Let a,b,,c and x elements in the group G. In each of the following solve for x in terms of a,b,c, and c.

Solve simultaneously x^2 a=bxc^-1 and acx=xac.

Answer:

16.

Step-by-step explanation:

The largest numerical value present in the set P is 10, then x is 4+10 = 14. Then the set P is

P = {14, 4, 6, 10, -2, 14, 6, 1, 14}.

Now, the range is the difference between the greatest and the smallest element in the set. The greatest number is 14 and the smallest is -2, then the range is

14-(-2) = 14+2 = 16.

Then the range of the set P is 16.

Answer:

5000kg = 5.5 short tons

Step-by-step explanation:

This can be solved as a rule of three problem.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

Unit conversion problems, like this one, is an example of a direct relationship between measures.

Each kg has 0.0011 short tons. How many short tons are there in 5000 kg. So?

1kg = 0.0011 short tons

5000kg - x short tons

[tex]x = 5000*0.0011[/tex]

[tex]x = 5.5[/tex] short tons.

5000kg = 5.5 short tons

To convert 5000 kg to short tons, multiply by the conversion factor and round up.

The conversion factor between kilograms and short tons is 0.00110231. To convert 5000 kg to short tons, we multiply by the conversion factor:

5000 kg * 0.00110231 = 5.51155 short tons

Rounding up to the nearest 100th gives us 5.52 short tons.

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Answer:

[tex]\mathbb{Z}\times \mathbb{Z}=\{(a,b)\lvert a, b\in \mathbb{Z}\}[/tex]

Step-by-step explanation:

In general, the Cartesian product of two sets [tex]A,B[/tex] is a new set defined by

[tex]A\times B=\{(a,b)\lvert a\in A,b\in B\}[/tex]

The pair [tex](a,b)[/tex] is ordered pair because the order is important, that is to say, in general [tex](a,b)\neq (b,a)[/tex].

One of the most important Cartesian products in mathematics is [tex]\mathbb{R}\times \mathbb{R}=\{(x,y) \lvert x,y \in \mathbb{R}\}[/tex] which is precisely the Cartesian Plane xy. The set [tex]\mathbb{Z}\times \mathbb{Z}[/tex] is a subset of [tex]\mathbb{R}\times \mathbb{R}[/tex] which is the set of all the points in the Cartesian plane whose coordinates are integers numbers. So, sketching the set [tex]\mathbb{Z}\times \mathbb{Z}[/tex] we have a picture as the shown below.

Answer: $59313.58

Step-by-step explanation:

Formula to find the accumulated amount of the annuity is given by :-

[tex]FV=A(\frac{(1+\frac{r}{m})^{mt})-1}{\frac{r}{m}})[/tex]

, where A is the annuity payment deposit, r is annual interest rate , t is time in years and m is number of periods.

Given : m= $2000 ; m= 1   [∵ its annual] ;   t= 10 years ;   r= 0.06

Now substitute all these value in the formula , we get

[tex]FV=(4500)(\frac{(1+\frac{0.06}{1})^{1\times10})-1}{\frac{0.06}{1}})[/tex]

⇒ [tex]FV=(4500)(\frac{(1.06)^{10})-1}{0.06})[/tex]

⇒ [tex]FV=(4500)(\frac{0.79084769654}{0.06})[/tex]

⇒ [tex]FV=(4500)(13.1807949423)[/tex]

⇒ [tex]FV=59313.5772407\approx59313.58 \ \ \text{ [Rounded to the nearest cent]}[/tex]

Hence, the accumulated amount of the annuity= $59313.58

The square of a prime number is not prime.

a) let x ∈ R, If x  ∈ {prime numbers}, then [tex]x^{2}[/tex]∉{prime numbers}

there says that if x is a real and x is in the set of the prime numbers, then the square of x isn't in the set of prime numbers.

b) Prove or disprove the statement.

ok, if x is a prime number, then x only can be divided by himself. Now is easy to see that [tex]x^{2}[/tex] = x*x can be divided by himself and x, then x*x is not a prime number, because can be divided by another number different than himself

Final answer:

Rephrased as an if-then statement, 'If p is a prime number, then p² is not prime.' It is proved through contradiction that the square of a prime number cannot be prime, as it would have a factor other than 1 and itself.

Explanation:

The question asks us to consider the statement: "The square of a prime number is not prime." Let's tackle this in two parts:

(a) If-Then Statement

An if-then statement using mathematical language and notation for the given statement might look like this: "If p is a prime number, then p² is not a prime number." Written symbolically, if we let p represent a prime number, then we can express the statement as: "If p is a prime, then p² is not prime," or more formally, "∀p ∈ Primes, (¬ Prime(p²))."

(b) Proof

We can prove this statement by contradiction. Assume the contrary: that there exists a prime number p such that p² is also prime. Since p is prime, p ≥ 2. So p² will be greater than p and have p as a factor, which contradicts the definition of a prime number (a prime number has no positive divisors other than 1 and itself). Hence, the square of a prime number cannot be prime, confirming the original statement.

For making mathematical induction, we need:

An [tex]n_0[/tex] for which the relation holds true

if its true for [tex]n_i[/tex], then, is true for [tex]n_{i+1}[/tex]

the relationship is not true for 1 or 2

[tex]1^3-1 = 0 < 4*1[/tex]

[tex]2^3-1 = 8 -1 = 7 < 4*2 = 8[/tex]

but, is true for 3

[tex]3^3-1 = 27 -1 = 26 > 4*3 = 12[/tex]

lets say that the relationship is true for n, this is

[tex]n^3 -1 \ge 4 n[/tex]

lets add 4 on each side, this is

[tex]n^3 -1 + 4 \ge 4 n + 4[/tex]

[tex]n^3 + 3 \ge 4 (n + 1)[/tex]

now

[tex](n+1)^3 = n^3 +3 n^2 + 3 n + 1[/tex]

[tex](n+1)^3 \ge n^3 + 3 n [/tex]

if [tex]n \ge 1[/tex] then [tex]3 n \ge 3[/tex] , so

[tex](n+1)^3 \ge n^3 + 3 n \ge n^3 + 3 [/tex]

[tex](n+1)^3 \ge  n^3 + 3 \ge 4 (n + 1)  [/tex]

[tex](n+1)^3  \ge 4 (n + 1)  [/tex]

and this is what we were looking for!

So, for any natural equal or greater than 3, the relationship is true.

Answer:

Step-by-step explanation:

Given that t A, B, and C are invertible matrices of the same

size.

To pr that [tex](ABC)^{-1} = C^{-1}B^{-1}A^{-1}.[/tex]

to pr that  [tex]( ABC) (C^{-1}B^{-1}A^{-1})=e[/tex]

LS=[tex]( AB(C (C^{-1})B^{-1}A^{-1})\\=ABIB^{-1}A^{-1})\\=A(BB^{-1})A^{-1})\\=AA^{-1})\\=I[/tex]

Thus proved

To show that the product of invertible matrices A, B, and C (ABC) is invertible and that its inverse is (ABC)-1 = C-1B-1A-1, we use the property that the inverse of a product is the product of inverses in reverse order, and the associativity of matrix multiplication.

A student asked to show that if A, B, and C are invertible matrices of the same size, the product ABC is invertible, and that the inverse of this product is (ABC)-1 = C-1B-1A-1.

To prove this, we can use the properties of matrix multiplication and inverses:

This completes the proof that the product ABC is invertible and its inverse is C-1B-1A-1

Answer:

The ball will take 6.3 seconds to reach the maximum height and hit the ground.

Step-by-step explanation:

a). When a ball was thrown upwards with an initial velocity u then maximum height achieved h will be represented by the equation

v² = u² - 2gh

where v = final velocity at the maximum height h

and g = gravitational force

Now we plug in the values in the equation

At maximum height final velocity v = 0

0 = (28)² - 2×(9.8)h

19.6h = (28)²

h = [tex]\frac{(28)^{2}}{19.6}[/tex]

  = [tex]\frac{784}{19.6}[/tex]

  = 40 meter

B). If the ball misses the building and hits the ground then we have to find the time after which the ball hits the ground that will be

= Time to reach the maximum height + time to hit the ground from the maximum height

Time taken by the ball to reach the maximum height.

Equation to find the time will be v = u - gt

Now we plug in the values in the equation

0 = 28 - 9.8t

t = [tex]\frac{28}{9.8}[/tex]

 = 2.86 seconds

Now time taken by the ball to hit the ground from its maximum height.

H = ut + [tex]\frac{1}{2}\times g\times (t)^{2}[/tex]

(17 + 40) = 0 + [tex]\frac{1}{2}\times g\times (t)^{2}[/tex]

57 = 4.9(t)²

t² = [tex]\frac{57}{4.9}[/tex]

t² = 11.63

t = √(11.63)

 = 3.41 seconds

Now total time taken by the ball = 2.86 + 3.41

                                                      = 6.27 seconds

                                                      ≈ 6.3 seconds

Therefore, the ball will take 6.3 seconds to reach the maximum height and hit the ground.

Closure Property for the set of integers for Subtraction

Take two integers,

A=6

B=9

A-B

=6-9

= -3

Which is also an Integer.Hence Closure Property is satisfied by Integers with respect to Subtraction.

⇒Closure Property for the set of integers for Division

Take two integers,

A=6

B=9

[tex]\rightarrow \frac{A}{B}\\\\=\frac{6}{9}\\\\=\frac{2}{3}-----\text{Not an Integer}[/tex]

Which is not an Integer.Hence Closure Property is not satisfied by Integers with respect to Division.

Answer:

The range represents the number of users each month for 24 months

Step-by-step explanation:

The range depends on the y axis. Looking at the y axis we see the number of users for each month. Looking at the x axis we see that it's 24 months not 20 months

From the graph, we can interpret the range represents the number of users each month for 24 months. Therefore, option B is the correct answer.

We need to find what can be interpreted from the range of the given graph.

The range is the difference between the smallest and highest numbers in a list or set. To find the range, first put all the numbers in order. Then subtract (take away) the lowest number from the highest.

From the given graph, we can see the x-axis represents the number of months and the y-axis represents the number of users.

So, the range represents the number of users each month for 24 months

20-15.8=4.2

Therefore, option B is the correct answer.

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Answer:

(a) Not a group

(b) Not a group

(c) Abelian group

Step-by-step explanation:

In order for a system <G,*> to be a group, the following must be satisfied

(1) The binary operation is associative, i.e., (a*b)*c = a*(b*c) for all a,b,c in G

(2) There is an identity element, i.e., there is an element e such that a*e = e*a = a for all a in G

(3) For each a in G, there is an inverse, i.e, another element a' in G such that a*a' = a'*a = e (the identity)

If in addition the operation * is commutative (a*b = b*a for every a,b in G), then the group is said to be Abelian

(a)  

The system <G,*> is not a group since there are no identity.  

To see this, suppose there is an element e such that  

a*e = a

then  

a-e = a which implies e=0

It is easy to see that 0 cannot be an identity.

For example  

2*0 = 2-0 = 2

Whereas

0*2 = 0-2 = -2

So 2*0 is not equal to 0*2

(b)

The system <G,*> is not a group either.

If A is a matrix 2x2 and the determinant of A det(A)=0, then the inverse of A does not exist.

(c)

The table of the operation G is showed in the attachment.

It is evident that this system is isomorphic under the identity map, to the cyclic group

[tex]\mathbb{Z}_{5}[/tex]

the system formed by the subset of Z, {0,1,2,3,4} with the operation of addition module 5, which is an Abelian cyclic group

We conclude that the system <G,*> is Abelian.

Attachment: Table for the operation * in (c)

Answer:

3/10

Step-by-step explanation:

If the probability of losing is 7/10, then the probability of winning is 3/10. To find out the probability of winning we just have to subtract the probability of losing from the total which is 10:

10/10 = total probability

7/10 = probability of losing

Probability of winning is 3/10.

Answer:

No, because these events are independent.

Step-by-step explanation:

When we roll one die (a fair die), we will have outcomes in the numbers from 1 to 6.

And when we flip a coin, we will have either a heads or a tails.

Now these two events are independent events as the occurrence of any result in one event is not affecting the result of second event,

Hence, these two are independent events. So, the answer is no, the occurrence of a particular outcome on the die will not affect the probability of a particular outcome on the coin.

Final answer:

The outcomes of a die roll and a coin toss are independent events, meaning the result of the die does not affect the result of the coin toss.

Explanation:

The occurrence of a particular outcome on the die does not affect the probability of a particular outcome on the coin because each event is independent of the other.

For example, if you roll a die and flip a coin, the result of the die does not influence the result of the coin toss as they are separate events.

Additionally, the concept of independence in probability states that the outcomes of the coin toss and die roll do not impact each other.

Answer:

A) [tex]\frac{2469}{20000}[/tex]

B)  [tex]\frac{54321}{100000}[/tex]

Step-by-step explanation:

To answer this question first we define what are rational numbers.

A rational number is a number that can be expressed in the form of a fraction [tex]\frac{x}{y}[/tex], where x and y are integers and y ≠ 0.

a) 0.12345

This can be written in fraction form as

[tex]\frac{12345}{100000}[/tex]

Now, we simplify this fraction to lowest form. Simplified form of the above fraction is [tex]\frac{2469}{20000}[/tex]. This simplified form was obtained by dividing both numerator an denominator by 5.

b) This number can be written in fraction form as [tex]\frac{54321}{100000}[/tex]

Now, we simplify this fraction.

The simplified form of the above fraction is [tex]\frac{54321}{100000}[/tex] as the numerator and denominator have no common factor.

Answer:

P = 0.42

Step-by-step explanation:

This probability problem can be solved by building a Venn like diagram for each probability.

I say that we have two sets:

-Set A, that is the probability of receiving this virus through the email.

-Set B, that is the probability of receiving it through the webpage.

The most important information in these kind of problems is the intersection. That is, that he virus enters the system simultaneously by both email and webpage with a probability of 0.17. It means that [tex]A \cap B[/tex] = 0.17.

By email only

The problem states that there is a 40 chance of receiving it through the email. It means that we have the following equation:

[tex]A + (A \cap B) = 0.40[/tex]

[tex]A + 0.17 = 0.40[/tex]

[tex]A = 0.23[/tex]

where A is the probability that the system receives the virus just through the email.

The problem states that there is a 40% chance of receiving it through the email. 23% just through email and 17% by both the email and the webpage.

By webpage only

There is a 35% chance of receiving it through the webpage. With this information, we have the following equation:

[tex]B + (A \cap B) = 0.35[/tex]

[tex]B + 0.17 = 0.35[/tex]

[tex]B = 0.18[/tex]

where B is the probability that the system receives the virus just through the webpage.

The problem states that there is a 35% chance of receiving it through the webpage. 18% just through the webpage and 17% by both the email and the webpage.

What is the probability that the virus does not enter the system at all?

So, we have the following probabilities.

- The virus does not enter the system: P

- The virus enters the system just by email: 23% = 0.23

- The virus enters the system just by webpage: 18% = 0.18

- The virus enters the system both by email and by the webpage: 17% = 0.17.

The sum of the probabilities is 100% = 1. So:

P + 0.23 + 0.18 + 0.17 = 1

P = 1 - 0.58

P = 0.42

There is a probability of 42% that the virus does not enter the system at all.

Answer:

Eli walked 12 feet down the hall of his house to get to the door. He continued in a straight line out the door and across the yard to the mailbox, a distance of 32 feet.

He came straight back across the yard 14 feet and stopped to pet his dog.

Part A : find the image attached.

Part B : [tex]12+32+14=58[/tex] feet

Part C : [tex]32-14=18[/tex] feet from his house.

Answer with Step-by-step explanation:

We are given that two matrices A and B are square matrices of the same size.

We have to prove that

Tr(C(A+B)=C(Tr(A)+Tr(B))

Where C is constant

We know that tr A=Sum of diagonal elements of A

Therefore,

Tr(A)=Sum of diagonal elements of A

Tr(B)=Sum of diagonal elements of B

C(Tr(A))=[tex]C\cdot[/tex] Sum of diagonal elements of A

C(Tr(B))=[tex]C\cdot[/tex] Sum of diagonal elements of B

[tex]C(A+B)=C\cdot (A+B)[/tex]

Tr(C(A+B)=Sum of diagonal elements of (C(A+B))

Suppose ,A=[tex]\left[\begin{array}{ccc}1&0\\1&1\end{array}\right][/tex]

B=[tex]\left[\begin{array}{ccc}1&1\\1&1\end{array}\right][/tex]

Tr(A)=1+1=2

Tr(B)=1+1=2

C(Tr(A)+Tr(B))=C(2+2)=4C

A+B=[tex]\left[\begin{array}{ccc}1&0\\1&1\end{array}\right]+\left[\begin{array}{ccc}1&1\\1&1\end{array}\right][/tex]

A+B=[tex]\left[\begin{array}{ccc}2&1\\2&2\end{array}\right][/tex]

C(A+B)=[tex]\left[\begin{array}{ccc}2C&C\\2C&2C\end{array}\right][/tex]

Tr(C(A+B))=2C+2C=4C

Hence, Tr(C(A+B)=C(Tr(A)+Tr(B))

Hence, proved.

Answer:

The probability that all will last at least 30,000 miles is 0.3164.

Step-by-step explanation:

Consider the provided information.

For a particular tire brand, the probability of wearing out before 30,000 miles is 0.25. Someone who buys a set of four of these tires,

That means the probability of not wearing out is 1-0.25 = 0.75

Now use the binomial distribution formula.

[tex]^nC_r (p^{n-r})(q^r)[/tex]

Substitute p = 0.25, q=0.75, r=4 and n=4

[tex]^4C_4 (0.25)^{4-4}(0.75)^4[/tex]

[tex]1 \times 0.75^4[/tex]

[tex]0.3164[/tex]

The probability that all will last at least 30,000 miles is 0.3164.

Answer:

(a) symmetric

(b) The mean, median, and mode may be the same value.

and, The mode is the tallest bar in a histogram.

(c) variance

Step-by-step explanation:

Data is said to be symmetric if it is not skewed. i.e. if it has peaked at the center of the distribution.

Skewness gives us an idea about the shape of the curve.

Kurtosis enables us to have an idea about the flatness and peakedness of the frequency.

Mode represents the highest repetition of the observation.

The mean, median, and mode can have the same value.

The histogram represents the frequency of observation by the length of the bar this is the same as Mode. Thus, the mode is the tallest bar in a histogram.

Variance is the square of standard deviation and it is defined as, the sum of the square of the distance of an observation from the mean.

So, Variance is the correct option.

Answer:

x = tex]5\frac{\textup{7}}{\textup{8}}\ in[/tex]

Step-by-step explanation:

Given:

Total height of the bench = 18 in

Depth of the stone bench = [tex]3\frac{\textup{3}}{\textup{8}}\ in[/tex] = [tex]\frac{\textup{27}}{\textup{8}}\ in[/tex]  = 3.375 in

Measure of stones = [tex]3\frac{\textup{1}}{\textup{2}}\ in[/tex] = [tex]\frac{\textup{7}}{\textup{2}}\ in[/tex]  = 3.5 in

measure of another stone =  [tex]5\frac{\textup{1}}{\textup{4}}\ in[/tex] = [tex]\frac{\textup{21}}{\textup{4}}\ in[/tex]  = 5.25 in

let the height of the third stone be 'x'

Now,

The total height of the bench = depth of bench + Measure of two stones + x

18 = 3.375 + 3.5 + 5.25 + x

or

x = 18 - 12.125

or

x = 5.875 in

or

x = tex]5\frac{\textup{7}}{\textup{8}}\ in[/tex]

Answer:

We can place everything with v(x) on one side of the equality, everything with x on the other side. This is done on the step-by-step explanation, and shows that the new equation is separable.

Step-by-step explanation:

We have the following differential equation:

[tex]xy' = y(ln x - ln y)[/tex]

We are going to apply the following substitution:

[tex]y = xv(x)[/tex]

The derivative of y is the derivative of a product of two functions, so

[tex]y' = (x)'v(x) + x(v(x))'[/tex]

[tex]y' = v(x) + xv'(x)[/tex]

Replacing in the differential equation, we have

[tex]xy' = y(ln x - ln y)[/tex]

[tex]x(v(x) + xv'(x)) = xv(x)(ln x - ln xv(x))[/tex]

Simplifying by x:

[tex]v(x) + xv'(x) = v(x)(ln x - ln xv(x))[/tex]

[tex]xv'(x) = v(x)(ln x - ln xv(x)) - v(x)[/tex]

[tex]xv'(x) = v(x)((ln x - ln xv(x) - 1)[/tex]

Here, we have to apply the following ln property:

[tex]ln a - ln b = ln \frac{a}{b}[/tex]

So

[tex]xv'(x) = v(x)((ln \frac{x}{xv(x)} - 1)[/tex]

Simplifying by x,we have:

[tex]xv'(x) = v(x)((ln \frac{1}{v(x)} - 1)[/tex]

Now, we can apply the above ln property in the other way:

[tex]xv'(x) = v(x)(ln 1 - ln v(x) -1)[/tex]

But [tex]ln 1 = 0[/tex]

So:

[tex]xv'(x) = v(x)(- ln v(x) -1)[/tex]

We can place everything that has v on one side of the equality, everything that has x on the other side, so:

[tex]\frac{v'(x)}{v(x)(- ln v(x) -1)} = \frac{1}{x}[/tex]

This means that the equation is separable.

Answer:

So you may know that that the time for downloading a file  of A (MB) is B (seconds), where A and B are numbers.

Then suppose if some file size is C, which is 10 times more than A, and you assume that the download speed sees this as downloading 10 times the A (MB) file. Thus the time for downloading this second file will be 10*B (seconds).

As, if                            [tex]C\ (MB) file=10*A\ (MB) file[/tex]

then                             [tex]C\ (MB)file =10*B\ (seconds)[/tex]

B(seconds) is the time required to download A (MB) file.

To find the specific circle passing through points (2,3), (-3,2), and (0,0), we started by substituting these coordinates in the general circle equation. This gave us a system of equations, which when solved, resulted in the coefficients for the circle equation. The equation of the circle we were seeking is (x^2 + y^2)/13 - (3x/32) + (15y/32) = 0.

In order to find the equation for a specific circle we need to plugin the coordinates for (x, y) in the general equation a(x^2+ y^2) + bx + cy + d = 0 which represent three points on the circle. Let's start to solve this system using the points (2,3), (−3,2), and (0,0).

Now, we have a system of three equations with three variables that we can solve. By subtracting the second from the first to eliminate d and simplifying, we obtain 5b + c = 0. Substituting d = 0 into the first and second equations, we get 13a + 2b + 3c = 0 and 13a - 3b + 2c = 0. Solving this system gives a = 1/13, b = -3/32 and c = 15/32.

Therefore, the equation of the circle is (x^2 + y^2)/13 - (3x/32) + (15y/32) = 0.

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The probability of selecting a cookie with either chocolate chips or pecans from a plate of 50 cookies is 36%.

The question asks about the probability of selecting a cookie with either chocolate chips or pecans from a plate of 50 cookies, with some overlapping flavors. Given the information:

Total cookies: 50

Chocolate chip cookies: 10

Pecan cookies: 14

Cookies with both flavors: 6

To calculate the probability of choosing a cookie with either chocolate chips or pecans (or both), we can use the formula:

P(C OR N) = P(C) + P(N) \'u2013 P(C AND N)

where:

P(C) is the probability of choosing a chocolate chip cookie,

P(N) is the probability of choosing a pecan cookie, and

P(C AND N) is the probability of choosing a cookie with both flavors.

Thus:

P(C) = 10/50 = 0.20,

P(N) = 14/50 = 0.28, and

P(C AND N) = 6/50 = 0.12.

Therefore:

P(C OR N) = 0.20 + 0.28 - 0.12 = 0.36 or 36%.

This is the probability that a randomly selected cookie from the plate will have either chocolate chips, pecans, or both.

Answer:

8 rental movies will make the costs of the two stores be the same

Step-by-step explanation:

- Mr Cosgrove is comparing movie rentals deals

- Netflix charge a flat rate of $8.50

- Blockbuster charge $4.50 plus a $0.50 per movie

- We need to find how many movie rental make the costs of the two

  stores will be the same

- Assume that the number of rental movies is m

∵ Blockbuster charge $4.50 plus a $0.50 per movie

∴ Blockbuster cost = 4.50 + 0.50 m

∵ The cost of Netflix = 8.50

- Equate the two costs to find the number of the rental movies

∵ 4.50 + 0.50 m = 8.50

- Subtract 4.50 from both sides

∴ 0.50 m = 4

- Divide both sides by 0.50

∴ m = 8 movies

8 rental movies will make the costs of the two stores be the same

Answer:

240

Step-by-step explanation:

Given:

The cost function for canoes, c(x) = 24,000 + 100x

Revenue function for canoes, R(x) = 200x

here, x is the number of canoes sold

Fixed cost = $24,000

Cost of production = $100

now at break even point, Revenue = cost

thus,

200x = 24,000 + 100x

or

100x = 24,000

or

x = 240 canoes

Hence, at breakeven, the number of canoes sold is 240

Answer: The proof is done below.

Step-by-step explanation:  Given that the square matrix  A is called orthogonal provided that [tex]A^T=A^{-1}.[/tex]

We are to show that the determinant of such a matrix is either +1 or -1.

We will be using the following result :

[tex]|A^{-1}|=\dfrac{1}{|A|}.[/tex]

Given that, for matrix A,

[tex]A^T=A^{-1}.[/tex]

Taking determinant of the matrices on both sides of the above equation, we get

[tex]|A^T|=|A^{-1}|\\\\\Rightarrow |A|=\dfrac{1}{|A|}~~~~~~~~~~~~~~~~~~~~[\textup{since A and its transpose have same determinant}]\\\\\\\Rightarrow |A|^2=1\\\\\Rightarrow |A|=\pm1~~~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}][/tex]

Thus, the determinant of matrix A is either +1 or -1.

Hence showed.

Sets and set operations are ways of organizing, classifying and obtaining information about objects according to the characteristics they possess, as objects generally have several characteristics, the same object can belong to several sets, an example is the subjects of a school , where students (objects) are classified according to the subject they study (set).

The intersection of sets is a new set consisting of those objects that simultaneously possess the characteristics of each intersected set, the intersection of two subjects will be those students who have both subjects enrolled.

The union of sets is a new set consisting of all the objects belonging to the united sets, the union of two subjects will be all students of both courses.

In this case there are three sets B, C and S of which we are given the following information:

Answer

n(BꓵSꓵC)=5

n(BꓵS)=10 – 5 = 5

n(BꓵC)=12 – 5 = 7

n[(BꓵC)ꓴ(BꓵS)ꓴ(CꓵS)]=21  – 5 – 5 – 7 = 4

n(B)=36 – 5 – 5 – 7 = 19

n(S)=30 – 5 – 5 – 4 = 16

n(C)=34 – 5 – 7 – 4 = 18

Answer:

They will meet again in 2014

Step-by-step explanation:

Once every year the Earth will go back through the point where it met with Jupiter. Let's write those years:

Earth = {2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016...}

Once every 12 years Jupiter will go back through the point where it met with the Earth. Let's write it down:

Jupiter= {2002, 2002+12=2014, 2014+12=2026, 2026+12=2038...}

The Earth and Jupiter will meet again for the first time (after 2002) the year they both go trought that point.

Earth = {2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016...}

Jupiter= {2002, 2014, 2026, 2038...}

As we can see, 2014 is the first year the'll meet again.

Answer with Step-by-step explanation:

The given differential equation is variable separable in nature and hence will be solved accordingly as follows:

[tex]\frac{dy}{dx}=\frac{y}{x}\\\\=\frac{dy}{y}=\frac{dx}{x}\\\\\int \frac{dy}{y}=\int \frac{dx}{x}\\\\ln(y)=ln(x)+ln(c)\\\\ln(y)=ln(cx)\\\\(\because ln(ab)=ln(a)+ln(b))\\\\\therefore y=cx[/tex]

where 'c' is constant of integration whose value shall be obtained using the given condition [tex]y(1)=-2[/tex]

Thus we have

[tex]-2=c\times 1\\\\\therefore c=-2\\[/tex]

Thus solution becomes

[tex]y=-2x[/tex]