Estimate the difference by first rounding each number to the nearest thousand. 18 000 - 2351 - 1987 - 2416 is about ?

The answer of the given question based on the trigonometric identity is , the correct answer is D. +-5/4.

A trigonometric identity is an equation that is true for all values of the variables in the equation, where the variables are angles of a right triangle. These identities are used to simplify trigonometric expressions and solve trigonometric equations. Some common trigonometric identities include the Pythagorean identity, the reciprocal identities, the quotient identities, the even/odd identities, and the sum/difference identities.

To find the value of secθ given that cscθ is 5/3, we can use the following trigonometric identity:

secθ = 1/cosθ

We can start by finding the value of cosθ using the given value of cscθ:

cscθ = 5/3

Reciprocal of cscθ is sinθ:

sinθ = 1/cscθ = 1/(5/3) = 3/5

We know that sinθ = 1/cscθ and cosθ = √(1 - sin²θ) from the Pythagorean identity.

Plugging in the value of sinθ, we get:

cosθ = √(1 - sin²θ) = √(1 - (3/5)²) = √(1 - 9/25) = √(16/25) = 4/5

Now, we can substitute the value of cosθ into the formula for secθ:

secθ = 1/cosθ = 1/(4/5) = 5/4

Therefore, the correct answer is D. +-5/4.

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A random sample 50 employees yielded a mean of 2.79 years that employees stay in the company and o- 76. We test for the nut hypothesis that the population mean is less or equal than.inst the alternative hypothesis that the population mean is greater than 3 At a significance leve 0.01, we have enough evidence that the average time is less than 3 years is true.

To determine whether the null hypothesis (population mean <= 3) can be rejected in favor of the alternative hypothesis (population mean > 3) at a significance level of 0.01, we can conduct a one-sample t-test.

The test statistic is calculated as follows:
t = (sample mean - hypothesized population mean) / (sample standard deviation / sqrt(sample size))
Plugging in the given values, we get:
t = (2.79 - 3) / (0.76 / sqrt(50))
t = -2.12
The degrees of freedom for this test is 49 (sample size - 1). Using a t-distribution table with 49 degrees of freedom and a one-tailed test at a significance level of 0.01, we find a critical value of 2.405. Since our calculated t-value (-2.12) is less than the critical value (-2.405), we can reject the null hypothesis in favor of the alternative hypothesis.

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A strategy to find the total area of water that will be sprayed by the four nozzles is to first find the total arc measure covered by the four nozzles and then find the fraction of the circle covered by the sprayed water.

The sum arc measure of the water spray in each sector, produced by all four nozzles, totals to 100 degrees. To calculate what portion of the circular fountain is covered by the sprayed water, divide this value by the circle's 360-degree total. For convenience, let us define r as the radius of the fountain. Then find the area of the circle.

Next, expand your knowledge of the coverage area further and multiply the fraction by the entire circular fountain's span to find the precise square footage. In turn, determining the wet sides' length requires assessing the circumference of the entire structure and multiplying it again by the fractional width measured earlier from the nozzle spray.

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The area of the square is increasing at a rate of 40 cm²/s when the area is 16 cm².

We are required to determine the rate at which the area of a square is increasing when each side is increasing at 5 cm/s and the area is 16 cm².

First, let's establish some variables:

Let s be the side length of the square
Let A be the area of the square
ds/dt is the rate at which the side length is increasing, which is given as 5 cm/s
dA/dt is the rate at which the area is increasing, which we need to find

The area of a square is given by the formula:

A = s².

Now, we can differentiate both sides with respect to time (t):

dA/dt = 2s * (ds/dt)

We know that the area A is 16 cm². Since A = s², we can find the side length s:

s² = 16

s = 4 cm

Now, plug the values of s and ds/dt into the equation we derived:

dA/dt = 2 * 4 * 5

dA/dt = 40 cm²/s

The rate of change for the area is 40 cm²/s.

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The length will be 375 m.

The width will be 250 m.

What is perimeter?

The complete length of a shape's edge serves as its perimeter in geometric terms. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet.

let the length be x and the width be w

The perimeter will be:

2x+3w=1500

thus

3w=(1500-2x)

w=(1500-2x)/3

w=500-2/3x

The area will be:

A=x*w

A=x(500-2/3x)

A=500x-(2/3)x²

The above is a quadratic equation; thus finding the axis of symmetry we will evaluate for the value of x that will give us maximum area.

Axis of symmetry:

x=-b/(2a)

from our equation:

a=(-2/3) and b=500

thus

x=-500/[2(-2/3)]

x=375

the length will be 375 m

The width will be 250 m

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The probability of having fewer than 15 no-shows among 100 advanced reservations is approximately 0.8508

In our case, np = 100 * 0.12 = 12 and n(1-p) = 100 * 0.88 = 88, so we meet the criteria for using the normal approximation.

Next, we'll use the normal distribution formula to find the probability that there will be fewer than 15 no-shows:

P(X < 15) = P(Z < (15 - 12) / 2.60) = P(Z < 1.15)

Here, Z is a standard normal variable with mean 0 and standard deviation 1. We can use a normal distribution table or calculator to find that P(Z < 1.15) is approximately 0.8749.

However, we need to include the correction for continuity since we're approximating a discrete binomial distribution with a continuous normal distribution.

The correction for continuity involves adjusting the boundaries of the interval by 0.5. In this case, we're interested in the probability of having fewer than 15 no-shows, so we'll adjust the upper boundary to 14.5:

P(X < 15) ≈ P(Z < (14.5 - 12) / 2.60) = P(Z < 1.04)

Using a normal distribution table or calculator, we can find that P(Z < 1.04) is approximately 0.8508.

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The inverse Laplace transforms of the given functions.
[tex]a) F(s) = (s^{n+1} * n!) / (s^{n+1})[/tex]

Simplify F(s).
F(s) = n! (since[tex]s^{n+1}[/tex] in the numerator and denominator cancels out)
Apply the inverse Laplace transform.

[tex]L^(-1){n!} = t^n * u(t)[/tex]

[tex](a): t^n * u(t)[/tex], where u(t) is the unit step function.
b) F(s) = (2s + 1) / (4s^2 + 4s + 5)
Rewrite F(s) in the standard form for inverse Laplace transforms of a quadratic denominator.
[tex]F(s) = (2s + 1) / (2s + 1)^2[/tex].

Apply the inverse Laplace transform using the property [tex]L^{-1}{1 / (s + a)^2} = t * e^{-a*t} * u(t).[/tex]
In our case, a = 1.
[tex]L^{-1}{(2s + 1) / (2s + 1)^2} = t * e^{-t}* u(t)[/tex]
[tex](b): t * e^(-t) * u(t),[/tex]  where u(t) is the unit step function.

The Laplace transform is a mathematical technique used to convert a function of time into a function of complex frequency.

The inverse Laplace transform is the reverse process, which is used to convert a function of complex frequency back into a function of time.

The inverse Laplace transform is defined as follows:

f(t) = (1/2πi) ∫γ [[tex]F(s) e^{st} ds[/tex] ]

where f(t) is the function of time, F(s) is the Laplace transform of f(t), γ is a contour in the complex s-plane that encloses all the poles of F(s), and i is the imaginary unit.

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The statement means that, on average, students who live within the city limits have longer bus rides to school compared to students who live in the suburbs.

The school board member is stating that the typical bus ride duration for students residing in the city limits is greater than the bus ride duration for students residing in the suburbs. This suggests that students living in urban areas, which are typically more densely populated, may have to travel longer distances to reach their schools compared to students living in suburban areas, where schools are usually located closer to residential areas. Factors such as urban sprawl, school district boundaries, and availability of public transportation could contribute to longer bus rides for city-dwelling students.

Therefore, the statement implies that there may be a disparity in bus ride durations between students living in the city limits and those living in the suburbs, with the former group likely experiencing longer travel times.

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On solving we got that, As a result, cylinder B now has a capacity of 10687.5 cubic units.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

Cylinders' volumes are inversely correlated to the square of their heights. Therefore, we can determine the ratio of two cylinders' volumes if we know the ratio of their heights.

The height ratio between cylinders A and B is 14:42, which may be written as 1:3.

Therefore, the volume ratio between cylinders A and B is:

(1/3^2 : 1^2 = 1/9 : 1 = 1 : 9

Therefore, cylinder B's volume is nine times that of cylinder A's.

Since cylinder A has a capacity of 1187.5, cylinder B has the following volume:

9 x 1187.5 = 10687.5

As a result, cylinder B now has a capacity of 10687.5 cubic units.

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The answers to the separate questions are as follows- 1) The probability that the sum is lesser than 7 = 0.69.2) The probability that the sum isn't separable by 3 or 4 = 0.69.

We assume that the dice are fair and have 6 sides numbered 1 to 6.

To calculate the probability of each event, we can use the formula

P( event) = number of outcomes in the event/ total number of possible outcomes

For illustration, the total number of possible issues is 6 × 6 = 36, since each die has 6 possible issues and the two dice are independent.

1) Event 1- The sum is lesser than 7

We can cipher the number of issues in this event by counting the number of ways to get a sum lesser than 7. There are 6 possible issues with a sum of 7( 1 6, 2 5, 3 4, 4 3, 5 2, 6 1), and 5 possible issues with a sum of 6( 1 5, 2 4, 3 3, 4 2, 5 1). thus, there are 36- 6- 5 = 25 issues with a sum lesser than 7. Therefore, the probability of this event is

P( sum> 7) = 25/ 36 = 0.69( rounded to two decimal places)

2) Event 2 -The sum isn't divisible by 3 and not divisible by 4

To cipher the number of issues in this event, we need to count the number of issues that aren't divisible by 3 and not separable by 4. There are 9 issues that are separable by 3( 1 2, 1 5, 2 1, 2 4, 3 3, 4 2, 4 5, 5 1, 5 4) and 3 issues that are divisible by 4( 1 3, 2 2, 3 1). There's 1 outgrowth( 3 3) that's divisible by both 3 and 4, so we must abate it from the aggregate. thus, there are 36- 9- 3 1 = 25 issues that aren't separable by 3 or 4. Therefore, the probability of this event is

P( sum not divisible by 3 or 4) = 25/ 36 = 0.69( rounded to two decimal places)

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The lines L1 and L2 will intersect at an intersection point (-1,3,-4) for L1 is (-1,1,-2), and the directional vector of L2 is (1,1,3).

We need to compare their directional vectors to determine the relationship between L1 and L2. The directional vector of L1 is (-1,1,-2), and the directional vector of L2 is (1,1,3).

Since these vectors are not scalar multiples of each other, the lines are not parallel.

To determine if they intersect or are skew, we can find the point of intersection using the system of equations formed by setting the equations of L1 and L2 equal to each other.

Solving this system of equations, we find that x = -1, y = 3, and z = -4.

Therefore, the lines intersect at the point (-1,3,-4).

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The marginal profit function at x = 1,500 is P'(1500) = 155 dollars per unit.

To find the marginal profit function, we first need to find the revenue and profit functions using the given price-demand and cost functions.

1. Price-demand function: p = 295 - (x/80)
2. Cost function: C(x) = 36,000 + 110x

First, find the revenue function, R(x). Revenue is the product of the price per unit and the number of units sold, so R(x) = px.

R(x) = (295 - (x/80))x

Next, find the profit function, P(x). Profit is the difference between revenue and cost, so P(x) = R(x) - C(x).

P(x) = (295 - (x/80))x - (36,000 + 110x)

Now, we'll find the derivative of the profit function with respect to x, which is the marginal profit function, P'(x).

P'(x) = d/dx[(295 - (x/80))x - (36,000 + 110x)]

Using the product rule and the constant rule, we get:

P'(x) = (295 - (x/80)) - x/80 + (-110)

Simplify the expression:

P'(x) = 295 - 2x/80 - 110

Now, evaluate the marginal profit function at x = 1,500.

P'(1500) = 295 - 2(1500)/80 - 110

Calculate the result:

P'(1500) = 295 - 30 - 110 = 155

Therefore, the marginal profit function at x = 1,500 is P'(1500) = 155 dollars per unit.

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Answer:

The surface area of a three dimensional shape is the total area of all of the faces. To find the surface area of a shape, we find the area of each face and add them together.

Step-by-step explanation:

The number of shots attempted by the player until he misses is considered a binomial equation because the probability of success is always constant.

Therefore, the particular criteria for forming a binomial equation is

Therefore, for the given case, the hockey player uses 21% of his shots and is requested to make his shots until he misses. The total number of shots attempted is observed.

Since each shot has only dual possible outcomes (success or failure), and probability of success is constant then this experiment meets all the  four characteristics of forming a  binomial experiment.

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Answer:

15×(-1)^10 -2*(-1)-3 = 14

We can say with 99% confidence that the true proportion of people who like dogs is within the range of 6% +/- 5% (i.e., between 1% and 11%).

The margin of error (MOE) is a measure of how much the results of a survey may differ from the true population values. It is affected by the sample size and the level of confidence of the survey.

To calculate the margin of error at the 99% confidence level for this poll, we can use the following formula:

MOE = z * (sqrt(p*q/n))

where:

z is the z-score corresponding to the confidence level. For a 99% confidence level, z = 2.576

p is the proportion of respondents who said they liked dogs, which is 0.06 in this case

q is the complement of p, which is 1 - 0.06 = 0.94

n is the sample size, which is 150 in this case

Plugging in the values, we get:

MOE = 2.576 * (sqrt(0.06*0.94/150)) = 0.049

Rounding to three decimal places, the margin of error is 0.049 or approximately 0.05.

Therefore, we can say with 99% confidence that the true proportion of people who like dogs is within the range of 6% +/- 5% (i.e., between 1% and 11%).

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The conditions on the constants a, b, c, and d for the differential equation to be exact are a = d and b = c.

And, Once we have established the given differential equation is exact, we can find its general solution by using the following formula:

∫Mdx + ∫(N - ∂∫M/∂y dy)dy = C,

where C is the constant of integration.

Now, For find the conditions on the constants a, b, c, and d such that the given differential equation is exact, we need to use the following theorem:

A necessary and sufficient condition for the differential equation

M dx + N dy = 0 to be exact is that,

⇒  ∂M/∂y = ∂N/∂x.

Hence, Using this theorem, we can find the conditions on a, b, c, and d as follows:

∂M/∂y = a, and ∂N/∂x = d.

Therefore, for the differential equation to be exact, we need;

⇒ a = d.

Similarly, ∂M/∂x = b, and ∂N/∂y = c.

Therefore, for the differential equation to be exact, we need,

⇒ b = c.

Hence, the conditions on the constants a, b, c, and d for the differential equation to be exact are a = d and b = c.

And, Once we have established the given differential equation is exact, we can find its general solution by using the following formula:

∫Mdx + ∫(N - ∂∫M/∂y dy)dy = C,

where C is the constant of integration.

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x = ±√(4t/ arctan(t/4)), we have two solutions for x, because we didn't specify which sign to take when taking the square root. Finally, we can check that this solution satisfies the differential equation and the initial condition.

To solve this differential equation using separation of variables, we need to separate the variables x and t on opposite sides of the equation and integrate each side separately. Here's how we can do it:

(t^2 + 16) dx/dt = x^2 + 16

Dividing both sides by (x^2 + 16), we get:

(t^2 + 16)/(x^2 + 16) dx/dt = 1

Now we can separate the variables:

(x^2 + 16)/(t^2 + 16) dx = dt

Integrating both sides:

∫(x^2 + 16)/(t^2 + 16) dx = ∫dt

To evaluate the integral on the left, we can use the substitution u = t/4, du = 1/4 dt:

∫(x^2 + 16)/(t^2 + 16) dx = 4∫(x^2 + 16)/(16u^2 + 16) dx

= 4∫(x^2 + 16)/(4u^2 + 4) dx

= 4∫(x^2/4 + 4)/(u^2 + 1) dx

= 4(x^2/4 arctan(u) + 4u) + C

= x^2 arctan(t/4) + 16t/4 + C

where C is the constant of integration. Now we can solve for x by plugging in the initial condition x(0) = 4:

x^2 arctan(0/4) + 16(0)/4 + C = 4^2

C = 16

So the particular solution is:

x^2 arctan(t/4) + 16t/4 + 16 = 16 + x^2 arctan(t/4)

Simplifying:

x^2 arctan(t/4) = 4t

x^2 = 4t/ arctan(t/4)

Therefore, x = ±√(4t/ arctan(t/4))

Note that we have two solutions for x, because we didn't specify which sign to take when taking the square root. Finally, we can check that this solution satisfies the differential equation and the initial condition.
To solve the differential equation (t² + 16) dx/dt = (x² + 16) with the initial condition x(0) = 4 using separation of variables, follow these steps:

1. Rewrite the equation as (t² + 16) dx = (x² + 16) dt.
2. Separate the variables: (1/(x² + 16)) dx = (1/(t² + 16)) dt.
3. Integrate both sides: ∫(1/(x² + 16)) dx = ∫(1/(t² + 16)) dt + C.
4. The antiderivatives are: (1/4)arctan(x/4) = (1/4)arctan(t/4) + C.
5. Apply the initial condition x(0) = 4: (1/4)arctan(4/4) = (1/4)arctan(0/4) + C, which simplifies to (1/4)(π/4) = C.
6. Solve for x(t): arctan(x/4) = arctan(t/4) + π.

The solution for x(t) is:
x(t) = 4 * tan(arctan(t/4) + π).

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Answer:

Step-by-step explanation:

For a system of equations, the solution is where the 2 lines intersect. They intersect at (-3,1).  But they only wan the y-coordinate, so it's the y part of the answer (x,y) x=-3  and y=1

So your answer is 1

We can say with 95% confidence that the true mean life of light bulbs in this shipment falls between 371.84 hours and 448.16 hours.

To construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment, we can use the following formula:

Confidence interval = sample mean +/- margin of error

where the margin of error is given by:

Margin of error = (critical value) x (standard deviation / sqrt(sample size))

Since we want a 95% confidence interval, the critical value is 1.96 (from the standard normal distribution table). Plugging in the given values, we get:

Margin of error = 1.96 x (108 / sqrt(81)) = 38.16

Therefore, the confidence interval estimate is:

410 +/- 38.16

or

(371.84, 448.16)

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The set of angles that does NOT produce a triangle is 43°, 62°, and 76°, as their sum is 181°

To determine if the given angles can form a triangle, we need to check if the sum of the three angles is equal to 180° (the sum of the angles of a triangle).

62°, 34°, and 84°:

Sum = 62° + 34° + 84° = 180°

26°, 48°, and 106°:

Sum = 26° + 48° + 106° = 180°

43°, 62°, and 76°:

Sum = 43° + 62° + 76° = 181°

34°, 67°, and 79°:

Sum = 34° + 67° + 79° = 180°

Answer:

500

Step-by-step explanation:

i think. i have down syndrome dont hate my answers.

Answer: 500

Step-by-step explanation:

To find the absolute extreme values of the function f(x) = 2[tex]x^{3}[/tex]- 24 on the interval [-3, 3], follow these steps:

1. Find the critical points by taking the derivative of the function and setting it to zero.
f'(x) = 6[tex]x^{2}[/tex]

2. Solve for x:
6[tex]x^{2}[/tex] = 0
x = 0

3. Evaluate the function at the critical point and the interval's endpoints:
f(-3) = 2[tex](-3)^{3}[/tex]- 24 = -90
f(0) = 2[tex](0)^{3}[/tex]- 24 = -24
f(3) = 2[tex](3)^{3}[/tex] - 24 = 90

4. Compare the function values and identify the absolute extremes:
Absolute minimum: f(-3) = -90
Absolute maximum: f(3) = 90

So, the absolute extreme values of the function f(x) = 2[tex]x^{3}[/tex] - 24 on the interval [-3, 3] are -90 (minimum) and 90 (maximum).

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The total square footage of the two-story home is 7,200 square feet.

To calculate the total square footage of the home, we need to multiply the exterior wall measurements of each floor and add them together.

Assuming both floors have the same dimensions, the square footage of one floor would be:

40 feet x 90 feet = 3,600 square feet

Since the house has two floors, we need to double this number:

3,600 square feet x 2 = 7,200 square feet

Therefore, the total square footage of the two-story home is 7,200 square feet.

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The acceleration vector a(t) is (-9 cos 3t)i + (-2 sin t)j.

If r(t) is the position vector of a particle in the plane at time t, and r(t) = (cos 3t) i + (2 sin t) j, you want to find the acceleration vector.

First, find the velocity vector by taking the derivative of the position vector with respect to time:

v(t) = dr(t)/dt = (-3 sin 3t) i + (2 cos t) j

Next, find the acceleration vector by taking the derivative of the velocity vector with respect to time:

a(t) = dv(t)/dt = (-9 cos 3t) i + (-2 sin t) j

The acceleration vector a(t) is (-9 cos 3t)i + (-2 sin t)j.

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The height of the cone needs to be 6 inches if it has the same width as the 3-inch wide, 2-inch tall cup to hold the same amount of ice cream

What is Volume of cone?

Volume of a cone = π r² h/3

Volume of cone = 1/3 * π * r² * h

Volume of cylinder = π * r² * h

Volume of cylinder = π * r² * h

π * (1.5)² * 2 = 4.5π

Volume of cone = 1/3 * π * r² * h

4.5π = 1/3 * π * (1.5)² * h

4.5π = 0.75π * h

h = 4.5π / 0.75π

h = 6

Therefore, the height of the cone needs to be 6 inches if it has the same width as the 3-inch wide, 2-inch tall cup to hold the same amount of ice cream

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(a) The probability of getting three heads and two tails in five coin tosses is 5/16

(b) The probability of getting the face ‘G’ two times in three rolls of a die as (6 + other + 6) is 5/216

(c) The probability that all four individuals in a telephone poll disapprove of the president's performance given that 46% of the population approve of his performance is 0.104.

(d) The probability of getting a full house when taking five cards from a deck is 0.00144 or approximately 0.14%.

(a) The probability of getting three heads and two tails in five coin tosses can be calculated as follows:

[tex]P(3 heads and 2 tails) = (5 choose 3) * (1/2)^3 * (1/2)^2 = 10/32 = 5/16[/tex]

(b) The probability of getting the face ‘G’ two times in three rolls of a die as (6 + other + 6) can be calculated as follows:

P(getting ‘G’ twice)[tex]= (1/6)^2 * (5/6)[/tex]

= 5/216

(c) The probability that all four individuals in a telephone poll disapprove of the president's performance given that 46% of the population approve of his performance is:

P(all four individuals disapprove) [tex]= (0.54)^4 = 0.104[/tex]

(d) The probability of getting a full house when taking five cards from a deck can be calculated as follows:

P(full house) = (13 choose 1) * (4 choose 3) * (12 choose 1) * (4 choose 2) / (52 choose 5) = 0.00144 or approximately 0.14%

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The relationship between the number of hours worked and the number of toys built can be represented by a linear equation y = 9x, where y is the number of toys built and x is the number of hours worked. The graph is attached below.

The graph representing this relationship is a straight line passing through the origin (0,0) with a slope of 9. The x-axis represents the number of hours worked, and the y-axis represents the number of toys built. As x increases, y increases proportionally at a rate of 9 units of y for every unit of x.

The slope of the line, which is the ratio of the change in y to the change in x, represents the rate of increase of the number of toys built per hour worked. In this case, the slope is 9, which means that the number of toys built increases by 9 for every additional hour worked.

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The function with the second derivative as f'' ( x ) = 3x + 4sin ( 2 ) is given by f ( x ) = ( 1/2 )x³ + 2x²sin(2) + 4x + 2

Given data ,

To find the function f(x) given the information about its second derivative and initial conditions, we can integrate the second derivative twice and apply the initial conditions to determine the constants of integration.

First, integrating f''(x) = 3x + 4 sin(2), we get:

f'(x) = 3/2 * x² + 4 * x * sin(2) + C1

where C1 is a constant of integration.

Next, integrating f'(x), we get:

f(x) = 1/2 * x³ + 4/2 * x² * sin(2) + C1 * x + C2

where C2 is another constant of integration

Now, we can apply the initial conditions to determine the values of C1 and C2

Given f(0) = 2, we have:

f(0) = 1/2 * 0³ + 4/2 * 0² * sin(2) + C1 * 0 + C2 = C2 = 2

So, C2 = 2

Given f'(0) = 4, we have:

f'(0) = 3/2 * 0² + 4 * 0 * sin(2) + C1 = C1 = 4

So, C1 = 4

Now, substituting the values of C1 and C2 into our expression for f(x), we get:

f(x) = ( 1/2 )x³ + 2x²sin(2) + 4x + 2

So, the function f(x) that satisfies the given conditions is:

f(x) = ( 1/2 )x³ + 2x²sin(2) + 4x + 2

Hence , the function is f ( x ) = ( 1/2 )x³ + 2x²sin(2) + 4x + 2

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The Solution of the equation is y(t) = L⁻¹{(s + 10 + [tex]e^-^$^\pi$^s[/tex] +  [tex]e^-^3^$^\pi$^s[/tex] ) / (s² + 10s + 29)}.

To use the Laplace transform to solve the initial-value problem y'' + 10y' + 29y = δ(t - π) + δ(t - 3π), y(0) = 1, y'(0) = 0, you'll first apply the Laplace transform to both sides, then solve for Y(s), and finally apply the inverse Laplace transform.

1. Apply the Laplace transform to both sides: L{y''} + 10L{y'} + 29L{y} = L{δ(t - π)} + L{δ(t - 3π)}.
2. Use the properties of Laplace transforms for derivatives and translations: s²Y(s) - sy(0) - y'(0) + 10(sY(s) - y(0)) + 29Y(s) =  [tex]e^-^$^\pi$^s[/tex]  + [tex]e^-^3^$^\pi$^s[/tex] .
3. Plug in the initial conditions: s²Y(s) - s + 10(sY(s) - 1) + 29Y(s) = [tex]e^-^$^\pi$^s[/tex] +  [tex]e^-^3^$^\pi$^s[/tex] .
4. Solve for Y(s): Y(s) = (s + 10 +  [tex]e^-^$^\pi$^s[/tex]  +  [tex]e^-^3^$^\pi$^s[/tex] ) / (s² + 10s + 29).
5. Apply the inverse Laplace transform: y(t) = L⁻¹{Y(s)}.

The main answer is y(t) = L⁻¹{(s + 10 + [tex]e^-^$^\pi$^s[/tex] +  [tex]e^-^3^$^\pi$^s[/tex] ) / (s² + 10s + 29)}.

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