct answer. The graph of the function is shown f(x)=sin(x). What is the domain of the function? PLATO USERS
Answer:
Domain: (−∞,∞),{x|x∈R}
Step-by-step explanation:
the domain will be Domain: (−∞,∞),{x|x∈R} because
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Answer: Option C: (−∞,∞)
Step-by-step explanation:
answer is for plato
A train arrives at a station and waits 3.5 minutes before departing. Another train arrives at the station 16 minutes later, repeating the cycle. Identify the probability that a train will not be at the station when you arrive.
Step-by-step explanation:
so we can safely say that one train will be there for 3.5 minutes and it will be gone for 16 minutes before another train comes and repeats the cycle.
we want how many minutes that the train will not be there over the total time that is possible so that would be
16/(16+3.5)
16/19.5
32/39
Hope that helps :)
The probability of the train not being at the station on arrival would be:
0.82
Find the probabilityGiven that,
The duration for which the train makes = 3.5 minutes
The time at which the second train arrives = 16 minutes later
We know that this cycle is repeated.
The duration for which the train is not there would be:
[tex]= 16/(16+3.5)[/tex]
[tex]= 16/19.5[/tex]
[tex]= 32/39[/tex]
[tex]= 0.82[/tex]
Thus, 0.82 is the probability.
Learn more about "Probability" here:
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how many ping pong balls do you need if you want to arrange them in the shape of an equilateral triangle with 23 rows
how do you add an equation vertically
"Vertically" probably refers to stacking the polynomials on top of each other and aligning them at equal powers of x. So kind of like what you might do when adding two numbers with more than 1 digit each.
For example, the sum 123 + 456 can be computed by adding digits in the same digits place:
… 1 2 3
+ 4 5 6
… 5 7 9
Then for this sum, you have
… - 3x ² - 6x + 8
+ 8x ² + 7x - 14
… 5x ² + x - 6
lol helpppppppppp ♀️
Answer:
The answer is 52.6038
Step-by-step explanation:
Answer:
Step-by-step explanation:
52.6038
3. Which of the following values is closest
to the diameter of a circle with an area of
314 square inches?
a. 20 inches
b. 10 inches
c. 100 inches
d. 31.4 inches
a ski resort charges $45 for an All day lived pass and $40 per day for renting boots and a snowboard. At a store, you can buy boots and a snowboard for $360. How many times must you go snowboarding at the ski resort for the cost of buying your own boots and snowboard to be less than renting them?
which of the following best describes a cylinder?
Answer:
The answer is C because a cylinder has a set of two parallel lines
Mrs. DeMarco wants to estimate the height of her garage
oor. Select the best benchmark for her to use.
the width of a paperclip
the length of a baseball bat
the height of a license plate
how far she can walk in 20 minutes
Answer:
length of baseball bat.
Step-by-step explanation:
she can measure the bat and times by how many times she could fit the bat there u know
PLS HELP
A triangle is shown.
Part A: What is the value of x, in degrees?
13.8
16
20
24
Part B: What is the measure of 2Ain degrees?
22
30
38
48
Answer:
A: x=16
B: ∠A=22
Step-by-step explanation:
A
The angles in a triangle always add up to 180. Therefore:
∠A+∠B+∠C=180
(2x-10)+(3x)+(4x+46)=180
2x+3x+4x+46-10=180
9x=180-36
x=144/9
x=16
B
∠A=2x-10
∠A=2(16)-10
∠A=32-10
∠A=22
Find equation to (-3,0),m=5
Answer:
Step-by-step explanation:
y=5x+15
Graph f(x) = 2 x + 3 + 2.
Answer:
HOPE IT'S HELPS YOU....HELP WILL GIVE BRANLIEST!
Answer:
Since it is vertical angles, 4x - 12 = 88.
Answer is 25
Step-by-step explanation:
88 + 12 = 100
100 / 4 = 25
PLEASE!!
What is the COMBINED surface area of the prisms shown below?
I need help in this question. I am not understanding what it is telling me so HELP
Answer: You solve it its easy subtraction :)
Step-by-step explanation:
$14,000 - $5,358 - $2,834 - $4,132 = answer
Graph logarithmic function g(x)=log3x
Answer:
Step-by-step explanation:
The logarithmic equation provided can be seen graphed in the image attached below. We can see from the graph that as the line approaches the +3 on the y-axis it moves infinitely towards positive infinity on the x-axis. Never truly touching the y-value of 3 but instead increasing less and less every time towards it.
Write 5-7 well-written sentences. REREAD FOR ERRORS AND FIX THEM!! If you could spend the perfect day, what would you do? Where would you be? Who would be with you? What do you see? Eat? Play? Use your senses to describe it? What do you see? Hear? Smell? Taste? Touch?
Answer:
If i could spend the perfect day I would go to the park and take a jog. I would also go to Mc donalds and get a vanilla flavored ice cream! I love the taste of vanilla, Next, i will watch my favorite movie on my phone. Then, i will go to the zoo with my friends and see my favorite animals! Lastly,i will go home and take a nap.Step-by-step explanation:
okay
What is the quotient of 15÷3?
5
Hope this helps you :D
Help me plsssssssssssssss
Hi
Answer:
1,900 i think is the answer pls dont sue me lol
Answer:
1900
Step-by-step explanation:
To the nearest hundred, so we have to look at the tens place. The tens place is 9, ninety, it's greater than 5, so we round up. therefore, 1894 rounded to the nearest hundred is 1900.
tion
4. Solve the equation below.
7.7 + d = 10.5
A 18.2
C -18.2
B -2.8
D 2.8
US
PLS HELPPP GEOMETRY!!!!!!
You are on a fishing boat that leaves its pier and heads east. After traveling for 24 miles, there is a report warning of rough seas directly south. The captain turns the boat and follows a bearing of S45°W for 13.4 miles.
a. At this time, how far are you from the boat's pier?
b. What bearing could the boat have originally taken to arrive at this spot?
Answer:
a-17.3 miles b-S57 E
Step-by-step explanation:
d^2=30^2+12^2-2(30)(12)
sin45
Help. I have no idea what to do
Answer: The answer is to ask your parents because something is definitely wrong with your teacher to give such a hard sun to solve like that.
Step-by-step explanation: sorry man that hard.
Trevor has an investment worth $6,774.50. He made his only deposit in it 22 years ago. Given that the investment yields 2.7% simple interest annually, how big was the initial deposit?
Answer:
4,250.00
Step-by-step explanation:
Brainliest?
Answer:
$4,250.00 Answer choice C
Step-by-step explanation:
Just took the test!
Explain:In (4a)^3
•c
Answer:
Step-by-step explanation:
(4a)^3
4^3 a^3
=12a
How many solutions are there in the equation 2x - 5 = 2x - 5?
Answer:
the equation 2(x - 5) = 2x + 3 has no solutionStep-by-step explanation:
Given the table below, what is the x-intercept of the linear function?
Answer:
-6
Step-by-step explanation:
Morgan bought a sofa for $216.00. The finance charge was $25 and she paid for it over 15 months.
Use the formula Approximate APR =(Finance Charge÷#Months)(12)Amount Financed
to calculate her approximate APR.
Round the answer to the nearest tenth.
9.4%
9.3%
1.2%
1.1%
Answer:
9.3%
Step-by-step explanation:
Took the test, so i checked the answers and the correct one was 9.3% have a beautiful day
graphing sucks
ugh. but anyway does anybody know how to graph using only an equation? please explain thoroughly. PLEASE SAVE MY LIFE!
- you can substitue x and y for 1, or 0 and graph each point
- use desmos and copy it
- depending on the type of equation find the x-intercept, y-intercept, vertex and then draw a line between them
I will give BRAINLIST Solve the quadratic function by factoring.
2x^2+7x-4=0
Someone please help I will give BRAINLIST please show work I need this for my test
Solve.
b + 2/9 = 1 3/5