Consider the IVP x' (t) = 2t(1 + x(t)), x(0) = 0. 1 (a) Find the first three Picard iterates x, x2, x3 for the above IVP (b) Using induction, or otherwise, show that n(t) = t + t^4/2! + t^6/3! +.... + t^2n/n!. What's the power series solution of the above IVP (ignore the problem of convergence)? 2 marks (c) Find the solution to the above IVP using variable separable technique.

(a) To find the first three Picard iterates for the given initial value problem (IVP) x'(t) = 2t(1 + x(t)), x(0) = 0, we use the iterative scheme:x(t) = 0, andx(t) = [0, t] 2s(1 + x(s)) ds.Using this scheme, we can calculate the following iterates:x(t) = 0,x(t) = [0, t] 2s(1 + x(s)) ds = [0, t] 2s(1 + 0) ds = [0, t] 2s ds = t,x(t) = [0, t] 2s(1 + x(s)) ds = [0, t] 2s(1 + s) ds.To evaluate x(t), we integrate the expression inside the integral:x(t) = [0, t] 2s + 2s ds = [s + 1/2 * s] evaluated from 0 to t = (t + 1/2 * t) - (0 + 0) = t + 1/2 * t.Therefore, the first three Picard iterates for the given IVP are:x(t) = 0,x(t) = t, andx(t) = t + 1/2 * t.(b) To show that n(t) = t + t^4/2! + t^6/3! + .... + t^(2n)/n!, we can use induction. The base case for n = 1 is true since a(t) = t, which matches the first term of the power series. a(t) = a(t) + t^(2k + 2)/(k + 1)! = t + t^4/2! + t^6/3! + .... + t^(2k)/k! + t^(2k + 2)/(k + 1)! = t + t^4/2! + t^6/3! + .... + t^(2k)/k! + t^(2k + 2)/(k + 1)! = t + t^4/2! + t^6/3! + .... + t^(2k)/(k! * (k + 1)/(k + 1)) + t^(2k + 2)/(k + 1)! = t + t^4/2! + t^6/3! + .... + t^(2k + 2)/(k + 1)!(c) To find the solution to the IVP x'(t) = 2t(1 + x(t)), x(0) = 0, using the variable separable technique, we rearrange the equation as:dx/(1 + x) = 2t dt.Now, we can integrate both sides:(1/(1 + x)) dx = 2t dt.Integrating the left side yields:ln|1 + x| = t + CLearn more about the Picard iterates here: brainly.com/question/31535547#SPJ11

Mathematics High School

Consider the IVP
x' (t) = 2t(1 + x(t)), x(0) = 0. 1
(a) Find the first three Picard iterates x₁, x2, x3 for the above IVP
(b) Using induction, or otherwise, show that än(t) = t² + t^4/2! + t^6/3! +.... + t^2n/n!. What's the power series solution of the above IVP (ignore the problem of convergence)? 2 marks
(c) Find the solution to the above IVP using variable separable technique.

Answers

Answer 1

(a) To find the first three Picard iterates for the given initial value problem (IVP) x'(t) = 2t(1 + x(t)), x(0) = 0, we use the iterative scheme:

x₁(t) = 0, and

xₙ₊₁(t) = ∫[0, t] 2s(1 + xₙ(s)) ds.

Using this scheme, we can calculate the following iterates:

x₁(t) = 0,

x₂(t) = ∫[0, t] 2s(1 + x₁(s)) ds = ∫[0, t] 2s(1 + 0) ds = ∫[0, t] 2s ds = t²,

x₃(t) = ∫[0, t] 2s(1 + x₂(s)) ds = ∫[0, t] 2s(1 + s²) ds.

To evaluate x₃(t), we integrate the expression inside the integral:

x₃(t) = ∫[0, t] 2s + 2s³ ds = [s² + 1/2 * s⁴] evaluated from 0 to t = (t² + 1/2 * t⁴) - (0 + 0) = t² + 1/2 * t⁴.

Therefore, the first three Picard iterates for the given IVP are:

x₁(t) = 0,

x₂(t) = t², and

x₃(t) = t² + 1/2 * t⁴.

(b) To show that än(t) = t² + t^4/2! + t^6/3! + .... + t^(2n)/n!, we can use induction. The base case for n = 1 is true since a₁(t) = t², which matches the first term of the power series.

aₖ₊₁(t) = aₖ(t) + t^(2k + 2)/(k + 1)!

= t² + t^4/2! + t^6/3! + .... + t^(2k)/k! + t^(2k + 2)/(k + 1)!

= t² + t^4/2! + t^6/3! + .... + t^(2k)/(k! * (k + 1)/(k + 1)) + t^(2k + 2)/(k + 1)!

= t² + t^4/2! + t^6/3! + .... + t^(2k + 2)/(k + 1)!

(c) To find the solution to the IVP x'(t) = 2t(1 + x(t)), x(0) = 0, using the variable separable technique, we rearrange the equation as:

dx/(1 + x) = 2t dt.

Now, we can integrate both sides:

∫(1/(1 + x)) dx = ∫2t dt.

Integrating the left side yields:

ln|1 + x| = t² + C₁

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Related Questions

A bathtub can hold 66 galons of water. The faucet adds 5.5 gallons of water to the bathtub evvery minute. Find out how many minutes it takes to fill the empty bathtub.

Answers

Answer:

It would take 12 minutes for the bathtub to fill up.

Answer:

66=5.5m

Step-by-step explanation:

two players, a and b, alternatively toss a fair coin (a tosses the coin first, then b tosses the coin, then a , then b .. .). the sequence o f heads and tails is recorded. i f there is a head followed by a tail (ht subsequence), the game ends and the person who tosses the tail wins. what is the probability that a wins the game?

Answers

When two players A and B alternately flip a fair coin, the probability E(T) is 4. (A tosses the coin first, then B tosses the coin, then A, then B and so on).

What is probability?

Probability theory, a subfield of mathematics, gauges the likelihood of an occurrence or a claim being true. An event's probability is a number between 0 and 1, where approximately 0 indicates how unlikely the event is to occur and 1 indicates certainty.

Two players, A and B, turn a fair coin in succession (A tosses the coin first, then B tosses the coin, then A, then B and so on). The order of the heads and tails is noted. If there is a head followed by a tail, the game is done, and the person who throws the tail wins (HT sub-sequence).

We must ascertain the game's estimated duration. Allow T coin tosses for the game to proceed. Find E (T).

$E(T) = first instance of HT in seriesE(T) =0.5×(1+2)+0.5×(1+E(T))\\0.5×E(T)=0.5×4\\E(T) =4$

As a result, when two players A and B alternately flip a fair coin, The E(T) is 4. (A tosses the coin first, then B tosses the coin, then A, then B and so on).

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Which expressions are equivalent to 2(4f+2g)2(4f+2g)2, left parenthesis, 4, f, plus, 2, g, right parenthesis ? Choose 3 answers: Choose 3 answers: (Choice A, Checked) A 8f+2g8f+2g8, f, plus, 2, g (Choice B) B 2f(4 + 2g)2f(4+2g)2, f, left parenthesis, 4, plus, 2, g, right parenthesis (Choice C) C 8f+4g8f+4g8, f, plus, 4, g (Choice D, Checked) D 4(2f+g)4(2f+g)4, left parenthesis, 2, f, plus, g, right parenthesis (Choice E) E 4f+4f+4g4f+4f+4g

Answers

Answer:

8f + 4g

Step-by-step explanation:

Given

2(4f + 2g)

Required

Determine the equivalent expression

2(4f + 2g)

Open the bracket

2 * 4f + 2 * 2g

Perform right operations on the above

8f + 4g

The expression cannot be further simplified; Hence the result of 2(4f + 2g) when simplified is 8f + 4g

Answer: 8f + 4g also 4(2f + g) and 4f + 4f+ 4g

Step-by-step explanation:

Pls Help, I will give 5 star and thanks, Plus Brain to correct answer, Plus extra points if correct!!

The table shows the relationship between the participants walking and running for the week's cross-country practices.

Walk (laps) 3 B 15

Run (laps) 5 10 D

Total (laps) A C 40

At this rate, how many laps will the participants walk if the total distance is 32 miles? How many miles will they run?

They will walk 7 laps and run 17 laps for a total of 32 miles.

They will walk 12 laps and run 20 laps for a total of 32 miles.

They will walk 14 laps and run 18 laps for a total of 32 miles.

They will walk 10 laps and run 22 laps for a total of 32 miles.

Answers

Using proportional relationships, we can say that They will walk 12 laps and run 20 laps for a total of 32 miles.

What is the direct proportional relationship?

In a direct proportional relationship, the output variable is found by the multiplication of the input variable and the constant of proportionality k, as follows:

y = kx.

Given that we know this, they walk 3/8 of the 8 miles that make up the complete distance. Run 5/8 of the route.

The following are the proportional relationships for the distances:

Walked = 3/8 x Total Distance.Ran = 5/8 x Total Distance.

For a total distance of 32 miles, the distances walked and run are given:

Walked: 3/8 x 32 = 3 x 4 = 12 miles = 12 laps.Ran: 5/8 x 32 = 5 x 4 = 20 miles = 20 laps.

therefore, They will walk 12 laps and run 20 laps for a total of 32 miles as per the proportional relation.

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Use the number line to find the indicated
distance.
-9-8-7-6-5
Find JK.
K
-4 -3
-2
01 2 3
M
5 6 7 8 9

Answers

Answer: 5

==================================================

Reason:

There are 5 spaces, or gaps, between points J and K

In other words, start at either J or K. Then move 5 units left or right to arrive at the other point.

-----------

Alternative method:

You can use absolute value and subtraction in two slightly different ways

|J - K| = |-8 - (-3)| = |-8 + 3| = |-5| = 5|K - J| = |-3 - (-8)| = |-3 + 8| = |5| = 5

Absolute value is used in distance problems to avoid a negative result. Negative distances aren't possible.

in a chi-squared test, if the null hypothesis is true, we expect the test statistic to be:

Answers

If the null hypothesis is true in a chi-squared test, then we expect the test statistic to be approximately equal to its expected value.

In a chi-squared test, the null hypothesis is the statement that there is no significant association between two variables. If the null hypothesis is true, then we expect the test statistic to be approximately equal to its expected value. The expected value is calculated using the degrees of freedom and the expected frequency of each category in the contingency table.

The chi-squared test statistic is calculated by subtracting the observed frequency from the expected frequency for each category and then squaring the result. These squared differences are then summed across all categories to calculate the chi-squared test statistic.

If the null hypothesis is true, we expect the test statistic to be close to its expected value. This is because when the null hypothesis is true, the observed frequencies should be close to the expected frequencies. Therefore, the squared differences should be small, resulting in a small test statistic.

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I had 6 cupcakes two days ago and 13 yesterday. Help me continue this pattern

Answers

U had 12 the next day and 18 after that

The next three days of the pattern would be 15 cupcakes, 17 cupcakes, and 19 cupcakes.

What is an arithmetic sequence?

An arithmetic sequence is defined as an arrangement of numbers that is in a particular order.

There were 6 cupcakes two days ago and 13 yesterday.

Adding a fixed number of cupcakes each day: One simple way to continue the pattern is to add a fixed number of cupcakes each day.

For example, if you add 2 cupcakes every day, the next three days of the pattern would be 15 cupcakes, 17 cupcakes, and 19 cupcakes.

Thus, the next three days of the pattern would be 15 cupcakes, 17 cupcakes, and 19 cupcakes.

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A watch is sold for $139 at a loss of $30. What was the original cost of the watch?

Answers

Answer:

$169

Step-by-step explanation:

Answer:

$169

Step-by-step explanation:

3 + 3 = 6

139

030

--------

169

the points L, M, N and O all lie on the same line segment, in that order, such that the ratio of LM : MN : NO is equal to 3 : 5 : 3. if LO = 11, find NO

Answers

Answer:

Step-by-step explanation:

Answer:

LM:MN:

NO =3:

Step-by-step explanation:

Olivia used her past health history and information about her doctor visits to create this table to compare health costs with and without insurance

What is the expected value of each option?

The expected value of healthcare without insurance is:

The expected value of healthcare with insured is:

Answers

Considering a discrete distribution, the expected values for each case are given by:

Without insurance: $437.25.With insurance: $1,636.40.

What is the mean of a discrete distribution?

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

Without insurance, the distribution is given by:

P(X = 1050) = 0.32.P(X = 225) = 0.45.

Hence:

E(X) = 0.32 x 1050 + 0.45 x 225 = $437.25.

With insurance, the distribution is given by:

P(X = 75) = 0.32.P(X = 72) = 0.45.

Adding to the cost of insurance, we have that:

E(X) = 1580 + 75 x 0.32 + 72 x 0.45 = $1,636.40.

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We traveled coast-to-coast, discovering new destinations and picking the top landmark in every state. Which did we choose for Arizona? 1. The Hoover Dam 2. Antelope Canyon 3. Havasupai Falls 4. Monument Valley

Answers

The correct response is 3. Havasupai Falls. We travelled from coast to coast, finding new places to visit and selecting the best landmark in each state. We decided on Havasupai Falls for Arizona.

Arizona's most well-known and current official moniker, "The Grand Canyon State," honours the state's most recognizable natural landmark, the Grand Canyon. Arizona is frequently referred to as the "Copper State," which indicates how abundant this mineral is there. More than 7.279 million people live in the stunning state of Arizona, which is situated in the southwest region of the country. Arizona is a great area to live and is frequently regarded as one of the best places in the nation to start a new life. It is known as The Grand Canyon State. Discover Arizona's snow in Flagstaff. Snow does fall in Flagstaff. plenty of snow! In its yearly assessment based on the state's Gross Domestic Product (GDP), population, and job growth, the American Legislative Exchange Council's most recent Rich States, Poor States research gave Arizona the highest ratings in the nation.

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ok so now that i have no kids left i need to get more and i have two places, at the park there is 14 kids playing but at the school there are 247 kids but i only have 500 bullets left what place should i go to

Answers

Answer: sir what-or ma'am but anyways to answer your question- start at the school so if the kids dodge a bullet- you have back-up

Step-by-step explanation:

Answer:

Bro the question is incomplete, share the complete question.

Step-by-step explanation:

Jack sells cars. Last year he sold 30 cars. The next year Jack sold 48 cars. What is the percent of change?

Answers

Answer:

60% increase

Step-by-step explanation:

Take the new minus the original

48-30 = 18

Divide by the original

18/30

Change to percent form

60% increase

What is the main benefit of this model? A. It helps make predictions about the number of sides in a triangle
B. It shows a pattern for determining the side lengths of a triangle.
C. It shows the shape and parts of a triangle.
D. It helps visualize the features of a triangle.

Answers

The main benefit is B

Answer:b

Step-by-step explanation:

If the temperature on a January day is at 0°F at 7:00 am increases to 14°F by noon and then
decreases by 20°F from noon to 10:00 pm, what is the temperature at 10:00 pm?
a) 17 °F
b) -6 °F
c) 5 °C
d) -10 °F

Answers

Answer:

-6 because if you minus 14 to 20 you will get -6

In a random sample of 100 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3444 with a standard deviation of $2504.
Construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns.
The lower bound is _____. (Round to the nearest dollar asneeded.)
The upper bound is ______. (Round to the nearest dollar asneeded.)
Interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns. Choose the correct answer below.
A. One can be 90% confident that the mean additional tax owed is greater than the upper bound.
B. One can be 90% confident that the mean additional tax owed is less than the lower bound.
C. One can be 90% confident that the mean additional tax owed is between the lower and upper bounds.

Answers

The true mean additional tax owed for estate tax returns is between approximately $3056 and $3832. This means option C is the correct answer: One can be 90% confident that the mean additional tax owed is between the lower and upper bounds.

Based on a random sample of 100 audited estate tax returns, the mean amount of additional tax owed was estimated to be $3444, with a standard deviation of $2504. Using this data, a 90% confidence interval for the mean additional amount of tax owed can be calculated. The lower bound of the confidence interval is approximately $3056, and the upper bound is approximately $3832. Therefore, one can be 90% confident that the true mean additional tax owed for estate tax returns falls between these two values.

To construct the 90% confidence interval, we can use the formula:

Confidence Interval = mean ± (critical value) * (standard deviation / sqrt(sample size))

Since the sample size is large (n = 100), we can assume a normal distribution and use the z-score critical value. The critical value for a 90% confidence interval is 1.645.

Plugging in the values, we have:

Confidence Interval = $3444 ± 1.645 * ($2504 / sqrt(100))

= $3444 ± 1.645 * ($2504 / 10)

= $3444 ± 1.645 * $250.4

= $3444 ± $411.86

Calculating the lower and upper bounds:

Lower bound = $3444 - $411.86 ≈ $3056

Upper bound = $3444 + $411.86 ≈ $3832

Therefore, we can say with 90% confidence that the true mean additional tax owed for estate tax returns is between approximately $3056 and $3832. This means option C is the correct answer: One can be 90% confident that the mean additional tax owed is between the lower and upper bounds.

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The graph below shows the hourly wage of Tina when she was working
summers from high school through college. How much did her salary
increase between 2001 and 2006? Do not include units in your answer.

Answers

Tina's salary increased upto 5 between 2001 and 2006.

What is Graph?

Graph is the pictorial representation of a mathematical relation, function in Cartesian or other mathematical plane.

Here in the given problem there is one graph in Cartesian Plane.

The graph shows the change in hourly wage of Tina with respect to year.

According to the graph,

Tina's hourly wage in 2001 is 5.50; in 2002 is 6.50; in 2003 is 7.00; in 2004 is 7.75; in 2005 is 8.50 and in 2006 is 10.50; in 2007 is 11.50.

Then the change in hourly wage of Tina in between 2001 and 2006 is = 10.50 - 5.50 = 5

Hence the change in hourly wage of Tina in between 2001 and 2006 is 5.

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David hikes 2 1/4 miles in 1/2 an hour. At his rate, how long will it take him to walk 18 miles

Answers

the answer should be 4 hours in 18 miles

I seriously hate my fat finger xd help

Answers

Answer:

They are equal to one another

Step-by-step explanation:

1/2 is equal to .5, so id you have 6.5 on one side and 6.5 on the other, they are equal.

Answer: =

6 $\frac{1}{2}$ = 50/100

=6.5

so 6.5 = 6 $\frac{1}{2}$

a watch cost $120 more than a bag. Janet used 1/3 of her money to buy the bag and was then short of $50 to buy the watch. How much did Janet has at first

Answers

Janet had $450 at first.

Let's assume that the bag cost x dollars, then the watch will cost (x + 120) dollars. Janet used 1/3 of her money to buy the bag, which means she spent (1/3)M = x dollars on the bag. This implies that she had 2/3M dollars left, where M is the amount of money she had at first.

Now, we know that Janet was short of $50 to buy the watch, which means she had (x + 120 - 50) = (x + 70) dollars left after buying the bag. This amount is equal to 2/3*M dollars. So, we can write the equation (2/3)*M = (x + 70).

Substituting x = (1/3)*M in the above equation, we get: (2/3)*M = (1/3)*M + 70. Solving for M, we get M = 450.

Therefore, Janet had $450 at first.

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The polygons are similar but not necessarily drawn to scale.

Answers

If two polygons are similar, then they have the same shape but not necessarily the same size.

They can be either enlarged or reduced in size or reflected. Hence, even though the polygons are similar, they do not have to be drawn to scale.

The term scale refers to the ratio of any two corresponding lengths in two geometric figures, so if two figures are drawn to scale, the ratio of their corresponding lengths in the drawing is the same as the ratio of their corresponding lengths in the actual figures.

Therefore, if two polygons are similar but not drawn to scale, the drawing may not be an accurate representation of the actual figures, as the scaling information may not be consistent.

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[ IS MY ANSWER RIGHT??? ]
[ IF NOT PLEASE PROVIDE DETAILS N THE REAL ANSWER. }

Answers

Yes you did!!!Good job

y-y1 = m(x-x1)

x1,y1 = point = 5,1

m = slope = -3

input the point/value

y-1 = -3 (x-5)

It is known that 2x-3/x = x + 1 What is the value of x^2 -x + 3

Answers

The value of the equation x² - x + 3 is 37/9.

We have,

We can start by multiplying both sides of the equation by x:

2x - 3/x = x + 1

2x - 3 = x^2 + x

Rearranging and simplifying, we get:

x^2 - x + 3 = (2x - 3) + x^2

x^2 - x + 3 = x^2 + 2x - 3

-x + 3 = 2x - 3

5 = 3x

x = 5/3

Now we can substitute x into the equation x^2 - x + 3:

x^2 - x + 3 = (5/3)^2 - 5/3 + 3

x^2 - x + 3 = 25/9 - 15/9 + 27/9

x^2 - x + 3 = 37/9

Therefore,

The value of x² - x + 3 is 37/9.

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Divide. Write your answer in simplest form.
7
-:6
3

Answers

Answer:

7/2

Step-by-step explanation:

rewrite the equation

7 / 6 / 3

7 /2

simplest doesn't mean answer fully just get to the simplest form where there are only absolute numbers

A hashing algorithm is a routine that converts a primary key value into a relative record number. T/F

Answers

True. A hashing algorithm is a mathematical routine used to generate a hash value from a primary key value.

A relative record number that may be used to retrieve the required record in a data structure is created using this hash value.

In order to facilitate quick access to the data, the hashing algorithm makes sure that the same hash value is generated for every primary key. How quickly the requested record can be found depends on how effective the hashing method is.

Also, it must be safe enough to prevent two main key values from producing the same hash value, as doing so might result in accessing the wrong records.

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Find three solutions of the equation. y=-2x-1 a. (–2, 3), (1, –3), (2, –4) c. (1, –3), (0, –1), (–1, 0) b. (–2, 3), (1, –3), (0, –1) d. (0, –1), (3, –9), (–2, 3)

Answers

The three solutions of the equation y = -2x - 1 are (–2, 3), (1, –3), and (2, –4).

To find the solutions, we substitute different values of x into the equation and solve for y.

For the first solution, when x is –2, we have y = -2(-2) - 1 = 3. Therefore, the first solution is (-2, 3).

For the second solution, when x is 1, we have y = -2(1) - 1 = -3. Thus, the second solution is (1, -3).

For the third solution, when x is 2, we have y = -2(2) - 1 = -4. Hence, the third solution is (2, -4).

These three points satisfy the equation y = -2x - 1, and therefore, they are the solutions of the equation. They represent coordinate pairs (x, y) where the equation holds true.

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A bag of marbles contains only yellow and purple marbles. The number of yellow marbles is five times as the purple marbles. The bag contains 138 total marbles. develop a system of equations to represent the situation, determine how many of each color of marble is in the bag
show how you checked your answers

Answers

First create a system of equations:
y = number of yellow marbles
p = number of purple marbles
y = 5p
y + p = 138
Now since we already have one of the variables (y) by itself in the first equation, we can substitute that into the second equation so we only have one variable (p) and it’s easier to solve.
(5p) + p = 138
Combine like terms
6p = 138
Divide both sides by 6
p = 23
Now that we have p, we can substitute it in the first equation so we only have one variable (y) and it’s easier to solve.
y = 5(23)
y = 115
So there’s are 23 purple marbles and 116 yellow marbles.
To check our answers, add together the purple and yellow marble numbers that we got to see if it equals 138 to see if they fit into the equations.
23 + 115 = 138
138 = 138
Also, we can check if there are 5 times as many yellow marbles as purple marbles.
115 = 5 x 23
115 = 115
Since we got a valid equation, the answers are correct.

Answer:

23 purple marbles

115 yellow marbles

Step-by-step explanation:

Let y represent the yellow marble

Let p represent the purple marbles

So, we have the equations

y = 5p

Therefore, y + p = 138

5p + p = 138

6p = 138

p = 23 marbles

Now let's put 23 in for the p in the equation to find the number of yellow marbles

y = 5(23)

y = 115 marbles

Can someone help me please?? It’s due on thursday ❤️ (rotate picture if u can)

Answers

Answer: The two companies never charge the same amount to rent a bicycle for the same number of hours.

How much is a 90 angle?.

Answers

A 90-degree angle, also known as a right angle, is an angle that measures exactly 90 degrees.

A 90-degree angle is an angle that measures exactly 90 degrees. It is also known as a right angle, since it is an angle that forms a right angle.

In geometry, angles are measured in degrees, and a full rotation is equal to 360 degrees. The angle formed by two intersecting lines is measured by the amount of rotation needed to bring one line to overlap the other. A 90-degree angle is formed by rotating one line by exactly 90 degrees, so that the two lines form a right angle.

A right angle is an angle that measures 90 degrees and it is represented by a small square at the angle's vertex. This square is a symbol of the angle's measure. In addition, the right angle is the only angle that can be used in geometry to create perpendicular lines which are lines that meet at 90 degrees.

In real-world applications, right angles are found in many things such as the corners of a room, the corners of a square or rectangle, and the corners of tools like a square, a ruler, and a protractor.

Therefore, In short, a 90-degree angle, also known as a right angle, is an angle that measures exactly 90 degrees. It's represented by a small square at the angle's vertex, and it's the only angle that can be used in geometry to create perpendicular lines.

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2013 people live on an island. Some of these people are truthtellers and the others are liars. The truthtellers always tell the truth whereas the liars always lie. Each day one of the people says 'when i have left the island the number of truthtellers will be the same as the number of liars. Then he leaves the island. After 2013 days there is no longer anybody living on the island. How many liars were living there to begin with?

Answers

There were 671 liars living on the island initially out of 2013 people living on the island.

Let's assume the number of truthtellers is x and the number of liars is y.

According to the given information, each day one person says that when they leave the island, the number of truthtellers will be equal to the number of liars.

This implies that the person speaking must be a liar.

On the first day, if the person speaking is a liar, then the number of liars on the island will increase by one (y+1) and the number of truthtellers will remain the same (x).

The equation can be written as:

x = y + 1

On the second day, if the second person speaking is also a liar, the number of liars will increase by one again (y+2) and the number of truthtellers will remain the same (x).

The equation becomes:

x = y + 2

We can generalize this pattern for each day:

x = y + k

After 2013 days, there is no one left on the island, so x + y = 0.

Substituting this into the equation, we get:

(y + k) + y = 0

Simplifying, we find:

2y + k = 0

Since y represents the number of liars, we want to find a value of y that satisfies this equation.

To do that, we need to find a value of k such that 2013 + k is divisible by 2.

By observing that 2013 is odd, we can conclude that k must be an odd number.

The smallest odd number that satisfies this condition is k = 1.

Substituting k = 1 into the equation, we get:

2y + 1 = 0

Solving for y, we find:

y = -1/2

However, the number of liars cannot be negative, so we can disregard this solution.

Therefore, there were 671 liars living on the island initially.

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