Verify that the function(s) solve the following differential equations (DES): a) y' = -5y; y = 3e-5x b) y' = cos(3x); y = į sin(3x) + 7 c) y' = 2y; y = ce2x , where c is any real number. d) y" + y' – 6y = 0 ; yı = (2x, y2 = (–3x e) y" + 16y = 0; yı = cos(4x), y2 = sin(4x)

Answer:

B. 10n-207

Step-by-step explanation:

Given,

The average number of points of 9 games = 23,

∵ Sum of observations = their average × number of observations

So, the total points for 9 games = 9 × 23 = 207,

Now, if the average of 10 games is n,

Then the total points for 10 games = 10n,

Hence, the number of points earned in 10th game = the total points for 10 games - the total points for 9 games

= 10n - 207

i.e. OPTION B is correct.

The correct option is (B) 10n-207. To raise the average points per game to 'n' after the 10th basketball game, one would need to score '10n - 207' points in the 10th game, based on the previous average of 23 points in the first 9 games.

To solve the problem, we need to use the formula for calculating an average. The average score after 10 games would be the total points scored divided by 10. If the first 9 games had an average of 23 points, the total points from these games is 9 × 23 = 207.

To get an average of n after the 10th game, the total points should be 10× n. Therefore, the points needed in the 10th game would be the difference between 10 × n and the sum score from the first 9 games (207). That gives us the equation 10n - 207 for the number of points needed in the 10th game.

Recall that for [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n\ge0}x^n[/tex]

Replace [tex]x[/tex] with [tex]-9x^2[/tex] and we get

[tex]\displaystyle\frac1{1-(-9x^2)}=\sum_{n\ge0}(-9x^2)^n=\sum_{n\ge0}(-9)^nx^{2n}[/tex]

Lastly, multiply this by [tex]x^3[/tex], so that

[tex]\boxed{f(x)=\displaystyle\sum_{n\ge0}(-9)^nx^{2n+3}}[/tex]

Answer:

(a) 0.09 (b) 0.322 (c) 0.0966

Step-by-step explanation:

Let's define first the following events

M: an applicant is a male

F: an applicant is a female

A: an applicant is accepted

E: an applicant is enrolled

S: the sample space

Now, we have a total of 7500 applicants, and from these applicants 4200 were male and 3300 were female. So,

P(M) = 0.56 and P(F) = 0.44, besides

P(A | M) = 0.3, P(E | A∩M) = 0.3, P(A | F) = 0.35, P(E| A∩F) = 0.3

(a) 0.09 = (0.3)(0.3) = P(A|M)P(E|A∩M)=P(E∩A∩M)/P(M)=P(E∩A | M)

(b) P(A) = P(A∩S) = P(A∩(M∪F))=P(A∩M)+P(A∩F)=P(A|M)P(M)+P(A|F)P(F)=(0.3)(0.56)+(0.35)(0.44)=0.322

(c) P(A∩E)=P(A∩E∩S)=P(A∩E∩(M∪F))=P(A∩E∩M)+P(A∩E∩F)=0.0504+P(E|A∩F)P(A|F)P(F)=0.0504+(0.3)(0.35)(0.44)=0.0966

Answer:

The required value for the blank is 21.8696

Step-by-step explanation:

Consider the provided information,

4,275.50= 391/2 of ____

Replace blank with x.

4,275.50 = 391/2 of x

Use sign of multiplication for "of".

[tex]4,275.50 = \frac{391}{2} \times x[/tex]

Solve the above expression for x.

[tex]x=\frac{4,275.50\times 2}{391} [/tex]

[tex]x=21.8696 [/tex]

Thus, the required value for the blank is 21.8696

The probability that no passenger is in their assigned seat is 0.6321.

Given data:

There are N passengers in a plane with N assigned seats (N is a positive integer), but after boarding, the passengers take the seats randomly.

The probability that no passenger is in their assigned seat is often referred to as the "surprising" or "paradoxical" result.

The problem is analyzed for a specific case with N = 3 passengers.

Passenger 1 sits in Seat 1: In this case, the remaining two passengers have a 1/2 chance of sitting in their assigned seats.

Passenger 1 sits in Seat 2: Again, the remaining two passengers have a 1/2 chance of sitting in their assigned seats.

Passenger 1 sits in Seat 3: In this case, the remaining two passengers will definitely sit in their assigned seats.

Now, calculate the probability for each scenario and find the overall probability that no passenger is in their assigned seat:

Scenario 1: Probability = 1/3 * 1/2 = 1/6

Scenario 2: Probability = 1/3 * 1/2 = 1/6

Scenario 3: Probability = 1/3

Overall Probability = Probability of Scenario 1 + Probability of Scenario 2 + Probability of Scenario 3

= 1/6 + 1/6 + 1/3

= 1/2

Thus, for N = 3, the probability that no passenger is in their assigned seat is 1/2.

Now, consider the case when N → ∞ (approaching infinity).

In this scenario, the probability can be calculated as follows:

Probability = 1 - 1/e

Hence, as N approaches infinity, the probability that no passenger is in their assigned seat approaches 1 - 1/e, or approximately 0.6321.

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Final answer:

The derangement problem questions the probability that no passenger sits in their assigned seat, which asymptotically approaches 1/e (approximately 0.3679) as the number of seats and passengers approaches infinity.

Explanation:

The student's question is about calculating the probability that no passenger is seated in their assigned seat on a plane where passengers sit randomly, especially when the number of passengers (and seats) approaches infinity. This is known as the derangement problem or a problem of calculating permutations where no element appears in its original position. A specific case of permutations without fixed points is termed as a derangement, and the probability of a derangement occurring in a permutation of N elements approaches 1/e where e is Euler's number, approximately equal to 2.71828. Through an approximation known as the asymptotic probability, when N → [infinity], this probability approaches 1/e, which is approximately 0.3679 or 36.79%.

Answer:

F(x)=[tex]\frac{3^x}{ln(3)}[/tex]+7cosh(x)+C

Step-by-step explanation:

The function is f(x)=3ˣ+7sinh(x), so we can define it as f(x)=g(x)+h(x) where g(x)=3ˣ and h(x)=7sinh(x).

Now we have to find the most general antiderivative of the function this means that we have to calculate [tex]\int\ {f(x)} \, dx[/tex] wich is the same as [tex]\int\ {(g+h)(x)} \, dx[/tex]

The sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. Then,

[tex]\int\ {(g+h)(x)} \, dx[/tex] = [tex]\int\ {g(x)} \, dx + \int\ {h(x)} \, dx[/tex]

1- [tex]\int\ {g(x)} \, dx =[/tex][tex]\int\ {3^x} \, dx = \frac{3^x}{ln(3)}+C[/tex] this is because of the rule for integration of exponencial functions, this rule is:

[tex]\int\ {a^x} \, dx =\frac{a^x}{ln(x)}[/tex], in this case a=3

2-[tex]\int\ {h(x)} \, dx =[/tex][tex]\int\ {7sinh(x)} \, dx =7\int\ {sinh(x)} \, dx =7cosh(x)+C[/tex] , number seven is a constant (it doesn´t depend of "x") so it "gets out" of the integral.

The result then is:

F(x)= [tex]\int\ {(h+g)(x)} \, dx=\int\ {h(x)} \, dx +\int\ {g(x)} \, dx[/tex]

[tex]\int\ {3^x} \, dx +\int\ {7sinh(x)} \, dx = \frac{3^x}{ln(3)} +7cosh(x) + C[/tex]

The letter C is added because the integrations is undefined.

To prove that V is a vector space we must prove that the sum define on it satisfy conmutativiy, asociativity and existence of the neutral element and inverses. Also, the scalar multiplication define on V must satisfy distributivity propertie with respect to the sum and viceversa, and an asosiativity too in the sense that [tex]x(y\cdot v)= (xy)\cdot v[/tex] for [tex]x,y\in \mathbb{R}[/tex] and [tex]v\in V[/tex]. One can prove with this that the neutral element for the sum is unique. But with your operations you  have two neutral elements for [tex](1;2)[/tex]

[tex](1;2)+(-1;3)=(0;0)[/tex]

and

[tex](1;5)+(-1;11)=(0;0)[/tex]

So, you dont have a vector space.

Final answer:

The set V, with its defined addition and scalar multiplication operations, does not fulfill essential vector space properties such as the existence of an additive identity, presence of additive inverses, and correct scalar multiplication effects on components. Therefore, V is not a vector space.

Explanation:

To determine if a set V, defined with specific addition and scalar multiplication operations, is a vector space, it must satisfy several properties commonly defined in linear algebra.

For V to be considered a vector space, the addition operation must be associative and commutative, there must be an additive identity (zero vector), each vector must have an additive inverse, scalar multiplication must be associative, there must be a multiplicative identity (1), and both operations must distribute over vector addition and scalar addition.

The defined operations on V are (x₁; y₁) + (x₂; y₂) = (x₁ + x₂; 0) for vector addition and c(x; y) = (cx; 0) for scalar multiplication. These operations fail to satisfy several vector space properties, including:

Due to these inadequacies, V does not meet the criteria for a vector space under the provided operations.

Annual is 1 year.

1 year has 12 months.

Multiply the monthly rate by 12:

3.15 x 12 = 37.80%

Answer: a) The average of this set of numbers is 1.98 s. b) the standard deviation is 0.02 s.

Step-by-step explanation: The average is calculated by adding all numbers found in the experiment divided by the number of values. [tex]average = \frac{Sumxi}{n}[/tex]

The standard deviation is given by the square root of the squared sum of the difference between each number and the average divided by the total number of values. For instance, std deviation = [tex]\frac{sqrt{(1.94-1.98)^{2}+(1.96-1.98)^2+(2.01-1.98)^2...}}{11}[/tex] and that is done with all 11 terms of data minus the average to find the standard deviation.

In excel, the average of a certain set of numbers (displayed in cells A1 to A5) can be found by the commands =average(A1:A5). The standard deviation can be found by the commands =stdedv.p(A1:A5) in which p is the population. You can decrease decimals by clicking on the icon displayed in the Ribbon.

Answer: n/2-3=13;n=32

Step-by-step explanation: the equation and solution correctly represents this sentence is n/2-3=13;n=32,

first we have to keep in mind the beginning of the exercise which is (three less) then we know that the 3 has a sign of subtraction(-3), when we meet this, we can know that this exercise has only two possible good answer.

Only the equations which have a (-3) inside will be good,the rest of equation has a 3 with a sum sign

N/2+3=13;n=5 ,

n/2+3=13;=20

then we can omit the past equations, and this let us this equations as possible.

n/2-3=13;n=32

n/2-3=13;n=8

after this we only need resolve the equation to get a correct result, only the equation with a correct result will be the correct answer,

then we proceed to clear n from each equation.

for this equation the result must be equal to 8 if we clear n.

n/2-3=13;n=8

n/2-3=13

we pass the three to the other side with sum sign

n/2=13 + 3

we resolve the sum

n/2=16

after we pass the 2 multiplying to the other side

n = 16  × 2

we resolve the product of the multiplication

n = 32

but this answer said the result must be equal to 8

n=8 ≠ n = 32

as this result is different, we can conclude that this is a bad answer,

the we get only one possibility

this equation

n/2-3=13;n=32

for this equation the result must be equal to 32 if we clear n.

n/2-3=13;n=8

n/2-3=13

we pass the three to the other side with sum sign

n/2=13 + 3

we resolve the sum

n/2=16

after we pass the 2 multiplying to the other side

n = 16  × 2

we resolve the product of the multiplication

n = 32

Answer:

C

Um I'm late but its not any of the others so....

Step-by-step explanation:

The statement to be proved using mathematical induction is:

We will begin the proof showing that the base case is satisfied (n=4).

[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex].

Then, 1 is true for n=4.

Now we will assume that the statement holds for some arbitrary natural number [tex]n\geq 4[/tex] and prove that then, the statement holds for n+1. Observe that

[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]

With this the inductive step has been proven and then, our statement is true,

For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]

Answer:

[tex]P \cap R=\{2,3,5,7\}[/tex]

Step-by-step explanation:

Given : Let P={ 1,2,3,5,7,9}, Q= { 1,2,3,4,5}, and R= {2,3,5,7,11}

To find : The value of [tex]P \cap R[/tex] ?

Solution :

The intersection of two sets is defined as the set of their common elements or the elements appear in both the sets.

The sets are P={ 1,2,3,5,7,9} and R= {2,3,5,7,11}

Common elements in P and R are {2,3,5,7}

So, [tex]P \cap R=\{2,3,5,7\}[/tex]

Therefore, The value of [tex]P \cap R=\{2,3,5,7\}[/tex]

Answer:

466mEq

Step-by-step explanation:

First, we need to know the concentration of NaCl in a normal saline solution, this is by definition 0.9%, meaning we have 0.9g of NaCl per 100ml of solution, we want to know how much NaCl we have in 3L (3000ml):

[tex]3000ml*\frac{0.9g}{100ml}=27g=27000mg[/tex]

So, we have 27000mg in 3L of normal saline solution.

Now, acording to our milliequivalent (mEq) equation ([tex]mEq=\frac{mg}{pE}[/tex]) where pE is de molecular mass of NaCl divided by their charges, in this case 1:

[tex]pE= \frac{23+35}{1}=\frac{58}{1} = 58[/tex]

Finally we substitute in the mEq formula:

[tex]mEq=\frac{mg}{pE}=\frac{27000}{58}=466mEq[/tex]

I hope you find this information useful! Good luck!

Answer:

Where are the sets of vectors?

Step-by-step explanation:

Which set of vectors?

Answer:

1600 ml.

Step-by-step explanation:

Let x represent amount of tube-feeding formula.

We have been given that a tube feeding formula contains 6 grams of protein per each 80 ml of the formula.

To solve our given problem, we will use proportions as:

[tex]\frac{x}{\text{120 gram}}=\frac{\text{80 ml}}{\text{ 6 grams}}[/tex]

[tex]\frac{x}{\text{120 gram}}*\text{120 gram}=\frac{\text{80 ml}}{\text{ 6 grams}}*\text{120 gram}[/tex]

[tex]x=\text{80 ml}*20[/tex]

[tex]x=\text{1600 ml}[/tex]

Therefore, the patient should get 1600 ml of tube feeding formula every day.

Answer:

Values for each variable are:

x = 19

y = -4

z = -12

Step-by-step explanation:

As we can remember the Gauss-Jordan elimination method consists of creating a matrix with all the equations of the system.  Remember that, if a variable does not appear in one of the equations, we give a value of 0 to its coefficient .  Each equation will constitute a line of the matrix. So, the matrix will look like this:

2   2   1   18

1    0   1    7

0   4  -3  20

For the Gauss-Jordan elimination we can multiply lines, add or subtract one line to another or we can rearrange the order at any given time. The goal is to get only 1s in the matrix diagonal, to determine the value of each variable.

Since we already have a line with a 1, we'll take that line as our starting point, and we'll rearrange it as our 1st line. By multiplying the 1st line for  2 and then subtracting the result to the second line:

1   0   1   7

0   2  -1  4

0   4  -3  20

Now, we multiply the second line by 2 and subtract the result to the third line

1   0   1   7

0   2  -1  4

0   0  -1  12

In order to get the value of Z all we have to do is multiply the third line by (-1).

1   0   1   7

0   2  -1  4

0   0  1  -12

Now, we add the third line to the second line.

1   0   1   7

0   2  0  -8

0   0  1  -12

Then, multiply the second line by a fraction 1/2, to get the value for Y

1   0   1   7

0   1  0  -4

0   0  1  -12

Finally, we subtract the third line to the 1st line to get the value for X

1   0   0  19

0   1  0  -4

0   0  1  -12

All we got left is to prove our answer is correct by replacing the variables in the system with the values found:

First equation

2(19) + 2(-4) + (-12) = 18

38 - 8 - 12 = 18

38 - 20 = 18

Second equation

19 + (-12) = 7

19 -12 = 7

Third equation

4(-4) - 3(-12) = 20

-16 + 36 = 20

Answer:

x = 50

y = -10

Step-by-step explanation:

x + y = 40

x + 10 = 60

60 - 10 =x

60 - 10 = 50

x = 50

50 + y = 40

40 - 50 = y

40 - 50 = -10

y = -10

Hey!

------------------------------------------------

Solve for x:

x + y = 40

x + y - y = 40 - y

x = 40 - y

50 - 10 = 40

50 + (-10) = 40

x = 50

y = -10

------------------------------------------------

Solve for x:

x + 10 = 60

x + 10 - 10 = 60 - 10

x = 50

------------------------------------------------

Hope This Helped! Good Luck!

Answer:

(a) 0.8512 (b) 0.8512 (c) 0.7024 (d) 0.0110

Step-by-step explanation:

The blood glucose follows a normal distribution N(μ=85;σ=24).

For every value of X, we can calculate the z-score (equivalent for a N(0;1)) and compute the probability.

(a) P(x>60)

z = (x-μ)/σ = (60-85)/24 = -1.0417

P(x>60) = P(z>-1.0417) = 0.8512

(b) P(x<110)

z = (x-μ)/σ = (110-85)/24 = 1.0417

P(x<110) = P(z<1.0417) = 0.8512

(c) P(60<x<110) = P(x<110)-P(x<60)

P(60<x<110) = P(z<1.0417) - P(z<-1.0417)

P(60<x<110) = 0.8512 - (1-0.8512) = 0.8512 - 0.1488 = 0.7024

(d) P(x>140)

z = (x-μ)/σ = (140-85)/24 = 2.2917

P(x>140) = P(z>2.2917) = 0.0110

Final answer:

Explanation of probabilities for different blood glucose levels using mean and standard deviation.

Explanation:

Probability calculations for blood glucose levels:

(a) x is more than 60: Calculate the z-score using the formula z = (x - μ) / σ. With x = 60, μ = 85, and σ = 24, find the probability using a standard normal distribution table.

(b) x is less than 110: Use the z-score formula with x = 110, μ = 85, and σ = 24 to determine the probability.

(c) x is between 60 and 110: Find the individual probabilities for x = 60 and x = 110, then subtract the two values to get the probability in this range.

(d) x is greater than 140: Similar to the previous steps, find the z-score for x = 140 and calculate the probability.

Answer:

[tex]Area=\frac{7}{12}[/tex]

Step-by-step explanation:

[tex]Area=\int\limits^a_b {f(x)} \, dx =\int\limits^4_3 {\frac{7}{x^{2}} } \, dx =-7*\frac{1}{x}=-7(1/4-1/3)=\frac{7}{12}[/tex]

Answer:

264,666.

Step-by-step explanation:

We have been given that a number has seven digits.

All digits are 6 except the hundred-thousands' digit.

Hundred thousands: 100,000.

The hundred thousand's digit is 2, so its value would be 200,000.

We have been given that thousands's digit is 4, so its value would be:

4 thousands: 4,000

The number with given hundred thousand's and thousands's digit would be 204,000.

Since all digits are 6 except the hundred-thousands' digit, therefore, our number would be 264,666.

Answer:

There was 2800 tons of sand in the mound initially.

Step-by-step explanation:

Let there be m tons of sand on the beach initially.

Throughout the day, 1200 tons of sand is added to the mound. So, total sand becomes = [tex]m+1200[/tex]

Two dump trucks come in and take 800 tons of sand each from the mound.

Means they took [tex]800\times2=1600[/tex] tons of sand

At the end of the day, the mound has 2,400 tons of sand.

This can be modeled as:

[tex]m+1200-1600=2400[/tex]

Solving for m:

[tex]m-400=2400[/tex]

[tex]=> m=2400+400[/tex]

m = 2800

Hence, there was 2800 tons of sand in the mound initially.

The initial amount of sand at the beach was 4000 tons.

To solve this problem, we can set up an equation based on the given information. Let's assume that the initial amount of sand at the beach is m tons. Throughout the day, 1200 tons of sand is added, so the total amount of sand becomes m + 1200 tons. Two dump trucks take 800 tons of sand each, so the amount of sand remaining is (m + 1200) - (2 * 800) tons. At the end of the day, the mound has 2400 tons of sand, so we can set up the equation (m + 1200) - (2 * 800) = 2400.

Simplifying the equation, we have m - 1600 = 2400, which can be further simplified to m = 4000.

Therefore, the initial amount of sand at the beach was 4000 tons.

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Answer:

The diagonals of parallelogram bisect each other.

Step-by-step explanation:

Consider the provided statement.

The parallelogram has few properties.

1: The opposite side of parallelogram are parallel by the definition of parallelogram.

2: The opposite sides and angles of a parallelogram are congruent.

3: The consecutive angles of a parallelogram are supplementary.

4: The diagonals of parallelogram bisect each other.

From the above properties of a parallelogram we can say that:

The diagonals of parallelogram bisect each other.

The diagonals of a parallelogram always bisect each other, dividing each other into two equal parts. However, unless the parallelogram is a rectangle or square, the diagonals themselves are not necessarily of equal length.

In a parallelogram, the diagonals bisect each other, meaning they intersect and divide each other into two equal parts. So, if you have a parallelogram ABCD, its diagonals AC and BD will intersect at a point E, making the two parts of each diagonal (AE and EC, BE and ED) equal in length. In other words, AE equals EC and BE equals ED. However, unless the parallelogram is a special case, like a rectangle or a square, the diagonals themselves are not of equal length. Hence, the diagonals of a parallelogram are always bisecting each other but are not necessarily equal in length.

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Answer:

d. uniform

Step-by-step explanation:

If the data is uniformly distributed i.e. it follows Uniform Probability Distributions then it can be visualized by a histogram. Then the shape of the histogram will be bell-shaped which means as the value of x is increases the value of y also increases for small values of y and it will decrease for a large value of y.

But, other types of probability distributions can't be visualized by a histogram.

Hence option (d) is correct.

Answer:

Lady purchased 12 trinkets costing $0.30 each and 8 trinkets costing $0.65 each.

Step-by-step explanation:

A lady buys total number of trinkets = 20

Cost of each trinket is either $0.30 or $0.65.

Let the number of trinkets is x she purchased for $0.30 and y for $0.65

Then x + y = 20 --------(1)

Since she spends total amount = $8.80

Then the equation will be

0.30x + 0.65y = 8.80 ---------(2)

We replace x = (20 - y) from equation (1) to equation (2)

0.30(20 - y) + 0.65y = 8.80

6 - 0.30y + 0.65y = 8.80

0.35y + 6 = 8.80

0.35y = 8.80 - 6

0.35y = 2.80

y = [tex]\frac{2.80}{0.35}[/tex]

y = 8

Now we put y = 8 in equation (1)

x + 8 = 20

x = 20 - 8

x = 12

Therefore, lady purchased 12 trinkets costing $0.30 each and 8 trinkets costing $0.65 each.

Answer:

120cm = 46.8 inches

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 cm has 0.39 inches. How many inches are there in 120 cm?

1cm - 0.39inches

120cm - x inches

[tex]x = 120*0.39[/tex]

[tex]x = 46.8[/tex] inches

120cm = 46.8 inches

Answer:

Step-by-step explanation:

There are a total of 2+4+3 = 9 "ratio units", so each one is worth ...

  $8,280/9 = $920

Multiplying the ratio by $920, we get ...

  cook : custodian : bus driver = $920 × (2 : 4 : 3) = $1840 : $3680 : $2760

The monthly salaries for the school cook, custodian, and bus driver are $1,840, $3,680, and $2,760, respectively.

To find the monthly salary for each person, we need to first determine the common ratio between their salaries. The given salaries are in the ratio 2:4:3. We can assign a constant to the ratio, such as 2x:4x:3x.

Next, we can set up an equation using the given information that the combined monthly salaries total $8,280:

2x + 4x + 3x = 8,280

9x = 8,280

To solve for x, we can divide both sides of the equation by 9:

x = 920

Now, we can find the monthly salary for each person by substituting x back into the ratio:

School cook's monthly salary: 2x = 2(920) = $1,840

Custodian's monthly salary: 4x = 4(920) = $3,680

Bus driver's monthly salary: 3x = 3(920) = $2,760

Answer:

The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.

Step-by-step explanation:

To find : How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?

Solution :

A 150-W light bulb is on for 20-hours.

The heat produced by bulb is given by,

[tex]H=150\times 20[/tex]

[tex]H=3000\ W-hr[/tex]

We know that,

[tex]1\ \text{W-hr}=3.4\ \text{BTU}[/tex]

Converting W-hr into BTU,

[tex]3000\ \text{W-hr}=3000\times 3.4\ \text{BTU}[/tex]

[tex]3000\ \text{W-hr}=10200\ \text{BTU}[/tex]

Therefore, The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.

Answer:

There is a period of 3 consecutive days in which the website was hits at least 60 times.

Step-by-step explanation:

A web site was hit 300 times over a period of 15 days.

To solve this question we will use the Pigeonhole Principle.

Here, n = 300 and k = 5

We will find [tex]\frac{n}{k}[/tex] to get that there is a hole with at least [tex]\frac{300}{5}=60[/tex] pigeons.

Hence, there is a period of 3 consecutive days in which the website was hits at least 60 times.

Answer:

Mean: 810.51

Standard deviation: 128.32

Step-by-step explanation:

First, we calculate the sample mean. We have 35 data samples, so we compute it as [tex]\displaystyle \frac{562 + .... + 753}{35}[/tex].

In general, given a set of n samples [tex]\{ X_1,...,X_n\}[/tex], we calculate the sample mean as

[tex]\displaystyle \bar{X} = \sum_{i=1}^n \frac{X_i}{n}[/tex].

For the standard deviation [tex]\sigma[/tex], we first begin by calculating it's square. It can be obtained from the formula

[tex]\displaystyle \sigma ^2 = \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{n-1}[/tex]

By taking square root after computing the right hand side, we attain the desired value.

The attached image shows the dot diagram of this sample. The bottom pink vertical line shows the mean, and the two horizontal pink lines have a lenght of [tex]\sigma[/tex].

The standard deviation means "The average expected distance a new sample will be from the mean".  That's why usually, data samples which are denser around the mean have smaller standard deviations (as opposed to distributions who have a lot of values far away from the mean, which will make the standard deviation grow bigger).

Proof :

First, it is important to have in mind that a number [tex] m \in \mathbb{Z} [/tex] is a multiple of [tex]n\in\mathbb{Z} [/tex] iff there exists [tex]k\in\mathbb{Z}[/tex] such that  [tex] m = n \cdot k[/tex].

Also, you have to prove a logical equivalence. To this end, it is possible to prove two logical implications.

Step-by-step explanation:

1.) Let x, y be integers such that x + 3y is a multiple of 7. You have to prove that 3x +2y is a multiple of 7.

In effect, by hypothesis there exists k [tex]\in\mathbb{Z}[/tex] such that  x + 3y = 7 k .   So, you get  

[tex]\begin{equation*} 4(x+3y) + (3x + 2y) = 7x + 14y = 7 (x + 2y) \ \mbox{(direct computations and factoring)}\end{equation*} [/tex].

Therefore, 4(x +3y) + (3x +2y) is a multiple of 7. Then,

[tex](3x + 2y) = 7 (x + 2y) - 4(x + 3y) = 7 (x+2y) - 4 \cdot 7 k = 7 (x + 2y -4k) \ \mbox{(factoring)}[/tex].

Given that x,y,k are integers, then x + 2y - 4k is an integer  and hence, 3x + 2y is a multiple of 7.

To finish, it remains to prove its reciprocal statement.

2.) Let x, y be integers such that 3x + 2y is a multiple of 7. You have to prove that  x +3y  is a multiple of 7. Reasoining as before ,   there exists q [tex]\in\mathbb{Z}[/tex] such that 3x + 2y = 7 \cdot  q.  Thus,

 [tex]$$ \begin{equation*} 2(3x+2y) + (x + 3y) = 7x + 7y = 7 (x + y) \ \mbox{direct computations and factoring} \\\end{equation*} $$[/tex] Thus, [tex] 2(3x +2y) + (x +3y)[/tex] is a multiple of 7.

On the other hand,  using the hypothesis [tex] $$ \begin{equation*}   (x + 3y) = 7 (x + y) - 2(3x + 2y) = 7 (x+y) - 2 \cdot 7 q = 7 (x + y -2q) \ \mbox{(factoring)} \end{equation*} $$    [/tex] .

Finally, thanks that [tex]x,y,q [/tex] are integer numbers, then [tex] x + y - 2q[/tex] is a integer number and therefore, [tex] 3x + 2y [/tex] is a multiple of 7.