In 1987, the General Social Survey asked, "Have you ever been active in a veteran's group? " For this question, 52 people said that they did out of 98 randomly selected people. The General Social survey randomly selects adults living in the US. Someone wanted to compute a 95% confidence interval for p. What is parameter?

This question involves operations on sets to identify specific members based on conditions. Part a) resolves to D. {2,4}, while part b) finds the solution to be B. {2,4,6,7,8,9,10,11}, highlighting the application of intersection, union, and complement operations in set theory.

To solve these problems, we need to understand the operations on sets such as intersection (A∩B), union (A∪B), and the complement of a set (Bc). For part a), we identify set A as {2,4,6,8,10}, B as {1,3,5}, and C as {1,2,3,4,5}. A∩(B∪C) means we're looking for the intersection of A with the union of B and C. Since B∪C = {1,2,3,4,5}, intersecting this with A gives us D. {2,4} as the answer.

For part b), (A∩Bc)∪(B∩C)c means we're looking at elements in A but not in B, combined with elements not in both B and C. Since Bc = {6,7,8,9,10,11} and (B∩C)c = {6,7,8,9,10,11}, union these two gives us answer B. {2,4,6,7,8,9,10,11}, by including A∩Bc = {2,4,6,8,10} and excluding duplicates when union with (B∩C)c.

Answer:

Two lines are parallel when they share the same slope.

Step-by-step explanation:

Two lines are parallel when they share the same slope.

The slope-intercept form of the equation of a line is: y=mx + b, where 'm' is the slope and 'b' the y-intercept.

If two equations have the same value for 'm', then those lines are parallel, for example:

y = 3x + 8 (Red line)

y = 3x + 5 (Blue line)

y = 3x - 10 (Green line)

All the equations stated above are parallel, to show that, I'm attaching the graph of the equations :).

Answer:

the answer is a

Step-by-step explanation:

i just know

For this case we must solve the following equation:

[tex]4 ^ {x + 1} = 21[/tex]

We find Neperian logarithm on both sides:

[tex]ln (4 ^ {x + 1}) = ln (21)[/tex]

According to the rules of Neperian logarithm we have:

[tex](x + 1) ln (4) = ln (21)[/tex]

We apply distributive property:

[tex]xln (4) + ln (4) = ln (21)[/tex]

We subtract ln (4) on both sides:

[tex]xln (4) = ln (21) -ln (4)[/tex]

We divide between ln (4) on both sides:

[tex]x = \frac {ln (21)} {ln (4)} - \frac {ln (4)} {ln (4)}\\x = \frac {ln (21)} {ln (4)} - 1\\x = 1,19615871[/tex]

Rounding:

[tex]x = 1.1962[/tex]

Answer:

x = 1.1962

Answer: [tex]x[/tex]≈[tex]1.196[/tex]

Step-by-step explanation:

Given the equation [tex]4^{(x + 1)} = 21[/tex] you need to solve for the variable "x".

Remember that according to the logarithm properties:

[tex]log_b(b)=1[/tex]

[tex]log(a)^n=nlog(a)[/tex]

Then, you can apply  [tex]log_4[/tex] on both sides of the equation:

[tex]log_4(4)^{(x + 1)} = log_4(21)\\\\(x + 1)log_4(4) = log_4(21)\\\(x + 1) = log_4(21)[/tex]

Apply the Change of base formula:

 [tex]log_b(x) = \frac{log_a( x)}{log_a(b)}[/tex]

Then you get:

[tex]x =\frac{log(21)}{log(4)}-1[/tex]

[tex]x[/tex]≈[tex]1.196[/tex]

Answer:

$1314.37

Step-by-step explanation:

We have to calculate final value i.e. balance to earn $100,000 annually from interest.

= [tex]\frac{100,000}{0.08}[/tex] = $1,250,000

Now, N = n × y  = 12 × 25 = 300

         I  = 8% =  APR = 0.08

        PV = 0  = PMT = 0

        FV = 1,250,000 = A

[tex]A=\frac{PMT\times [(1+\frac{apr}{n})^{ny}-1]}{\frac{apr}{n}}[/tex]

[tex]PMT=\frac{A\times (\frac{APR}{n})}{[(1+\frac{APR}{n})^{ny}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (\frac{0.08}{12})}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+0.006667)^{300}-1]}[/tex]

[tex]PMT=\frac{\frac{25000}{3}}{[1.006667^{300}-1]}[/tex]

[tex]PMT=\frac{\frac{25000}{3}}{6.340176}[/tex]

Monthly payment (PMT) = $1314.369409 ≈ $1314.37

$1314.37 is required monthly payment in order to $100,000 interest.

The likelihood of a randomly chosen woman having a red blood cell count higher than the typical range of 4.2 to 5.4 million cells per microliter, given that the counts are normally distributed with a mean of 4.577 and a standard deviation of 0.382 million cells, is approximately 0.0158 or 1.58% when expressed as a percentage.

The subject matter here is the use of statistics to understand biological phenomena, specifically the distribution of red blood cell counts in women. The question asks for the probability that a randomly selected woman has a red blood cell count above the normal range of 4.2 to 5.4 million cells per microliter, given that the counts are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells.

Firstly, to answer this question, we must establish the z-scores for the boundaries of our range. The z-score formula is Z = (X - μ) / σ, where X is the value we are evaluating, μ is the mean, and σ is the standard deviation. The upper boundary of our range is 5.4 million cells, so to find the z-score for this we substitute into the formula: Z = (5.4 - 4.577) / 0.382, which gives us a Z-score of approximately 2.15.

However, we are interested in the probability of a woman having a count above the normal range, so we need the area of the curve beyond this z-score. You can find this probability using standard normal distribution tables or a calculator, which suggests that the probability of having a count above 5.4 is approximately 0.0158, or 1.58% when expressed as a percentage and rounded to four decimal places.

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Final answer:

The most B. reasonable interpretation of a 90% chance of rain is that there is a 0.90 probability that it will rain somewhere in the region.

Explanation:

The most reasonable interpretation of a 90% chance of rain, according to the given weather forecasting website, is option B: There is a 0.90 probability that it will rain somewhere in the region at some point during the day. This means that there is a high likelihood that rain will occur in the region, but it does not guarantee that every part of the region will experience rain. It indicates that out of 100 instances, rain is expected in approximately 90 of them.

It is important to note that options A, C, and D are not reasonable interpretations because option A assumes that 100% of the region will get rain, option C assumes that it will rain for 90% of the day, and option D states that none of the interpretations are reasonable, which is not accurate.

Final answer:

The most reasonable interpretation of a 90% chance of rain in a weather forecast is that there is a 0.90 probability of rainfall somewhere in the specified region at some point during the day.

Explanation:

When a weather forecast indicates a 90% chance of rain, it means there is a 0.90 probability that it will rain somewhere in the specified region at some point during the day. Therefore, the correct interpretation based on the given options is B. There is a 0.90 probability that it will rain somewhere in the region at some point during the day. Interpretation A, suggesting that 90% of the region will get rain, is not accurate because the percentage given in a forecast refers to probability, not an area's coverage. Interpretation C, suggesting it will rain for 90% of the day, is also incorrect because the percentage does not refer to the duration of rain but to the probability of occurrence. Statement D is incorrect because B provides a reasonable interpretation.

Answer:(-6,312), (6,312)

Step by Step explanation:

10x^2-y=48

y=-48+10x^2

2(-48+10x^2)=16x^2+48

Solve the equation for x.

x=-6

x=6

y=-48+10(-6)^2

y=-48+10×6^2

y=312

y=312

Answer: a) 30% and b) 45%

Step-by-step explanation:

Since we have given  that

Probability that adults regularly consume coffee P(C) = 45% = 0.45

Probability that adults regularly consume carbonated soda P(S) = 40% = 0.40

Probability that adults regularly consume atleast one of these two products P(C∪S) = 55% = 0.55

a) What is the probability that a randomly selected adult regularly consumes both coffee and soda?

As we know that

P(C∪S ) = P(C) +P(S)-P(C∩S)

[tex]0.55=0.45+0.40-P(C\cap S)\\\\0.55=0.85-P(C\cap S)\\\\0.55-0.85=-P(C\cap S)\\\\-0.30=-P(C\cap S)\\\\P(C\cap S)=0.30=30\%[/tex]

b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?

P(C∪S)'=n(U)-P(C∪S)

[tex]\\P(C\cup S)'=100-55=45\%[/tex]

Hence, a) 30% and b) 45%

Hello There!

[tex]\frac{5}{100}[/tex] written as a decimal is 0.05

Step #1     5/100 is the same thing as 5/5 over 100/5

Step #2 you have a quotient of 1/20

Step #3 divide 1 by 20 and you get a quotient of 0.05

In the given problem, 0.05 is the fraction [tex]\frac{5}{100}[/tex] written as a decimal.

A fraction is a mathematical expression that represents a part or a division of a whole. It is used to represent numbers that are not whole numbers or integers. A fraction consists of two components:

1. Numerator: The numerator is the number on the top of the fraction. It represents the quantity or part of the whole being considered.

2. Denominator: The denominator is the number at the bottom of the fraction. It represents the total number of equal parts into which the whole is divided.

To convert the fraction [tex]\frac{5}{100}[/tex] to a decimal, you can simply divide the numerator, 5 by the denominator, 100.

5 [tex]\div[/tex] 100 = 0.05.

Therefore, [tex]\frac{5}{100}[/tex] is equal to 0.05 as a decimal.

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[tex]\bf 12~~,~~\stackrel{12+3}{15}~~,~~\stackrel{15+3}{18}~~,~~\stackrel{18+3}{21}~\hspace{10em}\stackrel{\textit{common difference}}{d=3} \\\\[-0.35em] ~\dotfill\\\\ n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=12\\ d=3 \end{cases} \\\\\\ a_n=12+(n-1)3\implies a_n=12+3n-3\implies a_n=3n+9[/tex]

Answer:

tₙ = 3(3 + n)

Step-by-step explanation:

Points to remember

nth term of an AP

tₙ = a + (n - 1)d

Where a - first term of AP

d - Common difference of AP

To find the nth term  

The given series is,

12,15,18,21 .....

Here a = 12 and d = 15 - 12 = 3

tₙ = a + (n - 1)d

  = 12 + (n - 1)3

  =12 + 3n - 3

  = 9 + 3n

  = 3(3 + n)

Therefore tₙ = 3(3 + n)

Answer:

Yes they do.

And yes they do follow a normal distribution.

Percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.

It's the probability curve of a continuous distribution that's most likely symmetric around the mean. On the Z curve, at Z=0, the chance is 50-50. A bell-shaped curve is another name for it.

We have a data of final exam scores of 20 Introductory.

a) Range of 1 standard deviation:

(77.7 – 8.44, 77.7 + 8.44)                [69.3, 86.1]

Range of 2 standard deviation:

(77.7 – 2(8.44), 77.7 + 2(8.44))            [60.8, 94.6]

Range of 3 standard deviation:

(77.7 – 3(8.44), 77.7 + 3(8.44))           [52.4, 103.0]

Number of data points lie within 1 standard deviation = 14

Percent of data points lie within 1 SD = (14/20)×100 = 70%

Number of data points lie within 2 SD = 19

Percent of data points lie within 1 SD = (19/20)×100 = 95%

Number of data points lie within 3 SD = 20

Percent of data points lie within 1 SD = (20/20)×100 = 100%

Because these percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed.

b)

Because the histogram in the graph is symmetric, and the normal probability plot reveals that the points are very close to a straight line, the data appears to follow a normal distribution.

Thus, percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.

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In this question (https://brainly.com/question/12792658) I derived the Taylor series for [tex]\mathrm{sinc}\,x[/tex] about [tex]x=0[/tex]:

[tex]\mathrm{sinc}\,x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}[/tex]

Then the Taylor series for

[tex]f(x)=\displaystyle\int_0^x\mathrm{sinc}\,t\,\mathrm dt[/tex]

is obtained by integrating the series above:

[tex]f(x)=\displaystyle\int\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}\,\mathrm dx=C+\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]

We have [tex]f(0)=0[/tex], so [tex]C=0[/tex] and so

[tex]f(x)=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]

which converges by the ratio test if the following limit is less than 1:

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+3}}{(2n+3)^2(2n+2)!}}{\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)^2(2n)!}{(2n+3)^2(2n+2)!}[/tex]

Like in the linked problem, the limit is 0 so the series for [tex]f(x)[/tex] converges everywhere.

The Taylor series for the function f(x) = ∫ sinc(t)dt based at 0 is derived from the Taylor series of sinc(x) by integrating it term by term, given in summation notation as ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞. The series converges for all real numbers (-∞, ∞).

In order to find the Taylor series for the function f(x) = ∫ sinc(t)dt based at 0, one can use the Taylor series for sinc(x) and integrate term by term. We know the Taylor series for sinc(x) is x - x³/3! + x⁵/5! - ..., so the Taylor series for f(x) can be written as x²/2 - x⁴/4*3! + x⁶/6*5! - ... . In summation notation, this is ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞.

The Taylor series for any function converges to the function itself within a certain interval called the radius of convergence. For the Taylor series of sinc(x), due to the nature of sine being bounded between -1 and 1, the series will converge for all real numbers (-∞, ∞).

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Answer:

The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is __-80___feet per second.

Step-by-step explanation:

The average change rate m is calculated using the following formula

[tex]m=\frac{f(t_2)-f(t_1)}{t_2-t_1}[/tex]

In this case [tex]f(t) = -16t^2 + 48t + 100[/tex],  [tex]t_2 = 5\ s\ \ , t_1=3\ s[/tex]

Then

[tex]f(t_2) = f(5) =-16(5)^2 + 48(5) + 100[/tex]

[tex]f(t_2) = -60[/tex]

[tex]f(t_1) = f(3) =-16(3)^2 + 48(3) + 100[/tex]

[tex]f(t_1) = 100[/tex]

Finally

[tex]m=\frac{(-60)-100}{5-3}[/tex]

[tex]m=-80[/tex]

Answer:

378.5 or just 378

Step-by-step explanation:

This is a linear model with x representing the number of generations that's gone by, y is the number of butterflies after x number of generations has gone by, and the 350 represents the number of butterflies initially (before any time has gone by.  When x = 0, y = 350 so that's the y-intercept of our equation.)

The form for a linear equation is y = mx + b, where m is the rate of change and b is the y-intercept, the initial amount when x = 0.

Our rate of change is 1.5 and the initial amount of butterflies is 350, so filling in the equation we get a model of y = 1.5x + 350.

If we want y when x = 19, plug 19 in for x and solve for y:

y = 1.5(19) + 350

y = 378.5

Since we can't have .5 of a butterfly we will round down to 378

The scale factor of this dilation is 2/3.

It is required to find  scale factor of this dilation.

Scale Factor is defined as the ratio of the size of the new image to the size of the old image.

In the figure showing 6 to 9 is 2/3 dilation and 10 to 15 is also a 2/3 dilation.

So, the scale factor of this dilation is 2/3.

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Option b is correct. The scale factor is [tex]1 \frac{1}{2}[/tex].

To determine the scale factor of the dilation from Triangle ABC to Triangle A'B'C', we need to compare the lengths of corresponding sides.

The side lengths of Triangle ABC are:

The side lengths of Triangle A'B'C' are:

The scale factor is calculated by dividing the lengths of the corresponding sides of the triangles. Let's use AB and A'B' for our calculation:

Scale Factor = A'B'/AB = 9/6 = 3/2 = 1 whole 1/2

Thus, the scale factor is [tex]1 \frac{1}{2}[/tex], which corresponds to option b.

Complete question:

What is the scale factor of this dilation? Triangle ABC to A'B'C'.

Triangle ABC with AB = 6, BC = 10, CA = 6

Triangle A'B'C' with A'B'= 9, B'C'= 15, C'A' =9

a. 2/3

b. [tex]1 \frac{1}{2}[/tex]

c. 3

d. 5

Answer:

take 47.99 x .20 = 9.598

$9.60 off

then take 47.99 - 9.60 = $ 38.39

take 38.39 x .06 = 2.3034

$ 2.30 (tax)

add 38.39 + 2.30 = $40.69 or $40.70 is the final purchase price

(the two amounts depends on your choice answer or how it is rounded)

Step-by-step explanation:

Answer: 640

Step-by-step explanation:

Since the two triangles are similar we can simply multiply the lesser triangle's area by a constant to get our answer.

Polygon FGHIJ is ABCDE with a scale change of 4

For the reason that we are dealing with area, we will multiply 40 by 4² in stead of just 4.

40 * 16 = 640

Answer:

B. 640

Step-by-step explanation:

got it right 2021

Answer:

should be 53 if im right

Answer:

Zero

Step-by-step explanation:

We are given the following expression and we are to determine what is the sign of its product:

[tex] 4 . 3 . ( - 3 . 2 ) . 0 [/tex]

One of the three terms in the expression is positive while one is negative. So if we start multiplying the two terms from the left side. we will get a negative number.

But when we will multiply it with zero, the whole product will become zero as anything times zero is always zero. Therefore, answer will be zero.

Answer:

it is negative

Step-by-step explanation:

a positive times a negative is a negative.

Answer:

63.16 in approx.

Step-by-step explanation:

Let the shorter leg be S.  Then the longer leg is L = 3S + 3.

The formula for the area of a triangle is A = (1/2)(base)(height).  Here, that works out to A = 84 in^2 = (1/2)(S)(3S + 3).

Simplifying, we get 168 in^2 = S(3S + 3), or

3S^2 + 3S - 168 = 0, or

 S^2  +  S  - 56   = 0.  This factors as follows:  (S - 8)(S + 7) = 0, so the positive root is S = 8.  We discard the negative root.

Thus, the shorter leg length is 8 and the longer leg length is 3(8) + 3, or 27.

According to the Pythagorean Theorem, the hypotenuse length is given by

L^2 = 8^2 + 27^2, or

L^2 = 64 + 729 = 793.

L = hypotenuse length = √793, or approx. 28.2 in.

Then the perimeter of the triangle is 8 + 27 + 28.2 in, or approx. 63.16 in

Final answer:

The probability of choosing a green ball from the three boxes, given their individual selection probabilities and color distributions, is calculated using the law of total probability. The overall probability of selecting a green ball is found to be 29/80, or roughly 36.25%.

Explanation:

The question asks for the probability of choosing a green ball from three different boxes, given their individual probabilities of being chosen and the distribution of red and green balls in each box. To solve this, we employ the law of total probability which combines the probability of each event (selecting a box) with the conditional probability of finding a green ball within that selected box.

Box 1: Probability of green ball = 5 green balls / (5 red + 5 green) = 1/2

Box 2: Probability of green ball = 3 green balls / (7 red + 3 green) = 3/10

Box 3: Probability of green ball = 4 green balls / (6 red + 4 green) = 2/5

The overall probability is calculated as: P(Green) = P(Box 1) * P(Green|Box 1) + P(Box 2) * P(Green|Box 2) + P(Box 3) * P(Green|Box 3) = (1/4) * (1/2) + (1/2) * (3/10) + (1/4) * (2/5) = 1/8 + 3/20 + 1/10 = 29/80.

Therefore, the probability that the ball chosen is green is 29/80 or approximately 36.25%.

Answer: There are 22 whole cheese sandwiches that can be prepared.

Step-by-step explanation:

Since we have given that

Number of slices of bread = 44

Number of slices of cheese = 75

According to question, a cheese sandwich consists of 2 slices of bread and 3 slices of cheese.

So, we need to find the number of whole cheese sandwiches that can be prepared.

Number of sandwich containing only slice of bread is given by

[tex]\dfrac{44}{2}=22[/tex]

Number of sandwich containing only slice of cheese is given by

[tex]\dfrac{75}{3}=25[/tex]

As we know that each sandwich should contain both slice of bread and slice of cheese.

So, Least of (22, 25) = 22

Hence, there are 22 whole cheese sandwiches that can be prepared.

Answer:

ab + 3a + 8b + 24

Step-by-step explanation:

(a + 8)(b + 3)

a(b + 3) + 8(b + 3)

ab + 3a + 8b + 24

For this case we have the following expression:

[tex]y^ 7 * y^ 9 =[/tex]

By definition of multiplication of powers of the same base, we have to put the same base and add the exponents, that is:

[tex]a ^ n * a ^ m = a ^ {n + m}[/tex]

So:

[tex]y ^ 7 * y ^ 9 = y ^{7 + 9} = y ^ {16}[/tex]

Answer:

[tex]y^{16}[/tex]

Two boys working together can paint a fence in 5 hours with a work rate of 0.2 fences per hour. Adding one more boy increases this work rate to 0.3 fences per hour. This would allow them to complete the painting of the fence in approximately 3.3 hours.

This problem can be solved using the concept of work rate. The work rate is defined as the amount of work done per unit time.

In this case, two boys can paint a fence in 5 hours. So, their combined work rate is 1 fence per 5 hours, or 0.2 fences per hour.

When we add another boy to the group, we increase the total work rate by 50% as now there are 3 boys. So, their combined work rate becomes 0.2 fences/hour + (0.2 fences/hour) * 50% = 0.3 fences/hour.

To find out how long it would take these three boys to paint the fence, we divide the total work (1 fence) by the total work rate (0.3). So, 1 fence divided by 0.3 fences/hour = approximately 3.3 hours. That's how long it would take three boys to paint the fence.

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Answer:

  f(x) = x^4 +x^3 -2x^2 -8x +9

Step-by-step explanation:

You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...

  f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant

The second antiderivative is ...

  f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant

Then the constants can be found from ...

  f(0) = q = 9

  f(1) = 1 + 1 - 2 +p + 9 = 1

  p = -8

The solution is ...

  f(x) = x^4 +x^3 -2x^2 -8x +9

_____

The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.

Final answer:

To find f(x) given f''(x) = 12x² + 6x − 4, one must integrate twice and use the initial conditions f(0) = 9 and f(1) = 1 to solve for the constants. The final function is f(x) = x⁴ + x³ - 2x² - 8x + 9.

Explanation:

The question asks to find the antiderivative f(x) given its second derivative f''(x) =  12x² + 6x − 4, and two initial conditions, f(0) = 9, and f(1) = 1. To solve for f(x), we first integrate the second derivative twice to get the original function.

Integrating f''(x), we get:

f'(x) = ∫( 12x² + 6x - 4)dx = 4x³ + 3x² - 4x + C

We then integrate f'(x) to find f(x):

f(x) = ∫(4x³ + 3x² - 4x + C)dx = x⁴ + x³ - 2x² + Cx + D

Using the initial conditions:

Therefore, the original function is f(x) =  x⁴ + x³ - 2x² - 8x + 9.

Sin is the measure of the opposite leg over the hypotenuse from the given angle:

opposite/hypotenuse

We must find the sin of Angle A, and in order to do so we must find the opposite leg and hypotenuse:

opposite leg/hypotenuse

8/10

Simplify:

8/10 = 4/5

Hence, the sin of <A is 4/5

For this case we have by definition, the sine of an angle is given by the leg opposite the angle on the hypotenuse of the triangle. Then, according to the figure we have:

[tex]Sin (A) = \frac {8} {10}[/tex]

Simplifying we have to:

[tex]Sin (A) = \frac {4} {5}[/tex]

Answer:

Option B

Answer: 1.8446

Step-by-step explanation:

Given claim : [tex]\mu>\mu_0,\text{ where }\mu_0=135[/tex]

Sample size : [tex]n=23[/tex]

Sample mean : [tex]\overline{x}=140[/tex]

Standard deviation : [tex]\sigma = 13[/tex]

The test statistic for population mean is given by :-

[tex]z=\dfrac{x-\mu_0}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

[tex]\Rightarrow\ z=\dfrac{140-135}{\dfrac{13}{\sqrt{23}}}\\\\\Rightarrow\ z=1.84455058589\approx1.8446[/tex]

Hence, the value of test statistic =  1.8446

I guess the "5" is supposed to represent the integral sign?

[tex]I=\displaystyle\int_1^4\ln t\,\mathrm dt[/tex]

With [tex]n=10[/tex] subintervals, we split up the domain of integration as

[1, 13/10], [13/10, 8/5], [8/5, 19/10], ... , [37/10, 4]

For each rule, it will help to have a sequence that determines the end points of each subinterval. This is easily, since they form arithmetic sequences. Left endpoints are generated according to

[tex]\ell_i=1+\dfrac{3(i-1)}{10}[/tex]

and right endpoints are given by

[tex]r_i=1+\dfrac{3i}{10}[/tex]

where [tex]1\le i\le10[/tex].

a. For the trapezoidal rule, we approximate the area under the curve over each subinterval with the area of a trapezoid with "height" equal to the length of each subinterval, [tex]\dfrac{4-1}{10}=\dfrac3{10}[/tex], and "bases" equal to the values of [tex]\ln t[/tex] at both endpoints of each subinterval. The area of the trapezoid over the [tex]i[/tex]-th subinterval is

[tex]\dfrac{\ln\ell_i+\ln r_i}2\dfrac3{10}=\dfrac3{20}\ln(ell_ir_i)[/tex]

Then the integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac3{20}\ln(\ell_ir_i)\approx\boxed{2.540}[/tex]

b. For the midpoint rule, we take the rectangle over each subinterval with base length equal to the length of each subinterval and height equal to the value of [tex]\ln t[/tex] at the average of the subinterval's endpoints, [tex]\dfrac{\ell_i+r_i}2[/tex]. The area of the rectangle over the [tex]i[/tex]-th subinterval is then

[tex]\ln\left(\dfrac{\ell_i+r_i}2\right)\dfrac3{10}[/tex]

so the integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac3{10}\ln\left(\dfrac{\ell_i+r_i}2\right)\approx\boxed{2.548}[/tex]

c. For Simpson's rule, we find a quadratic interpolation of [tex]\ln t[/tex] over each subinterval given by

[tex]P(t_i)=\ln\ell_i\dfrac{(t-m_i)(t-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+\ln m_i\dfrac{(t-\ell_i)(t-r_i)}{(m_i-\ell_i)(m_i-r_i)}+\ln r_i\dfrac{(t-\ell_i)(t-m_i)}{(r_i-\ell_i)(r_i-m_i)}[/tex]

where [tex]m_i[/tex] is the midpoint of the [tex]i[/tex]-th subinterval,

[tex]m_i=\dfrac{\ell_i+r_i}2[/tex]

Then the integral [tex]I[/tex] is equal to the sum of the integrals of each interpolation over the corresponding [tex]i[/tex]-th subinterval.

[tex]I\approx\displaystyle\sum_{i=1}^{10}\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt[/tex]

It's easy to show that

[tex]\displaystyle\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt=\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)[/tex]

so that the value of the overall integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)\approx\boxed{2.545}[/tex]

The question asks to approximate the given integral using three numerical methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. These methods use simple geometric shapes to estimate the area under the curve. Due to the complexity of the integral in question, assistance from computer software or a graphing calculator will likely be necessary.

The question is about using numerical methods to approximate a given integral using three methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. All of these methods are used to approximate the definite integral of a function over an interval. They divide the interval into n subintervals and then use simple geometric shapes to approximate the area under the curve of the function.

To compute these, you would follow these steps: 1. For the Trapezoidal Rule, average the end points and multiply by the width of each interval. 2. For the Midpoint Rule, evaluate the function at the midpoint of each interval, multiply by the width of each interval. 3. For Simpson's Rule, apply the specific weighted average formula that gives more weight to the midpoint

Please note, however, that due to the complexity of the integral of ln(t), you would likely need to use computer software or a graphing calculator to perform these approximations. Please consult with your teacher for the best approach based on what resources are available to you.

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