(a+4)-[12-(a+2)-(a+2)

Substitute y = tan(x) and dy = sec²(x) dx to transform the integral to

[tex]\displaystyle \int_0^{\frac\pi2} \cot(x) \ln(\sec(x)) \, dx = \frac12 \int_0^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy[/tex]

and we split the integral at y = 1.

Examining the one over [0, 1], expand into partial fractions :

[tex]\dfrac1{y(1+y^2)} = \dfrac1y - \dfrac y{1+y^2}[/tex]

Then

[tex]\displaystyle \int_0^1 \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \int_0^1 \frac{\ln(1+y^2)}y \, dy - \int_0^1 \frac{y \ln(1+y^2)}{1 + y^2} \, dy[/tex]

In the integral over [1, ∞), substitute y = 1/z :

[tex]\displaystyle \int_1^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \int_0^1 \frac{z \ln\left(1+\frac1{z^2}\right)}{1+z^2} \, dz \\\\ = \int_0^1 \frac{z \ln\left(1+z^2\right)}{1+z^2} \, dz - 2 \int_0^1 \frac{z \ln(z)}{1+z^2} \, dz[/tex]

Some terms cancel and we're left with

[tex]\displaystyle \int_0^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \frac12 \int_0^1 \frac{\ln(1+y^2)}y \, dy - \int_0^1 \frac{y \ln(y)}{1+y^2} \, dy[/tex]

Use the series expansions of ln(1 + y) and 1/(1 - y) - both are valid for |y| < 1.

[tex]\displaystyle \int_0^1 \frac{\ln(1+y^2)}y \, dy = - \sum_{n=1}^\infty \frac{(-1)^n}n \int_0^1 y^{2n-1} \, dy = -\frac12 \sum_{n=1}^\infty \frac{(-1)^n}{n^2} = \frac{\pi^2}{24}[/tex]

[tex]\displaystyle \int_0^1 \frac{y \ln(y)}{1+y^2} \, dy = \sum_{n=0}^\infty (-1)^n \int_0^1 y^{2n+1} \ln(y) \, dy = -\frac14 \sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^2} = -\frac{\pi^2}{48}[/tex]

Putting everything together, we have

[tex]\displaystyle \int_0^{\frac\pi2} \cot(x) \ln(\sec(x)) \, dx = \frac12\times\frac{\pi^2}{24} - \left(- \frac{\pi^2}{48}\right) = \boxed{\frac{\pi^2}{24}}[/tex]

Step-by-step explanation:

1)

Take 12x as a common factor

12x(3x-2)

12 since both 36 and 24 are divisible by 12

2)

Take 12x as a common factor again

12x(x-2)

3)

Take 15x as a common factor

15x(3x²-2)

3x²-2 can be written as the difference of 2 squares

15x(√3 x+√2)(√3 x-√2)

Answer:

[tex]12x(3x-2); 12x(x-2); 15x(3x^2-2)[/tex]

Step-by-step explanation:

Look at the coefficients of the first espression. 36 and 24 are both multiples of 12. We can collect 12x from both and we get

[tex]36x^2-24x=12x(3x-2)[/tex]

Second, 12 and 24 are obviously one twice the other, the common factor is again 12x:

[tex]12x^2-24x = 12x(x^2-2)[/tex]

Third polynomial: this time the greatest common divisor of 45 and 30 is 15. Let's collect 15x from either

[tex]45x^3-30x=15x(3x^2-2)[/tex]

Answer: A

Step-by-step explanation:

Answer:

The answer is A;
✔ m∠2 + m∠3

3x⁴ = 3x³ • x, and

3x³ (x - 2) = 3x⁴ - 6x³

Subtract this from the numerator to get a remainder of

(3x⁴ - 8x³ - x² + 9x + 5) - (3x⁴ - 6x³) = -2x³ - x² + 9x + 5

-2x³ = -2x² • x, and

-2x² (x - 2) = -2x³ + 4x²

Subtract this from the previous remainder to get a new remainder of

(-2x³ - x² + 9x + 5) - (-2x³ + 4x²) = -5x² + 9x + 5

-5x² = -5x • x, and

-5x (x - 2) = -5x² + 10x

Subtract this from the last remainder to get a new one of

(-5x² + 9x + 5) - (-5x² + 10x) = -x + 5

-x = -1 • x, and

-1 (x - 2) = -x + 2

This gives a new remainder of

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

3 does not divide x, so we're done.

So, we have

(3x⁴ - 8x³ - x² + 9x + 5) / (x - 2) = 3x³ - 2x² - 5x - 1 + 3/(x - 2)

Answer:

[tex]\sf\fbox\red{Answer:-}[/tex]

gud noon!

dopamine is responsible for how we feel pleasure

[tex]\small\fbox{\blue{\underline{mαrk \; mє \; вrαínlíєѕt \; plєαѕє ♥}}}[/tex]

A real-world situation to model the number line is: It was colder on Friday night than Saturday morning because 0 is greater than -1

See attachment for the number line

From the figure, we have the following points

0 and -1

The number line can be modelled by temperature where a colder temperature is represented by 0, while the other temperature is -1 degrees Celsius

Hence, a complete scenario is:

On a Friday night, the measure of the temperature is 0 degrees Celsius. On Saturday morning, the temperature is -1 degree Celsius. It was colder on Friday night than Saturday morning because 0 is greater than -1

Read more about number lines at:

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It can be noted that the diameter will be 4.26 inches.

From the information given, the area has been given as 14.18 square inches.

Therefore, the diameter will be calculated thus:

πr²= 14.18

3.14 × r² = 14.8

r² = 14.18/3.14

r² = 4.5

r = 2.13

Therefore, the diameter will be;

= 2r

= 2 × 2.13

= 4.26 inches.

Learn more about diameter on:

https://brainly.com/question/23220731

Answer:

he Formula h = -16t^2 + vt + s; where

h = height after t seconds

t = time in seconds

v = upward velocity

s = initial height (t=0)

also note that

-16t^2 represents the downward force of gravity for t seconds

+vt = the upward force for t seconds

:

the height when he hits the water = 0 therefore:

0 = -16t^2 + 8t + 24

usually written

-16t^2 + 8t + 24 = 0

we can simplify this, divide by -8, then we have

2t^2 - t - 3 = 0

this will factor to

(2t - 3)(t + 1) = 0

the positive solution

2t = 3

t = 3/2

t = 1.5 seconds to hit the water

-

Form the actual equation

Find vertex as vertex is maximum

x coordinate

y coordinate

Max height is 25m

Time to reach is 0.25s

Answer:

Plug the x values into the equation and the answers are the y values

Answer: 13.71

Step-by-step explanation:

Hundredths is the second place

Answer:

13.71

Step-by-step explanation:

The hundredths place is the second decimal, and, since the decimal after that is a 1, you round down and get 13.71

Answer:

[tex]A. 10\sqrt{2}[/tex]

Step-by-step explanation:

The triangle is a 45-45-90 triangle, which has the property that the side across from the 90°, in this case, x, is equal to one of the other sides multiplied by [tex]\sqrt{2}[/tex].

So:

10 x [tex]\sqrt{2}[/tex] = [tex]10\sqrt{2}[/tex]

The value of x in the given triangle is 10√2. So, correct option is A.

In triangle ABC, since angle B is a right angle, we can use the Pythagorean theorem to find the length of side AC (denoted by x).

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, AB and BC are the two sides adjacent to the right angle:

AB² + BC² = AC²

Substitute the given values:

10² + 10² = x²

100 + 100 = x²

200 = x²

Now, to find the value of x, take the square root of both sides:

x = √200

Simplify the square root:

x = √(100 * 2)

x = 10√2

So, the value of x in the triangle is 10√2.

The correct answer is A: 10√2.

To learn more about triangle click on,

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#SPJ2

Complete question is;

Researchers from the Educational Testing Service (ETS) found that providing immediate feedback to students answering openended questions can dramatically improve students’ future performance on exams (Educational and Psychological Measurement, Feb. 2010). The ETS researchers used questions from the Graduate Record Examination (GRE) in the experiment. After obtaining feedback, students could revise their answers. Consider one of these questions. Initially, 50% of the students answered the question correctly. After providing immediate feedback to students who answered incorrectly, 70% answered correctly. Consider a bank of 100 open-ended questions similar to those on the GRE. a. In a random sample of 20 students, what is the probability that more than half initially answer the question correctly? b. Refer to part a . After providing immediate feedback, what is the probability that more than half of the students answer the question correctly?

Answer:

A) 41.2%

B) 95.2%

Step-by-step explanation:

This is a binomial probability distribution problem, so we will use the formula;

P(k) = (n!/(k!(n - k)!) × p^(k) × (1 - p)^(n-k)

A) Initially, 50% of the students answered the question correctly.

Thus, p = 0.5

Also,n = 20

Now, the probability that more than half initially answer the question correctly would be:

P(k > 10)

This can be expressed as;

P(k > 10) = P(11) + P(12) + P(13) + P(14) + P(15) + P(16) + P(17) + P(18) + P(19) + P(20)

P(11) = (20!/(11!(20 - 11)!) × 0.5^(11) × (1 - 0.5)^(20-11) = 0.1602

Similarly,

P(12) = (20!/(12!(20 - 12)!) × 0.5^(12) × (1 - 0.5)^(20-12) = 0.1201

P(13) = (20!/(13!(20 - 13)!) × 0.5^(13) × (1 - 0.5)^(20-13) = 0.0739

Using online binomial probability calculator we can get the remaining values which are;

P(14) = 0.037

P(15) = 0.0148

P(16) = 0.0046

P(17) = 0.0011

P(18) = 0.0002

P(19) = 0.00002

P(20) = 0

Thus;

P(k > 10) = 0.1602 + 0.1201 + 0.0739 + 0.037 + 0.0148 + 0.0046 + 0.0011 + 0.0002 + 0.00002 + 0 ≈ 0.41192 = 41.2%

B) After providing immediate feedback, we are told that 70% answered correctly.

Thus; p = 70% = 0.7

Similar to A above and using online binomial probability calculator, we have;

P(11) = 0.0654

P(12) = 0.1144

P(13) = 0.1643

P(14) = 0.1916

P(15) = 0.1789

P(16) = 0.1304

P(17) = 0.0716

P(18) = 0.0278

P(19) = 0.0068

P(20) = 0.0008

Thus;

P(k > 10) = 0.0654 + 0.1144 + 0.1643 + 0.1916 + 0.1789 + 0.1304 + 0.0716 + 0.0278 + 0.0068 + 0.0008 = 0.952 = 95.2%

Answer:

Options A & C are False while Options B & D are true.

Step-by-step explanation:

The definition of least squares regression line is given as;

The Least Squares Regression Line is a line that normally causes the vertical distance from the data points to the regression line to be as small as possible.

The term “least squares” is derived from the fact that the best line of fit is one that minimizes the the sum of squares of the errors which is also called the variance.

From this definition above, we can see that options A and C are false while option B is the perfect definition for the least line of squares.

Now, the point (x-bar,y-bar ) is known as the mean.

However, every least squares must pass through the middle point of the given data. The mean is the middle point of data in regression analysis. Thus, we can infer that the least-squares regression line always passes through the mean which is point (x-bar,y-bar ). Thus, option D is true.

Check the picture below.

the ball is higher than the building when y = 0, or namely when s(t) = 0, any other time is above it, and hmmm when it's that?

[tex]\stackrel{s(t)}{0}~~ = ~~-16t^2+64t+400\implies 0=-16(t^2-4t-25) \\\\[-0.35em] ~\dotfill\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ 0=\stackrel{\stackrel{a}{\downarrow }}{1}t^2\stackrel{\stackrel{b}{\downarrow }}{-4}t\stackrel{\stackrel{c}{\downarrow }}{-25} \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ t= \cfrac{ - (-4) \pm \sqrt { (-4)^2 -4(1)(-25)}}{2(1)}\implies t=\cfrac{4\pm\sqrt{116}}{2}[/tex]

[tex]t=\cfrac{4\pm 2\sqrt{29}}{2}\implies t= \begin{cases} 2+\sqrt{29}~~\textit{\Large \checkmark}\\ 2-\sqrt{29} \end{cases}~\hfill \boxed{(0~~,~~2\sqrt{29})}[/tex]

notice, we can't use the 2-√29 because is a negative value and "t" is never negati ve, also notice the interval notation, we have parentheses on both sides.

Answer:

220

Step-by-step explanation:

GIven:

Joyce paid $143.00 for an item at the store that was 35 percent off the original price.

To Find:

What was the original price?

Solution:

Since that was 35 percent off the original price

Thus, 100 - 35 = 65

Equation:

143 = 0.65x

DIvide both sides by 0.65

143/0.65 = 0.65x/0.65

x = 220

Check Answer:

220 x 65% = 143

~Lenvy~

The value '0' is between the points above sea level and points below sea level. '0' being neither negative nor positive means it cannot be below or above sea level, rather '0' meters represents sea level.

Hope that helps!

Answer:

A) 3

Step-by-step explanation:

Ques: 30/(4√3 + 3√2) = 4√3 - a√2

Rationalise the denominator.

→ 30/(4√3 + 3√2) × (4√3 - 3√2)/(4√3 - 3√2) = 4√3 - a√2

→ (120√3 - 90√2)/(48 - 18) = 4√3 - a√2

→ (120√3 - 90√2)/30 = 4√3 - a√2

→ 120√3 - 90√2 = 120√3 - 30a√2

→ 90√2 = 30a√2

On solving we get,

→ a = 3

Ques: 30/(4√3 + 3√2) = 4√3 - a√2

→ 30 = 4√3(4√3) + 4√3(-a√2) + 3√2(4√3) + 3√2(-a√2)

Solve the brackets,

→ 30 = 48 - 4a√6 + 12√6 - 6a

→ - 18 = - 4a√6 - 12√6 - 6a

Put like terms on one side,

→ - 18 + 6a = - 4√6 (a + 3)

Take the common,

→ 3(a - 3) = - 2√6 (a + 3)

→ 3 (a - 3) = 2√6 (-a - 3)

→ [3(a - 3)]/2√6 = - a - 3

Rationalise the denominator,

→ 2√6(3a - 9)/24 = - a - 3

→ (3a√6 - 9√6)/12 = - a - 3

→ √6(a - 3) = - 4a - 12

Do squaring on both sides,

→ [√6(a - 3)]² = - (4a + 12)²

→ 6(a² + 9 - 6a) = - (16a² + 144 + 96a)

Solve the brackets,

→ 6a² + 54 - 36a =  - 16a² - 144 - 96a

→ 22a² - 132a + 198 = 0

→ 11a² - 66a + 99 = 0

→ a² - 6a + 9 = 0

Split the middle term,

→ a² - 3a - 3a + 9 = 0

→ a(a - 3) -3(a - 3) = 0

→ (a - 3)(a - 3) = 0

On comparing we get,

→ a = 3 (key to solve this and similar questions is rationalisation of the denominator)

Hence, the correct option is option A) 3.

Answer:

I only

Step-by-step explanation:

dy/dx = x + 1 / y -3

(y - 3)dy = (x + 1) dx

integrate

1/2y^2 - 3x = 1/2x^2 + x

1/2y^2 = 1/2x^2 + 4x

y^2 = x^2 + 8x

y = (x^2 + 8x)^1/2

Answer:

Kindly check explanation

Step-by-step explanation:

Given the data : 5, 9, 10, 11, 15

s = √[(ΣX - m)² / (n - 1)]

n = number of sample = 5

m = mean

m = ΣX/n

m = (5 + 9 + 10+ 11 + 15) / 5

m = 50/5 = 10

s =√((5-10)^2 + (9-10)^2 + (10-10)^2 + (11-10)^2 + (15-10)^2) / 5-1

= 3.601

(B) Multiply each data value by 5 to obtain the new data set 25, 45, 50, 55, 75. Compute s.

m = ΣX/n

m = (25 + 45 + 50 + 55 + 75) / 5

m = 250/5 = 50

s =√((25-50)^2 + (45-50)^2 + (50-50)^2 + (55-50)^2 + (75-50)^2) / 5-1

= 18. 028

(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c?

The standard deviation changes by almost c times multiplied by the initial value.

3.601 * 5 = 18.005 which is almost equivalent to 18.028

(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s 3.1 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calcula- tions?

No

Given 1 mile 1.6 kilometers, what is the standard deviation in kilometers?

If 1 mile = 1.6km

s = 3.1 miles

s in kilometers = (1.6 * 3.1) = 4.96km

Answer:

31-32

Hyposis: SHe is the mother.

Conclusion: She's not.

Step-by-step explanation:

Answer:

23

Step-by-step explanation:

8+15=23

in all, he moved 8 feet to the water, then another 15 feet below the water. this means that he ended up (8+15) feet from where he started.

Answer:

23

Step-by-step explanation:

Answer: A class of 449 students went on a field trip. They took 11 vehicles, some cars and some buses. Find the number of cars and the number of buses they took if each car holds 3 students and each bus holds 55 students.

Step-by-step explanation: they took 8 buses.

55 x 8 = 440

3 x 3 = 9

440 + 9 = 449

Answer: X minus 1 over 2

Step-by-step explanation:

Use an inverse function calculator

Answer:− -19s

Step-by-step explanation:

Move  

Answer:

c

Step-by-step explanation:

Answer:

~c~

Step-by-step explanation:

Answer:

17cm

Step-by-step explanation:

The perimeter of a square is the 4 sides added up, so we can just divide 4 by 68 to find out what one length is.

68/4= 17

We can prove this by adding it again.

17+17+17+17=68

Answer:

Step-by-step explanation: becuase if the ratio is 2:3 then the ratio reduced to 1 would be 2/3:1

Answer:

(First Option): 2 over 3 tablespoon of vinegar is mixed with every tablespoon of olive oil

Step-by-step explanation

For every 2 tablespoons of vinegar, it was mixed with 3 tablespoons of olive oil. This leads to a ratio of 2:3, and that rules out option 2 and 3. To get the unit rate we divide 2 tablespoons of vinegar by 3 tablespoons of olive oil. This means that there is 2/3 tablespoons of vinegar for every tablespoon of olive oil and leads it to be answer choice 1.

Answer:

2 × 2 x 2 x 2 x 2 or 25

Answer:

2*2*2*2*2, which can be also written as [tex]2^5[/tex].

Answer:

interior alternative angles

x=--4°

Step-by-step explanation:

please mark me as brainlest

112

suplementry angle add to 180 so 180_68=112