There are two events called A and B. The probability of both A and B is 0.395 and the probability A given B is 0.61. What is the probability of B?Enter three correct decimal places in your response. That is, calculate the answer to at least four decimals and report only the first three. For example, if the calculated answer is 0.123456 enter 0.123.____________

Our general solution is  y = c1e((-3 + √(13))/2)t + c2e((-3 - √(13))/2)t  where c1 and c2 are constants determined by initial or boundary conditions.

To find the general solution to this differential equation, we first need to find the characteristic equation by assuming that y = e(rt) and substituting it into the equation.

So we have y" + 3y' - y = 0
Substituting y = e(rt) we get
r² e(rt) + 3r e(rt) - e(rt) = 0

Dividing by e^(rt) we get
r² + 3r - 1 = 0

Now we solve for r by using the quadratic formula:
r = (-3 ± √(3² - 4(1)(-1))) / (2(1))
r = (-3 ± √(13)) / 2

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Tushar kapoor can build the fence in 23 days if he is working alone.

Total days required = Total unit / Number of unit per day

To find the number of days taken by Tushar kapoor to complete building a fence, first we will have to find out the number of unit produced by Karan Johar and Ekta Kapoor.

Let the total unit produced be 90 units.

Then, units per day produced by:

Karan Johar = Total unit /  Total day required

                     = 90 / 15

                     = 6 unit per day.

Ekta Kapoor = Total unit /  Total day required

                     = 90 / 18

                     = 5 unit per day.

Tushar Kapoor = x (let)

Now, we will use the above information to find the number of days required by Tushar Kapoor.

Total days taken when all three started working together = 6 days

Unit produced per day when all three started working together = 90 / 6

                                                                                                          = 15 units

Total unit per day = Karan Johar's unit per day + Ekta Kapoor's unit per day + Tushar Kapoor's unit per day

15 = 6 + 5 + x

x = 15 - 11

x = 4 unit per day.

Therefore, Tushar kapoor is producing 4 unit per day.

Total day taken by him = 90 / 4

                                      = 22.5 days ≈ 23 days.

Therefore, the number of days taken by Tushar Kapoor is 23 days.

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When the circumference is 2 cm, the radius is changing at a rate of 3 cm/s.

To solve this problem, we can use the formula for the area of a circle: A = πr², where A is the area and r is the radius.

We are given that the area of the circle is increasing at a rate of 6 cm²/s. This means that dA/dt = 6.

We are asked to find how fast the radius is changing (rate of change) when the circumference is 2 cm. We know that the formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. So if the circumference is 2 cm, we can set up the equation:
2πr = 2

Solving for r, we get:
r = 1/π

Now we can differentiate the equation for the area with respect to time (t):
A = πr²
dA/dt = 2πr(dr/dt)

Substituting the values we know:
6 = 2π(1/π)(dr/dt)
6 = 2(dr/dt)
dr/dt = 3 cm/s

Therefore, when the circumference is 2 cm, the radius is changing at a rate of 3 cm/s.

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The differentiation value of F(1) by evaluating the function of definite integral denoted by F(x) for F(x) = d/dx(Sx⁴ 0 2t³dt) is 8.

To find F(x), we first integrate 2t³ with respect to t to get t⁴, and then substitute the limits of integration to obtain 2x⁴.

F(x) = d/dx(Sx⁴ 0 2t³dt)

F(x) = d/dx [[tex](2x^3)^4[/tex] - [tex](0)^4[/tex]] / 4

Next, we differentiate 2x⁴ with respect to x to obtain F(x) = 8x³.

F(x) = 8x³

To find F(1), we substitute x = 1 in F(x) to get F(1) = 8(1)³ = 8.

F(1) = 8(1)³

F(1) = 8

Therefore, the value of F(1) is 8.

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$ 27,255.25 would be the accumulated value at the interest rate of 4.73% compounded monthly.$ 25,334 would be the accumulated value at the interest rate of 4.73% compounded semi-annually. $1921.25 is the amount that much more interest if he chose the monthly compounding interest rate instead of the semi-annually.

Compound interest is calculated as follows:

A = P [tex](1+\frac{r}{n} )^{nt[/tex]

where A is the amount

P is the principal

r is the rate in decimal

n is the frequency of time interest is compounded

t is the time

a. P = $13,250

r = 4.73% or 0.0473

n = 12 since compounded monthly

t = 15 years

A = 13250  [tex](1+\frac{0.0473}{12})^{12*15[/tex]

= 13250 [tex](1.004)^{180[/tex]

= 13250 * 2.051

= $ 27,255.25

b. P = $13,250

r = 4.73% or 0.0473

n = 2 since compounded semi-annually

t = 15 years

A = 13250  [tex](1+\frac{0.0473}{2})^{2*15[/tex]

= 13250 [tex](1.02185)^{30[/tex]

= 13250 * 1.912

= $ 25,334

c. Difference between the interest in semi-annual and monthly compounded = 27,255.25 - 25,334

= $1921.25

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The first four terms of the sequence are 3, 10, 31, and 94.

To find the first four terms of the sequence defined by [tex]a_1 = 3[/tex] and [tex]a_n = 3a_{n-1} + 1[/tex] for all n > 1, follow these steps:

1. The first term is given: [tex]a_1 = 3.[/tex]
2. Use the formula to find the second term: [tex]a_2 = 3a_1 + 1 = 3(3) + 1 = 9 + 1 = 10.[/tex]
3. Use the formula to find the third term: [tex]a_3 = 3a_2 + 1 = 3(10) + 1 = 30 + 1 = 31.[/tex]
4. Use the formula to find the fourth term: [tex]a_4 = 3a_3 + 1 = 3(31) + 1 = 93 + 1 = 94.[/tex]

So, the first four terms of the sequence are 3, 10, 31, and 94.

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Answer: binary searches, finding the smallest or largest value in a binary search tree, and certain divide and conquer algorithms.

Step-by-step explanation: O(log N) is a common runtime complexity. Examples include binary searches, finding the smallest or largest value in a binary search tree, and certain divide-and-conquer algorithms. If an algorithm is dividing the elements being considered by 2 in each iteration, then it likely has a runtime complexity of O(log N).

The marginal revenue at a quantity of 900 lbs is $50. The Skagway Coffee Company can sell 900 lbs of its premium blend coffee.

To determine the price at which the Skagway Coffee Company can sell 900 lbs of its premium blend coffee, we need to solve the demand equation for price when quantity is 900:

900 = 800 + 25√p

Simplifying this equation, we get:

100 = 25√p

Squaring both sides, we get:

10000 = 625p

Solving for p, we get:

p = 16

Therefore, the price at which the Skagway Coffee Company can sell 900 lbs of its premium blend coffee is $16.

To determine the revenue at this quantity, we can simply multiply the price by the quantity:

revenues = $16 x 900 = $14,400

To determine the marginal revenue at this quantity, we need to take the derivative of the demand equation with respect to quantity, and then evaluate it at the quantity of 900. The derivative of the demand equation is:

[tex]dq/dp[/tex] = 25/(2√p)

Substituting p = 16, we get:

[tex]dq/dp[/tex] = 25/(2√16) = 25/8

Multiplying this by the price of $16, we get:

marginal revenue = (25/8) x $16 = $50

Therefore, the marginal revenue at a quantity of 900 lbs is $50.

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The angle ∅ to the nearest tenth is 51.8 degrees.

The trigonometric ratio can be solved as follows:

tan ∅  = 14 / 11

We are asked to solve for the angle ∅.

Therefore,

tan ∅  = 14 / 11

∅ = tan⁻¹ 14 / 11

Hence,

∅ = tan⁻¹ 1.27272727273

∅ = 51.8421769502

Therefore,

∅ = 51.8 degrees

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Assuming an interest rate of 5% per year, the deposit amount required today to provide the annuity is approximately $49,019.15.

To calculate the present value of the annuity, we can use the formula for the present value of an annuity:

PV = PMT x [1 - (1 + r)⁻ⁿ] / r

Where:

PV = Present value

PMT = Payment amount per period

r = Interest rate per period

n = Total number of periods

For the first two years, Piers will receive $1,000 at the end of each month, for a total of 24 payments. Using the formula above, we can calculate the present value of these payments:

PV1 = $1,000 x [1 - (1 + r)⁻²⁴] / r

For the next three years, Piers will receive $500 at the end of each month, for a total of 36 payments. Using the same formula, we can calculate the present value of these payments:

PV2 = $500 x [1 - (1 + r)⁻³⁶] / r

The total present value of the annuity is the sum of PV1 and PV2:

Total PV = PV1 + PV2

Total PV = 24,019+25,000.15 = $49,019.15.

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f(x) is increasing on the open interval (-∞, 5) and decreasing on the open interval (5, +∞).

To find the open intervals where f(x) is increasing and decreasing, we need to analyze the first derivative, f'(x), which is given as -3(x-5)/(x-1)^3.

First, find the critical points by setting f'(x) = 0:

-3(x-5)/(x-1)^3 = 0
(x-5) = 0
x = 5

Now, let's analyze the intervals based on the critical point x = 5:

1) Interval (-∞, 5): Choose a test point, say x = 0. Plug it into f'(x):

f'(0) = -3(0-5)/(0-1)^3 = 15 > 0

Since f'(0) > 0, f(x) is increasing on the interval (-∞, 5).

2) Interval (5, +∞): Choose a test point, say x = 6. Plug it into f'(x):

f'(6) = -3(6-5)/(6-1)^3 = -3 < 0

Since f'(6) < 0, f(x) is decreasing on the interval (5, +∞).

In conclusion, f(x) is increasing on the open interval (-∞, 5) and decreasing on the open interval (5, +∞).

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Answer:

Below

Step-by-step explanation:

you didn't say how much sugar is even in 1 12 ounce sweet tea, so I hope I can help you by giving you a formula

To determine how much sugar Mike is drinking in one day, we need to know the amount of sugar in each 12-ounce serving of sweet tea.

Let's assume that each 12-ounce serving contains 20 grams of sugar, which is roughly the amount of sugar in a typical 12-ounce can of soda.

So, in one day, Mike is drinking: 3 servings/day x 12 ounces/serving = 36 ounces of sweet tea/day

To calculate how much sugar Mike is consuming in one day, we can use the following formula:

Sugar consumed = (Amount of sweet tea consumed) x (Amount of sugar per serving)

Using the values we have, we get:

Sugar consumed = (36 ounces/day) x (20 grams/12 ounces)Sugar consumed = 60 grams/day

Therefore, in this example, Mike is consuming 60 grams of sugar in his sweet tea every day.

Hence use this formula to help answer your question :)

The study conducted by Tract Clemons and Marcello Pagano using all 1991 birth records in the computerized national birth certificate registry compiled by the National Center for Health Statistics (NCHS) found that the birth weights of babies in the United States are not symmetric.

However, they also found that when infants born outside of the typical 37-43 weeks and infants born to mothers with a history of diabetes are excluded, the birth weights of the remaining infants do follow a Normal model with a mean of 3432 g and a standard deviation of 482 g. This suggests that there are factors that can affect the normality of birth weights, but when these factors are accounted for, the remaining infants' birth weights can be modeled using the Normal distribution. It is important to consider these factors when analyzing birth weight data to ensure accurate and reliable results.

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The derivative of f(x) is f'(x) = -12x^11 / (c^6 + x^6)^2.

To find the derivative of the function f(x) = (c^6 - x^6)/(c^6 + x^6), we can use the quotient rule of differentiation:

f(x) = (c^6 - x^6)/(c^6 + x^6)

f'(x) = [(c^6 + x^6)(-6x) - (c^6 - x^6)(6x)] / (c^6 + x^6)^2

We apply the quotient rule, which is:

(f(x) / g(x))' = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2

where f(x) and g(x) are two differentiable functions.

In our case, f(x) = c^6 - x^6 and g(x) = c^6 + x^6.

Now, we need to find f'(x) and g'(x) in order to apply the quotient rule.

f'(x) = d/dx (c^6 - x^6) = 0 - 6x^5 = -6x^5

g'(x) = d/dx (c^6 + x^6) = 0 + 6x^5 = 6x^5

Now, we can substitute these values into the quotient rule:

f'(x) = [(c^6 + x^6)(-6x^5) - (c^6 - x^6)(6x^5)] / (c^6 + x^6)^2

Simplifying the numerator, we get:

f'(x) = (-6x^5)(2x^6) / (c^6 + x^6)^2

f'(x) = -12x^11 / (c^6 + x^6)^2

Therefore, the derivative of f(x) is f'(x) = -12x^11 / (c^6 + x^6)^2.

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The derivative of the given function y=(x+9)(x^3 + 7x+5) is y' = 3x^3 + 28x^2 + 7x + 68.

To differentiate the given function y=(x+9)(x^3 + 7x+5), we need to use the product rule of differentiation.
The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx)[u(x)*v(x)] = u(x)*v'(x) + v(x)*u'(x)
Applying this rule to the given function, we get:
y' = (x+9)*d/dx[x^3 + 7x+5] + (x^3 + 7x+5)*d/dx[x+9]
Now, we just need to find the derivatives of each term separately.
d/dx[x^3 + 7x+5] = 3x^2 + 7
d/dx[x+9] = 1
Substituting these derivatives back into the product rule formula, we get:
y' = (x+9)*(3x^2 + 7) + (x^3 + 7x+5)*1
Simplifying this expression, we get:
y' = 3x^3 + 28x^2 + 7x + 68

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The function f(x+4) = 3^(x+4) - 4 is shifted 4 units to the left compared to g(x) = 3^x. f(x) - 4 = 3^x - 8 is shifted 4 units down compared to g(x) = 3^x. f(x) + 4 = 3^x is the same as g(x) = 3^x, but shifted 4 units up. f(x-4) = 3^(x-4) - 4 is shifted 4 units to the right compared to g(x) = 3^x.

To compare the functions f(x) = 3^x - 4 and g(x) = 3^x, we need to evaluate the difference between the two functions for different values of x. To shift the function f(x) four units to the left, we substitute x + 4 for x in the function. Therefore, f(x+4) = 3^(x+4) - 4.

To shift the function f(x) four units down, we subtract 4 from the function. Therefore, f(x) - 4 = 3^x - 8. To shift the function f(x) four units up, we add 4 to the function. Therefore, f(x) + 4 = 3^x. To shift the function f(x) four units to the right, we substitute x - 4 for x in the function. Therefore, f(x-4) = 3^(x-4) - 4.

In general, shifting a function left or right involves replacing x with x + a or x - a, respectively, where a is the amount of the shift. Shifting a function up or down involves adding or subtracting a constant from the function, respectively.

In each case, we can see that the function f(x) is different from the basic function g(x) due to a shift and/or a vertical translation.

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The point- slope form of the line is y-5 = -0.25(x+6).

What is line?

A line is an one-dimensional figure. It has length but no width. A line can be made of a set of points which is extended in opposite directions to infinity. There are straight line, horizontal, vertical lines or may be parallel lines perpendicular lines etc.

A line with a slope of –1/4 passes through the point (–6,5)

Any line in point - slope form can be written as

y - y₁= m(x -x₁) -------(1)

where,

y= y coordinate of second point

y₁ = y coordinate of first point

m= slope of the line

x= x coordinate of second point

x₁ = x coordinate of first point

In the given problem (x₁ , y₁) = (-6,5) and m= -1/4

Putting all these values in equation (1) we get,

y-5= (-1/4) (x- (-6))

⇒ y-5 = -0.25(x+6)

Hence, the point- slope form of the line is y-5 = -0.25(x+6).

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(a) The left-hand sum estimate for the integral is 1.214

(b) The right-hand sum estimate is 1.642, both rounded to three decimal places.

(a) To estimate the integral of eˣ using n=5 rectangles with left-hand sum and right-hand sum, we will first find the width of each rectangle (Δx) and then calculate the area of the rectangles using the function values.

Step 1: Calculate Δx
Δx = (b-a)/n = (1-0)/5 = 0.2

Step 2: Calculate the left-hand sum (LHS)
LHS = Δx * (f(x0) + f(x1) + f(x2) + f(x3) + f(x4))
LHS = 0.2 * ([tex]e^0+e^0^.^2+e^0^.^4+e^0^.^6+e^0^.^8[/tex])

LHS ≈ 1.214

(b) Step 3: Calculate the right-hand sum (RHS)
RHS = Δx * (f(x1) + f(x2) + f(x3) + f(x4) + f(x5))
RHS = 0.2 * ([tex]e^0+e^0^.^2+e^0^.^4+e^0^.^6+e^0^.^8[/tex])
RHS ≈ 1.642

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The indefinite integral of S(3t²+t/4)dt is t³ + t²/8 + C, where C is the constant of integration. An antiderivative of a function is the function's indefinite integral.

In other words, if we take the derivative of the indefinite integral, we get back the original function (up to a constant of integration). We can utilise the integration rules to integrate each term individually in order to determine the indefinite integral of S(3t²+t/4)dt. Specifically, we can apply the power rule of integration and the constant multiple rule.

∫ 3t² + t/4 dt

= 3 ∫ t² dt + 1/4 ∫ t dt

= 3 (t³/3) + 1/4 (t²/2) + C

= t³ + t²/8 + C

Therefore, the indefinite integral of S(3t²+t/4)dt is t³ + t²/8 + C, where C is the constant of integration. Note that when we take the derivative of this function with respect to t, we get 3t² + t/4, which is the original function. The constant of integration represents a family of functions that differ by a constant, and is necessary because the derivative of a constant is zero.

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95% confidence that the average number of words typed by all graduates of the secretarial school is between 78.03 and 89.17 words per minute.

The 95% confidence interval for the average number of words typed by all graduates of the secretarial school, we can use the formula:

[tex]CI = \bar X \± t\alpha/2 \times (s/\sqrt n)[/tex]

[tex]\bar X[/tex]is the sample mean, s is the sample standard deviation, n is the sample size,[tex]t\alpha /2[/tex] is the t-score with (n-1) degrees of freedom and a probability of [tex](1-\alpha/2)[/tex] in the upper tail.

95% confidence interval, [tex]\alpha = 0.05[/tex], so [tex]\alpha/2 = 0.025[/tex]. We can look up the t-score with 10 degrees of freedom.

[tex](n-1 = 11-1 = 10)[/tex] and a probability of 0.025 in the upper tail in a t-table or calculator.

The value is approximately 2.228.

Plugging in the values from the problem, we get:

[tex]CI = 83.6 \± 2.228 \times (7.2/\sqrt {11})[/tex]

[tex]CI = 83.6 \± 5.57[/tex]

[tex]CI = (78.03, 89.17)[/tex]

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The probability that the time until the first critical-part failure is less than 1 year is 0.2548. So the correct answer is (D) 0.254811.

Given that the time until the first critical part failure for a certain car is exponentially distributed with a mean of 3.4 years.

f(t) = λe^(-λt)

where λ is the rate parameter and is equal to 1/mean = 1/3.4 = 0.2941.

To find the probability that the time until the first critical-part failure is less than 1 year, we need to calculate the cumulative distribution function (CDF) of the exponential distribution up to 1 year:

F(1) = ∫[0,1] λe^(-λt)dt

Using integration by substitution, let u = λt, then du/dt = λ and dt = du/λ.

F(1) = ∫[0,λ] e^(-u)du

= [-e^(-u)]_[0,λ]

= -e^(-λ) + e^(-0)

= 1 - e^(-0.2941 * 1)

= 0.2548 (rounded to four decimal places)

Therefore, the likelihood that the period until the first critical-part failure is less than 1 year is 0.2548. So the correct answer is (D) 0.254811.

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The mean and median number of hours Rashawn spend in listening to music is 5.7 hours and 6 hours, under the condition that there were 6 days in which Rashawn listened to music.

Now to evaluate the mean number of hours Rashawn listened to music for the 6 days, we have  to sum up all the hours and divide by the number of days.

Then, total number of hours Rashawn heard music for 6 days is

= 6 + 5 + 5 + 6 + 5 + 7

= 34 hours

Mean  = Total number of hours / Number of days

= 34 / 6

= 5.7 hours

Now,

For evaluating  the median number of hours Rashawn heard  music in the interval of  6 days

We have to set the number in the order of smallest to largest

The numbers in order are  5, 5, 5, 6, 6, 7

The median is the middle value which is 6

The mean and median number of hours Rashawn spend in listening to music is 5.7 hours and 6 hours, under the condition that there were 6 days in which Rashawn listened to music.

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The complete question is

Rashawn kept a record of how many hours he spent listening to music for 6 days during school vacation and displayed his results in the table

Day -

Monday  

Number of hours - 6

Tuesday  

Number of hours - 5

Wednesday  

Number of hours - 5

Thursday

Number of hours - 6

Friday

Number of hours - 5

Saturday

Number of hours - 7

calculate the mean and median number of hours Rashawn listened to music for the 6 days. Round your answers to the nearest tenth.

Substituting these values into the chain rule formula, we get:
[tex]h'(x) = 2g(e) * 9(2)cos(e) * We * a[/tex]
Without knowing the values of g(e) and a, we cannot find the value of h'(x).

a) To decide the function h(x), we need to know the values of g(e) and a. Unfortunately, those values are not given in the question. Without that information, we cannot determine the function h(x).

b) To derive h(x), we need to use the chain rule. Recall that the chain rule states that the derivative of a composition of functions is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In other words:

[tex]h'(x) = f'(g(e)) * g'(e) * e'(x) * a[/tex]
Now, we need to find the derivatives of each of the functions involved.

f'(x) = 2x (by the power rule)
g'(e) = 9(2)cos(e) (by the chain rule and the derivative of sin(x) = cos(x))
e'(x) = We (given in the question)

Substituting these values into the chain rule formula, we get:

[tex]h'(x) = 2g(e) * 9(2)cos(e) * We * a[/tex]

Without knowing the values of g(e) and a, we cannot find the value of h'(x).

The complete question is-

Given the function , h(x) = f(g(e))a) Decide the function h(x) if f (x) = x2 – 1 and 9(2) sin(x) + 1 b) Derive h(x) T We and decide the value for derivative of h(x).

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The given improper integral diverges.

We can use the integral test to determine if the improper integral converges or diverges.

Consider the function f(x) = x^2 ln(x).

Taking the derivative of f(x), we get:

f'(x) = 2x ln(x) + x

Setting f'(x) = 0 to find the critical points, we get:

2 ln(x) + 1 = 0

ln(x) = -1/2

x = e^(-1/2)

Note that f(x) is positive and decreasing for x > e^(-1/2). Therefore, we have:

∫1 [infinity] In(x) /x^2 dx = ∫e^(-1/2) [infinity] x^2 ln(x) /x^2 dx

= ∫e^(-1/2) [infinity] ln(x) dx

Since ln(x) is an increasing function, we know that:

∫e^(-1/2) [infinity] ln(x) dx is divergent by the integral test.

Therefore, the original improper integral:

∫1 [infinity] In(x) /x^2 dx

is also divergent by comparison to the divergent integral.

In conclusion, the given improper integral diverges.

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The expression that would result in a rational number would be C ( 5  1/9 ) ( - 0. 3 bar ) .

The laws of maths state that when two rational numbers multiply themselves, the result would be a rational number. We can therefore check if these numbers are rational.

Convert the mixed fraction (5 1/9) to an improper fraction:

5 1/9 = (5 x 9 + 1) / 9 = 46 / 9

This is an improper fraction.

- 0. 3 bar in fraction form is :

= - 1 / 3

It is rational.

This therefore means that ( 5  1/9 ) ( - 0. 3 bar ) gives a rational number.

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The integral of ∫1 to 0 (X-8/x²-7x+10 dx ) is undefined because the natural logarithm function is not defined for negative values.

First, we need to factor the denominator to get (x-2)(x-5). Then, we can use partial fraction decomposition to write the integrand as A/(x-2) + B/(x-5), where A and B are constants to be determined.

Multiplying both sides of the equation by (x-2)(x-5) and setting x = 2 and x = 5 gives the equations A = 2 and B = -1.

So, the integrand can be written as 2/(x-2) - 1/(x-5).

Integrating with respect to x, we get:

∫1 to 0 (X-8/x²-7x+10 dx ) = ∫1 to 0 (2/(x-2) - 1/(x-5) dx)

= 2ln|x-2| + ln|x-5| evaluated from x = 0 to x = 1.

= 2ln(-1) + ln(-4) - 2ln(-3) - ln(-4)

= undefined

The integral is undefined because the natural logarithm function is not defined for negative values.

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Answer:

71.6

Step-by-step explanation:

358 / 5 = 71.6

Using cylindrical coordinates,

(a) The volume of the region E is 96π/5 units.

(b) The centroid of E is (0, 0, 24/5) units.

For part (a), we first need to find the limits of integration. Since the paraboloid and the cone intersect at z = 16, we have the limits of integration for z as 2√(x² + y²) ≤ z ≤ 48 - x² - y². For ρ, we have 0 ≤ ρ ≤ 2z/√(2) = z√2, and for θ, we have 0 ≤ θ ≤ 2π. Thus, the integral to find the volume is:

V = ∫∫∫ E dV = ∫∫∫ zρ dz dρ dθ

where E is the region given in the problem. Evaluating this integral gives V = (256/15)π.

For part (b), we use the formulas for the centroid in cylindrical coordinates:

x = (1/M) ∫∫∫ E ρ cosθ z dV

y = (1/M) ∫∫∫ E ρ sinθ z dV

z = (1/2M) ∫∫∫ E (ρ² - z²) dV

where M is the mass of the solid. Since the density is constant, M is proportional to the volume, so we can find the centroid by evaluating the integrals without dividing by M. Evaluating these integrals gives (x, y, z) = (0, 0, 16/5).

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Use cylindrical coordinates.

(a) Find the volume of the region E that lies between the paraboloid z = 48 - x² - y² and the cone z = 2 √(x² + y²).

(b) Find the centroid of E (the center of mass in the case where the density is constant) (x, y, z) = ______.

The difference between the linear and projectile motion is that "Linear-motion" refers to motion of object in a "straight-line", while "projectile-motion" refers to motion of an object that is thrown into air, in a curved path.

In linear motion, the object moves along a straight line, with its velocity and acceleration aligned in the same direction. The object's speed and direction may change, but its motion remains linear.

In projectile motion, the object moves along a curved path under the influence of gravity. The object is launched into the air with an initial velocity, and then gravity causes it to follow a parabolic path until it lands back on the ground. The motion of the object is influenced by both its initial velocity and the force of gravity.

In both linear and projectile motion, the "horizontal-component" of velocity will usually remain constant because there is no external force acting on the object in horizontal direction, and thus no acceleration.

Therefore, the object will continue to move at a constant velocity in the horizontal direction, as long as there is no external force acting on it.

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