For each of the methods we've learned so far:
(a) integration.
(b) e^rt,
(c) separation of variables,
(d) Laplace transform,
state whether the method works for the given problem. Briefly explain why (it works or fails).

Our assumption that two consecutive elements of the Fibonacci sequence can be divisible by 13 is false, and the proposition is proved.

We begin with the fact that the Fibonacci sequence is an increasing sequence; that is, each element is greater than the one before it.

The sequence is defined as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, and so on.

Now, let us prove the proposition by contradiction. Assume that there are two consecutive elements of the Fibonacci sequence that are divisible by 13. Then there must be some integer n such that:

F(n) = 13m, andF(n + 1) = 13p,

where m and p are integers.

Since the Fibonacci sequence is defined recursively by F(n) = F(n-1) + F(n-2), we can obtain an expression for F(n-2) in terms of F(n) and F(n-1):

F(n-2) = F(n) - F(n-1)Since F(n) and F(n-1) are both divisible by 13, their difference must also be divisible by 13.

Therefore, F(n-2) is divisible by 13 as well. However, this contradicts the well-ordering principle, which states that every non-empty set of positive integers has a least element.

Since the Fibonacci sequence is an increasing sequence, there must be some least element k such that F(k) is divisible by 13 but F(k-1) is not. But this means that F(k-2) cannot be divisible by 13, which contradicts our assumption that both F(n) and F(n-1) are divisible by 13.

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Answer:

6

Step-by-step explanation:

You can memorize these for the smaller shapes

for equilateral polygons:

triangle: 60

square: 90

pentagon: 108

hexagon: 120

Otherwise, I believe the formula is the total degrees in a polygon (s-2)*180 where s is the number of sides.

So the measure of each angle on average would be (s-2)*180/s.

There (s-2)*180/s = 120

180(s-2) = 120s

180s-360 = 120s

60s = 360

s = 6

Rounding to the nearest tenth, the  580000 milligrams is approximately 20.5 ounces using conversion.

To convert from milligrams to ounces, we use the conversion factor 1 ounce = 28349.5 milligrams. This means that there are 28349.5 milligrams in one ounce.

To convert 580000 milligrams to ounces, we multiply the given value by the conversion factor:

580000 milligrams * (1 ounce / 28349.5 milligrams)

The milligram unit cancels out, leaving us with the result in ounces:

580000 / 28349.5 ≈ 20.4597 ounces

Since we are asked to round the answer to the nearest tenth, we round 20.4597 to one decimal place, which gives us 20.5 ounces. This means that 580000 milligrams is approximately equal to 20.5 ounces.

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Answer:

a = 8

Step-by-step explanation:

Given F is proportional to a then the equation relating them is

F = ka ← k is the constant of proportion

To find k use the condition F = 36 when a = 9

36 = 9k ( divide both sides by 9 )

4 = k

F= 4a ← equation of proportion

When F = 32 , then

32 = 4a ( divide both sides by 4 )

8 = a

Answer:

Step-by-step explanation:

E is just that Vertex/Angle.

\(sin\) is a measure of an angle.

You should learn SOHCAHTOA

\(sin = opposite/hypotenuse\\cos(ine) = adjacent/hypotenuse\\tan(gent)=opposite/adjacent\)

The opposite side is relative to the angle.

In this case, the opposite of E is 20

The hypotenuse is 29

so,

\(sin(E)= \frac{20}{29}\)

Answer:

it would be 2/20

Step-by-step explanation:

add all of the menus that she received together for the denominator and put the number of Chinese menus she received for the numerator

The kind of roots does the equation f(x) is option (C) 3 real roots is the correct answer.

A polynomial equation is a sum of constants and variables. A polynomial is an expression consisting of indeterminate's (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

For the given situation,

The equation \(f(x) = x^3 - x^2 - x + 1\)

The roots of the polynomial equation can be found as follows,

Step 1:

Substitute the value of x in which the equation f(x) equals zero.

⇒ Put \(x = 1\), the equation becomes

⇒ \(f(1) = 1^3 - 1^2 - 1 + 1\)

⇒ \(f(1) = 0\)

Thus \((x - 1)\) is the one root of the equation.

Step 2:

The other roots can be found by framing the quadratic equation on dividing the equation \(f(x) = x^3 - x^2 - x + 1\) by \((x - 1)\)

On dividing \(f(x) = x^3 - x^2 - x + 1\) by \((x - 1)\) we get the quotient,

⇒ \((x^{2} -1)\)

Step 3:

Now factorize the quadratic equation, \((x^{2} -1)\)

It can be expand using the identity,

⇒ \((x^{2} -1^{2} ) = (x+1)(x-1)\)            [∵ (a^2 - b^2) = (a+b)(a-b) ]

Thus the factors of the equation are \((x-1)(x+1)(x-1)\). So the roots are \(1,-1,1\).

Hence we can conclude that the kind of roots does the equation f(x) is option (C) 3 real roots is the correct answer.

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The total interest income reported over the life of the bond investment would be $7,500.

To calculate the total interest income, we need to determine the annual interest payment and multiply it by the number of years until maturity. In this case, the bond has a par value of $50,000 and pays interest at a 3% annual rate. The annual interest payment can be calculated by multiplying the par value by the interest rate, which gives us $50,000 * 0.03 = $1,500.

Since the bond has a maturity period of 5 years, we can multiply the annual interest payment by the number of years to obtain the total interest income. Therefore, the total interest income is $1,500 * 5 = $7,500.

Adelphi Company purchased the bonds at 101, which means they bought them at a premium. The straight-line amortization method assumes that the premium or discount is amortized evenly over the life of the bond. However, since the premium was not mentioned explicitly in the question, we can assume that it does not affect the calculation of the total interest income.

In conclusion, over the life of the bond investment, Adelphi Company will report a total interest income of $7,500.

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Answer:

3-6x>3-6x

Step-by-step explanation:

12x>0

x>12/0=0

This years tuition is $1900

Let us assume that the next year tuition fee be x dollars.

The next year's tuition will be $2000 which is $600 more than this year's tuition fee.

Chana will pay $2000 which is 80 percent of her college tuition.

Using percentage formula we get an equation,

x × (80/100) = 2000

we solve this equation for x.

x = (2000 × 100)/80

x = 2500

If the next year's tuition fee is $600 more than this year's tuition fee, then this year's tuition fee would be,

2500 - 600 = 1900 dollars

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Usually there are 52 Sundays in a year approximately.

There are a total of 52 weeks in a year if we take a rough estimate.

Every week has one sunday in it. So, going by this logic, each year will have about 52 sundays in it.

Now, one thing that we should notice here is that is not certain that there will be definitely 52 sundays. This number of 52 can increase also by a number of one or two because it is possible that the year might not starting from sunday. Because when we do 365/7 we get 52.14 so, this decimal place might include one or two Sundays in it.

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Answer:

464 feet

Step-by-step explanation:

The Sine of the angle of inclination gives the increase in altitude.

This is because the Sine of an angle is equal to the Opposite side divided by the Hypotenuse side. i.e

Sin Ф = Opposite  ÷ Hypotenuse

Sin 5° = O ÷ 5328

0·08716 = O ÷ 5328

Make O the subject and multiply every fraction by 5328

(0·08716/1 × 5328)= (O/5328 × 5328)

464·388 = O

∴ The increase in altitude is 464·388 feet

This is supposed to be to the nearest foot so we round off

464·388 becomes 464 feet

Answer:

Hi! In order to determine how much money Aiden will have in his account after 8 months, let's figure out how much he deposits each month.

He started with $250. After one month, he had $586; after two months, he had $922; after three months, he had $1,258. Each month, there is a gain of $336.

Therefore, after four months, Aiden will have $1,594. After five months, he will have $1,930. After six months, he will have $2,266. After seven months, he will have $2,602. Finally, after eight months, he will have $2,938.

Hope this helps!

Answer:

33

Step-by-step explanation:

because a triangle measure 90 degrees so you just subtract the amount that you add and got

Answer: 11 square root of 2

The length of each side of the square is (3x-2) units

The dimensions of the rectangle are (5x + 4y) and (5x -4y) units

From the question, we are to factor out the given expression for the area of the square

The expression is

9x² − 12x + 4

Factoring

9x² − 12x + 4

9x² −6x -6x + 4

3x(3x -2) -2(3x-2)

(3x -2)(3x -2)

Hence, the length of each side of the square is (3x-2) units

Part B:

The given expression is

25x² − 16y²

Factoring

25x² − 16y²

By difference of two squares, we get

(5x + 4y)(5x -4y)

Hence, the dimensions of the rectangle are (5x + 4y) and (5x -4y) units

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Answer:

3 and 5.

Step-by-step explanation:

Let the 2 numbers be x and x + 2, then:

3(x + 2)^2 = 10x + 45

3x^2 + 12x + 12 - 10x - 45 = 0

3x^2 + 2x - 33 = 0

3x^2 - 9x + 11x - 33 = 0

3x(x - 3) + 11(x - 3) = 0

(3x + 11)(x - 3) = 0

x = 3, x = -11/3

We are only interested in the 3 as we are looking for an integer.

So, the answer is 3 and 3+2 = 5.

Answer:

c = 99 and b = 81

The statement "If g(x) > f(x), and if ∫g(x) dx is divergent, then ∫f(x) dx is also divergent" is false.

The divergence or convergence of an integral depends on the behavior of the function being integrated, not the relationship between two different functions.

The given statement suggests that if g(x) is greater than f(x) and the integral of g(x) diverges, then the integral of f(x) must also diverge. However, this is not necessarily true. The divergence or convergence of an integral depends on the properties of the function being integrated.

Consider a scenario where g(x) and f(x) are both positive functions. If ∫g(x) dx diverges, it means that the integral does not have a finite value. However, f(x) could still have a finite integral if it is bounded or has certain properties that lead to convergence. Therefore, the divergence of ∫g(x) dx does not imply the divergence of ∫f(x) dx.

In conclusion, the relationship between two functions and the divergence or convergence of their integrals are not directly connected, so the statement is false.

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The precentile for a student in the class who studies 8 hours a week is 5.74%.

To find the percentile for a student who studies 8 hours a week, we need to first standardize the student's study time by subtracting the mean study time and dividing it by the standard deviation of study time. This will give us the student's z-score. We can then look up the student's z-score in a standard normal table to find the percentile.

The student's z-score is (8 - 9.92) / 4.54 = -1.67.

Looking up -1.67 in a standard normal table, we find that the percentile is about 5.74%. This means that about 5.74% of the students in the class studied for less than 8 hours per week.

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As per the given standard deviation, the percentile for a student in the class who studies 8 hours a week is 50.80%

The term standard deviation in math is known as a measure which shows how much variation from the mean exists.

Here we have given that in one statistics course students reported studying an average of 9.92 hours a week with a standard deviation of 4.54.

Through this details, we have identified that the value of the following are given that,

=> weekly study time = 10 hours

=> average study time = 9.92 hours

=> standard deviation = 4.54 hours

Then the formula to calculate the Test statistic is written as

=> z = (weekly study time - average study time)/standard deviation

Apply the values on it, then we get

=> z = (10 - 9.92)/4.54

=> z = 0.08/4.54 = 0.02

While we take the value of z table, then we get the Cumulative area of the test statistic is 0.5080.

And the percentile is calculated by multiply the value by 100 as,

=> 0.5080 × 100 = 50.80

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Answer:

f(x) = -1/x^2 + 4e^x + 5x + 1 + 1/x - 4e^(-1)

Step-by-step explanation:

We can find the function f(x) by integrating f'(x) with respect to x:

â«f'(x) dx = â«(2/(x^3) + 4e^x + 5) dx

f(x) = -1/x^2 + 4e^x + 5x + C

To find the constant C, we can use the given initial conditions:

f(-1) = 1 = -1/(-1)^2 + 4e^(-1) - 5 + C

C = 1 + 1/1 - 4e^(-1)

f(x) = -1/x^2 + 4e^x + 5x + 1 + 1/x - 4e^(-1)

Therefore, the function f(x) is:

f(x) = -1/x^2 + 4e^x + 5x + 1 + 1/x - 4e^(-1)

A correlation coefficient is consistently a number between 1 and - 1 .

The correlation coefficient is a factual measure that demonstrates the strength and heading of the straight connection between two factors. It goes from - 1 to 1, where - 1 addresses a completely bad correlation (i.e., as one variable expands, different declines), 0 addresses no correlation, and 1 addresses an entirely sure correlation (i.e., as one variable builds, the other likewise increments).

The indication of the correlation coefficient demonstrates the heading of the relationship, while the extent of the coefficient shows the strength of the relationship. A correlation coefficient of 0 shows no straight connection between the factors, while values nearer to - 1 or 1 demonstrate a more grounded direct relationship.

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Answer:

Step-by-step explanation:

Since ∆ ABC is a triangle the sum of it's angles add up to 180°

In order to find x we must first find \(\angle\) AED

Let \(\angle\) AED = E

To find \(\angle\) AED add up all the angles in the triangle and equate it to 180°

That's

35 + 90 + E = 180

125 + E = 180

E = 180 - 125

E = 55°

E and x lie on a straight line

So we have

x + 55 = 180°

x = 180 - 55

Hope this helps you

There are 54,912 different combinations for a three of a kind hand in poker.

To calculate the number of combinations for a three of a kind hand in poker, we need to consider the following:

Choose the rank of the three-of-a-kind: There are 13 ranks in a standard deck of cards, so we have 13 options.

Choose the three cards of the chosen rank: There are 4 cards of each rank in a deck, so we can choose 3 cards from the 4 available. This can be calculated as a combination, denoted as "4 choose 3," which is equal to C(4, 3) = 4.

Choose the ranks for the remaining two cards: We have 12 remaining ranks to choose from, excluding the rank of the three-of-a-kind. We need two different ranks, so this can be calculated as a combination, denoted as "12 choose 2," which is equal to C(12, 2) = 66.

Choose the two cards of the chosen ranks: For each of the chosen ranks, there are 4 cards available. So, we can choose 1 card from each rank, resulting in 4 × 4 = 16 possible combinations.

To find the total number of combinations, we multiply the individual choices:

Total combinations = 13 × 4 × 66 × 16 = 54,912 combinations.

Therefore, there are 54,912 different combinations for a three of a kind hand in poker.

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Answer:

-4

Step-by-step explanation:

Answer:

-4

Step-by-step explanation:

g(x) tells us what the variable is and what variable should be replaced with a value.

g(1) tells us the value that will replace x.

Plug in 1 to the equation:

-5(1) + 1 = g(1)    (multiply)

-5 + 1 = g(1)       (combine like terms)

-4 = g(1)

Therefore, g(1) is equal to -4

Step-by-step explanation:

I know that both sides of the cats, males and females, receive the same # of new cats.

I write an equation, (not an inequality though):

3((34+2x)/5)=2((13+2x)/5)

From here on, I solve for x. I get x = -38. I know we can't have negative pairs and we're only bringing in more cats, and not taking any away. I add 38, and try to see what percentage of cats are male, and I get fairly close, getting to about 62.5% I think. Then, I divide 38 by 2, getting 19, and see what else I can get. I try to find a value between the 38 and lesser 19 so as to find a value which will give me a male cat percentage of 60%. I get about 57.5% with 19 plugged in, which is when I see that hey, the differences are the same. What I then do is find the average of 19 + 38, which is 57.5. I round that to 58, and I get the right answer.

I see that perhaps my solution wasn't really orthodox or anything, but with a little bit of trial and error and a bit of my creative thinking, I guess I managed to solve the problem. What worries me though is this is the first problem in the "questions" section of the inequalities section of AoPS, so I'm definitely struggling to deal with the concepts.

Answer:

part 1 is $27.72

part 2 is $33.88

Step-by-step explanation:

part 1

8% + 25%= 33%

                     -----

                       100

33x84 divided by 100= $27.72

part 2

11x56 divided by 100x5.50=33.88

hope this helps!

The total quantity produced when p = 5 is 26 in the long run.

In the short run, each firm will produce where its marginal cost equals the market price, which we'll denote as p.

So, for Type A firms, the short-run supply curve is:

q4(p) = (p-2)/2

And for Type B firms, the short-run supply curve is:

qB(p) = (p-2)/6

The short-run market supply is simply the sum of the quantities supplied by each type of firm at a given price level. So, if there are 10 Type A firms and 6 Type B firms, the short-run market supply at a price level of p is:

Qs(p) = 10q4(p) + 6qB(p)

= 10(p-2)/2 + 6(p-2)/6

= 8p - 20

The total quantity produced when p=5, we can substitute p=5 into the market supply equation we just derived:

Qs(5) = 8(5) - 20

= 20

So, the total quantity produced when p=5 is 20.

The long run, firms can enter or exit the market, and as a result, the number of firms in each market may change.

In this case, we'll assume that the number of Type A and Type B firms can adjust freely in the long run, and that each firm earns zero economic profit in the long run.

If there are zero economic profits, each firm's total revenue (TR) must equal its total cost (TC), which means that price must equal average total cost (ATC).

For Type A firms, ATC4(q) = TC4(q)/q = 1/q + 2 + q, so:

5 = 1/q + 2 + q

Rearranging and solving for q, we get:

q4 = 2

So each Type A firm will produce 2 units of output in the long run.

For Type B firms, ATCB(q) = TCB(q)/q = 6/q + 2 + 3q, so:

5 = 6/q + 2 + 3q

Rearranging and solving for q, we get:

qB = 1

So each Type B firm will produce 1 unit of output in the long run.

If there are 10 Type A firms and 6 Type B firms, the total quantity produced in the long run when p=5 is:

Qlr = 10q4 + 6qB

= 10(2) + 6(1)

= 26

So, the total quantity produced when p = 5 is 26 in the long run.

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Answer:

i think  

3 ,  

12

12 ,  

48

48 ,  

192

192  ,  

768

768

Answer:

A: Isosceles B: Scalene C: Isosceles, Equilateral D: Scalene

Step-by-step explanation:

A: 2 angles are equal so the sides are (Isosceles)

B: None of the sides are equal (Scalene)

C: All the sides are equal (Isosceles & Equilateral)

D: None of the angles are equal, so none of the sides are (Scalene)