Jasmine deposited $400 in a bank that paid her 2. 15% interest every year. Assuming no deposits or withdrawals were made. How much money will she have in 5 years? round to nearest.




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To apply the Integral Test, we need to find a function f(x) that is continuous, positive, and decreasing such that f(x) = 1/(5√x).

Taking the integral of f(x) from 1 to infinity, we get:

∫1 to infinity (1/(5√x)) dx = 2/5

Since this integral is a finite number, the series is convergent by the Integral Test.
To determine whether the series is convergent or divergent, we will apply the Integral Test as requested. The given series is:

Σ (from n=1 to infinity) of (1 / (5√n))

First, let's consider the function f(x) = 1 / (5√x). This function is positive, continuous, and decreasing for x ≥ 1, which are the necessary conditions for applying the Integral Test.

Now, we evaluate the improper integral:

∫ (from x=1 to infinity) of (1 / (5√x)) dx

To solve this integral, we'll first rewrite the integrand:

1 / (5√x) = 1 / (5x^(1/3))

Now integrate:

∫(1 / (5x^(1/3))) dx = (3/2) * (1/5) * x^(2/3) + C = (3/10) * x^(2/3) + C

Evaluate the improper integral:

lim (t -> infinity) [∫(from x=1 to t) of ((3/10) * x^(2/3)) dx]

= lim (t -> infinity) [(3/10) * (t^(2/3) - 1)]

Since the exponent (2/3) is less than 1, the limit converges to a finite value:

lim (t -> infinity) [(3/10) * (t^(2/3) - 1)] = -(3/10)

Since the improper integral converges, by the Integral Test, the given series is convergent as well.

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The parameters of the function S(t)=31,500(1.034)t are the initial number of digital subscriptions, which is 31,500, and the monthly growth rate, which is 3.4%.

The given function S(t)=31,500(1.034)t is a exponential growth function that models the number of digital subscriptions S as a function of t months after the launch of the advertising campaign. The parameters of the function are the initial number of digital subscriptions, which is 31,500, and the monthly growth rate, which is 3.4%.

The initial value of 31,500 represents the number of digital subscriptions at the start of the advertising campaign. This means that the campaign began with 31,500 digital subscribers.

The monthly growth rate of 3.4% represents the rate at which the number of digital subscriptions is increasing each month due to the advertising campaign. This means that for each month after the launch of the campaign, the number of digital subscribers is increasing by 3.4% of the previous month's total.

For example, after one month, the number of digital subscribers would be:

S(1) = 31,500(1.034)1 = 32,687

After two months, the number of digital subscribers would be:

S(2) = 31,500(1.034)2 = 33,912

And so on...

Therefore, the initial value and monthly growth rate are important parameters that help us understand how the number of digital subscriptions is changing over time due to the advertising campaign.

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The work required to pump the liquid out of the bucket is approximately 2.482 x 10⁷ Joules.

To create a bucket by rotating around the y-axis, we will use the formula for volume of revolution:

V = π ∫[a,b] (f(y))² dy where f(y) is the function being rotated, and a and b are the limits of integration.

In this case, the limits of integration are y = 0 and y = 3, and the function being rotated is y = 4 ln(x - 4), or x = e⁽y/⁴⁾ + 4.

So, we have:

V = π ∫[0,3] ((e⁽y/⁴⁾ + 4))² dy

V = π ∫[0,3] (e⁽y/²⁾ + 8e⁽y/⁴⁾+ 16) dy

V = π (2e⁽³/²⁾ + 32e⁽³/⁴⁾ + 48)

Now, to find the work required to pump the liquid out of the bucket, we need to use the formula:

W = ∫[h1,h2] ρgV(y) dy

where h1 is the height of the liquid (2 meters in this case), h2 is the height of the top edge of the bucket, ρ is the density of the liquid (860 kg/m^3), g is the acceleration due to gravity (9.8 m/s²), and V(y) is the volume of the liquid at height y.

To find V(y), we need to first find the radius of the bucket at height y.

The radius is given by: r(y) = e⁽y/⁴⁾+ 4

So, the volume of the liquid at height y is:

V(y) = π(r(y))² (h2 - y)

Plugging in the values, we have:

W = ∫[0,2] 860×9.8×π((e⁽y/⁴ + 4)²)×(2-y) dy

W = 2.482 x 10⁷J

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The expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

To find an expression for the area of a rectangle inscribed in a circle of radius 9, we need to use the terms "rectangle" and "circle."

First, let's denote the length of the rectangle as "n" and the width as "w." Since the rectangle is inscribed in a circle with a radius of 9, its diagonal is equal to the diameter of the circle, which is 2 * 9 = 18.

Using the Pythagorean theorem for the right triangle formed by half of the diagonal, the length, and the width, we can write the equation:

(1/2 * 18)^2 = n^2 + w^2
81 = n^2 + w^2

Now, we need to express w in terms of n. To do this, we'll isolate w from the equation:

w^2 = 81 - n^2
w = √(81 - n^2)

The area of the rectangle can be calculated as the product of its length and width:

Area = n * w
Area = n * √(81 - n^2)

So, the expression for the area of the rectangle inscribed in a circle of radius 9, in terms of n, is:

Area = n * √(81 - n^2)

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The table of values to show the effect of the transformation is

x f(x) g(x)

-2 4   -4

-1 1      -1

0 0     0

1 1       -1

2 4     -4

From the question, we have the following parameters that can be used in our computation:

f(x) = x²

Also, we have

g(x) is a reflection of f(x)across the x-axis

This means that

g(x) = -f(x)

So, we have

g(x) = -x²

Using the above as a guide, we have the following:

x f(x) g(x)

-2 4   -4

-1 1      -1

0 0     0

1 1       -1

2 4     -4

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The current scores of both Jenna and Kera in the game they were playing are

Jenny's current score is -15.

Kera's current score is 8.

In the given problem, we are given two players, Jenny and Kera, playing the game.

Jenny has -10 points, which means she already has negative points. He has since lost 5 more points. So his current score would be:

-10 - 5 = -15

So now Jenny has a score of -15.

Kera, meanwhile, has 22 points and 14 points to lose. So his current score would be:

22 - 14 = 8

So now Kera has 8 points.

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The probability that a person has a disease given that they test positive, when 0.4% of the population has the disease and the test is positive with probability 0.99 if they have the disease and 0.03 if they don't have it, is 0.116 or about 11.6%.

Let D be the event that the person has the disease and T be the event that the person tests positive. We need to calculate P(D|T), the probability that the person has the disease given that they test positive.

Using Bayes' theorem, we have

P(D|T) = P(T|D) * P(D) / P(T)

where P(T|D) is the probability of testing positive given that the person has the disease, P(D) is the prior probability of having the disease, and P(T) is the total probability of testing positive, which can be calculated as

P(T) = P(T|D) * P(D) + P(T|D') * P(D')

where P(T|D') is the probability of testing positive given that the person does not have the disease, and P(D') is the complement of P(D), which is the probability of not having the disease.

Substituting the given values, we get

P(D|T) = (0.99 * 0.004) / [(0.99 * 0.004) + (0.03 * 0.996)]

= 0.116

Therefore, the probability that the person has the disease given that they test positive is 0.116 or about 11.6%.

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Answer:

[tex]\left(x\:+\:1\right)^2\:+\:y^2=25[/tex]

Step-by-step explanation:

The equation of a circle with radius r and center at (a, b) is given by
(x - a)² + (y - b)² = r²

Let's first find the radius

The circle intersects the x axis at two points (-6, 0) and (4, 0)

The diameter is therefore the absolute difference between the x values:
|-6 - 4| same as |4 - (-6)| = 10

The radius r = 5  (half of diameter)

Now, let's find the center point of the circle. This will lie midway between (-6, 0) and (4, 0)

Midpoint  (xm, ym) between two points(x1, y) and (x2, y2) :
xm = (x1 + x2)/2 = (-6 + 4)/2 = -1
ym = (y1 + y2)/2 = (0 + 0)/2 = 0

So the center (a, b) = (-1, 0) with a = -1, b = 0

The equation of the circle therefore is
(x - a)² + (y - b)² = r²
( x - (-1) )² + (y - 0)² = 25

(x + 1)² + y² = 25

To find a and b take any point (x, y) and plug these

Answer:

  u = -2.34  or  u = 18.34

Step-by-step explanation:

You want to solve u² -16u = 43 by completing the square.

To complete the square, add the square of half the coefficient of the linear term to both sides.

  u² -16u +(-16/2)² = 43 +(-16/2)²

  u² -16u +64 = 107 . . . . . . . simplify

  (u -8)² = 107 . . . . . . . . . write as a square

  u -8 = ±√107 . . . . . . square root

  u = 8 ± √107  . . . add 8

  u = -2.34  or  u = 18.34 . . . . . find the decimal values

<95141404393>

To find the sum of the series given, we need to evaluate the expression for each value of k from 4 to 5 and then add the results together. The expression is 2(K²- 2K). Let's calculate the sum:

For k = 4:
2(4² - 2*4) = 2(16 - 8) = 2(8) = 16

For k = 5:
2(5² - 2*5) = 2(25 - 10) = 2(15) = 30

Now, we add the results together:
Sum = 16 + 30 = 46

So, the sum of the series is 46.

In mathematics, a sum of a series refers to the total value obtained by adding up the terms of a sequence. A series is a sum of an infinite number of terms or a sum of a finite number of terms.

For example, the sum of the series 1 + 2 + 3 + 4 + 5 is:

1 + 2 + 3 + 4 + 5 = 15

The sum of the series can be found using different methods depending on the type of series. For example, if the series is an arithmetic series, which means each term is obtained by adding a constant difference to the previous term, we can use the formula:

Sn = n/2 [2a + (n - 1)d]

Where Sn is the sum of the first n terms of the series, a is the first term, d is the common difference, and n is the number of terms in the series.

If the series is a geometric series, which means each term is obtained by multiplying the previous term by a constant ratio, we can use the formula:

Sn = a(1 - r^n) / (1 - r)

Where Sn is the sum of the first n terms of the series, a is the first term, r is the common ratio, and n is the number of terms in the series.

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Answer:

pls mrk me brainliest (⁠´⁠(⁠ｪ⁠)⁠｀⁠）

The probability of randomly selecting an orange hat is 1/12 or 0.0833.

In the sample, there are total of 4 pieces of clothing and 3 colors.

The possible outcomes is:

= 4 x 3

= 12

So, the outcomes when randomly selecting a piece of clothing and a color is 12.

From 12 possible outcomes, there is only one outcome where you end up with an orange hat.

So, the probability of randomly selecting an orange hat is:

= 1/12

= 0.0833.

Correct question "You're at a clothing store that dyes your clothes while you wait. You get to pick from 4 pieces of clothing (shirt, pants, socks, or hat) and 3 colors (purple, blue, or orange). If you randomly pick the piece of clothing and the color, what is the probability that you'll end up with an orange hat?"

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The complete conditions are

By definition a quadrilateral is a shape that has four sides and four angles

Next, we test the conditions

Two pairs of parallel sides and at least two right angles

This is true because quadrilaterals like rectangles and squares have two pair of parallel sides and right angles

One pair of parallel sides and no right angles

This is also true because quadrilaterals like trapezoid have one pair of parallel sides and may or may not have right angle

One pair of parallel sides and three right angles

This is false because a quadrilateral cannot be drawn with this condition

No parallel sides and four right angles

This is false because a quadrilateral cannot be drawn with this condition

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If the triangle given by the matrix [235-102] is scaled by a factor of 2, the side lengths of the triangle will be doubled. The angle measures of the triangle will remain the same since scaling does not affect the angles.

When a triangle is scaled by a factor of 2, the side lengths will be doubled, but the angle measures will remain the same. Here's a step-by-step explanation:

1. The original matrix is [2 3 5; -1 0 2].

2. Apply the scale factor of 2 to each of the side lengths by multiplying the matrix by 2: [4 6 10; -2 0 4].

3. The side lengths of the triangle have been doubled, but the angle measures remain the same.

So, after scaling the triangle by a factor of 2, the side lengths will be doubled while the angle measures will stay the same.

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Answer: Let's denote the number of cubic centimeters of the 20% salt solution as x and the number of cubic centimeters of the 60% salt solution as y.

We know that the total volume of the mixture is 100 cubic centimeters, so we have:

x + y = 100

We also know that the final solution should be a 30% salt solution. This means that the amount of salt in the final solution should be 0.3 times the total volume of the solution:

0.3(100) = 0.20x + 0.60y

where 0.20x represents the amount of salt in the 20% salt solution and 0.60y represents the amount of salt in the 60% salt solution.

We now have two equations with two unknowns:

x + y = 100

0.20x + 0.60y = 30

We can solve for x and y by using any method of linear equations, such as substitution or elimination.

Here, we will use substitution. Solving the first equation for x, we get:

x = 100 - y

Substituting this expression for x in the second equation, we get:

0.20(100 - y) + 0.60y = 30

Simplifying and solving for y, we get:

20 - 0.20y + 0.60y = 30

0.40y = 10

y = 25

So, we need 25 cubic centimeters of the 60% salt solution.

To find the amount of the 20% salt solution, we can substitute this value of y back into either equation:

x + y = 100

x + 25 = 100

x = 75

So, we need 75 cubic centimeters of the 20% salt solution.

Therefore, we need to mix 75 cubic centimeters of the 20% salt solution and 25 cubic centimeters of the 60% salt solution to obtain 100 cubic centimeters of a 30% salt solution.

The expression represented by the given number line is f(x) = -k(x+2.5)(x-3) where k > 0.

The expression represented by the given number line can be determined by identifying the values that correspond to the endpoints of each arrow and the direction of the arrow.

Starting from the left endpoint, the arrow goes from -2.5 to -1. This means that the expression is positive between -2.5 and -1. To determine the exact expression, we need to know the interval of the arrow.

The arrow starts at 0 and ends at 3, which means the expression is positive between 0 and 3. Finally, the arrow goes from 3 to -2.5, which means the expression is negative between 3 and -2.5.

Putting all of this information together, we can write the expression as:

f(x) = k(x+2.5)(x-3)

where k is a constant that determines the overall scale of the expression. Since the expression is positive between -2.5 and -1, we know that k must be negative. Since the expression is negative between 3 and -2.5, we know that k must be positive.

Therefore, the expression represented by the given number line is:

f(x) = -k(x+2.5)(x-3) where k > 0.

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Based on the above, the proportion of the population who said yes is  78%.

To be able to calculate the population proportion who said yes,  you  have to divide the number of students who said "Yes" by the total amount or number of students in the  whole population:

Hence it will be:

Population proportion who said yes = 741/950

= 0.78

= 78%

So, the proportion of the population who said yes is 0.78 or 78%.

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See text below



At a high school with 950 total students, the true opinions of the entire student body on whether they approve of the student council president are shown below. Follow the directions below to determine a confidence interval for a sample of size 125.

Population Yes 741, Population No 209,  Population Size 950

Population proportion who said yes: ---

The windshield sector area is 706.86 cm².

To calculate the sector area of the windshield wiper, we need to use the formula for the area of a sector of a circle. The formula is:

A = (θ/360°) x πr²

where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.

In this problem, we are given that the windshield wiper has a length of 45 cm, which means that the radius of the circle traced by the wiper is 45 cm/2 = 22.5 cm.

We are also given that the wiper creates a central angle of 120° in one wipe. Substituting these values into the formula, we get:

A = (120°/360°) x π(22.5 cm)²

A = (1/3) x π x (22.5 cm)²

A ≈ 706.86 cm²

Therefore, the sector area of the windshield wiper is approximately 706.86 square centimeters.

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The volume of one smaller box is 1 cubic foot.

The volume of one smaller box is calculated by raising the length of its sides to the power of 3, so the volume of one smaller box is:

Volume (smaller box) = (1 foot)³ = 1 cubic foot

The number of smaller boxes that fit into a shipping container is obtained by multiplying the number of boxes that fit along each dimension, so the number of smaller boxes that fit into a shipping container is:

Number of smaller boxes = 20 boxes (width) × 40 boxes (length) × 20 boxes (height) = 16,000 boxes

To calculate the volume of the large shipping container, we need to multiply the length, width, and height of the container. Since each dimension is given in terms of the number of boxes it can hold, we need to multiply each dimension by the length of one smaller box (1 foot) to obtain the dimensions in feet. Therefore, the volume of the large shipping container is:

Volume of the large shipping container = 20 boxes (width) × 1 foot (length of one smaller box) × 40 boxes (length) × 1 foot (length of one smaller box) × 20 boxes (height) × 1 foot (length of one smaller box) = 16,000 cubic feet

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The mathematical probabilities of p values lies between range from 0 to 1 and  using technology, p-value is 0.050086.

Sample  = n = 28.

t = 2.051

P value by the means of the technology is 0.050086.

The likelihood of receiving outcomes from a statistical hypothesis test that are at least as severe as the actual results, provided the null hypothesis is true, is known as the p-value in statistics. The p-value provides the minimal level of significance at which the null hypothesis would be rejected as an alternative to rejection points. The alternative hypothesis is more likely to be supported by greater evidence when the p-value is lower.

P-value is frequently utilised by government organisations to increase the credibility of their research or reports. The U.S. Census Bureau, for instance, mandates that any analysis with a p-value higher than 0.10 be accompanied by a statement stating that the difference is not statistically significant from zero.

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Complete question:

A significance test was performed to test H o : u = 2 versus the alternative He: u # 2. A sample of size 28 produced a standardized test statistic of t = 2.051. Assume all conditions for inference are met. Using Table B, the P-value falls between           and         . (Do not round) Using technology the P-value is          . (Round to 4 decimal places)

The total area of the wrapping paper is 72.5 in².

In the given figure (attached below), we have two shapes one is a triangle and the other one is a rectangle. To find the total area of the wrapping paper we have to add the area of the rectangle part and the area of the trianglular part.

Total area = Area of the rectangular part + area of the triangular part.

Area of the rectangular part = length x breadth

from the below figure, length = 15 in

                                     breadth = 4 in

So, area of the rectangular part = 15 in x 4 in = 60 in²

Similarly, area of the triangular part = 1/2 x base x height

from the below figure, base of the triangle = 15 in -10 in = 5 in

                                      height of the triangle = 9 in - 4 in = 5 in

So, area of the triangular part = 1/2 x 5 in x 5in =  12.5 in²

Now, the total area of the wrapping paper = area of the rectangular part + area of the triangular part = 60 in² + 12.5 in² = 72.5 in².

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If the point starts at the π/2 position, the equation for the blade of the wind turbine will be sin(θ + π/2) = sin(θ - π/2)cos(β) + cos(θ - π/2)sin(β).

The equation for the blade of the wind turbine is given by the expression sin(θ) = sin(θ - β)cos(α) + cos(θ - β)sin(α), where θ represents the angle of the blade, β represents the angle between the wind direction and the blade, and α represents the pitch angle of the blade.

If the point starts at the π/2 position, we need to substitute θ + π/2 for θ in the equation. This gives us sin(θ + π/2) = sin(θ - β + π/2)cos(α) + cos(θ - β + π/2)sin(α).

Using trigonometric identities, we can simplify this expression to sin(θ + π/2) = cos(θ - β)cos(α) - sin(θ - β)sin(α).

Finally, substituting β for (π/2 - β) in the above equation, we get sin(θ + π/2) = sin(θ - π/2)cos(β) + cos(θ - π/2)sin(β). This is the required equation for the blade of the wind turbine if the point starts at the π/2 position.

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Answer: A

Step-by-step explanation:

It's A because our function of f is multiplied by 3.

Since our y intercept is 1, and we are multiplying the function f by 3,

our new y intercept is 3, meaning it is A.

Another way to check this is by using the other two points on your graph.

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Answer:

Step-by-step explanation:

Your original function has points at

0, 1

1, 2

2,4  

if you stretched it by 3, multiply your y by 3

new function:

0, 3

1, 6

2, 12

To find the first term and common ratio of a geometric sequence, we can use the formula for the nth term:

a_n = a_1 * r^(n-1)

We are given the eighth and eleventh terms, so we can set up two equations:

a_8 = a_1 * r^(8-1) = 81920

a_11 = a_1 * r^(11-1) = 5242880

After dividing the second with by the first equation, we get:

(a_1 * r^(11-1)) / (a_1 * r^(8-1)) = 5242880 / 81920

Simplifying, we get:

r³ = 64

Doing the root of cube both sides, we get:

r = 4

Substituting this into the first equation, we get:

a_1 * 4^(8-1) = 81920

a_1 * 4^7 = 81920

a_1 = 5

Therefore, the first term is 5 and the common ratio is 4.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r). The formula for the nth term of a geometric sequence is:

an = a * r^(n-1)

Given that the 8th term (a8) is 81,920 and the 11th term (a11) is 5,242,880, we can set up the following equations:

81920 = a * r^(8-1) => 81920 = a * r⁷ (1)

5242880 = a * r^(11-1) => 5242880 = a * r¹⁰ (2)

Now, we need to find the values of a (the first term) and r (the common ratio). Divide equation (2) by equation (1):

(5242880 / 81920) = (a * r¹⁰) / (a * r⁷)

64 = r^3

Now, we can find the common ratio r:

r = 4 (since 4³ = 64)

Next, substitute r back into equation (1) to find the first term a:

81920 = a * 4⁷

a = 81920 / 16384

a = 5

So, the first term (a) is 5, and the common ratio (r) is 4.

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On performing an ANOVA test the p-value obtained is less than the chosen significance level, hence it is verified that the sample standard deviations use of ANOVA allow the the means to compare the population means. Since ANOVA test shows significant difference therefore, it suggests that these factors play a role in influencing the evaluation of the product.

To verify that the sample standard deviations use of ANOVA allows means to compare the population means, discuss the terms ANOVA, sample standard deviation, population means.

1. ANOVA (Analysis of Variance): ANOVA is a statistical method used to compare the means of multiple groups to determine if there's a significant difference between them.

2. Sample Standard Deviation: Sample standard deviation is a measure of how spread out the values in a sample are. It helps estimate the population standard deviation, which is necessary for calculating the F statistic in ANOVA.

a. Calculate the sample means and standard deviations for each group.
b. Perform an ANOVA test using calculated means and standard deviations.
c. Interpret results: If p-value obtained from the ANOVA test is less than the chosen significance level (e.g., 0.05), it means there is a significant difference between population means.

Regarding the effect of the subject's gender and attractiveness of the confederate on the evaluation of the product, if the ANOVA test shows a significant difference, it suggests that these factors play a role in influencing the evaluation of product. You can further analyze the data by performing post-hoc tests to identify which specific groups differ significantly.

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Answer:

  area ≈ 9.219 square units

Step-by-step explanation:

You want the approximate area under the curve y = -1/2x² +x +5 on the interval [1.5, 4] using 5 trapezoids.

The interval can be divided into 5 intervals of width ...

  (4 -1.5)/5 = 2.5/5 = 0.5

The "bases" of each trapezoid will be the function values at the ends of the intervals, for example, at x=1.5 and x=2. The "height" of each trapezoid is the width of the sub-interval, 0.5.

The area formula for a trapezoid applies:

  A = 1/2(b1 +b2)h

  A = 1/2(f(x) +f(x +0.5))·0.5 . . . . . for x = 1.5, 2, 2.5, 3, 3.5

The sum of the areas is computed in the attachment as ...

  area under the curve = 9.21875

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Additional comment

The value of the integral is 445/48 ≈ 9.2708333...

Answer:

Step-by-step explanation: Solution:

Total cost: 150
150-30=120
120 divided by  5   = 24
They visited 24 times in a month.

The difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.

(a) The difference in costs between a hovercraft and a boat can be expressed as follows:

Cost difference = Hovercraft cost - Boat cost

Cost difference = (-1012 + 350t + 400) - (30 + 140t)

Cost difference = -1012 + 350t + 400 - 30 - 140t

Cost difference = 220t - 642

Therefore, the difference in costs between a hovercraft and a boat can be expressed as 220t - 642.

(b) To find the difference in the year 2010, we need to substitute t = 10 in the expression we derived in part (a):

Cost difference = 220t - 642

Cost difference = 220(10) - 642

Cost difference = 2200 - 642

Cost difference = 1558

Therefore, the difference between the costs of a hovercraft and a boat in the year 2010 was approximately $1,558.

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The first four nonzero terms for cos(20x²) are:
1
The first four nonzero terms for ln(1 - 4y) are:
-4y + 8y² - 32y³ +...

To find the first four nonzero terms of the Taylor series for the function cos(20x²) about 0,

we need to find the first few derivatives of the function, and evaluate them at x = 0.

f(x) = cos(20x²)
f'(x) = -40x * sin(20x²)
f''(x) = -40(40x² * cos(20x²) + 20sin(20x²))
f'''(x) = 40(1600x³ * sin(20x²) + 120x * cos(20x²))

Now, evaluate these at x = 0:
f(0) = cos(0) = 1
f'(0) = 0 (since sin(0) = 0
f''(0) = -40(0) = 0
f'''(0) = 0 (since cos(0) = 1

The first four nonzero terms for cos(20x²) are:
1

Now, let's find the first four nonzero terms of the Taylor series for the function f(y) = ln(1 - 4y) about 0.

f(y) = ln(1 - 4y)
f'(y) = -4 / (1 - 4y)
f''(y) = 16 / (1 - 4y)²
f'''(y) = -96 / (1 - 4y)³

Evaluate these at y = 0:
f(0) = ln(1) = 0
f'(0) = -4 / (1) = -4
f''(0) = 16 / (1)² = 16
f'''(0) = -96 / (1)³ = -96

The first four nonzero terms for ln(1 - 4y) are:
-4y + 8y² - 32y³ +...

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