The number of online buyers in Western Europe is expected to grow steadily in the coming years. The function below for 1 Sr59, gives the estimated buyers as a percent of the total population, where tis measured in years, with t1 corresponding to 2001. Pt) 27.4 14.5 In(t) (a) What was the percent of online buyers in 2001 (t-1)? % How fast was it changing in 2001? /yr (b) What is the percent of online buyers expected to be in 2003 (t-3)? % How fast is it expected to be changing in 2003? %/yr

The Integral test, which is also known as Cauchy's criterion, is a method that determines the convergence of an infinite series by comparing it with a related definite integral.

In a series, the terms can either be decreasing or increasing. When the terms are decreasing, the Integral test is used to determine convergence, whereas when the terms are increasing, the Integral test can be used to determine divergence. For example, consider the series\[S = \sum\limits_{n = 1}^\infty {\frac{{\ln (n + 1)}}{{\sqrt n }}} \]. Now, we'll apply the Integral test to determine the convergence of the above series. We first represent the series in the integral form, which is given as\[f(x) = \frac{{\ln (x + 1)}}{{\sqrt x }},\] and it's integral from 1 to infinity is given as \[I = \int\limits_1^\infty {\frac{{\ln (x + 1)}}{{\sqrt x }}} dx\]. Next, we'll find the integral of f(x), which is given as \[I = \int\limits_1^\infty {\frac{{\ln (x + 1)}}{{\sqrt x }}} dx\]\[u = \ln (x + 1),\] so, the equation can be rewritten as \[I = \int\limits_0^\infty {u^2 e^{ - 2u} du}\]\[I = \frac{1}{{\sqrt 2 }}\int\limits_0^\infty {{y^2}e^{ - y} dy}\]\[I = \frac{1}{{\sqrt 2 }}\Gamma (3)\]. The given series [infinity] 3 cos(n) n n = 1 is a converging series because the Integral test is applied to determine its convergence.

The Integral test helps to determine the convergence of a series by comparing it with a related definite integral. The Integral test is only applicable when the terms of the series are decreasing. If the series fails the Integral test, then it's necessary to use other tests to determine the convergence or divergence of the series. The Integral test is a simple method for determining the convergence of an infinite series. Therefore, the series [infinity] 3 cos(n) n n = 1 is a converging series. The Integral test is applied to determine the convergence of the series and it is only applicable when the terms of the series are decreasing.

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Answer:

The answer would be 4,450

Step-by-step explanation:

:)

By mathematical induction, we have proved that the recursive algorithm in part (a) correctly computes the sum of the first n positive integers.

(a) Here is a recursive algorithm that computes sn = 1 + 2 + 3 + ... + n using the formula s1 = 1, sn = sn-1 + n for all n ≥ 2:

Algorithm Sum(n):

Input: A positive integer n.

Output: The sum of the first n positive integers.

If n = 1, return 1.

Otherwise, return Sum(n-1) + n.

(b) To prove that the algorithm in part (a) is correct, we will use mathematical induction.

Base case: When n = 1, the algorithm returns 1, which is indeed the sum of the first 1 positive integer. So the algorithm is correct for the base case.

Induction hypothesis: Assume that the algorithm is correct for some positive integer k, i.e., Sum(k) = 1 + 2 + 3 + ... + k.

Induction step: We will show that the algorithm is also correct for k+1. By the recursive formula, we have

Sum(k+1) = Sum(k) + (k+1)

By the induction hypothesis, we know that Sum(k) = 1 + 2 + 3 + ... + k. So we can substitute it in the above equation to get

Sum(k+1) = (1 + 2 + 3 + ... + k) + (k+1)

Simplifying the right side gives

Sum(k+1) = 1 + 2 + 3 + ... + (k+1)

which shows that the algorithm is correct for k+1 as well.

Therefore, by mathematical induction, we have proved that the recursive algorithm in part (a) correctly computes the sum of the first n positive integers.

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Answer:

17, 34, 51, 68, 85, 102, 119, 136, 153, 170

hope this helps

Answer:

The multiples of 17 are: 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, 578, 595, 612, 629, 646, 663, 680, 697, 714, 731, 748, 765, 782, 799, 816, 833, 850, 867, 884, 901, 918, 935, 952, 969, 986.

Step-by-step explanation:

Easiest way to solve this is to probably grab a piece of paper and just add 17 until you find whatever you’re looking for!

The probability of selecting a defective roofing sheet from the finished products of ABC is 2.45%. If a randomly selected defective sheet is found, the probability that it was made by machine B3 is approximately 54.55%.

To determine the probability of selecting a defective roofing sheet from the finished products of ABC, we need to calculate the weighted average of the defect rates for each machine.

The weighted average defect rate is calculated by multiplying the defect rate of each machine by its corresponding proportion of production and summing the products.

Weighted Average Defect Rate = (0.02 * 0.30) + (0.03 * 0.45) + (0.02 * 0.25)

                            = 0.006 + 0.0135 + 0.005

                            = 0.0245

Therefore, the probability of selecting a defective roofing sheet from the finished products of ABC is 2.45%.

To calculate the probability that a randomly selected defective sheet was made by machine B3, we need to consider the proportion of defective sheets contributed by B3 out of the total defective sheets.

The proportion of defective sheets from B3 = (0.02 * 0.30) / 0.0245

                                            = 0.006 / 0.0245

                                            ≈ 0.245

The probability that a defective sheet was made by machine B3 can be obtained by dividing the proportion of defective sheets from B3 by the overall probability of selecting a defective sheet.

Probability of sheet made by B3 = (0.245 / 0.0245) * 100

                              ≈ 54.55%

Therefore, if a randomly selected defective roofing sheet is found, the probability that it was made by machine B3 is approximately 54.55%.

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The three angles of a triangle equal 180

Third angle = 180 - 31.8 - 89.8 = 58.4

Answer: 58.4

If a pack of 50 pencils cost $5.45 then a single pencil costs $0.109.  

Ratio basically compares quantities, that means it shows the value of one quantity with respect to the other quantity.

If a and b are two values, their ratio will be a:b,

Given that,

Cost of a pack of 50 pencils = $5.45.

To find the cost of one pencil,

Use ratio property,

50 pencil costs = $5.45

1 pencil costs = 5.45 / 50 =  $0.109

Cost of one pencil is $0.109.  

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Answer:

\(\frac{6}{5}\)

Step-by-step explanation:

Note that the denominator (bottom number) are the same when you are adding. Simply add the numerator (top number) together:

\(\frac{2}{5} + \frac{4}{5} = \frac{2 + 4}{5} = \frac{6}{5}\)

As it asks for an improper fraction. Simply leave it as it is.

\(\frac{6}{5}\) is your answer.

~

In summary: Sequence 1 has no term out of increasing order. Sequence 2 has the first term out of increasing order at position 2. Sequence 3 has the first term out of increasing order at position 7. Sequence 4 has the first term out of increasing order at position 4. Sequence 5 has the first term out of increasing order at position 3.

1 5 78 99 101 202 400: This sequence is in increasing order throughout, so there is no term that breaks the increasing pattern.

5 2 7 90 85 80 72: The sequence starts with 5, then decreases to 2 (out of increasing order) at position 2.

5 6 9 10 14 21 20: The sequence is increasing until position 6, where it reaches 21. However, the next term, 20, is lower than the previous term, 21 (out of increasing order) at position 7.

3 7 10 9 8 14 17: The sequence starts with 3 and increases until position 3 (10). However, at position 4, the next term, 9, is lower than the previous term, 10 (out of increasing order).

5 77 25 45 22 94 58 99: The sequence is increasing until position 2, where it reaches 77. However, at position 3, the next term, 25, is lower than the previous term, 77 (out of increasing order).

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Answer:

23.7

Step-by-step explanation:

Answer:

79

Step-by-step explanation:

90-11=79

Answer:

79

Step-by-step explanation:

To find the range, subtract the biggest number and the smallest number. In this case, it would be 90-11 which is 79.

Answer: \(4760\frac{39}{50}\)

Step-by-step explanation:

The quotient of the division  42,847 / 9 in mixed numbers is 4760 39/50.

The number system is a way to represent or express numbers.

A decimal number is a very common number that we use frequently.

Since the decimal number system employs ten digits from 0 to 9, it has a base of 10.

Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.

Given the division,

42847 / 9

42847 / 9 = 4760.777778

4760.777778 ⇒ 4760.78 (rounded to the nearest tenth).

4760.78  = 4760 + 0.78

⇒ 4760 + 78/100

⇒  4760 + 39/50

⇒  4760  39/50

Hence "The quotient of the division  42,847 / 9 in mixed numbers is 4760 39/50".

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Answer:

Một trục số đi từ âm 6 đến dương 6. Các điểm ở âm 6, âm 2, 5.

Step-by-step explanation:

Answer:

d

Step-by-step explanation:

The required function is

y = x(x + 3)(x + 1)(x - 6)

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here,

The function has zeroes at x = -3, -1, 0 and 6

The required function is

y = (x - (-3))(x - (-1))(x - 0)(x - 6)

y = x(x + 3)(x + 1)(x - 6)

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Answer:

F(3)=23

Step-by-step explanation:

F(3)= 7(3+1)-5

F(3)=7(4)-5

F(3)=28-5

F(3)= 23

you always want to do the problem in the parenthesis first. and you replace a with 3

Answer:  intersection of the lines drawn from each vertex of the triangle and perpendicular to its opposite side D. intersection of the lines drawn to the midpoint of each side of the triangle to its opposite vertex.

Step-by-step explanation:

Answer:

67°.

Step-by-step explanation:

1) m∠PMR=m∠LMN=3x+19°;

2) m∠LMN+m∠LMP=180°, it can be written as 3x+19+9x-31=180;

3) if to solve the equation 3x+19+9x-31=180, then x=16;

4) m∠PMR=3x+19=48+19=67°.

The equation used in writing a partial fraction decomposition is given as:

Negative 15 x + 10 / (5 x - 2) = a / (5 x - 2) + b / (5 x - 2).

To determine the values of a and b, we can use a system of equations. This system of equations consists of two equations, one for a and one for b. The equation for a is: Negative 15 x + 10 = a * (5 x - 2). The equation for b is: 0 = b * (5 x - 2). We can solve these equations simultaneously to get the values of a and b.

For example, if x = 1, then the equation for a becomes: Negative 15 + 10 = a * (5 - 2), which simplifies to -5 = a * 9. Therefore, a = -5/9. The equation for b becomes: 0 = b * (5 - 2), which simplifies to 0 = b * 3. Therefore, b = 0.

In summary, the system of equations used to determine the values of a and b is:

Negative 15 x + 10 = a * (5 x - 2) and 0 = b * (5 x - 2). By solving these equations simultaneously, we can get the values of a and b.

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Answer:

B!!!

Right on Edge

Step-by-step explanation:

By finding two consecutive square numbers such that our number is between them we can find a good estimation of the square root of our number.

Let's say that we want to find the square root of a number x such that x is not a perfect square.

To get an estimation, we just need to find two consecutive square numbers such that one is smaller than x and the other is larger than x, and the square root of x will be bounded by the square roots of these square numbers.

Let's suppose that x = 21

The square number smaller than 21 is 16, which is 4^2

The square number larger than 21 is 25, which is 5^2

Then we can write:

√16 < √21 < √25

if we simplify this, we get:

4 < √21 < 5

So we have a estimation of the square root of 21.

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By taking the quotient between the total volume of paint and the volume needed to fill a container we can see that 16 containers can be filled.

We know that the art teacher has 12 + 4/5 gallons of paint, and we know that it requires 4/5 gallons to fill a single container.

Then the number of containers that can be filled is equal to the quotient between the total volume of paint and the volume needed fill a single container, it gives:

(12 + 4/5)/(4/5) = 16

So 16 containers can be totally filled.

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In this scenario, a player can place a bet of $4.00 on the outcome of rolling two dice. If the sum of the roll is 7, the player wins $1600, otherwise, they lose $4.00.

To analyze the possible outcomes of rolling two dice, we need to consider all the combinations of numbers that can appear on each die. Each die has six sides, numbered from 1 to 6. When rolling two dice, we can have a total of 36 different outcomes (6 possibilities for the first die multiplied by 6 possibilities for the second die).

Now, we need to determine the sum of each pair of numbers. For example, if the first die shows a 1 and the second die shows a 6, the sum is 7. We can list all the possible outcomes and their corresponding sums:

(1, 1) - Sum: 2

(1, 2) - Sum: 3

(1, 3) - Sum: 4

(1, 4) - Sum: 5

(1, 5) - Sum: 6

(1, 6) - Sum: 7

(2, 1) - Sum: 3

(2, 2) - Sum: 4

(6, 6) - Sum: 12

Out of these 36 outcomes, there are six combinations that result in a sum of 7. These combinations are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). If the sum is 7, the player wins $1600, and if the sum is any other number, the player loses $4.00.

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The simplified product of \(2\sqrt{5x^{3} }\) and -3\(\sqrt{10x^{2} }\) is -30\(x^{5/2} \sqrt{2}\).

Given Two expressions: -3\(\sqrt{10x^{2} }\) and 2\(\sqrt{5x^{3} }\).

We have to multiply both the expressions and it can be done as under:

-3\(\sqrt{10x^{2} }\) *2\(\sqrt{5x^{3} }\)

Firstly we have to multiply -3 with 2 to get

=-6\(\sqrt{10x^{2} }\sqrt{5x^{3} }\)

Then we have to find square root of x cube and x square which is x to the power 3/2 and x to the power 1.

=\(-6x^{3/2} x\sqrt{10}\sqrt{5}\)

Now we have to multiply both the numbers in the root to get the answer;

=-6\(x^{5/2} \sqrt{50}\)

Square root of 50 is 5 root 2.

=-6*5\(\sqrt{2}\)\(x^{5/2}\)

=-30\(\sqrt{2}\)\(x^{5/2}\)

Hence the simplified product is -30\(\sqrt{2}\)\(x^{5/2}\).

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Answer:

No, it is not possible

Step-by-step explanation:

Required

Is it possible to have a line segment with: \(m = \infty\)

To answer this question, we will interpret the steepness as slope.

So, we have:

\(m = \infty\)

The interpretation of a line with: \(m = \infty\) is:

It starts from (x,y_1) and ends at (x,y_2)

When the slope (m) is then calculated, we have:

\(m = \frac{y_2 - y_1}{x_1 - x_1}\)

\(m = \frac{y_2 - y_1}{0}\)

\(m = \infty\)

What this means is that, the line has no end points.

A line segment as described here means a line that has endpoints (i.e. a finite starting point and a finite finish point)

So, we've established that: \(m = \infty\) means no endpoints and a line segment has end points, then we can conclude that it is not possible to create a line segment with \(m = \infty\)

Steepness of a  line is the rise or fall of it at a sharp angle.The Is it not possible to create a line segment with infinite steepness, as the slope of the such line will be infinite whether a line is plotted with finite points.

Steepness of a  line is the rise or fall of it at a sharp angle. The slope of the line describe the steepness of a line.

A infinite steepness refers to a vertical line on the graph. When the line with infinite steepness plotted on the graph it is only parallel to the y axis. The change on the x axis is zero and does not move in the x axis, when a line has infinite steepness.

To understand it better suppose a line is passed through the point \((x_1,y_1)\) and \((x_2,y_2)\) points. For such line the slope of the line can be given as,

\(m=\dfrac{y_2-y_1}{x_2-x_1} \)

The line with infinite steepness does not move in the x axis. The coordinate of the x axis same for the line. Thus,

\(x_2=x_1\)

The slope for such line is,

\(m=\dfrac{y_2-y_1}{x_1-x_1} \)

\(m=\dfrac{y_2-y_1}{0} \)

\(m=\infty\)

Thus the slope of the line is infinite. But the line With no finite point is not possible to plot.

Hence the Is it not possible to create a line segment with "infinite steepness, as the slope of the such line will be infinite whether a line is plotted with finite points.

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Exact value of the definite integral is 320. Comparing the results: Exact value of the definite integral = 320, Trapezoidal Rule approximation (n = 4) = 340, Simpson's Rule approximation (n = 4) ≈ 246.6667.

What is trapezoid?

A trapezoid is a quadrilateral (a polygon with four sides) that has one pair of parallel sides. The parallel sides are called the bases of the trapezoid, while the non-parallel sides are called the legs.

To approximate the value of the definite integral ∫[0, 4] 5x * x^2 dx using the Trapezoidal Rule and Simpson's Rule, we need to specify the value of n, which represents the number of subintervals.

Let's calculate the approximations using n = 4 for both methods:

Trapezoidal Rule:

Using n = 4, we divide the interval [0, 4] into four subintervals of equal width: h = (4 - 0) / 4 = 1.

The approximated value using the Trapezoidal Rule is given by:

\(T_4 = (h/2) * [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)]\)

Plugging in the values:

\(T_4 = (1/2) * [f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)]\\\\= (1/2) * [5(0)(0^2) + 2(5)(1)(1^2) + 2(5)(2)(2^2) + 2(5)(3)(3^2) + 5(4)(4^2)]\\\\= (1/2) * [0 + 10 + 80 + 270 + 320]\\\\= (1/2) * 680\\\\= 340\)

Simpson's Rule:

Using n = 4, we divide the interval [0, 4] into four subintervals of equal width: h = (4 - 0) / 4 = 1.

The approximated value using Simpson's Rule is given by:

\(S_4 = (h/3) * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\)

Plugging in the values:

\(S_4 = (1/3) * [f(0) + 4f(1) + 2f(2) + 4f(3) + f(4)]\\\\= (1/3) * [5(0)(0^2) + 4(5)(1)(1^2) + 2(5)(2)(2^2) + 4(5)(3)(3^2) + 5(4)(4^2)]\\\\= (1/3) * [0 + 20 + 40 + 360 + 320]\\\\= (1/3) * 740\\\\= 246.6667\)

Exact value of the definite integral:

∫[0, 4] 5x * \(x^2\) dx = [(5/4) * \(x^4\)] evaluated from 0 to 4

\(= (5/4) * 4^4 - (5/4) * 0^4\\\\= (5/4) * 256 - (5/4) * 0\\\\= 320 - 0\\\\= 320\)

Comparing the results:

Exact value of the definite integral = 320

Trapezoidal Rule approximation (n = 4) = 340

Simpson's Rule approximation (n = 4) ≈ 246.6667

As we can see, the Trapezoidal Rule approximation is slightly greater than the exact value, while Simpson's Rule approximation is less than the exact value.

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Answer:

its the first one

Step-by-step explanation:

To find the area of the region interior to r = 9 + 7sin(θ) below the polar axis, we can integrate the function from the lower bound of θ to the upper bound of θ and take the absolute value of the integral.

To find the area of the region formed by two petals of r = 8sin(3θ), we can integrate the function over the appropriate range of θ and take the absolute value of the integral. To find dy/dx for the given parametric equations x = t^(1/3) and y = 3 - t, we differentiate y with respect to t and x with respect to t and then divide dy/dt by dx/dt.

To find the area of the region interior to r = 9 + 7sin(θ) below the polar axis, we need to evaluate the integral ∫[lower bound to upper bound] |1/2 * r^2 * dθ|. In this case, the lower bound and upper bound of θ will depend on the range of values where the function is below the polar axis. By integrating the expression, we can find the area of the region. To find the area of the region formed by two petals of r = 8sin(3θ), we need to evaluate the integral ∫[lower bound to upper bound] |1/2 * r^2 * dθ|.

The lower bound and upper bound of θ will depend on the range of values where the function forms the desired shape. By integrating the expression, we can calculate the area of the region. To find dy/dx for the parametric equations x = t^(1/3) and y = 3 - t, we differentiate both equations with respect to t. Taking the derivative of y with respect to t gives dy/dt = -1, and differentiating x with respect to t gives dx/dt = (1/3) * t^(-2/3). Finally, we can find dy/dx by dividing dy/dt by dx/dt, resulting in dy/dx = -3 * t^(2/3).

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Answer:

156 cm^2

Step-by-step explanation: