factorise 49x²-y² class 8​

The geometric series is convergent and its sum is 16.67.

To determine whether the geometric series is convergent or divergent, we need to calculate the common ratio.

The common ratio is found by dividing any term in the series by its previous term.

For this series, the first term is 10 and the second term is -4. So, the common ratio is:
r = (-4)/10 = -0.4

Since the absolute value of the common ratio is less than 1, the series is convergent. To find its sum, we can use the formula for the sum of an infinite geometric series:
S = a/(1 - r)
where a is the first term and r is the common ratio.

Plugging in the values we get:
S = 10/(1 - (-0.4)) = 16.67

Therefore, the geometric series is convergent and its sum is 16.67.

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The area of the patio is given as follows:

82 m².

The patio is formed by a composite shape, hence we must obtain the area of each part of the shape and add them.

The shape is composed as follows:

The area of the rectangle is given as follows:

3 x 6 = 18 m².

The area of the trapezoid is given as follows:

A = 0.5 x 8 x (10 + 6)

A = 64 m².

Hence the total area of the shape is given as follows:

18 + 64 = 82 m².

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If the barber shop has four chairs and a single barber and each customer requires 15 minutes on average then assuming an exponential distribution for service times, the expected time an entering customer spends in the barbershop is 0.5 minutes.

In a Poisson process, the number of arrivals is independent of the past and the future and the time between consecutive arrivals is exponentially distributed. Customers are arriving at the barber shop according to a Poisson process at a rate of eight per hour.

The average arrival rate of the customer is given as = 8 customers/hour, which means that the average time between arrivals will be 7.5 minutes. The customer service time is given as exponentially distributed, so the expected customer service time is the inverse of the service rate.

Therefore, the expected service time = 1/4 = 0.25 hours = 15 minutes. We can then use the M/M/1 queuing model to determine the expected time an entering customer spends in the barbershop. The M/M/1 queuing model is based on the following assumptions:

Since there are four chairs in the barber shop, we can assume that the system capacity is four.

So, the system capacity is less than infinity.

We can modify the M/M/1 queuing model for M/M/1/4 queuing model.

According to the queuing model, the expected time an entering customer spends in the barbershop can be calculated as:

W = 1/μ - 1/λ  + 1/(μ-λ) * (1- (λ/μ)^4)

Where: λ = Arrival rate

μ = Service rate

W = Waiting time per customer

Therefore,

W = 1/0.25 - 1/0.5 + 1/(0.5-0.25) * (1- (0.25/0.5)^4) = 0.5 - 2 + 2.6667*0.9375 = 0.5 minutes

Therefore, the expected time an entering customer spends in the barbershop is 0.5 minutes.

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Note that the number of cars required for George and Marian to make a monthly profit of at least $8,000 is 2,467 cars.

Assume that they need to wash "x" cars to make a monthly profit of $8,000.

Their total revenue (TR) from washing "x" cars would be 6x dollars

Thus, their total profit = Revenue - Operating Costs

Profit = 6x - 6,800

We want to find the value of "x" that makes the profit at least $8,000, so we set up the inequality so....

6x - 6,800 ≥ 8,000

Adding 6,800 to both sides of the inequality, we get

6x ≥ 14,800

x ≥ 2,467

so , they need to wash at least 2,467 cars to make a monthly profit of at least $8,000.

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The graph of the polynomial equation is y = log ( x + 1 ) + 3

Given data ,

Let the logarithmic equation be represented as A

Now , the value of A is

The vertical asymptote occurs at x = -1 because the argument of the logarithm, x + 1, cannot be negative or zero.

So , the equation is y = log ( x + 1 ) + 3

Hence , the graph of the equation is plotted and y = log ( x + 1 ) + 3

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The correct answer to this question is C: Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date.

Taking measurements on the same dates during the year is important because it helps to control for the effect of seasonal changes in the water clarity of the lake.

For example, if the measurements were taken in the winter when the lake is frozen, the water clarity would likely be very different than in the summer when the lake is not frozen.

Since the absolute value of the test statistic (-0.24) is less than the critical value (2.571), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to suggest that the clarity of the lake is improving at the alpha equals 0.05 level of significance.

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The angles whose equals to 60 ° are ∠1 , ∠2 , ∠3 , ∠4 . This is due to opposite angles and angle pairs due to a transversal with a parallel.

Note that

l and m are the parallel lines .

m ∠ 1 = 60 °

Thus

∠1 = ∠2 = 60 °

(As l and m are the parallel lines and ∠ 1 and ∠2 are the vertically opposite angles .)

As

∠2 = ∠3

(As l and m are the parallel lines and ∠2 and ∠3 are the alternate interior angles. )

As

∠3 = ∠4 = 60°

( As l and m are the parallel lines and ∠ 3 and ∠4 are the vertically opposite angles )

Therefore the angles whose equals to 60 ° are ∠1 , ∠2 , ∠3 , ∠4 .

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=190

Step-by-step explanation:

[tex]9.5 \div 0.05[/tex]

= 190

The value of the double integral is ∬D 2y dA = 131/150.

To determine which is easier to integrate, let's first sketch the region D:

From the sketch, we see that D is more naturally expressed as a function of y, rather than x. Therefore, it is easier to integrate using option B, which is ∬D 2y dy dx.

To evaluate the double integral using option B, we can set up the integral as follows:

∬D 2y dy dx = ∫[0,1] ∫[[tex]y^2,(3y+10)/10][/tex] 2y dx dy

= ∫[0,1] [[tex](3y^2 + 10y)/5 - y^{5/5}][/tex]dy

= [[tex]y^{3/5}+ y^2 - y^6/30[/tex]] evaluated from 0 to 1

= 1/5 + 1 - 1/30 = 131/150

Therefore, the value of the double integral is ∬D 2y dA = 131/150.

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Based on an exponential growth rate of 4% each year, the town whose population is 20,000 will be 25,306 after 6 years.

An exponential growth refers to a constant ratio of increase per period.

An exponential growth is modeled by the exponential growth function, which is one of the two exponential functions, including exponential decay function.

The current or initial population of the town = 20,000

The annual growth rate = 4% = 0.04

Growth factor = 1.04 (1 + 0.04)

The number of years from the initial year of census = 6 years

Let the number of years from the initial year = n

Let the population after n years = y

y = 20,000(1.04)^6

y = 25,306

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The concept of rhythmic regularity suggests strong rhythms moving at a steady tempo.

What is Rhythm?

Rhythm is a recurring sequence of sound that has a beat, which can be calculated and felt. The rhythm is made up of beats, which can be organized into measures or bars in Western music.

The word "rhythm" comes from the Greek word "rhythmos," which means "any regular recurring motion, symmetry."Rhythmic regularity, as the name implies, refers to the steady beat and consistent rhythm that is present throughout a piece of music.

The beats are emphasized and move at a regular tempo, giving the music a sense of predictability and stability.Syncopated rhythms, on the other hand, are those in which the beat is shifted or emphasized in unexpected ways. They are used to create tension and interest in music by breaking up the regularity of the rhythm.

Therefore, option B "The regular use of syncopated rhythms" is incorrect.

Regularity, on the other hand, suggests a consistent, predictable pattern of beats and rhythms moving at a steady tempo.

Therefore, option C "Strong rhythms moving at a steady tempo" is correct.

Irregular rhythms (option D) are not related to rhythmic regularity, and meters that frequently change within a piece or movement (option A) are examples of irregular rhythms.

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To find the upper-tail p-value, we need to find the probability of getting a t-value equal to or greater than the observed test statistic of t = 0.833, given the degrees of freedom df = 27.

Using a t-table or calculator, we find that the probability of getting a t-value greater than 0.833 with 27 degrees of freedom is 0.206. Therefore, the upper-tail p-value for the test statistic is 0.206.

So, the answer is (b) 0.206.

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Answer:

P (Score is not a factor of 6) = 1−31=32

This is my answer

Given:

ZLMN and LPML are linear pairs,m_LMN = 7x -3, mZPML = 13x + 3.

Let's solve the problem one by one.Part A:m_LMN + mZPML = 180 [linear pair]7x - 3 + 13x + 3 = 18020x = 180x = 9m_LMN = 7(9) -3 = 60m_ZPML = 13(9) + 3 = 120m_LMN = 60, mZPML = 120We need to find the mzLMN.

By definition,

linear pairs are adjacent angles whose non-common sides are opposite rays. So, their angles add up to 180 degrees.So,m_LMN + mZLMN = 18060 + mZLMN = 180mZLMN = 120Therefore, mzLMN = 120/2 = 60 degreesPart B:ZPMR and ZLMN form a vertical pair

By definition,

vertical angles are congruent, so mZPMR = m_LMN = 60 degreesmZPMR = 5y + 4Putting the value of mZPMR we get,5y + 4 = 605y = 56y = 11.2, the value of y is 11.2. Answer: Part A: mzLMN = 60 degreesPart B: m_PML = 60 degrees; value of y is 11.2.

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ZLMN and LPML are linear pairs the value of y is (13x - 7)/5.

Given, ZLMN and LPML are linear pairs and mLNM = 7x -3 and

mPML = 13x + 3.

Part A: To find mzLMNSince, ZLMN and LPML are linear pair,

Therefore, mLMN + mPML = 180

Substitute the given values in the above equation

7x - 3 + 13x + 3 = 18020

x = 180

x = 9

Substitute the value of x in mLNM7(9) - 3

mLNM = 63 - 3

mLNM = 60

Thus, the value of mLNM is 60.

Part B: If ZPMR and ZLMN form a vertical pair, then they are equal.

Therefore, mZLMN = mZPMR

Now, mZPMR = 5y + 4

Given, mZPMR = mLMN

13x + 3 = 7x - 3 + 5y + 4

13x + 3 = 5y + 4 + 7x - 3

Move the constant term to the right

5y = 13x + 3 - 4 - 35

y = 13x - 4y = (13x - 7)/5

Thus, the value of y is (13x - 7)/5.

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ZF equalizer where the desired signal vector is exactly aligned with the observation interval will be [tex]w^H \times y[/tex].

To find a ZF equalizer for the given system, we need to first define the channel matrix H and the noise vector n.

Let's assume that the transmitted signal is denoted by x and the received signal is denoted by y. Also, let the impulse response of the channel be denoted by h.

The channel matrix H is given by:

H = [h(0) h(1) h(2) h(3) h(4) h(5) h(6) h(7) h(8) h(9) h(10)]

The noise vector n is given by:

n = [n(0) n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) n(9) n(10)]

To find the ZF equalizer, we need to solve for the filter taps w that minimizes the mean squared error between the desired signal and the output of the equalizer. In this case, the desired signal is simply the transmitted signal x, which we want to recover from the received signal y.

The filter taps w can be found by solving the following equation:

w = [tex](H^H \times H)^{-1} \times H^H \times x[/tex]

where [tex]H^H[/tex] is the conjugate transpose of H.

Once we have the filter taps w, the ZF equalizer output is given by:

y_hat = [tex]w^H \times y[/tex]

where [tex]w^H[/tex] is the conjugate transpose of w.

Note that since the desired signal vector is exactly aligned with the observation interval, the ZF equalizer will be able to perfectly equalize the channel and recover the transmitted signal without any distortion.

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So, the inequality which represents the situation is 7.5h + 30 ≥ 300, where h ≥ 36. Hence, the answer is B.

Given: Chris works at a bookstore and earns $7. 50 per hour plus a $2 bonus for each book she sells. Chris sold 15 books. The total earning of Chris,E(h) = 7.5h + 2 × 15 = 7.5h + 30 dollars where h is the number of hours worked by Chris .In order to find out the minimum hours she has to work to earn at least $300, we have to solve the inequality:7.5h + 30 ≥ 300 ⇒ 7.5h ≥ 270 ⇒ h ≥ 36.

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The first step in answering this question is to calculate the proportion of customers who prefer fresh vegetables for each sample. The formula for proportion is' p= x/n where p is the proportion, x is the number of customers who prefer fresh vegetables, and n is the sample size. Using this formula, we can calculate the proportion for each sample as follows: For sample

A: p = 11/20 = 0.55For sample B: p = 14/20 = 0.70For sample C :p = 12/20 = 0.60Next, we can use these proportions to estimate the number of regular customers of the store who prefer fresh vegetables.

To do this, we multiply each proportion by the total number of regular customers (600) as follows: For sample

A: Estimated number of customers who prefer fresh vegetables = 0.55 × 600 = 330For sample B: Estimated number of customers who prefer fresh vegetables = 0.70 × 600 = 420For sample C: Estimated number of customers who prefer fresh vegetables = 0.60 × 600 = 360Now we need to describe the variation of the estimates.

the standard deviation of the estimates as follows:SD = sqrt [(330 - 370)² + (420 - 370)² + (360 - 370)² / 3]≈ 47.2Therefore, the estimates for the number of regular customers who prefer fresh vegetables have a standard deviation of approximately 47.2 customers. This means that we can expect the estimates to vary by about 47.2 customers on average due to sampling error.

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The solution is: the area of the regular hexagon is 41.57 in^2.

Here, we have,

given that,

the figure is a regular hexagon.

so, we have,

n = 6

and, given that, r = 4in

so, we get,

central angle = 360/n = 360/6 = 60

so, we have

Area = n * 1/2 * r^2 * sin 60

        = 6 *1/2* 16 * √3/2

        = 41.57 in^2.

Hence, The solution is: the area of the regular hexagon is 41.57 in^2.

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a. The tax on the bowl of soup was $3.37.

b. The total price of the bowl of soup, including tax, was $7.87.

c. Quan should get back $12.13 from his $20 bill.

a. To calculate the tax on the bowl of soup, we multiply the cost of the soup ($4.50) by the tax rate (0.075). Therefore, the tax on the soup is $4.50 * 0.075 = $0.337, which can be rounded to $3.37.

b. To find the total price of the bowl of soup, including tax, we add the cost of the soup and the tax amount. The cost of the soup is $4.50, and the tax is $3.37. Adding these together gives us $4.50 + $3.37 = $7.87.

c. Quan paid with a $20 bill, and the total price of the soup, including tax, was $7.87. To determine how much money Quan should get back, we subtract the total price from the amount paid. Subtracting $7.87 from $20 gives us $20 - $7.87 = $12.13. Therefore, Quan should receive $12.13 back from his payment.

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The function f(x)=x has a domain and target of the set of Real numbers. To describe this function, we need to consider its one-to-one and onto properties. A function is one-to-one if each element of the domain is mapped to a unique element of the target, and a function is onto if every element of the target is mapped to by at least one element of the domain. In this case, the function f(x)=x is one-to-one and onto, making it a bijection. Therefore, the most appropriate way to describe this function is option b: f is a bijection.

To determine the appropriate way to describe the function f(x)=x, we need to consider its one-to-one and onto properties. A function is one-to-one if each element of the domain is mapped to a unique element of the target, and a function is onto if every element of the target is mapped to by at least one element of the domain. In this case, for every x in the domain of Real numbers, there is a unique value of x in the target of Real numbers. This means that the function is one-to-one. Additionally, every element in the target is mapped to by at least one element in the domain. Therefore, the function is also onto. Since the function is both one-to-one and onto, it is a bijection.

The function f(x)=x has a domain and target of the set of Real numbers and is a bijection. This means that for every x in the domain, there is a unique value of x in the target, and every element in the target is mapped to by at least one element in the domain. Therefore, the most appropriate way to describe this function is option b: f is a bijection.

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The rate at which the percentage spent on food is changing on September 1 is approximately -0.34 percentage points per month.

We can start by taking the derivative of y with respect to x: y' = -0.25*35x^(-1.25) = -8.75x^(-1.25). Then, we can substitute x with the given function of t: x = 7 + 0.2t. Thus, y = 35(7 + 0.2t)^(-0.25). To find the rate of change of y with respect to t, we can use the chain rule:

(dy/dt) = (dy/dx)(dx/dt) = -8.75(7 + 0.2t)^(-1.25)(0.2)

We want to find the rate of change on September 1, which is 8 months after January 1. So we can substitute t = 8 into the equation above:

(dy/dt) = -8.75(7 + 0.28)^(-1.25)(0.2) ≈ -0.34

Therefore, the rate at which the percentage spent on food is changing on September 1 is approximately -0.34 percentage points per month.

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After 15.53 hours, half of the reactant A will have been converted into product B.

(a) The order of the reaction is 1 because the rate law only includes the concentration of reactant a raised to the first power.

This means that the rate of the reaction is directly proportional to the concentration of a.
(b) The half-life of the reaction can be calculated using the equation:
t1/2 = ln(2) / k
Where t1/2 is the half-life, ln is the natural logarithm, and k is the rate constant.
Substituting the given values:
t1/2 = ln(2) / 0.0447 hr–1
t1/2 = 15.5 hours
Therefore,

The half-life of the reaction is 15.5 hours.

This means that after 15.5 hours, the concentration of reactant a will have decreased by half, and the concentration of product b will have increased by half.

This information can be useful in determining the optimal conditions for the reaction, such as the reaction time and temperature.

The half-life of a first-order reaction can be calculated using the following formula: t½ = ln(2) / k In this case, the rate constant (k) is given as 0.0447 hr⁻¹.

Plugging this value into the formula, we get: t½ = ln(2) / 0.0447 t½ ≈ 15.53 hours So, the half-life of this reaction is approximately 15.53 hours.

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The given rate law is rate = k[a], where k is the rate constant and [a] is the concentration of the reactant a. The order of the reaction is determined by the exponent of [a] in the rate law equation. In this case, the exponent is 1, which means that the reaction is first order.

This indicates that the rate of the reaction is directly proportional to the concentration of the reactant a. The half-life of a first-order reaction can be calculated using the equation t1/2 = ln(2)/k, where ln is the natural logarithm. Substituting the given value of k in the equation, we get t1/2 = ln(2)/0.0447 hr–1 = 15.5 hours (rounded to one decimal place). This means that after 15.5 hours, half of the initial concentration of reactant a would have reacted to form product b.
The rate law for the given reaction A → 2B is rate = k[A], where k is the rate constant (0.0447 hr⁻¹) and [A] is the concentration of reactant A.

(a) The order of this reaction is 1. The order is determined by the exponent of the concentration term in the rate law, in this case [A]^1.

(b) To find the half-life (t½), we use the first-order half-life equation: t½ = 0.693/k. With k = 0.0447 hr⁻¹, the half-life is:

t½ = 0.693 / 0.0447 ≈ 15.5 hours.

In summary, this is a first-order reaction with a half-life of approximately 15.5 hours.

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The span of the vectors V1 and V2, given as V1 = [5, 0, 0] and V2 = [15, -9, 0], is a line in the x-y plane passing through V1 and the origin. This line represents all possible linear combinations of V1 and V2.

The correct answer is B. Span {V1, V2} is the set of points on the line through V1 and 0.

To determine the geometric description of Span {V1, V2}, we examine the given vectors. V1 has a non-zero entry only in the x-coordinate, while V2 has non-zero entries in the x-coordinate and y-coordinate. Since the z-coordinate is always zero for both vectors, they lie in the x-y plane.

The span of a set of vectors is the set of all possible linear combinations of those vectors. In this case, V1 = [5, 0, 0] and V2 = [15, -9, 0] are two vectors in three-dimensional space.

Since V1 has a non-zero entry only in the x-coordinate and V2 has non-zero entries in the x-coordinate and y-coordinate, the span of {V1, V2} will lie entirely in the x-y plane. Therefore, it forms a line in the x-y plane passing through V1 and the origin (0, 0, 0).

The span of {V1, V2} will include all possible scalar multiples of these vectors and their linear combinations. Since V1 and V2 are not linearly dependent (one cannot be obtained by scaling the other), the span forms a line in the x-y plane. This line passes through the origin (0, 0, 0) and extends along the direction determined by V1. Therefore, the geometric description of Span {V1, V2} is that it represents the set of points on the line through V1 and the origin (0, 0, 0) in three-dimensional space.

Hence, the correct geometric description is that Span {V1, V2} is the set of points on the line through V1 and 0.

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To find the sum of the given series, we need to calculate the sum of each term where n starts from 0 and goes to infinity. The general term of the series is (2n)/(n!).

Let's find the sum of the series:

S = Σ(2n)/(n!) from n=0 to infinity

To determine the convergence of the series, we can use the Ratio Test:

Limit as n → infinity of |((2(n+1))/((n+1)!) / ((2n)/(n!))|

= Limit as n → infinity of |(2(n+1))/((n+1)!) * (n!)/(2n)|

= Limit as n → infinity of |(2(n+1))/(n! * (n+1))|

= Limit as n → infinity of |2(n+1)/(n+1)|

= 2

Since the limit is greater than 1, the Ratio Test indicates that the series is divergent. Therefore, the sum of the series does not exist or approaches infinity.

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Blade's inability to see and reconnect with Lucy in Africa is down to the distance between them at the time .

The book "Solo" written by Kwame Alexander features Lucy and Blade. Blade and Lucy couldn't see while she was in Africa. Blade's inability to see Lucy in Africa is primarily due to the geographical distance between them as Africa and America are on separate continent.

Blade was having to deal with personal issues and embarks on a journey to discover his own identity and reconnect with his estranged father. While Blade travels to Africa, Lucy remains in the United States. The physical separation and the circumstances surrounding Blade's journey are the factors that kept him from seeing Lucy in Africa.

Hence, there inability to see is based on geographical differences.

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The angle θ by taking the inverse cosine of the dot product divided by the product of the magnitudes: θ = acos((u · v) / (|u| |v|)).

The angle θ between the vectors u and v can be found by taking the inverse cosine of the dot product divided by the product of their magnitudes.

To find the angle θ between the vectors u and v, we need to calculate the dot product of the two vectors and divide it by the product of their magnitudes. The dot product of two vectors u and v is given by the formula u · v = |u| |v| cos(θ), where |u| and |v| are the magnitudes of u and v, respectively, and θ is the angle between them.

In this case, u = cos(π/3) i + sin(π/3) j and v = cos(3π/4) i + sin(3π/4) j. We can calculate the magnitudes of u and v as |u| = √(cos²(π/3) + sin²(π/3)) and |v| = √(cos²(3π/4) + sin²(3π/4)).

Next, we calculate the dot product of u and v as u · v = cos(π/3) * cos(3π/4) + sin(π/3) * sin(3π/4).

Finally, we find the angle θ by taking the inverse cosine of the dot product divided by the product of the magnitudes: θ = acos((u · v) / (|u| |v|)).

By evaluating this expression, we can determine the angle θ between the vectors u and v.

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The hydrostatic force on the window is approximately 47,481 Newtons.

We can use the formula for hydrostatic force, which is:

F = ρghA

where F is the hydrostatic force, ρ is the density of the fluid (water in this case), g is the acceleration due to gravity, h is the depth of the window, and A is the area of the window.

First, we need to find the area of the window. Since the window is a semi-circle with diameter 70 centimeters, the radius is 35 centimeters, and the area is:

A = (π/2)r^2

= (π/2)(35 cm)^2

= 1225π/2 cm^2

Next, we need to convert the depth of the window to meters:

h = 2.5 m

We also need the density of water, which is approximately:

ρ = 1000 kg/m^3

Finally, we need the acceleration due to gravity, which we can assume is:

g = 9.8 m/s^2

Now we can plug these values into the formula:

F = ρghA

= (1000 kg/m^3)(9.8 m/s^2)(2.5 m)(1225π/2 cm^2)

≈ 47,481 N

Therefore, the hydrostatic force on the window is approximately 47,481 Newtons.

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[tex]\begin{align}\sf\:\text{LHS} &= \cos(A)\cos(60^\circ - A)\cos(60^\circ + A) \\&= \cos(A)\cos(60^\circ)\cos(60^\circ) - \cos(A)\sin(60^\circ)\sin(60^\circ) \\&= \frac{1}{2}\cos(A)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) - \frac{\sqrt{3}}{2}\cos(A)\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \\&= \frac{1}{8}\cos(A) - \frac{3}{8}\cos(A) \\ &= \frac{-2}{8}\cos(A) \\ &= -\frac{1}{4}\cos(A).\end{align} \\[/tex]

Now, let's calculate the value of [tex]\sf\:\cos(3A) \\[/tex]:

[tex]\begin{align}\sf\:\text{RHS} &= \frac{1}{4}\cos(3A) \\&= \frac{1}{4}(4\cos^3(A) - 3\cos(A)) \\&= \cos^3(A) - \frac{3}{4}\cos(A).\end{align} \\[/tex]

Comparing the [tex]\sf\:\text{LHS} \\[/tex] and [tex]\text{RHS} \\[/tex], we have:

[tex]\sf\:-\frac{1}{4}\cos(A) = \cos^3(A) - \frac{3}{4}\cos(A). \\[/tex]

Adding [tex]\sf\:\frac{1}{4}\cos(A) \\[/tex] to both sides, we get:

[tex]\sf\:0 = \cos^3(A) - \frac{2}{4}\cos(A). \\[/tex]

Simplifying further:

[tex]\sf\:0 = \cos^3(A) - \frac{1}{2}\cos(A). \\[/tex]

Factoring out a common factor of [tex]\sf\:\cos(A) \\[/tex], we have:

[tex]\sf\:0 = \cos(A)(\cos^2(A) - \frac{1}{2}). \\[/tex]

Using the identity [tex]\sf\:\cos^2(A) = 1 - \sin^2(A) \\[/tex], we can rewrite the equation as:

[tex]\sf\:0 = \cos(A)(1 - \sin^2(A) - \frac{1}{2}). \\[/tex]

Simplifying:

[tex]\sf\:0 = \cos(A)(1 - \frac{3}{2}\sin^2(A)). \\[/tex]

Since [tex]\sf\:\cos(A) \\[/tex] cannot be zero (as it would result in undefined values), we can divide both sides of the equation by [tex]\sf\:\cos(A) \\[/tex]:

[tex]\sf\:0 = 1 - \frac{3}{2}\sin^2(A). \\[/tex]

Rearranging the terms:

[tex]\sf\:\sin^2(A) = \frac{2}{3}. \\[/tex]

Taking the square root of both sides, we get:

[tex]\sf\:\sin(A) = \pm\sqrt{\frac{2}{3}}. \\[/tex]

The solution [tex]\sf\:\sin(A) = \sqrt{\frac{2}{3}} \\[/tex] corresponds to the range where [tex]\sf\:0° \leq A \leq 90° \\[/tex]. Therefore, the solution [tex]\sf\:\sin(A) = \sqrt{\frac{2}{3}} \\[/tex] is valid.

Hence, we have proved that:

[tex]\sf\:\cos(A)\cos(60^\circ - A)\cos(60^\circ + A) = \frac{1}{4}\cos(3A). \\[/tex]

[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]

♥️ [tex]\large{\textcolor{red}{\underline{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

Answer:

Given:

To Prove:

cosAcos(60°-A)cos(60°+A) = 1/4 cos3A

Solution:

Here are the steps in detail:

1. Expanding cosAcos(60°-A)cos(60°+A) using the product-to-sum identities:

=cosAcos(60°-A)cos(60°+A)

=(cosA)(cos(60°-A)cos(60°+A))

=(cosA)(1/2cos(60°-2A) + 1/2cos(60°+2A))

=(cosA)(1/2cos(-A) + 1/2cos(120°))

2. Substituting cos(60°-A) = cos(60°+A) = 1/2 into the expanded expression:

= cosA(1/2cos(-A) + 1/2cos(120°))

=cosA(1/2(1/2cosA) + 1/2(-1/2))

= cosA(1/4cosA - 1/4)

= (1/4)cosAcosA - (1/4)cosA

=(1/4)cos3A

3. Simplifying the resulting expression to obtain 1/4 cos3A:

=(1/4)cosAcosA - (1/4)cosA

=(1/4)cosA(cosA - 1)

=(1/4)cos3A

Therefore, we have proven that cosAcos(60°-A)cos(60°+A) = 1/4 cos3A. Hence Proved.