Make up a sequences that have (a) 3,3,3,3,... as its second differences. (b) 1, 2,3,4,5,... as its third differences (c) 1, 2, 4,8,16,... as its 100th differences.

Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.

Given:

Let’s consider the probability of Deshawn not doing basketball after school:Probability of Deshawn not doing basketball after school = 100% - Probability of Deshawn doing basketball after school= 100% - 20% = 80%

Similarly, let’s consider the probability of Deshawn not talking to friends after school: Probability of Deshawn not talking to friends after school = 100% - Probability of Deshawn talking to friends after school= 100% - 45% = 55% Probability of Deshawn doing neither basketball nor talking to friends after school:Probability of Deshawn not doing basketball after school * Probability of Deshawn not talking to friends after school= 80% * 55% = 44%

The probability of Deshawn doing either basketball or talking to friends after school is 65%, and the probability of Deshawn doing neither basketball nor talking to friends after school is 44%, which is greater than 35% which is Dans mistake. Hence, Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.

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The x-intercepts are (0, 0) and (4, 0), and the y-intercept is (0, 0).

By graphing linear equatiοns, yοu can explain the relatiοnship between twο variables visually. We can easily see what happens tο οne variable as the οther grοws by using a graph. The value οf the x variable rises as we mοve tο the right οn a graph.

[tex]y = (3/2)x^2 - 6x[/tex] is the given equatiοn.

We can use the fοrmula tο find the x-cοοrdinate(s) οf the vertex οf this parabοla:

x = -b/2a

where a and b are the cοefficients οf the equatiοn's x² and x terms, respectively.

In this case, a = 3/2 and b = -6, resulting in:

x = -(-6)/(2*3/2) = 4

As a result, the vertex's x-cοοrdinate is 4.

Tο find the y-cοοrdinate οf the vertex, enter this value οf x intο the fοllοwing equatiοn:

[tex]y = (3/2)(4)^2 - 6(4) = -12[/tex]

As a result, the parabοla's vertex is at the pοint (4, -12).

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Answer: 96

Step-by-step explanation:

M = 180 - 84 = 96

m<k = 96

The subsets B. The 3×3 matrices with all zeros in the third row. C. The diagonal 3×3 matrices, and F. The symmetric 3×3 matrices are subspaces of M3(R).

A subspace of a vector space is a portion of that space that meets the three criteria of closure under addition, closure under scalar multiplication, and the presence of the zero vector. If two vectors from the subspace are added, the resultant vector will still be in the subspace because of closure under addition. If a vector from the subspace is multiplied by any scalar, the resultant vector will still be in the subspace, according to the concept of closure under scalar multiplication.

The conditions of a subspace are: closure under addition, closure under scalar multiplication, and contains the zero vector.

For all the options we have:

A: The 3 x 3 matrices in reduced row-echelon form (A): As this subset is not closed under addition, M3(R), it is not a subspace of M3(R).

B. The 3 x 3 matrices with all zeros in the third row: Due to its closure under addition and scalar multiplication as well as the presence of the zero vector, this subset is a subspace of M3(R).

C. The diagonal 3 x 3 matrices: This subset, which is closed under addition and scalar multiplication and contains the zero vector, is a subspace of M3(R).

D. The invertible 33 matrices: Because this subset is not closed under addition, M3(R), it is not a subspace of M3(R).

E. The 3 x 3 matrices that are not invertible Due to the fact that it is not closed under scalar multiplication, this subset is not a subspace of M3(R).

F. The symmetric 3x 3 matrices: This subset, which is closed under addition and scalar multiplication and contains the zero vector, is a subspace of M3(R).

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The induced emf in a region of the brain when a conducting loop is held near a person's head and the current in the loop is changed rapidly, is equal to -525 V.

The induced emf can be calculated using Faraday's law of electromagnetic induction, which states that the emf induced in a loop of wire is equal to the rate of change of magnetic flux through the loop.

The magnetic flux (Φ) is equal to the product of the magnetic field (B) and the area (A) through which it passes. Therefore, the induced emf (ε) is given by:

ε = -dΦ/dt ⇒ -B dA/dt.

Where the negative sign indicates that the emf is induced in a direction that opposes the change in magnetic flux.

In this problem, the magnetic field changes at a rate of 3.00 × 10^4 T/s over an area of 1.75 × 10^-2 m^2. Therefore, the induced emf is:
Plugging in our values, we get:
E = (-3.00 10^4 T/s)(1.75 10^(-2) m^2)/(1 s)
E = -525 V

Therefore, the induced emf, in this case, is -525 V. Here, the negative sign shows that the emf is induced in a direction that opposes the change in magnetic flux

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Answer:

5/7

Step-by-step explanation:

Make all the denominators 42.

26/42, 27/42, 28/42, 29/42

The pattern is the numerator increases by one each time.

30/42 = 5/7

Hope this helps!

Using proportions, the real height of the telephone mast is estimated to be 9 meters.

A proportion is a fraction of a total amount, and equations are constructed using these fractions and estimates to find the desired measures in the problem using basic arithmetic operations like multiplication and division. Because the telephone box and the mast are similar figures in this problem, their side lengths are proportional.

The following proportional relationship is established as a result:

x / 1.5 cm = 10.8 cm / 1.8 cm.

The relationship's left side can be simplified as follows:

6 = x / 1.5 cm.

The estimate is then calculated using cross multiplication, as shown below:

6 x 1.5 cm = 9.5 cm².

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The given family of functions is y = c1ex + c2e−x which is the general solution of the differential equation y'' − y = 0 on the indicated interval which is (−∞, ∞).

       Now, we are required to find a member of the family that is a solution to the initial-value problem which is

                             y(0) = 0 and y′(0) = 5.

     The differential equation is y'' − y = 0

      The characteristic equation is r2 − 1 = 0r2 = 1r1 = 1 and r2 = −1

     The general solution of the differential equation is y = c1ex + c2e−x

      Let us solve for the constants by using the given initial conditions:

                    At x = 0,y(0) = c1e0 + c2e0 = 0 + 0 = 0y(0) = 0

                    means c1 + c2 = 0or c1 = -c2At x = 0, y′(0) = c1ex |x=0 + c2e−x |x=0(d/dx)(c1ex + c2e−x) |x=0y′(0) = c1 - c2 = 5c1 - c2 = 5c1 - (-c1) = 5c1 + c1 = 5c1 = 5/2c1 = 5/2

                    Let's replace c1 = 5/2 in c1 = -c2, c2 = -5/2

                The solution of the initial-value problem y = (5/2)ex − (5/2)e−x is a member of the family y = c1ex + c2e−x that is a solution of the initial-value problem y(0) = 0 and y′(0) = 5.

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The sampling approach that was used in the statement "Kane randomly sampled 250 nurses from urban areas and 250 from rural areas from a list of licensed nurses in Wisconsin to study their attitudes toward evidence-based practice" is Stratified random sampling.

Stratified random sampling is a method of sampling that is based on dividing the population into subgroups called strata. Stratified random sampling is a statistical sampling method that involves the division of the population into subgroups or strata, and a sample is then drawn from each stratum in proportion to the size of the stratum. It's a sampling method that ensures the representation of all population strata in the sample, making it more effective than simple random sampling.

Stratified random sampling is used when there are variations in the population that are likely to influence the outcome of the study. The stratified random sampling method is used to ensure that these differences are reflected in the sample. In this way, the results of the study are more representative of the entire population than they would be if a simple random sample were used.

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The weight of the package is 56 using arithmetic operations.

Mathematicians use multiplication to calculate the product of two or more integers. It is a fundamental operation in mathematics that is frequently used in everyday living. When we need to combine sets of similar sizes, we use multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are compounded are referred to as the factors. Repeated adding of the same number is made easier by multiplying the numbers.

let weight of package be= x

x*5/7=40

x=(40*7)/5

x=56

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Answer:

Step-by-step explanation:

Let's use "n" to represent the number of slices in each pizza. Then the equation to describe the situation is:

999n = 727272

To solve for "n", we divide both sides by 999:

n = 727272/999

Using a calculator or long division, we get:

n ≈ 728.56

Therefore, each pizza has approximately 728 slices.

Answer :

1. We are given with a parallelogram whose opposite sides are :-

We know that opposite sides and angles of the parallelogram are equal.

PS = RQ

[tex] \implies \sf \: (-1 + 4x) = (3x + 3) \\ [/tex]

[tex] \implies \sf \: 4x - 3x = 3 + 1\\ [/tex]

[tex] \implies \sf \: x = 4 \\ [/tex]

Length of RQ is (3x + 3)

Substituting value of x = 4 in RQ.

[tex] \implies \sf \: (3x +3) \\ [/tex]

[tex] \implies \sf \: 3(4) + 3 \\ [/tex]

[tex] \implies \sf \: 12 + 3 \\ [/tex]

[tex] \implies \sf \: 15 \\ [/tex]

Therefore, Length of RQ will be 15

2. Find [tex] \sf \angle G [/tex]

We are given with :-

Since, Opposite angles of parallelogram are also equal :

[tex]\angle G = \angle E \\ [/tex]

[tex]\implies \sf \: 5x -9 = 3x + 11 \\ [/tex]

[tex]\implies \sf \: 5x - 3x = 11 +9 \\ [/tex]

[tex] \implies \sf \: 2x = 20 \\ [/tex]

[tex] \implies \sf \: x = \dfrac{20}{10} \\ [/tex]

[tex] \implies \sf \: x = 10 \\ [/tex]

Since [tex] \sf \angle G [/tex] is 5x - 9.

Substituting value of (x = 10) in [tex] \sf \angle G [/tex]

[tex] \implies \sf \: 5(10) - 9 \\ [/tex]

[tex] \implies \sf \: 50 -9 \\ [/tex]

[tex] \implies \sf \: 41 \\ [/tex]

Therefore, [tex] \sf \angle G [/tex] is 41.

Answer:

19. QR = 15 units.

20. m ∡G=41°

Step-by-step explanation:

The properties of a parallelogram are:

Question No. 19.

We know that the opposite side of parallelogram is equal, So

PS=QR

-1+4x=3x+3

First of all we will find the value of x.

-1 + 4x = 3x + 3

Subtracting 3x from both sides, we get:

x - 1 = 3

Adding 1 to both sides, we get:

x = 4

SO, QR=3*4+3=15

Side QR is 15 units.

Question No. 20.

We know that the opposite angle of parallelogram is equal, So

EF=GD

3x+11=5x-9
Subtracting 3x from both sides:

3x + 11 - 3x = 5x - 9 - 3x

11 = 2x - 9

Next, we can add 9 to both sides to isolate the variable term on one side:

11 + 9 = 2x - 9 + 9

20 = 2x

Finally, we can solve for x by dividing both sides by 2:

20/2 = 2x/2

x=10

Now

m ∡G=5x-9=5*10-9=41°

therefore m ∡G=41°

On solving the question we can say that Therefore, the solutions to the inequality given inequality are: x < 4 or x > 6.

An inequality in mathematics is a relationship between two expressions or values ​​that are not equal. Imbalance therefore leads to inequality. An inequality establishes a connection between two values ​​that are not equal in mathematics. Equality is different from inequality. The inequality sign () is most commonly used when two values ​​are not equal. Various inequalities are used to contrast values, no matter how small or large. Many simple inequalities can be solved by changing both sides until only variables remain. But many things contribute to inequality.

two inequalities

4x - 6 < 10

4x < 16

x < 4

2x - 4 > 8

2x > 12

x > 6

Therefore, the solutions to the given inequality are:

x < 4 or x > 6.

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Answer: 0000

Step-by-step explanation:

Answer:

When a point is reflected over the x-axis, the x-coordinate remains the same, but the y-coordinate is multiplied by -1.

So, the coordinates of X' are (9, -5) since the x-coordinate remains the same and the y-coordinate is multiplied by -1.

Similarly, the coordinates of Y' are (-3, -6) and the coordinates of Z' are (-8, -4).

Therefore, X′ is (9,−5), Y′ is (−3,−6), and Z′ is (−8,−4).

Maria's capital gain is 55.21%. Rounded to the nearest tenth of a percent, this is 55.2%.

To determine Maria's capital gain as a percent, we need to calculate the difference between the selling price and the purchase price, and then express this difference as a percentage of the purchase price.

The purchase price for 1,000 shares of stock was:

$35.50 x 1,000 = $35,500

The selling price for 1,000 shares of stock was:

$55.10 x 1,000 = $55,100

The capital gain is the difference between the selling price and the purchase price:

$55,100 - $35,500 = $19,600

To express this gain as a percentage of the purchase price, we divide the capital gain by the purchase price and multiply by 100:

($19,600 / $35,500) x 100 = 55.21%

In summary, to calculate the percent capital gain from the purchase and selling price of a stock, we simply divide the difference between the two prices by the purchase price and multiply by 100.

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Answer: 4x(x - 4)

Step-by-step explanation:

4x² - 16x = 4x(x - 4)

Now we can see that the expression inside the parentheses can also be factored:

x - 4 = (x - 4)

So the fully factorized expression is:

4x² - 16x = 4x(x - 4) = 4x(x - 4)

Answer:

ㅤ

- 4x( x + 4 )

ㅤ

Step-by-step explanation:

ㅤ

[tex]\large{\pmb{\sf{ - 4x^{2} - 16x}}}[/tex]

ㅤ

[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{4}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\pmb{\sf{\leadsto{- 4(x^{2} + 4x)}}}}[/tex]

ㅤ

[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{x}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\purple{\boxed{\pmb{\sf{\leadsto{- 4x(x + 4)}}}}}}[/tex]

━━━━━━━━━━━━━━━━━━━━━━

[tex]\star \: {\large{\underline{\underline{\pink{\mathfrak{More:-}}}}}} \: \star[/tex]

ㅤ

[tex]\large{\dashrightarrow}[/tex] Two positive always makes positive sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] Two negatives always makes positive sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] A positive and a negative always makes negative sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of two positives is always positive with a positive sign.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of two negatives is always positive with a negative sign.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of a positive and a negative is always negative with the sign of whose number is greater.

m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

The degree of rotation between two lines or two planes around a central point is measured by an angle. Typically, it is expressed in radians or degrees. Angles are used in many mathematical and scientific uses, including trigonometry, physics, and engineering, where they are crucial in determining the shape and characteristics of geometric figures. Angles come in four different varieties: acute (less than 90 degrees), right (exactly 90 degrees), oblique (more than 90 degrees), and straight (exactly 180 degrees).

given

Because a triangle's total sides equal 180 degrees, we have:

R, S, and T add up to 180.

Inputting the numbers provided yields:

m∠R + 72 + 32 = 180

Simplifying the equation:

m∠R = 76

m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

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Answer:

x = 8500

y = 15000

Step-by-step explanation:

small vans: x

large vans: y

A: 5x + 2y = 72500

B: 2x + 6y = 107000

5x + 2y = 72500 => y = (72500 - 5x)/2

2x + 6(72500 - 5x)/2 = 107000

2x + 217500 - 15x = 107000

15x - 2x = 217500 - 107000

13x = 110500

x = 110500/13 = 8500

y = (72500 - 5x)/2 = y = (72500 - 5x8500)/2 = 15000

Answer:

We see that each fraction is in simplest form, and they add up to 1, so this confirms that 168 is the highest number of similar bouquets that Fatima can make without having any flowers left over.

Step-by-step explanation:

Yeah, I guess what that person said ^^ ??

Answer:

Step-by-step explanation: i hope this help if not let me know so i can fix it

Jake is presently [tex]0[/tex] years old and the year is [tex]2036[/tex], which is not a relevant response or there may be some missing details in the question.

A span of time that is equivalent to one year on the Calender but starts at a different period. A cycle of 365 and 366 day split into twelve month starting in January and ending in December.

Every monthly on the calender has four complete weeks since every month has at least 28 days. A few month have a few more days, but these extra days don't add up to a full week, therefore they aren't counted.

Let's first find the current year, given that the year in which their ages will sum up to [tex]16x[/tex] is[tex]2036[/tex].

[tex]2036 - (y - 2036) = 2*2036 - y[/tex]

Simplifying this expression, we get:

[tex]2*2036 - y = 4072 - y[/tex]

[tex]2y = 4072[/tex]

[tex]y = 2036[/tex]

Now, let's find Jake's current age by subtracting his birth year from the current year [tex]y - (2036 - 7x)[/tex]

Since their ages sum up to 16x in 2036, we have:

[tex]x + 7x = 16x[/tex]

[tex]8x = 16x[/tex]

[tex]x = 0[/tex]

This means that Jake is currently [tex]0[/tex] years old, which is not a meaningful answer.

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Answer:

Step-by-step explanation: In the problem, they tell us that

dL / dt = 7 cm/s (the rate at which the length is changing) and

dw / dt = 8 cm/s (the rate at which the width is changing)

Want dA/dt (the rate at which the area is changing) when L = 7 cm and w = 5 cm

The equation for the area of a rectangle is:

A = L·w, so will need the product rule when taking the derivative.

dA/dt = L (dw/dt) + w (dL/dt)

Now just plug in all of the given numbers:

dA/dt = (7)(7) + (5)(8) = 49+40 = 89  cm²/s

The particular solution to the differential equation is given by: y(t) = y_ h(t) + y_ p(t) = c exp (-5/2 t) + t + 3Here, y_ h(t) is the homogeneous solution, which we don't need to find for this problem.

To solve the given differential equation, we'll need to use the method of undetermined coefficients. In this method, we assume that the particular solution to the differential equation has the same form as the forcing term. Here's how we can solve the given differential equation: Identify the forcing term and its derivatives. The forcing term is given by: f(t) = 68 - 20tWe can find its first derivative as follows: f'(t) = -20We can find its second derivative as follows: f''(t) = Guess the form of the particular solution We assume that the particular solution has the same form as the forcing term.

Since the forcing term is a first-degree polynomial, we assume that the particular solution also has the form of a first-degree polynomial: y_ p(t) = At + B Here, A and B are constants that we need to determine. Find the derivatives of the assumed form of the particular solution. Here are the first and second derivatives of the assumed form of the particular solution: y_ p(t) = At + B ==> y_ p'(t) = A ==> y_ p''(t) = 0. Substitute the assumed form of the particular solution and its derivatives into the differential equation Substituting y_ p(t), y_ p'(t), and y_ p''(t) into the differential equation, we get:8A + 20(At + B) = 68 - 20t Simplifying the above equation, we get: (8A + 20B) + (20A - 20)t = 68Comparing the coefficients of t and the constant terms on both sides,

we get two equations:8A + 20B = 68 (1)20A - 20 = 0 (2)Solving equation (2) for A, we get: A = 1 Substituting A = 1 into equation (1), we get:8 + 20B = 68Solving for B, we get: B = 3. Write the particular solution to the differential equation Substituting A = 1 and B = 3 into the assumed form of the particular solution, we get :y_ p(t) = t + 3Therefore, the particular solution to the differential equation is given by: y(t) = y_ h(t) + y_ p(t) = c exp (-5/2 t) + t + 3Here, y_ h(t) is the homogeneous solution, which we don't need to find for this problem.

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It pays to purchase the information for any value of c less than $13.25 then we buy the information for a price less than $13.25, we can increase our expected payoff above a loss of $1.

To calculate it Let C1 be the event that the first coin is selected and C2 be the event that the second coin is selected. Then, we have:

P(C1) = P(C2) = 1/2 (since one of the two coins is randomly chosen)

P(H|C1) = 0.6 (probability of getting heads when the first coin is flipped)

P(H|C2) = 0.3 (probability of getting heads when the second coin is flipped)

Let's consider the case when we do not buy the insider's information. Then, our expected payoff can be calculated as follows:

E(X) = P(C1) * P(H|C1) * (10) + P(C1) * P(T|C1) * (-10) + P(C2) * P(H|C2) * (10) + P(C2) * P(T|C2) * (-10)

= (1/2) * (0.610 - 0.410 + 0.310 - 0.710)

= -1

Therefore, if we do not buy the insider's information, our expected payoff is a loss of $1.

Now, let's consider the case when we buy the insider's information for an amount c.

If we buy the information, we will know which coin was selected and we can bet accordingly to maximize our expected payoff.

If we know that the first coin was selected, we should bet on heads since it has a higher probability of occurring. If we know that the second coin was selected, we should bet on tails since it has a higher probability of occurring.

Therefore, if we buy the insider's information, our expected payoff can be calculated as follows:

E(X|buying information) = P(C1) * P(H|C1) * (10-c) + P(C1) * P(T|C1) * (-c) + P(C2) * P(H|C2) * (-c) + P(C2) * P(T|C2) * (10-c)

= (1/2) * (0.6*(10-c) - 0.4c + 0.3(-c) - 0.7*(10-c))

= -0.2c + 1.5

To find the values of c for which it pays to purchase the information, we need to solve the inequality:

E(X|buying information) > E(X)

-0.2c + 1.5 > -1

Solving for c, we get:

c < 13.25

Therefore, it pays to purchase the information for any value of c less than $13.25. If we buy the information for a price less than $13.25, we can increase our expected payoff above a loss of $1.

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Answer:

Step-by-step explanation:

[tex]\angle R+\angle S+ \angle T =180[/tex]       (angle sum of a triangle is 180°)

[tex]2x+11+3x+23+x+42=180[/tex]

                              [tex]6x+76=180[/tex]

                                      [tex]6x=104[/tex]

                                        [tex]x=17.667[/tex]

[tex]\text{So we get: } \angle R= 46.33,\angle S=76,\angle T=59.667[/tex]

In ascending order:

   [tex]\angle R= 46.33,\angle T=59.667,\angle S=76[/tex]

The height of the triangle is 1.12 units (rounded to two decimal places).

The height of a triangle is the perpendicular distance from the base of the triangle to the opposite vertex. In other words, it is the length of the line segment that is perpendicular to the base and passes through the opposite vertex.

We can use the formula for the area of a triangle:

Area = (1/2) * base * height

And since we know the values of the base (b) and the area (a), we can rearrange the formula to solve for the height (h):

h = (2a) / b

Plugging in the values of a and b:

h = (2 * 14) / 25

h = 28 / 25

Therefore, the height of the triangle is 1.12 units (rounded to two decimal places).

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The value of ( sin 30° + cos 30° ) - ( sin 60° + cos 60° ) is 0.

(sine 30 + cos 30 ) - (sin 60 + cos 60 )

= ([tex]\frac{1}{2}[/tex] + [tex]\frac{\sqrt{3} }{2}[/tex]) - ([tex]\frac{\sqrt{3} }{2}[/tex] + [tex]\frac{1}{2}[/tex])

= [tex]\frac{1}{2}[/tex]  + [tex]\frac{\sqrt{3} }{2}[/tex] - [tex]\frac{\sqrt{3} }{2}[/tex] - [tex]\frac{1}{2}[/tex]

= [tex]\frac{\sqrt{3} }{2}[/tex] - [tex]\frac{\sqrt{3} }{2}[/tex]

= 0

Trigonometry is a branch of mathematics that focuses on the relationships between angles and the sides of triangles. It involves the study of trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant, and their various properties and applications. These functions relate the angles of a right triangle to the lengths of its sides, and they can be used to solve a wide range of problems in fields such as engineering, physics, astronomy, and navigation.

Trigonometry also includes the study of trigonometric identities, which are mathematical expressions that are true for all values of the variables involved. These identities can be used to simplify complex trigonometric expressions and to prove other mathematical theorems.

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The star Proxima Centauri is 4.2 light-years away from Earth, making it the sun's nearest rival. The word "nearest" means "nearest" in Spanish.

"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."

We always count the unit or amount value first and then calculate the more or less amount value.

For this reason, this procedure is called a unified procedure.

Many set values ​​are found by multiplying the set value by the number of sets.

A set value is obtained by dividing many set values ​​by the number of sets.

Hence, The star Proxima Centauri is 4.2 light-years away from Earth, making it the sun's nearest rival. The word "nearest" means "nearest" in Spanish.

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Given that a 0.625 kg basketball is dropped from a height of 3 meters straight down. The coefficient of restitution between the ball and the ground is e = 0.87. At 0.15 seconds after the basketball is dropped, a 58.5-gram tennis ball is dropped from the same 3-meter height straight onto the basketball. If the coefficient of restitution between the tennis ball and the basketball is e = 0.9,

    determine how high the tennis ball bounces above the ground after the collision. Neglect the size of the balls.

The balls meet at a height of 0.5073 m above the ground. Hence the height that the tennis ball bounces above the ground is to be calculated.

Given that the coefficient of restitution between the ball and the ground is e = 0.87The coefficient of restitution between the tennis ball and the basketball is e = 0.9

The coefficient of restitution(e) is defined as the ratio of the relative velocity of separation and relative velocity of approach between two objects.

When a ball falls from height, it gains potential energy.

Potential energy (PE) = mghWhere, m = mass of the object, g = acceleration due to gravity, h = height

PE of Basketball = mgh= 0.625 kg * 9.81 m/s² * 3 m= 18.4 Joules

Initial kinetic energy (KE) = PE of Basketball

KE of Basketball = 18.4 J

Let the velocity of the basketball before collision be u1 and the velocity of the tennis ball before collision be u2. After the collision, let the velocity of the basketball be v1 and the velocity of the tennis ball be v2.

Using the coefficient of restitution (e) we can find the velocity of the balls after collision

   v1 - v2 = -e(u1 - u2)

Initial momentum (P) = Final momentum (P)

before the collision  P = m1u1 + m2u2

after collision  P = m1v1 + m2v2P = (0.625 kg * u1) + (0.0585 kg * u2)P

                           = (0.625 kg * v1) + (0.0585 kg * v2)

        Using the above two equations and the given coefficients of restitution, we can find the velocity of the balls after collisionv1 = (m1u1 + m2u2 + e * m2 * (u2 - u1)) / (m1 + m2)v2 = (m1u1 + m2u2 + e * m1 * (u1 - u2)) / (m1 + m2)

     Here, m1 = mass of basketball

                     = 0.625 kg,

                m2 = mass of tennis ball

                      = 0.0585 kg,

  u1 = 0, u2 = 0P = m1v1 + m2v2 => v1 + v2 = P / (m1 + m2)

Also given that the time taken by the balls to meet is 0.15 seconds

Let h be the height to which the tennis ball bounces after the collision.

When the tennis ball bounces to height h, it gains potential energy.

KE + PE = Total energy

             = Constant Using the principle of conservation of energy we can find the height to which the tennis ball bounces after collision(1/2) * m2 * v2² + (1/2) * m2 * g * h

              = (1/2) * m2 * u2² + (1/2) * m2 * g * 0(1/2) * m2 * v2²

              = (1/2) * m2 * u2² - (1/2) * m2 * g * h(1/2) * v2²

              = (1/2) * u2² - g * h

Substituting the values of u1, u2, m1, m2, e, t and g, we get:

v1 = (0 + 0.0585 kg * 9.81 m/s² + 0.9 * 0.0585 kg * (0 - 0)) / (0.625 kg + 0.0585 kg)v1

   = 1.96 m/sv2

    = (0.625 kg * 0 + 0 + 0.9 * 0.625 kg * (0 - 0)) / (0.625 kg + 0.0585 kg)v2

    = 5.5 m/s

Here, u1 = u2 = 0.

Using the above equation, we can find the height to which the tennis ball bounces after collision (1/2) * (0.0585 kg) * (5.5 m/s)² = (1/2) * (0.0585 kg) * (0 m/s)² - (0.0585 kg) * 9.81 m/s² * h12.83 J

                = -0.286 J - 0.572 h0.572 h

                = -12.83 J / 2h

                = -12.83 J / (2 * 0.572)

                = 11.2 m

Hence the height to which the tennis ball bounces above the ground after the collision is 11.2 m.

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