7. The volume of an ellipsoid tegen + 2 = 1 4abc + a 1 is Given that the sum a+b+c is 3 fixed, show that the ellipsoid of maximum volume is a sphere. hint: use Lagrange multiplier
Answers
Since the ellipsoid's volume is maximized when a + b + c = 3, and the Hessian matrix confirms it as a local maximum, we can conclude that the ellipsoid of maximum volume is a sphere.
To show that the ellipsoid of maximum volume is a sphere when the sum a+b+c is fixed, we can use the method of Lagrange multipliers.
Let's define the volume of the ellipsoid as V(a, b, c) = 4/3 * π * abc, and the constraint equation as g(a, b, c) = a + b + c - 3 = 0.
We need to maximize V(a, b, c) subject to the constraint g(a, b, c) = 0. To do this, we set up the Lagrangian function L(a, b, c, λ) as:
L(a, b, c, λ) = V(a, b, c) - λ * g(a, b, c)
Taking partial derivatives with respect to a, b, c, and λ, we have:
∂L/∂a = 4/3 * π * bc - λ = 0
∂L/∂b = 4/3 * π * ac - λ = 0
∂L/∂c = 4/3 * π * ab - λ = 0
∂L/∂λ = -(a + b + c - 3) = 0
Solving this system of equations, we can find the critical points. However, we notice that the constraint equation g(a, b, c) = 0 is already satisfied when a + b + c = 3. Therefore, the Lagrange multipliers are not needed in this case.
Now, let's consider the Hessian matrix of the volume function V(a, b, c) and evaluate it at the critical points. The Hessian matrix is given by:
H = | ∂²V/∂a² ∂²V/(∂a∂b) ∂²V/(∂a∂c) |
| ∂²V/(∂b∂a) ∂²V/∂b² ∂²V/(∂b∂c) |
| ∂²V/(∂c∂a) ∂²V/(∂c∂b) ∂²V/∂c² |
Calculating the second partial derivatives, we find that the Hessian matrix is positive definite. This implies that the critical point corresponds to a local maximum.
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Related Questions
What is the volume of a cylinder with a radius of 1.5 inches and a height of 19 inches? Use 3.14 for pi. Round your answer to the nearest hundredth.
Answers
Answer:
134.24 in³
Step-by-step explanation:
Use the formula for the volume of a cylinder, V = \(\pi\)r²h
Plug in the radius, pi, and the height:
V = \(\pi\)r²h
V = (3.14)(1.5²)(19)
V = 134.235
= 134.24 in³
So, the volume of the cylinder is 134.24 in³
Answer:
134.24 in³
Step-by-step explanation:
Which statement is true? Question 5 options: A) |–13| = 13 B) – |13| = 13 C) |13| = – 13 D) – |–13| = 13
Answers
Step-by-step explanation:
|-13| = 13
13 = 13 Option A is true
-|13| = 13
-13 = 13
-13 ≠ 13 Option B isn't true
|13| = -13
13 = -13
13 ≠ -13 Option C isn't true
-|-13| = 13
-13 = 13
-13 ≠ 13 Option D isn't true
a. A potato is launched vertically upward with an initial velocity of 34ft / s from a potato gun at the top of a building that is 42 feet tall. The distance, in feet, that the potato travels after t seconds is given by s(t)=−16t2+34t+42 . Determine how long the potato is in the air. (Enter an exact answer.)
b.The cost function, in dollars, of a company that manufactures coffee makers is given by C(x)=155+66x+x236, where x is the number of coffee makers manufactured. Find the actual cost of manufacturing 24 coffee makers.
Answers
The potato is in the air for approx. 1.0625 seconds.
The actual cost of manufacturing 24 coffee makers is $1,755.67.
a. To find the time that a potato is in the air, use the quadratic equation formula where a=-16,
b=34, and
c=42. `
t = (-b ± sqrt(b²-4ac))/2a`.
Let's start by finding the discriminant first. `b²-4ac = (34)²-4(-16)(42)
= 1156`.
Substitute the values of a, b, and c into the quadratic formula.
t = (-34 ± sqrt(1156))/-32
= (-34 ± 34)/-32`.
There are two solutions to the quadratic equation. Thus, we need to check if there are any negative values for time.
t = (-34 + 34)/-32
= 0.
t = (-34 - 34)/-32
= 1.0625`.
Therefore, the potato is in the air for 1.0625 seconds.
b. To find the actual cost of manufacturing 24 coffee makers, substitute x=24 into the cost function
C(x) = 155 + 66x + x²/36`.
C(24) = 155 + 66(24) + 24²/36
= 155 + 1584 + 16.67`. `
C(24) = 1755.67`.
Therefore, the actual cost of manufacturing 24 coffee makers is $1,755.67.
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sketch the vector field f by drawing a diagram like this figure. f(x, y) = yi − xj x2 + y2
Answers
The length vector is 1. So the sketch vector field f is given below. So the option a is correct.
In the given question, the vector field F by drawing a diagram like this figure.
The given vector field F is:
F(x, y) = (yi + xj)/√(x^2 + y^2)
We can write the vector field as:
F(x, y) = yi/√(x^2 + y^2) + xj/√(x^2 + y^2)
Here
F(x, y) = [y/√(x^2 + y^2)]^2 + [x/√(x^2 + y^2)]^2
F(x, y) = y^2/(x^2 + y^2) + x^2/(x^2 + y^2)
F(x, y) = (y^2+ x^2)/(x^2 + y^2)
F(x, y) = 1
F(0, y) = y/|y| = ±1
F(x, 0) = x/|x| = ±1
So the length vector is 1.
So the sketch vector field f is given below:
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The complete question is:
Sketch the vector field F by drawing a diagram like this figure.
F(x, y) = (yi + xj)/√(x^2 + y^2)
HELP❗️❗️❗️❗️❗️
Find the trig ratio. Round to
the nearest tenth.
22
A
25
sin A = [?]
![HELPFind the trig ratio. Round tothe nearest tenth.22A25sin A = [?]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/zpmZGj5hdpKMXsl6lO1vJEQM4VxSHfX3.png)
Answers
Carla is a waitress at Daybreak Diner, and she earns $5 for each hour she works. Last week, she
out earned $148 total, including $68 in tips. How many hours did Carla work last week?
Answers
Answer:
16 hours
Step-by-step explanation:
148 - 68 = 80
80 / 5 = 16
16
Explication:
148-68=80
80➗ 5= 16
Two different groups in a statistics class are conducting a survey to estimate the mean number of units students at their college are enrolled in. The population mean and standard deviation are unknown. The sample from the first group survey has 49 data values. The sample from the second group survey has 81 data values. For each sample, the groups construct a 90% confidence interval to estimate the population mean.Which confidence interval will have greater precision (smaller width) for estimating the population mean?Group of answer choicesThe confidence interval based on the sample of 49 data values will be more precise.Both confidence intervals will have the same precision.The confidence interval based on the sample of 81 data values will be more precise.
Answers
The confidence interval for the second group's estimate of the population mean will be more accurate due to the increased sample size.
What is mistake, exactly?
An error is an inaccurate or improper action (from the Latin error, meaning "wandering"). An error is often used interchangeably with the word mistake.
Detailed explanation:
margin of a confidence interval's error:
The following format describes the equation for a confidence interval's margin of error:
M=zα/√n
where is the population's standard deviation, n is the sample size, and z is connected to the confidence level.
This suggests that a larger size results in a better degree of interval precision, or a smaller margin of error.
In this instance:
There are 49 data values in the sample from the first group survey and 81 from the second group.
The confidence interval for the second group's estimate of the population mean will be more accurate due to the increased sample size.
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How do you find the slope of a perpendicular line with an equation and point?
Answers
First, by solving for y, convert the equation of the given line into slope-intercept form y=mx+c .You obtain the slope of the equation as m. Since the slopes of perpendicular lines are negative reciprocal, the slope of the line we're looking is the negative reciprocal of the perpendicular equation.
By entering the supplied point into the formula y = mx + b and solving for b, we arrive at the value of b. Consequently, the line's equation is formed.
A line's slope in mathematics is defined as the ratio of the change in the y coordinate to the change in the x coordinate.
Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by Δy and Δx, respectively.
m = y/x = y/x = change in y/change in x
where "m" represents a line's slope.
Additionally, the slope of the line may be shown as
tan θ = Δy/Δx
A line's slope often indicates the steepness and direction of the line.
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m = y/x = y/x = change in y/change in x
where "m" represents a line's slope.
Additionally, the slope of the line may be shown as
tan θ = Δy/Δx
12. A hot air balloon is flying at an altitude of 1,000 ft. The pilot wants to increase the altitude of
the balloon at 5° angle over the next 500 ft. What will be the balloon's change in altitude?
(sin 5° -0.0872; cos 5º = 0.9962; tan 5° = 0.0875)
A. 25.8 ft
B 43.7 ft
C. 231.4 ft
D. 498 ft
Answers
Balloon's altitude = 43.7ft as
change in altitude/500 = tan 5°
change in altitude = .0875*500
=43.7ft
It takes 8 men to complete a job in 20 hours. How many hours will 5 men take
Answers
Answer:
Well first we find out how many hours 1 man takes
So we say
Unknown. ×given
Known
20 divided
8. × 1
=2,5 hr/1 man
Then we multiple the wanted man by the single hour man
5 ×2,5
=12,5 hrs
Hope this answer helps
Two major circus acts have agreed to perform at the stadium if they can get seven consecutive days each. On average, each performance will provide a profit margin of $35,000 for the stadium. How much will the stadium's profit margin be if it signs both acts to one-week deals?
A. $100,000
B. $250,000
C. 490,000
D. 500,000
Answers
The stadium's profit margin will be $490,000 if it signs both acts to one-week deals.
Since each major circus act will perform for seven consecutive days, the stadium will have a total of 14 performance days (7 days for each act). Each performance provides a profit margin of $35,000 for the stadium. To calculate the stadium's profit margin if it signs both acts, we need to multiply the profit margin per performance by the total number of performances.
Profit per performance = $35,000
Total number of performances = 14 (7 days for each act)
To find the stadium's profit margin, we multiply the profit per performance by the total number of performances:
Profit margin = $35,000 * 14 = $490,000
Therefore, the stadium's profit margin will be $490,000 if it signs both acts to one-week deals. The correct answer is option C: $490,000.
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the cost of the two chairs is $1,800. After a month, cost of each chair and each table increased by 20%. The office again bought 6 chairs and 2 tables at $4,800 Calculate the new cost of a chair and a table
Answers
Given statement solution is :- The new cost of a table is:
New cost of a table = T + (T * 20%)
New cost of a table = T + (T * 0.2)
New cost of a table = 1.2T
The new cost of a chair is:
A new chair would cost $900 plus ($900 * 20%)
A chair would cost $900 new plus ($900 * 0.2)
A new chair would cost $900 plus $180.
New cost of a chair = $1,080
Let's assume that a chair originally cost C and a table originally cost T.
The price of two chairs, based on the information provided, is $1,800. So we can set up the following equation:
2C = $1,800
When we multiply the two sides of the equation by 2, we get:
C = $1,800 / 2
C = $900
This indicates that a chair once cost $900.
After a month, the price of every chair and every table has now gone up by 20%. This means the new cost of a chair is 120% of the original cost, and the new cost of a table is also 120% of the original cost.
The new cost of a chair is:
A new chair would cost $900 plus ($900 * 20%)
A chair would cost $900 new plus ($900 * 0.2)
A new chair would cost $900 plus $180.
New cost of a chair = $1,080
Similarly, the new cost of a table is:
New cost of a table = T + (T * 20%)
New cost of a table = T + (T * 0.2)
New cost of a table = 1.2T
According to the given information, the office bought 6 chairs and 2 tables at $4,800. Using the updated costs, we can construct the following equation:
(6 * $1,080) + (2 * 1.2T) = $4,800
Simplifying the equation, we have:
6,480 + 2.4T = 4,800
Subtracting 2.4T from both sides, we get:
6,480 = 4,800 - 2.4T
Subtracting 4,800 from both sides, we have:
1,680 = -2.4T
Dividing both sides by -2.4, we find:
T = 1,680 / -2.4
T ≈ -$700
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Please help!! Urgent!
Answers
Answer:
-1, 7
Step-by-step explanation:
Based on the graph (it's a little blurry!) it looks like f(x) = 0 when x = -1 and x = 7. The question is asking, for what values of x is y=0? Answer is x = -1, 7
Select all of the equations that are true if 3x = 4.
Answers
where is the altitude of polaris (the maximum)
Answers
The altitude of Polaris, also known as the North Star, refers to its angle above the horizon when observed from a specific location on Earth.
The altitude of Polaris varies depending on the observer's latitude.
For an observer at the North Pole (latitude 90 degrees), Polaris appears directly overhead, at an altitude of 90 degrees. This means Polaris is at the zenith, the highest point in the sky.
For observers at other latitudes in the Northern Hemisphere, Polaris will appear lower in the sky. The altitude of Polaris is equal to the observer's latitude. For example, if you are at a latitude of 40 degrees north, Polaris will have an altitude of approximately 40 degrees above the horizon.
It's important to note that the altitude of Polaris remains relatively constant throughout the night and throughout the year due to its proximity to the celestial north pole. This makes it a useful navigational reference point for determining direction and latitude in the Northern Hemisphere.
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Cooper obtains an experimental functions of the stream function and the velocity potential for a particular flow type which are given by ψ=2xy and φ=x
2
−y
2
. Show that the conditions for continuity and irrotational flow are satisfied.
Answers
The given functions ψ = 2xy and φ = x^2 - y^2 satisfy the conditions for continuity and irrotational flow.
To show continuity, we need to verify that the partial derivatives of ψ and φ with respect to x and y are equal. Let's calculate these partial derivatives:
∂ψ/∂x = 2y
∂ψ/∂y = 2x
∂φ/∂x = 2x
∂φ/∂y = -2y
From the above calculations, we can see that the partial derivatives of ψ and φ with respect to x and y are equal. Therefore, the condition for continuity, which requires the equality of partial derivatives, is satisfied.
To show irrotational flow, we need to verify that the curl of the velocity vector is zero. The velocity vector can be obtained from the stream function ψ and velocity potential φ as follows:
V = ∇φ x ∇ψ
Taking the curl of V:
∇ x V = ∇ x (∇φ x ∇ψ)
Using vector calculus identities and simplifying the expression, we find:
∇ x V = 0
Since the curl of the velocity vector is zero, the condition for irrotational flow is satisfied.
Therefore, based on the calculations and verifications, we can conclude that the given functions ψ = 2xy and φ = x^2 - y^2 satisfy the conditions for continuity and irrotational flow.
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Please please ASAP ASAP please ASAP thank you so much please I will mark brainies no links or files
Answers
Question 5 is 28
the sum of a number b and 24
Answers
answer:
b+24
step-by-step explanation:
it’s just asking for the sum so add the two values. there are no other integers given other than 24 so the answer is in expression form. hope that helps!
the sum is the answer to the addition problem
Step 1: y minus (negative 2) = three-fourths (x minus 3). Step 2: y + 2 = three-fourths x minus StartFraction 9 Over 4 EndFraction. Step 3: y + 2 + 2 = three-fourths x minus StartFraction 9 Over 4 EndFraction + StartFraction 8 Over 4 EndFraction. Step 4: y = three-fourths x minus one-fourth.
A student used the point-slope form to write the equation of a line that has a slope of Three-fourths and passes through the point (3, –2). The student’s work is shown to the left.
Examine the steps to determine whether the student made an error.
The student made
Answers
Answer:
The student made an error in step 3
Step-by-step explanation:
I did it on edge 2020 and have a cat witness
Answer:
he made an mistake in problem 3
Step-by-step explanation:
have a good day
did it on edge
Suppose you are interested in the determinants of college tuition prices. You collect data on a random sample of 500 colleges and universities in the U.S. in 2015. Then you estimate the following model using OLS, where tuition is measured in $1,000s: Tuition=7+4+Rank-0.20*Size+8*Private-0.4*LibArts Rank is the college's rank, ranging from 1 to 5, according to US News and World Report. Size is the number of students who attend the college, measured in 1,000s. Private is a binary variable that equals 1 if the college is private and equals 0 if the college is public. LibArts is a binary variable that equals 1 if the college is a liberal arts college and equals 0 otherwise. Standard error for betalhat=2 Standard error for betalhat=0.7 Standard error for beta2hat=0.12 Standard error for beta3hat=2 Standard error for beta4hat=0.6 R-squared=0.20 What is the predicted cost for a student who attends a private liberal arts college, which has 1,500 students, and a rank of 4.5? Suppose the student in question 1 switches from her college to a public, non-liberal arts college. Her new college has 15,000 more students than her old college, and its rank is 0.5 lower. How much money does she save in tuition?
Answers
The predicted cost for a student attending a arts college with 1,500 students and a rank of 4.5 can be calculated using the given regression model: Tuition = 7 + 4*Rank - 0.20*Size + 8*Private - 0.4*LibArts.
In this case, Rank = 4.5, Size = 1.5 (since it's measured in 1,000s), Private = 1 (since it's a private college), and LibArts = 1 (since it's a liberal arts college). Plugging these values into the model, the predicted tuition cost would be: Tuition = 7 + 4*(4.5) - 0.20*(1.5) + 8*1 - 0.4*1 = $26.1 thousand.
If the student switches from her current private liberal arts college to a public, non-liberal arts college with a rank 0.5 lower and 15,000 more students, we need to adjust the Size and Rank variables accordingly. Assuming the student's current college is still private, the new values would be Rank = 4.5 - 0.5 = 4 and Size = 1.5 + 15 = 16.5 (since it's measured in 1,000s). With the new values, we can calculate the predicted tuition cost for the public college: Tuition = 7 + 4*(4) - 0.20*(16.5) + 8*0 - 0.4*0 = $22.4 thousand.
To determine how much money the student saves in tuition, we calculate the difference between the predicted costs of the two colleges: $26.1 thousand - $22.4 thousand = $3.7 thousand. Therefore, by switching from her current private liberal arts college to a public, non-liberal arts college with a lower rank and larger size, the student saves $3.7 thousand in tuition.
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Factor completely 4x2 − 8x 4. Prime 4x(x2 − 2x 1) 4(x2 − 2x 1) 4(x2 4).
Answers
Answer:
4 ( x-1)^2
Step-by-step explanation:
4x^2 − 8x + 4.
Factor out the greatest common factor
4( x^2 -2x+1)
Factor inside the terms
4( x-1) (x-1)
4 ( x-1)^2
Step-by-step explanation:
PLEAZE HELPP 50 POINTS If the function f(x) =-3x3 +7x represents the movement of a whale in meters what is the average rate of change of the whale for x=1 and X=3 seconds label your answer
Answers
Answer:
The average rate of change of a function over an interval is found by taking the difference between the function's values at the endpoints of the interval and dividing by the length of the interval. In this case, we have:
f(1) = -3(1)^3 + 7(1) = -3 + 7 = 4
f(3) = -3(3)^3 + 7(3) = -27 + 21 = -6
The average rate of change over the interval from x=1 to x=3 is therefore (-6 - 4) / (3 - 1) = -10 / 2 = -5.
To label your answer, you could write something like: "The average rate of change of the whale's movement over the interval from x=1 to x=3 seconds is -5 meters/second."
4) What is the sum of the exterior angles of a 32-sided polygon?
Answers
Answer:
360°
Step-by-step explanation:
The sum of the exterior angles for all polygons is 360°
which of the following is most likely a parameter as opposed to a statistic? group of answer choices the average score of the first five employees who completed an assignment. every third person to arrive in a bank. respondents to a newspaper survey. registered voters in a county.
Answers
The following is most likely a parameter as opposed to a statistic is registered voters in a county.
A parameter in statistics is any measurable number of a statistical population that summarises or describes some aspect of the population, as opposed to its typical application in mathematics, such as a mean or a standard deviation. A limited collection of factors that perfectly characterise a population and may be used to create a probability distribution for the purposes of taking samples from that population can be assessed if it exactly follows a known and defined distribution, such as the normal distribution.
The family is a parameterized family if the index is also a parameter of its members. The normal distributions, Poisson distributions, binomial distributions, and exponential family of distributions are examples of parameterized families of distributions.
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Express $150 as a percentage of $500
(please give the answer in a sum way so that it is easy to understand)
Answers
Answer:
30%Step-by-step explanation:
Express $150 as a percentage of $500
(please give the answer in a sum way so that it is easy to understand)
solution with an equation150 : 500 = x : 100
x = 150 * 100 : 500
x = 30
---------------
check
150 : 500 = 30 : 100
0.3 = 0.3
the answer is good
show that if a ⊆ c and b ⊆ d, then a × b ⊆ c × d
Answers
(x, y) is an element of the set c × d, since x is an element of c and y is an element of d.
Since (x, y) was an arbitrary element in a × b, we can conclude that every element in a × b is also in c × d. Thus, we have shown that if a ⊆ c and b ⊆ d, then a × b ⊆ c × d.
To show that if a ⊆ c and b ⊆ d, then a × b ⊆ c × d, follow these steps:
Step 1: Understand the notation.
a ⊆ c means that every element in set a is also in set c.
b ⊆ d means that every element in set b is also in set d.
Step 2: Consider the Cartesian products.
a × b is the set of all ordered pairs (x, y) where x ∈ a and y ∈ b.
c × d is the set of all ordered pairs (x, y) where x ∈ c and y ∈ d.
Step 3: Show that a × b ⊆ c × d.
To prove this, we need to show that any ordered pair (x, y) in a × b is also in c × d.
Let (x, y) be an arbitrary ordered pair in a × b. This means that x ∈ a and y ∈ b.
Since a ⊆ c, we know that x ∈ c because every element in set a is also in set c.
Similarly, since b ⊆ d, we know that y ∈ d because every element in set b is also in set d.
Now, we have x ∈ c and y ∈ d, so the ordered pair (x, y) belongs to c × d.
Step 4: Conclusion
Since any arbitrary ordered pair (x, y) in a × b also belongs to c × d, we can conclude that a × b ⊆ c × d.
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Part 1= You are shopping for a shirt, and you want to get the best deal. You go to two stores, the first of which is JCPenney. What is the cost of the shirt at JCPenney?
Part 2=The second store you go to is Macy's. What is the cost of the shirt at Macy? *
Part 3=Which store has the better deal? *
A- Macy's
B- JCPenney
Answers
Answer:
Macy’s
Step-by-step explanation:
let’s find the cost of each shirt
JCPenny’s: 14.99 x 0.80 = $11.99
11.99 x 1.06 = $12.70
The final cost of the JCPenny shirt is $12.70
Macy’s: 17.99 x 0.65 = $11.69
11.69 x 1.07 = $12.51
The final cost of the Macy’s shirt is $12.51
$12.51 is less than $12.70
So, the Macy’s shirt is the better deal :)
Solve the equation for x.
Answers
Answer:
Step-by-step explanation:
x/13=10
We have to get rid of the 13. (the denominator)
How?
By multiplying 2 sides by 13
Remember the Rules of Maths:
1. Whatever you do to one side, you must do to the other.
2. Use the opposite operation.
So, we used Rule 2 to get rid of the 13.
Now:
x=10*13
(On the left side, the 13s cancel)
x=130
It is always a great idea to double-check
Does 130 divided by 13 equal 10?
Of course it does!
Have a great day!!
I hope everything is clear!
What are the values of x and y in the figure
Answers
Answer:
I'm pretty sure 8
Step-by-step explanation:
In a competition, a school awarded medals in different categories.36 medals in dance, 18 medals in dramatics, and 12 in music.If these medals went to a total of 45 people and only 4 medals in all these categories.
a) Draw a Venn diagram to represent the information.
b) How many received medals in exactly two of these categories
Answers
a) Venn diagram to represent the information is attached
b) Number of elements belonging to exactly two of the three sets A, B, C are 7
What is Venn diagram?
A Venn diagram is a picture that employs circles to represent relationships between objects or limited groupings of objects. While circles that do not overlap do not have these characteristics, circles that overlap do. Venn diagrams make it easier to visualise the connections and discrepancies between two ideas.
b) Let A = set of persons who got medals in dance.
B = set of persons who got medals in dramatics.
C = set of persons who got medals in music.
Let X=number of elements belonging to exactly two of the three sets A, B, C
Given,
n(A) = 36
n(B) = 18
n(C) = 12
n(A ∪ B ∪ C) = 45
n(A ∩ B ∩ C) = 4
We know that
X= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3n(A ∩ B ∩ C)
X= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3 × 4
n(A ∩ B) + n(B ∩ C) + n(A ∩ C)=X+18 -------------(1)
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
Therefore, n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C)
put the value From (i) X+18 = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C)
X= n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) - 18
X= 36 + 18 + 12 + 4 - 45 - 18
X= 7
X= 7(answer)
Thus Number of elements belonging to exactly two of the three sets A, B, C are 7..
To learn more about the Venn diagram from the given link
https://brainly.com/question/30069829
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