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The solutions of provide logarithmic equations are present in below :

1) x = 9 ; 2) x = 0 ; 3)p = 7/9 ; 4) k= 2 ; 5) a= -1 ; 6) r = -1/2 ; 7) n = 2 ; 8) x = 1/35 ; 9) m = -2 ; 10) x = 84 ; 11) v = -6, -6 ; 12) x = -4, -7

The logarithmic number is associated with exponent and power, such that if xⁿ = m, then it is equal to logₓ m = n. That is exponential value are inverse of logarithm values. Some basic properties of logarithmic numbers:

Now, we solve each logarithm equation one by one. Assume that 'log' is the base-10 logarithm where absence of base.

1) log (5x) = log (2x + 9)

Exponentiate both sides

=> 5x = 2x + 9

=> 3x = 9

=> x = 9

2) log (10 − 4x) = log (10 − 3x)

Exponentiate both sides,

=> 10 - 4x = 10 - 3x

simplify, => x = 0

3) log (4p − 2) = log (−5p + 5)

Exponentiate both sides,

=> 4p - 2 = - 5p + 5

simplify, => 9p = 7

=> p = 7/9

4) log (4k − 5) = log (2k − 1)

Exponentiate both sides,

=> 4k - 5 = 2k - 1

simplify, => 2k = 4

=> k = 2

5) log (−2a + 9) = log (7 − 4a)

Exponentiate both sides,

=> - 2a + 9 = 7 - 4a

simplify, => 2a = -2

=> a = -1

6) 2log₇( −2r) = 0

=> log₇( −2r) = 0

using the property, log꜀a = n <=> cⁿ = a

=> ( 7⁰) = - 2r

=> -2 × r = 1 ( since a⁰ = 1 )

=> r = -1/2

7) −10 + log₃(n + 3) = −10

=> log₃(n + 3) = −10 + 10 = 0

using the property, log꜀a = n <=> cⁿ = a

=> 3⁰ = n + 3

=> 1 = n + 3

=> n = 2

8) −2log₅ ( 7x ) = 2

=> log₅ 7x = -1

=> 5⁻¹ = 7x

=> x = 1/35

9) log( −m) + 2 = 4

=> log( −m) = 2

Exponentiate both sides,

=> -m = 2

=> m = -2

10) −6log₃ (x − 3) = −24

simplify, log₃ (x − 3) = 4

=> (x - 3) = 3⁴ ( since log꜀a = n <=> cⁿ = a )

=> x - 3 = 81

=> x = 84

11) log₁₂ (v²+ 35) = log₁₂ (−12v − 1)

=> log₁₂ (v²+ 35) - log₁₂ (−12v − 1) = 0

Using the quotient property of logarithm,

[tex]log_{12}( \frac{v²+ 35}{-12v-1}) = 0 [/tex]

[tex]\frac{v²+ 35}{-12v - 1} = {12}^{0} = 1 [/tex]

[tex]v²+ 35 = −12v − 1[/tex]

[tex]v²+ 35 + 12v + 1 = 0[/tex]

[tex]v²+12v + 36 = 0[/tex]

which is a quadratic equation, and solve it by middle term splitting method,

[tex]v²+ 6v + 6v + 36= 0[/tex]

[tex]v(v + 6) + 6(v + 6)= 0[/tex]

[tex](v + 6) (6 + v)= 0[/tex]

so, v = -6, -6

12) log₉(−11x + 2) = log₉ (x²+ 30)

=> log₉ (x²+ 30) - log₉(−11x + 2) = 0

Using the quotient property of logarithm,

[tex]log₉(\frac{x²+ 30 }{−11x + 2}) = 0[/tex]

[tex] \frac{x²+ 30}{-11x + 2} ={9}^{0} = 1 [/tex]

=> x² + 30 = - 11x + 2

=> x² + 11x + 30 -2 = 0

=> x² + 11x + 28 = 0

Factorize using middle term splitting,

=> x² + 7x + 4x + 28 = 0

=> x( x + 7) + 4( x + 7) = 0

=> ( x + 4) (x+7) = 0

=> either x = -4 or x = -7

Hence, required solution is x = -4, -7.

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Complete question:

kuta software infinite algebra 2 logarithmic equationsSolve each equation.

1) log 5x = log (2x + 9)

2) log (10 − 4x) = log (10 − 3x)

3) log (4 − 2) = log (−5 p + 5)

4) log (4k − 5) = log (2k − 1)

5) log (−2a + 9) = log (7 − 4a)

6) 2log₇ −2r = 0

7) −10 + log₃(n + 3) = −10

8) −2log₅ 7x = 2

9) log −m + 2 = 4

10) −6log₃ (x − 3) = −24

11) log₁₂ (v²+ 35) = log₁₂ (−12v − 1)

12) log₉(−11x + 2) = log₉ (x²+ 30)

The probability that the second child born to the couple will have blood type AB is 1/2.

Blood grouping is the categorization of blood into groups depending on the type of antigen and antibody present on the surface of red blood cells (RBCs).

The ABO blood grouping system, which includes A, B, AB, and O blood types, is the most well-known blood grouping system.

Blood group Rh (Rhesus) grouping system is another well-known blood grouping system that can be determined.

Therefore, the probability that the second child born to the couple will have blood type AB is 1/2.

50 percent of the offspring of this couple will inherit the gene for the ABO blood type A from the mother, while the remaining 50% will inherit the gene for the blood type B from the father.

The probability of inheriting the gene for blood type AB is thus 1/2 for a child born to these parents.

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Answer:

Second choice
∠BCA ≅ ∠DCA

Step-by-step explanation:

This logically follows from the fact that both ∠BCA and ∠DCA are right angles from the previous step. And the reason given is "All right angles are ≅)

So they are congruent

In the following question, option B, Juan McDonald is willing to pay $900 for a new iPad. he offers to pay $800 for an iPad at the apple store. it costs apple $700 to produce this iPad. A voluntary economic transaction between "Juan and Apple" will occur because it would be better off due to the transaction.

What is a voluntary economic transaction? Voluntary economic transactions involve willing buyers and willing sellers, and they usually involve the exchange of goods and services in return for money. In a free-market economy, people have the freedom to make voluntary economic transactions, and government intervention is minimal. What are the benefits of voluntary economic transactions? The benefits of voluntary economic transactions include the following: They are beneficial to both parties involved in the transaction.

The buyer obtains what they require, while the seller obtains the money they require. The exchange of goods and services for money encourages individuals to create, produce, and sell more. The economy is stimulated by increased economic activity, which leads to more job creation, more employment opportunities, and more revenue for the government. Juan McDonald is willing to pay $900 for a new iPad. He offers to pay $800 for an iPad at the Apple store. It costs Apple $700 to produce this iPad.A voluntary economic transaction between Juan and Apple will occur because it would be better off due to the transaction. So, the answer is option B: will; both Juan and Apple.

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After 5 hours, the tank will have 50 gallons of water.

To solve this problem, use the following equation:

Gallons = Rate x Time

In this case, the rate is 5 gallons per hour and the time is 5 hours, so the equation becomes:

Gallons = 5 x 5

Therefore, the tank will have 50 gallons of water after 5 hours.

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Answer:

Step-by-step explanation:

The 90% confidence interval for [tex]$\mu_W - \mu_M$[/tex] is approximately [tex]$-13.2 \pm 27.1$[/tex], or (-40.3, 13.9).

The answer is (c) [tex]13.2 \pm 33.1[/tex], which is not correct.

Which is the confidence interval?

A confidence interval is a range around a measurement that conveys how precise the measurement is.

We can use the two-sample t-interval formula to find the confidence interval for the difference in means:

[tex]$\bar{x}_W - \bar{x}M \pm t{\alpha/2, \nu} \sqrt{\frac{s_W^2}{n_W} + \frac{s_M^2}{n_M}}$[/tex]

where [tex]$\bar{x}_{W}$[/tex] and [tex]$\bar{x}M$[/tex] are the sample means, [tex]$s_W$[/tex] and [tex]$s_M$[/tex] are the sample standard deviations, [tex]$n_W$[/tex] and [tex]$n_M$[/tex] are the sample sizes, [tex]$\nu$[/tex] is the degrees of freedom, and [tex]$t{\alpha/2, \nu}$[/tex] is the critical value from the t-distribution with [tex]$\nu$[/tex]degrees of freedom and a level of significance of [tex]$\alpha=0.1$[/tex] (since we want a 90% confidence interval, which corresponds to a 10% level of significance).

Plugging in the values, we get:

[tex]$\bar{x}_W - \bar{x}M \pm t{0.05/2, 23} \sqrt{\frac{s_W^2}{n_W} + \frac{s_M^2}{n_M}}$[/tex]

[tex]$\begin{aligned} &= 213.6 - 226.8 \pm t_{0.025, 23} \sqrt{\frac{45.3^2}{14} + \frac{49.4^2}{11}} \ &= -13.2 \pm 2.074 \times 13.102 \ &= -13.2 \pm 27.115 \end{aligned}$[/tex]

Therefore, the 90% confidence interval for [tex]$\mu_W - \mu_M$[/tex] is approximately [tex]$-13.2 \pm 27.1$[/tex], or (-40.3, 13.9).

The answer is (c) [tex]13.2 \pm 33.1[/tex], which is not correct.

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complete question:

A phlebotomist measures the cholesterol levels of a sample of 25 people between

the ages of 35 and 44 years old. Here are summary statistics for the samples.

Cholesterol levels (mg per 100mL)

Women 35-44 years old

Men 35-44 years old

Sample mean

tilde x_{W} = 213.6

overline x_{M} = 226.8

Sample standard deviation

s_{W} = 45.3

s_{M} = 49.4

Sample size

n_{W} = 14

nu = 11

Assume that the conditions for inference have been met. Let mu_{w} ^ (- mu_{u}) be the difference in mean

cholesterol levels (in milligrams per 100 ml) in women and men of those ages.

Which of the following is a 90% confidence interval for mu_{W} ^ (- mu_{M})

Use a calculator to calculate the interval. (State your calculator settings for partial credit)

(a) - 13.2 plus/minus 33.1

(b) -13.2±29.7

(c) 13.2 plus/minus 33.1

(d) 13.2 plus/minus 29.7

(e) 13 plus/minus 33.1

Answer:

-13.2+- 33.1

Step-by-step explanation:

As we have defined the Matlab function called myta which takes four arguments in the form myta(f,n,a,b).

The purpose of the function is to find the nth Taylor polynomial of the function f(x) at x = a and evaluate it at x = b. Then, it should return the absolute value of the difference between this value and f(b).

Now that we have the nth Taylor polynomial of f(x) at x = a, we can evaluate it at x = b and calculate the absolute difference between this value and f(b).

function result = myta(f,n,a,b)

   syms x; % define x as symbolic variable

   g = f(a); % initialize g as f(a)

   for i=1:n % iterate from 1 to n

       deriv = diff(f,x,i-1); % calculate the ith derivative of f

       term = deriv*(x-a)^(i-1)/factorial(i-1); % calculate the ith term of the Taylor series

       g = g + term; % add the ith term to g

   end

   result = abs(g - f(b)); % calculate the absolute difference between g(b) and f(b)

end

This code calculates the absolute difference between g(b) and f(b) using the "abs" function and assigns it to the output variable "result".

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Answer:

14

Step-by-step explanation:

scores  10 10 12 13 14 15 18 19 19     Median score is the one in the middle

a) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

Exponents are mathematical notation used to indicate that a quantity is being multiplied by itself a certain number of times. The exponent is usually written as a superscript to the right of the base number.For example, in the expression 2³, the base number is 2 and the exponent is 3. This means that 2 is being multiplied by itself three times, resulting in a value of 8. Exponents can also be negative or fractional, indicating that the base number is being divided by itself a certain number of times.

In the given question,

a) The value of -42 is different from the value of (-4)². The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The value of -23 is different from the value of (-2)³. The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

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In summary, when repeated samples of size n are taken from a normal population with an unknown variance, the statistic t follows the t-distribution with n-1 degrees of freedom.

If repeated samples of size n are taken from a normal population with an unknown variance, then the statistic t follows the t distribution with n-1 degrees of freedom. When a sample of size n is taken from a normal population with a known variance and mean, the sample mean follows a normal distribution with a mean of μ and a variance of σ2/n. This distribution is known as the sampling distribution of the mean.

However, when the variance is unknown, the sampling distribution of the mean can no longer be calculated using the normal distribution. In this scenario, the sample mean is calculated using a t-distribution instead of a standard normal distribution.The t-distribution is similar to the standard normal distribution. However, it is more spread out and flatter than the standard normal distribution. As a result, the t-distribution has thicker tails than the standard normal distribution, which makes it more suitable for analyzing small samples of data.

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Answer:

Step-by-step explanation:

(52 divided by 80) multiplied by 100
that will give you the answer

On the material was used, the 100-meter flex of 14 AWA wire's entirety would range. If the silver and copper cable are in 0.0013 and 0.0014, aluminum spans 0.002 and 0.004, and iron is between 0.007 and 0.008

A force that works to slow something down or halt its progress: The air/wind drag slowed the vehicle down. the extent to which a material obstructs an electric charge from passing through it: There is little reluctance in copper.

The resistance (R) of a wire is given by the formula:

R = (ρ x L) / A

where:

ρ is the resistivity of the material

L is the length of the wire

A is the wire's cross-sectional size.

The cross-sectional area (A) of a wire with diameter d is given by the formula:

A = π x (d/2)²

where pi is a number in mathematics. (approximately equal to 3.14).

For a 100-meter length of 14 AWA wire with diameter 0.163 cm, we first need to convert the diameter to meters:

d = 0.163 cm = 0.00163 m

The cross-sectional area of the wire is then:

A = π x (0.00163/2)² = 2.087 x 10⁻⁶ m²

Using this value of A and the given resistivities, we can calculate the resistance for each material:

For copper:

R_copper = (1.67 x 10⁻⁸ O•m x 100 m) / (2.087 x 10⁻⁶ m²) = 0.00134 Ω

For silver:

R_silver = (1.59 x 10⁻⁸ O•m x 100 m) / (2.087 x 10⁻⁶ m²) = 0.00128 Ω

For aluminum:

R_aluminum = (2.65 x 10⁻⁸ O•m x 100 m) / (2.087 x 10⁶ m²) = 0.00199 Ω

For iron:

R_iron = (9.71 x 10⁻⁸ O•m x 100 m) / (2.087 x 10⁻⁶ m²) = 0.00735 Ω

Therefore, the overall resistance of the 100-meter length of 14 AWA wire made of these materials would depend on which material was used. If copper or silver were used, the resistance would be relatively low, around 0.0013-0.0014 Ω. If aluminum were used, the resistance would be higher, around 0.002 Ω. If iron were used, the resistance would be much higher, around 0.007 Ω.

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Answer: First one (the one you selected in the photo)

Step-by-step explanation:

If you need help draw a number line.

-8 is not equal to 8, as one is negative and the other is positive.

On a number line, -8 is to the left, meaning it is less, so the second one is false.

This means the first one is correct.

The value of z that should be used when making a 93% confidence interval for p is Z=1.81 that is option D.

The z score is a measure of how many standard deviations a raw score is below or above the population mean. It will be positive if the value is more than the mean and negative if it is less than the mean. It is often referred to as the standard score. It represents the number of standard deviations an entity has from the mean.

To utilise a z-score, both the mean and the population standard deviation must be known. The z score calculates the likelihood of a score occuring within a standard normal distribution. It also allows us to compare scores from various samples. A table for the values of, representing the values of the cumulative distribution function of the normal distribution, is known as a

At 93% confidence, the significance level is,

ɑ= 1-0.93

ɑ= 0.07

Divide alpha by 2,

ɑ/2= 0.07/2= 0.035

Now, from the ‘normal probability’ table, the z value corresponding to the inverse probability of 0.035 is 1.81.

As a result, the z value is 1.81 at 93% confidence level.

z= 1.81

The value for z score is 1.81

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Answer:

y= (5/3)x -4/3

Step-by-step explanation:

Slope obtained from the two points of the graph.

m=5/3

Points:  A(-1,-3).  B(2,2)

m=(y2-y1)/(x2-x1)

m=(2-(-3))/(2-(-1))

m=5/3

Replacing in the equation form: y=mx+b

y=(5/3)x+b

2=(5/3)2+b

2=10/3 +b

2-10/3= b

6/3 - 10/3 = b

b= -4/3

Joining all the terms we obtain:

y= (5/3)x -4/3

Answer: The slope is 5/3

Step-by-step explanation:

the poin at (2,2) is 5 up and 3 right of the point at (

-1,-3)

The solution of the system goes to infinity as t gets large.

Consider the given system of differential equations: dx/dt = 3x + 4ydy/dt = −x − 2yNow, find the solution of the system:(2)   dx/dt + 2 dy/dt = 7xObserve the second equation and multiply it by 2:2  dy / dt = −2x − 4y(3)   dx/dt + 2 dy/dt = 3x − 4ySubstitute the value of dy/dt from (3) into (2):(2)   dx/dt + 3x − 4y = 7xSimplify this equation:dx/dt − 4x = 4yTake the laplace transform of both sides: Laplace{dx/dt − 4x} = Laplace{4y} ⇒ sX(s) − x(0) − 4X(s) = 4Y(s) ⇒ X(s) = {4Y(s)}/{s − 4}Now, find the Laplace transform of y(s):(1)   dy/dt = −x − 2y ⇒ Laplace{dy/dt} = −Laplace{x} − 2Laplace{y} ⇒ sY(s) − y(0) = −X(s) − 2Y(s) ⇒ Y(s) = {−X(s)}/{s + 2}Substitute the value of X(s) in the above equation, we get:Y(s) = {−4Y(s)}/[(s − 4)(s + 2)]Simplify this equation:Y(s) = {A}/{s − 4} + {B}/{s + 2}Now, find the values of A and B:Multiplying the above equation by (s − 4) and (s + 2), we get:Y(s) = A(s + 2) + B(s − 4)Simplify this equation:Y(s) = (A + B)s + 2A − 4BAs per the equation (3), we can say A + B = 0, 2A − 4B = −4On solving, we get A = 2, B = −2Therefore, the value of Y(s) is:Y(s) = {2}/{s − 4} − {2}/{s + 2}Now, substitute the value of X(s) and Y(s) in the following equation:dx/dt − 4x = 4yThe solution of the above differential equation is:x(t) = {2}/{3}e^{4t} − {5}/{3}e^{-2t}The above is the general solution of the given system of differential equations.Now, find what happens to the system as t gets large?As t gets large, the term {5}/{3}e^{-2t} will tend to zero and {2}/{3}e^{4t} will tend to infinity.

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Answer:

-1, 2, 5 , 8 , 11   or  2, 5, 8, 11 , 14

Step-by-step explanation:

Assume n starts from 0

a(0) = 3 (0) -1 = -1
a(1) = 3 (1) -1 = 2
a(2) = 3 (2) -1 = 5
a(3) = 3 (3) -1 = 8
a(4) = 3 (4) -1 = 11

notice If we Assume n starts from 1

a(1) = 3 (1) -1 = 2
a(2) = 3 (2) -1 = 5
a(3) = 3 (3) -1 = 8
a(4) = 3 (4) -1 = 11
a(5)= 3 (5) -1 = 14

Notice a sequence can start from any N. But the most common ones are n=0 or n=1

After running the given code, Karel will be at Street 2 and Avenue 2, facing North.

Here is a step-by-step explanation of what the code does:

   move(); - Karel moves one block east, to Street 1 and Avenue 2.

   turnLeft(); - Karel turns left to face north.

   putBall(); - Karel puts a ball at Street 1 and Avenue 2.

   turnLeft(); turnLeft(); turnLeft(); - Karel turns left three times to face south.

   move(); - Karel moves one block south to Street 2 and Avenue 2.

   turnLeft(); - Karel turns left to face east.

So after executing this code, Karel will be at Street 2 and Avenue 2, facing North.

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when we Roll a fair 6 66-sided die. then the probability of P (not 5) is 65/66

     The concept used a die is a tool used for games and gambling. The die is a cube with six faces, each of which has a different number of dots from one to six. The roll of a die is a random event, which means that it is unpredictable, and each roll is independent of any other roll.

 Let's solve the question that is P(not 5)

                 Since the die is fair, we have 66 sides on the die. Each of the sides has a probability of 1/66 of being rolled. P(not 5) means that we need to determine the probability of rolling a number that is not 5.

           We know that a 66-sided die has 65 numbers other than 5.

     Therefore, the probability of rolling a number that is not 5 is 65/66.

            P(not 5) = 65/66

   So, the probability of not rolling a 5 is p(not 5) = 65/66.

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Answer:

[tex]x = 4\sqrt{3}[/tex]

Step-by-step explanation:

The triangle is right-angled

In a right-angled triangle, the following equality holds

[tex]\tan x = \dfrac{\text{Side Opposite x}}{\text{Side adjacent to x}}\\\\\tan 30 = \dfrac{4}{x}\\\\[/tex]

[tex]\tan 30 = \dfrac{1}{\sqrt{3}}\\\\\dfrac{1}{\sqrt{3}} = \dfrac{4}{x}\\\\[/tex]

Cross multiply:
[tex]1 \times x = 4 \times \sqrt{3}\\\\x = 4\sqrt{3}[/tex]

The differential equation by variation of parameters, subject to the initial conditions y(0) can be found by homogeneous equation and by integrating it.

The solution to the differential equation by variation of parameters, subject to the initial conditions y(0), can be found by following these steps:
1. Separate the equation into homogeneous and particular parts.
2. Solve the homogeneous equation for its particular solution.
3. Construct a new auxiliary equation from the particular solution, integrating the particular solution with respect to the parameters.
4. Solve the auxiliary equation, again with respect to the parameters, and then integrate the result with respect to x.
5. Substitute the particular solution and the new parameters back into the original equation to solve it completely.
6. Substitute the initial condition y(0) to solve for the constants of integration.

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It makes sense that 0 is the domain of y = √x because the square root of any number can never be negative, and 0 is the smallest positive number. Therefore, any values of x less than 0 would result in an undefined answer.

Answer:

[tex]\textsf{$\boxed{\checkmark}\;\;y$-value\;of\;vertex\;is\;$-1$}[/tex]

[tex]\textsf{$\boxed{\checkmark}$\;\;Minimum\;value\;occurs\;at\;$y = -1$}[/tex]

Step-by-step explanation:

Given equation:

[tex]y=x^2+8x+15[/tex]

As the given equation is quadratic with a positive leading coefficient, it is a parabola that opens upwards. Therefore, its vertex is its minimum point. This means that the minimum value of the range is the y-value of the vertex.

The x-value of the vertex of a parabola in the form y = ax² + bx + c is x = -b/2a. Therefore, the x-value of the vertex of the given equation is:

[tex]\implies x=\dfrac{-8}{2(1)}=-4[/tex]

To find the y-value of the vertex, substitute x = -4 into the equation:

[tex]\begin{aligned}\implies y&=(-4)^2+8(-4)+15\\&=16-32+15\\&=-16+15\\&=-1\end{aligned}[/tex]

Therefore, the minimum y-value of the function is y = -1, so the range is y ≥ -1.

Therefore, the following are true statement about the given equation:

Over the range [5, 7], the definite integral of f(x) = 1 / (x² + 10x + 25) is around -1/60.

To find the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7], we can use the following formula:

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

First, we need to find the antiderivative of f(x):

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx

To do this, we can use a technique called partial fraction decomposition:

1 / (x² + 10x + 25)

= A / (x + 5) + B / (x + 5)²

Multiplying both sides by the denominator (x² + 10x + 25), we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

A + B = 1

A + 1 = 1

A = 0

Therefore, the partial fraction decomposition of f(x) is:

1 / (x² + 10x + 25) = 1 / (x + 5)²

Now we can find the antiderivative:

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx = ∫ 1 / (x + 5)² dx

Using the substitution u = x + 5, du = dx, we get:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

where C is the constant of integration.

Now we can evaluate the definite integral over the interval [5, 7]:

∫[5,7] f(x) dx = F(7) - F(5)

∫[5,7] f(x) dx = [-1 / (7 + 5) + C] - [-1 / (5 + 5) + C]

∫[5,7] f(x) dx = [-1 / 12 + C] - [-1 / 10 + C]

∫[5,7] f(x) dx = -1 / 12 + C + 1 / 10 - C

∫[5,7] f(x) dx = -1 / 60

Therefore, the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7] is approximately -1/60.

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The value of m∠JKL is 34°

An angle formed by a tangent and a secant intersecting outside a circle is equal to one-half the difference of the measures of the intercepted arcs.

Based on the theorem above, we can say:

m∠JKL = 1/2 * (159 - 91)

m∠JKL = 1/2 * 68

m∠JKL = 34°

Therefore, the value of m∠JKL in the circle is 34°.

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At steady-state, the Laplace transform of the output is given by [tex]lim s->0 sC(s)[/tex]. If this limit exists, then the steady-state error is zero.

The transfer function of a plant with transfer function G(s) = 74109 is given.

We proceed in the following manner:

Step 1: Write the open-loop transfer function.[tex]G(s)D(s) = 74109 * (s+ci)/(s+c1) * K/(s+di)[/tex]

Step 2: Write the characteristic equation.(s + 6) (s + 3) (s2 + 3s + 9) = 0

Step 3: Compare the coefficients of the open-loop transfer function with the characteristic equation.

c1 + ci + di = 9c1ci + c1di + ci(di + 3) + K * 74109 = 27c1ci(di + 3) + K * 74109 * c1 = 81K * 74109 * di = 6 * 3 * 9 * (-74109)

Step 4: Solve for the parameters of the controller.The solution is obtained by solving the above equations.c2 = 18ci = 54co = 162di = 9

Step 5: Show that if the reference input to the system of the above exercise is a step of amplitude A, the steady-state error will be zero.The steady-state error can be calculated using the final value theorem. If the system is subjected to a step input of amplitude A, then the Laplace transform of the input is A/s.

The output of the system is given by[tex]C(s) = G(s)D(s) R(s)[/tex], where R(s) is the Laplace transform of the reference input. The steady-state error is given by the difference between the input and the output at steady-state. This can be calculated using the final value theorem.

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An assignment of probabilities to events in a sample space must obey all of the following: They must obey the addition rule for disjoint events, They must sum to 1 when adding over all events in the sample space, and The probability of any event must be a number between 0 and 1, inclusive. Hence, the correct option is All of the above.

Probability is the branch of mathematics that deals with the likelihood of a random event occurring. Probability is concerned with quantifying the probability of different results in a certain event.

The possibility that a specific event will occur is calculated using probability. Probability is calculated using several methods in mathematics, including axioms, probability spaces, events, random variables, and expectation values.

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The half-life of an isotope is the time by which there is a 50% probability that decay has occurred if Cobalt-60 has a half-life of 5.27 years then mean time to decay is 7.65 years , standard deviation of the decay time is 3.82 years , the 99th percentile is  36.4 years and  the mean and standard deviation of the total time the experiment will last is 6.61 years.

(a) The mean time to decay can be found using the formula: [tex]mean = half-life / ln(2)[/tex].

Therefore, for cobalt-60 with a half-life of 5.27 years, the mean time to decay is:

[tex]mean = 5.27 / ln(2) \approx7.65 years[/tex]

(b) The standard deviation of the decay time can be found using the formula:

standard deviation =    [tex]half-life /(\ sqrt{(ln(2))}).[/tex]

Therefore, for cobalt-60 with a half-life of 5.27 years, the standard deviation of the decay time is:

standard deviation = [tex]5.27 / (\sqrt{(ln(2))}) \approx 3.82 years[/tex]

(c) The 99th percentile can be found using the cumulative distribution function (CDF) of the exponential distribution. For cobalt-60 with a half-life of 5.27 years, the CDF is:

[tex]CDF(t) = 1 - e^{(-t/5.27)}[/tex]

Setting the CDF equal to 0.99 and solving for t, we get:

[tex]0.99 = 1 - e^{(-t/5.27)}[/tex]

[tex]e^{(-t/5.27)} = 0.01[/tex]

[tex]-t/5.27 = ln(0.01)[/tex]

[tex]t = -5.27 * ln(0.01)[/tex]

[tex]t\approx 36.4 years[/tex]

Therefore, the 99th percentile of the decay time for cobalt-60 is approximately 36.4 years.

(d) Three cobalt-60 atoms, then the total time the experiment will last is the sum of the decay times of each atom. Since the decay times are independent and identically distributed, the mean and standard deviation of the total time can be calculated by adding the means and variances of each individual decay time.

The mean of the total time is:

[tex]mean(total) = mean(atom1) + mean(atom2) + mean(atom3)[/tex]

[tex]mean(total) = 7.65 + 7.65 + 7.65[/tex]

[tex]mean(total) = 22.95 years[/tex]

The variance of the total time is:

[tex]variance(total) = variance(atom1) + variance(atom2) + variance(atom3)[/tex]

[tex]variance(total) = (3.82)^2 + (3.82)^2+ (3.82)^2[/tex]

[tex]variance(total) \approx43.67 years[/tex]

The standard deviation of the total time is the square root of the variance:

standard deviation(total) [tex]= \sqrt{(variance(total))}[/tex]

[tex]standard deviation(total) \approx6.61 years[/tex]

Therefore, the mean and standard deviation of the total time for observing three cobalt-60 atoms until they decay are approximately 22.95 years and 6.61 years, respectively.

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The volume of the sphere which is equivalent to the lung capacity is approximately =2,571 cm³

To calculate the volume of a sphere the formula used = V = 4/3 πr³

Radius = 8.5 cm

First cube the radius = 8.5³ = 614.125

The, multiply r³ by π = r³×π = 614.125× 3.14= 1928.3525

Take this answer and multiply it by 4 = 4×1928.3525= 7713.41

Last, divide this answer by 3 = 7713.41/3 = 2571.136666

Therefore the volume of the balloon = 2,571 cm³(approximately)

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