The height of a student above the ground on a circular Ferris Wheel can be modeled by the equation h = a sin (bt + c) + d, where h is the height in metres after t seconds.
The diameter of the wheel is 22 metres and the student starts at the bottom of the Ferris Wheel at t = 0 and at a height of 4 metres above the ground.
The Ferris Wheel reaches a maximum height for the first time after 45 seconds.
The value of d in the equation is

The value of t'(a) to the nearest tenth is 8.5. The first derivative of y(x) is y'(x) = 5e^x(2-x)/(2x^(3/2)). The first derivative of t(x) is t'(x) = [-5/4 x^(-7/4)sin(x/2)cos(x/2) - 2x^(-5)cos(x/2)]/[x^(5/4)cos(x/2)x^(-2)+8x+6]. The first derivative of r(x) is r'(x) = = [-2x^2 + 6x + 2 - sin x(2x+1)]/((x^2+x+4)^2 ln 6).

To find t'(a), we need to take the derivative of t(x) with respect to x and then evaluate it at x=a. Using the product rule and chain rule, we have:

t'(x) = 0.08e^x(x^2)' + 0.08e^x(2x)(ln 0.08)' = 0.08e^x(x^2+2xln0.08)

Substituting x=a=5, we get:

t'(a) = 0.08e^5(5^2+2(5)ln0.08) ≈ 8.54

Therefore, the value of t'(a)  is 8.5.

To find y'(x), we use the quotient rule:

y'(x) = [(5e^x)(√x) - (5ex/2)(1/2x^(3/2))]/x

Simplifying this expression, we get:

y'(x) = 5e^x(2-x)/(2x^(3/2))

Therefore, the first derivative of y(x) is 5e^x(2-x)/(2x^(3/2)).

To find t'(x), we use the chain rule and product rule:

t'(x) = [ln(x^(5/4)cos(x/2)x^(-2)+8x+6)]'

= [1/(x^(5/4)cos(x/2)x^(-2)+8x+6)] * [(5/4)x^(1/4)(cos(x/2)x^(-2))' + (-2x^(-3))(cos(x/2)x^(-2)) + 8 + 0]

= [1/(x^(5/4)cos(x/2)x^(-2)+8x+6)] * [-5/4 x^(-3/4)cos(x/2)x^(-2)sin(x/2) - 2x^(-3)cos(x/2)x^(-2)]

= [-5/4 x^(-7/4)sin(x/2)cos(x/2) - 2x^(-5)cos(x/2)]/[x^(5/4)cos(x/2)x^(-2)+8x+6]

Therefore, the first derivative of t(x) is  [-5/4 x^(-7/4)sin(x/2)cos(x/2) - 2x^(-5)cos(x/2)]/[x^(5/4)cos(x/2)x^(-2)+8x+6].

To find the first derivative of r(x), we use the quotient rule and chain rule:

r'(x) = [(1/log6)((2x+1)(x^2+x+4)' - (3x^2-sin x)(x^2+x+4))/((x^2+x+4)^2)]/ln(6)

= [(2x+1)(2x+1+1) - (3x^2-sin x)(2x+1)]/((x^2+x+4)^2 ln 6)

= [4x^2+6x+2 - 6x^2 - sin x(2x+1)]/((x^2+x+4)^2 ln 6)

= [-2x^2 + 6x + 2 - sin x(2x+1)]/((x^2+x+4)^2 ln 6)

Therefore, the first derivative of r(x) is  [-2x^2 + 6x + 2 - sin x(2x+1)]/((x^2+x+4)^2 ln 6).

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Answer:

D

Step-by-step explanation:

You are given the two sides and the abgle inbetween.

The closure of voltage-gated sodium channels allow the membrane potential of a neuron return to its original resting potential d)  d. Neuronal membranes are far more permeable to potassium than sodium when the membrane is at rest. As a result, the resting potential is close to the equilibrium potential of potassium.

The correct answer is d. Neuronal membranes are far more permeable to potassium than sodium when the membrane is at rest. As a result, the resting potential is close to the equilibrium potential of potassium.

The resting potential of a neuron is primarily determined by the balance of ion concentrations and the permeability of the neuronal membrane to different ions. At rest, the membrane is highly permeable to potassium ions (K+), which allows potassium to freely move in and out of the cell. This permeability is mainly due to the presence of leak channels for potassium.

In the absence of voltage-gated potassium channels, the closure of voltage-gated sodium channels would still allow sodium ions (Na+) to leak into the neuron, as stated in option a. However, the key point is that the leak of sodium alone is not sufficient to bring the membrane potential back to its original resting potential.

The primary factor in restoring the resting potential is the high permeability of the neuronal membrane to potassium ions. As the membrane potential depolarizes during an action potential, voltage-gated sodium channels open, allowing sodium ions to rush into the cell and contribute to depolarization. However, once the action potential reaches its peak and the voltage-gated sodium channels close, the high permeability of the membrane to potassium ions allows potassium to quickly move out of the cell, repolarizing the membrane and restoring the resting potential.

Option d correctly explains that the resting potential is close to the equilibrium potential of potassium because of the high permeability of the membrane to potassium ions. This permeability allows potassium to flow out of the cell, counteracting the depolarizing effects of sodium influx and bringing the membrane potential back to its resting state.

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To find the average value of a function from a graph, you can follow these steps:

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The value of m and n from the obtained probability is equal to 3 and 14 respectively.

As given in the question,

Nine tiles numbered 1, 2, .....9

Number of odd numbered tiles are 1,3,5,7,9 = 5

Number of even numbered tiles are 2,4,6,8 = 4

Number of players =3

Possibility to choose 3 tiles by first player out of 9 tiles = ⁹C₃

                                                                                            = 84 ways

Possibility to choose 3 tiles by second player out of 6 tiles = ⁶C₃

                                                                                                 =20 ways

Possibility to choose 3 tiles by third player out of 3 tiles = ³C₃

                                                                                             = 1 way

Total number of ways = 84× 20 ×1

                                     = 1680

To have a sum of 3 tiles is odd number at least one tile should be odd.

Chances of first player all the 3 tiles are 0dd out of 5 = ⁵C₃

Next two players must 2 even tiles and 1 odd tile.= ²C₁

One player can be selected in ³C₁ way

Remaining 4 even tiles distributed equally = (4!)(2!)/ (2!)² ( 2!)

Number of favourable outcomes =  ⁵C₃ × ²C₁ ׳C₁  × (4!)(2!)/ (2!)² ( 2!)

                                                       = 10 × 2 × 3 × 6

                                                       = 360

Probability (m/n) =  360 /1680

                            = 3/14

m = 3 and n = 14.

Therefore, from the probability that all three players obtain an odd sum is m/n , the value m = 3 and n = 14.

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Answer:

Positive

Step-by-step explanation:

\(\frac{2}{5}-(-\frac{5}{6})\)

Remember, a number minus the negative of a number would be equal to that first number plus that second number but not negative. In other words:

a - (-b) = a + b

So:

\(\frac{2}{5}-(-\frac{5}{6})=\frac{2}{5}+\frac{5}{6}\)

The sum of two positive numbers would be a positive number. Because of this, we can see that the value would be positive.

I hope you find my answer and explanation helpful. Happy studying. :)

Answer:

B. x =  -3 or x = 2

Step-by-step explanation:

Answer:

A least degree polynomial, having rational coefficients and a leading coefficient of 2, with,-4, 0, 2, and 4 as the zeros of the polynomial is;

f(x) = 2·x⁴ - 4·x³ - 32·x² + 64·x

Step-by-step explanation:

The given parameters of the polynomial are;

The leading coefficient of the polynomial = 2

The zeros of the polynomial = -4, 0, 2, 4

We note that zeros of -4, and 4 gives a factor of the form, (x² - 4²)

For a zero of the polynomial equal to 0, one of the factors of the polynomial is equal to 'x'

To have a leading coefficient of 2, we can add '2' as a factor of the polynomial

Therefore, we can have the factors of the polynomial as follows;

(x² - 4²)·2·x×(x - 2) = 0

From the above equation, using a graphing calculator, we get the following possible polynomial;

(x² - 4²)·2·x×(x - 2) = 2·x⁴ - 4·x³ - 32·x² + 64·x = 0

Therefore, a polynomial, function of least degree that has rational coefficients, a leading coefficient of 2,and the zeros, -4, 0, 2, and 4 is  f(x) = 2·x⁴ - 4·x³ - 32·x² + 64·x.

By using algebraic equation, it was found that Hiroshi spends 30 minutes on his math homework.

To find the amount of time Hiroshi spends on his math homework, we can set up an algebraic equation using the information given.

Let's assume Hiroshi spends "x" minutes on his math homework. We know that one-fourth of his total homework time is spent on math. Since he spends 30 minutes on history homework and 60 minutes on English homework, the total homework time is 30 + 60 + x.

Now, we can set up the equation:

1/4 * (30 + 60 + x) = x

To solve this equation, we can start by simplifying the left side:

1/4 * (90 + x) = x

Next, we can distribute 1/4 to the terms inside the parentheses:

(1/4) * 90 + (1/4) * x = x

Simplifying further, we get:

90/4 + x/4 = x

To eliminate the fraction, we can multiply the entire equation by 4:

90 + x = 4x

Now, we can solve for x by bringing all the x terms to one side:

90 = 4x - x

Combining like terms:

90 = 3x

Finally, divide both sides of the equation by 3 to solve for x:

x = 30

So, Hiroshi spends 30 minutes on his math homework.

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Best way to solve this is by using

\( \sqrt{s(s - a)(s - b)(s - c)} \)

\(where \: s = \frac{a + b + c}{2} \)

s=(12+8+17)/2

=18.5

using the formulae

area =43.5

In a binomial distribution, the number of ways of arranging all trials to be failures is 1. This means that there is only one way to have all trials result in failures, as each trial can only have one possible outcome - a failure.

1. In a binomial distribution, each trial can result in either a success or a failure. Let's assume that there are n trials in total.

2. To arrange all trials to be failures, we need to ensure that no trial results in a success. Therefore, for each trial, there is only one possible outcome - a failure.

3. The number of ways of arranging all trials to be failures can be calculated using combinations. In this case, we need to select 0 successes from n trials. The number of ways to do this is given by the combination formula: C(n, 0) = 1, where C represents the combination.

Therefore, the number of ways of arranging all trials to be failures in a binomial distribution is given by the combination formula, C(n, k) = n! / (k! * (n-k)!).

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The number of ways of arranging all trials to be failures in a binomial distribution is always 1. This is because there is only one way to have no successes in a set of trials.

Regardless of the number of trials or the probability of success, the number of ways of arranging all trials to be failures in a binomial distribution is always 1.

The number of ways of arranging all trials to be failures in a binomial distribution can be determined using the formula for the binomial coefficient.
The binomial coefficient, often denoted as nCk or n choose k, represents the number of ways to choose k items from a set of n items, without considering their order. In the context of a binomial distribution, n represents the total number of trials and k represents the number of successes (in this case, 0).
To find the number of ways of arranging all trials to be failures, we can use the binomial coefficient formula:
nCk = n! / (k!(n-k)!)
In this case, since we want all trials to be failures, k is equal to 0. Thus, the formula simplifies to:
nC0 = n! / (0!(n-0)!) = n! / (0! * n!) = 1

For example, if we have 5 trials and we want all of them to be failures, there is only one possible arrangement: FFFFF, where F represents a failure.

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The roots of the given equation is real and equal.

Given , 5\(x^{2} \\\) - 10x+ k=0

The quadratic equation is b² - 4ac = 0

Here, a= 5, b= -10, c= k

substitute in b² - 4ac = 0

(-10)² - 4 * 5* k =0

100 - 20k =0 , let this be equation (1)

100 = 20k

k = \(\frac{100}{20}\)

k = 5.

now, substitute  k= 5 in equation (1)

100 -20k = 0

100 - 20*5 = 0

100 - 100 = 0

Therefore, the given equation is real and equal .

The correct question is 5x² - 10x + k =0

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Answer:

0.375

Step-by-step explanation:

Answer:

decimal is 0.375.

Step-by-step explanation:

just divide 8 by 3

Hicksian demand functions are:x1** = 2P₁x₂ ; x₂** = P₂²

Utility function: u(x1+x2) = .5ln(x1) + .25ln(x₂) .The Marshallian demand functions are: x1* = - and x₂* = P₂.

The indirect utility function is found by substituting Marshallian demand functions into the utility function and solving for v(P₁, P₂, Y).u(x1*,x2*) = v(P₁,P₂,Y) ⇒ u(-, P₂) = v(P₁,P₂,Y) ⇒ .5ln(-) + .25ln(P₂) = v(P₁,P₂,Y) ⇒ v(P₁,P₂,Y) = - ∞ (as ln(-) is not defined)

Thus the indirect utility function is undefined.

Minimum expenditure function can be derived from the Marshallian demand function and prices of goods:

Exp = P₁x1* + P₂x2* = P₁(-) + P₂P₂ = -P₁ + P₂²

Minimum expenditure function is thus:

Exp = P₁(-) + P₂²

Hicksian demand functions can be derived from the utility function and prices of goods:

H1(x1, P1, P2, U) = x1*H2(x2, P1, P2, U) = x2*

Hicksian demand functions are:

x1** = 2P₁x₂

x₂** = P₂²

If there are no restrictions on the amount of money the consumer can spend, the Hicksian demand functions for x1 and x2 coincide with Marshallian demand functions.

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The given system with two coupled subsystems has an undamped natural frequency of 6.714 rad/s and a damping ratio of 0.3001.

The given system consists of two coupled subsystems: a rotational system and a field-controlled motor system. The rotational system is described by the equation of motion 30 dtdt​ + 10w = T(t), where T(t) is the torque applied by an electric motor. The motor system is modeled by the equation 0.001 dtdi​ + 5i = v(t), where i is the field current in amperes and v(t) is the voltage applied to the motor.

The damping ratio of the combined system can be determined by dividing the sum of the two time constants by the undamped natural frequency, i.e. ζ = (τ1 + τ2)ωn​. Given the time constants of the rotational and motor systems as 3 seconds and 0.001 seconds respectively, and the undamped natural frequency as ωn​ = 10 rad/s, we can calculate the damping ratio as ζ = (3 + 0.001) x 10 / 10 = 0.3001.

The combined system's undamped natural frequency is determined by solving the characteristic equation of the system, which is given by (30I + 10ωs)(0.001s + 5) = 0, where I is the identity matrix. This yields the roots s = -0.1667 ± 6.714i. The undamped natural frequency is therefore ωn​ = 6.714 rad/s.

In summary, the given system with two coupled subsystems has an undamped natural frequency of 6.714 rad/s and a damping ratio of 0.3001.

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Yes, X is binomial because it meets the four criteria for a binomial distribution. X, representing the number of games the team wins, is a binomial random variable.

Yes, X is binomial because it meets the four criteria for a binomial distribution:
1) there are a fixed number of trials (12 games),
2) each trial is independent (the outcome of one game does not affect the outcome of another game),
3) there are only two possible outcomes for each trial (win or lose), and
4) the probability of success (winning) is constant for each trial (assuming the team's ability does not change throughout the season).

Yes, X is a binomial random variable because it meets the criteria for a binomial experiment. The criteria are:

1. Fixed number of trials (n): The soccer team plays 12 games in its regular season.
2. Two possible outcomes: Each game can result in either a win or a loss.
3. Independent trials: The outcome of each game is independent of the outcomes of the other games.
4. Constant probability of success (p): The probability of winning a game remains the same for each game played.

Therefore, X, representing the number of games the team wins, is a binomial random variable.

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Answer:

625 is the correct answer :)

Step-by-step explanation:

hope this help  

The newspaper has 625 readers.

The percentage is calculated by dividing the required value by the total value and multiplying by 100.

Example:

Required percentage value = a

total value = b

Percentage = a/b x 100

Example:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

20% = 20/100 = 1/5

10% = 10/100 = 1/10

We have,

8% = 50 people

Multiply 100/8 on both sides,

100% = 100/8 x 50 people

100% = 625 people

Thus,

There are 625 readers.

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\(ANSWER\)

9 to the second power means :)

\(9 {}^{2} \\ = 9 \times 9 \\ = 81\)

~hope it helps~

The unit rate for each shirt if 24 shirts cost $259.36 as required to be determined is; $10.81.

It follows from the task content that the unit rate is to be determined if 24 shirts cost $259.36.

On this note, it follows that; the unit rate is given by the quotient;

Unit rate = Total cost / No. of units

Unit rate = 259.36 / 24

Unit rate = 10.80667.

Ultimately, the unit rate of the shirts as required is; $10.81.

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The value of the function f(x) = 3x - 5 when x = 2 is f(2) = 1

A function is a mathematical equation that shows the relationship between two variables.

Given the function f(x) = 3x - 5, we desire to find f(2).

To find f(2), we substitute x = 2 into the equation. So, we have that

f(x) = 3x - 5. So substituting x = 2 into the equation, w ehave

f(2) = 3(2) - 5

f(2) = 6 - 5

f(2) = 1

So, the value of the function f(x) = 3x - 5 when x = 2 is f(2) = 1

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The health club has 2 employees who work on lead generation. Each employee contacts leads for 20 hours a week and is paid $20 per hour. Approximately 8% of the leads become members and pay a one-time fee of $100. Material costs are $190 per week, and overhead costs are $1,100 per week. To analyze the productivity of the operation, we need to calculate the multifactor productivity in fees generated per dollar of input. The owner is also considering purchasing a new software program that would allow each employee to contact 20 more leads per week, but it would increase material costs by $260 per week. We need to calculate the new multifactor productivity if the software is purchased and determine how it would affect productivity.

a. To calculate the multifactor productivity, we need to determine the total fees generated and the total input costs. The total fees generated per week can be calculated as 8% of the total number of leads contacted multiplied by the one-time fee of $100, which is (0.08 * 200) * $100 = $1,600. The total input costs per week are the sum of employee wages, material costs, and overhead costs, which is (2 employees * 20 hours/week * $20/hour) + $190 + $1,100 = $2,490. Therefore, the multifactor productivity is $1,600 / $2,490 = 0.64.

b. If the owner purchases the software program and each employee can contact 20 more leads per week, the total number of leads contacted per week by both employees will be 2 * (200 + 20) = 440. The new material costs per week will be $190 + $260 = $450. The overhead costs remain the same at $1,100. The total input costs per week become (2 employees * 20 hours/week * $20/hour) + $450 + $1,100 = $1,650. The new multifactor productivity is $1,600 / $1,650 = 0.97.

c. The new multifactor productivity after purchasing the software program has increased to 0.97 from the previous value of 0.64. The change in productivity can be calculated as ((0.97 - 0.64) / 0.64) * 100 = 51.6%. Therefore, purchasing the software program would increase productivity by approximately 51.6%.

By analyzing the multifactor productivity and the impact of purchasing the software program, the owner can make an informed decision about whether the investment is worthwhile considering the potential increase in productivity.

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The line perpendicular to the line y=-3/4x+5 that includes the point (-3,-3) is y=-4/3x-7.

The general equation of a straight line is y=mx+c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis.

The given equation of a line is y=-3/4x+5 and the coordinate point is (-3, -3)

Here, slope of a line is -3/4.

The slope of a line perpendicular to given line is m1=-1/m2

Now, the slope of a line is -4/3

Substitute m= -4/3 and (x, y) =(-3, -3) in y=mx+c, we get

-3=-4/3 (-3)+c

c=-7

Substitute m= -4/3 and c=-7 in y=mx+c, we get

y=-4/3x-7

Therefore, the line perpendicular to the line y=-3/4x+5 that includes the point (-3,-3) is y=-4/3x-7.

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Answer:

10%=1500

so 100% = 15000

90%= 100% - 10%

15000-1500=13500

90%=13500

Answer:w

Step-by-step explanation:

The value of the linear equation after adding is -5x -8y = -5.

According to the statement

We have to find that the value of the linear equation.

So, For this purpose, we know that the

The linear equation, statement that a first-degree polynomial—that is, the sum of a set of terms, each of which is the product of a constant and the first power of a variable—is equal to a constant.

From the given information:

The given equations are:

4x-4y = -2

-9x-4y = -3.

Then

Now, add these equations then

4x-4y = -2 + -9x-4y = -3.

4x-9x-4y-4y = -2-3

-5x -8y = -5

After addition ,the value becomes -5x -8y = -5.

So, The value of the linear equation after adding is -5x -8y = -5.

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The amount of money we can make is $0.05.

We have,

P= $710

R= 5.1%

T= 12 year

Compounded Continuously:

A = P\(e^{rt\)

A = 710.00(2.71828\()^{(0.051)(12)\)

A = $1,309.32

Compounded Daily:

A = P(1 + r/n\()^{nt\)

A = 710.00(1 + 0.051/365\()^{(365)(12)\)

A = 710.00(1 + 0.00013972602739726\()^{(4380)\)

A = $1,309.27

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∠1 and ∠6 and ∠4 and ∠5  are adjacent angles in the figure.

What is meant by adjacent angles?

When an angle's vertex and side are shared by another angle, the two angles are said to be neighboring angles. The vertex of an angle is the point at which the rays that make up its sides come to an end. When two adjacent angles share a vertex and a side, they can be either complimentary or supplementary angles.

                                  The two items are next to one another if one object is next to the other. He waited in a room next to it. There were separate doors despite the schools being close by. The sum of supplementary neighboring angles is always 180. The reason for this is that all angles on a straight line add up to 180, and the two angles are adjacent to one another on the line.

In the figure,

  adjacent angles are

∠1 and ∠6 ,

∠4 and ∠5

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