\( \begin{aligned} \frac{\sin x+\tan x \cos x}{\tan x} & =\frac{\sin x+\sin x}{\tan x} \\ & =\frac{2 \sin x}{\tan x}\end{aligned} \)

We can use a proportion to solve the problem:

Let x be the number of minutes it would take Mika to eat 35 hot dogs

Then we can set up the proportion:

21 hot dogs / 6 minutes = 35 hot dogs / x minutes

To solve for x, we can cross-multiply:

21 hot dogs * x minutes = 6 minutes * 35 hot dogs

21x = 210

x = 10

Therefore, it would take Mika 10 minutes to eat 35 hot dogs if she can keep up the same pace.

Answer:

24 antelopes, 36 monkeys

Step-by-step explanation:

6 : 7 means that for every 6 antelopes, there are 7 monkeys, so

a/6 = m/7   (1)

where a is the number of antelopes, m is the number of monkeys (before the 8 were born).

After 8 monkeys are born, the new ratio becomes:

a/2 = (m+8)/3    (2)

You can solve a and m for these two equations:

Rewrite (1) as 7a = 6m (cross product)

Rewrite (2) as 3a = 2m+16  (cross product)

=> 2m = 3a-16 (isolate 2m)

=> 6m = 9a-48 (multiply by 3)

Plug in (2) into (1):

7a = 9a-48 =>

-2a = -48

a = 24 antelopes

m = (a/6)*7 = 28 monkeys BEFORE the 8 newborns, so now there are

24 antelopes and 28+8=36 monkeys

Goldstar Bank will have $42,000 in excess reserves as a result of this deposit. These excess reserves can be used to make additional loans or investments.

When a customer makes a deposit in a bank, the bank is required to hold a certain percentage of that deposit as reserves. The reserve requirement is set by the central bank and is typically a percentage of the bank's total deposits.

In this case, the customer has deposited $50,000 into Goldstar Bank and the reserve requirement is 16 percent. This means that Goldstar Bank must hold 16 percent of the deposit as reserves and can loan out the remaining amount.

The amount of reserves that Goldstar Bank must hold is:

16% of $50,000 = 0.16 x $50,000 = $8,000

This means that Goldstar Bank can loan out:

$50,000 - $8,000 = $42,000

However, since the question asks for the number of excess reserves Goldstar Bank will have, we need to subtract the required reserves from the total reserves.

The total reserves after the deposit will be:

Required reserves = $8,000

Excess reserves = Total reserves - Required reserves = $50,000 - $8,000 = $42,000

Complete question:

Suppose that initially Goldstar Bank is completely "loaned up." A customer then deposits $50,000 into the bank. The reserve requirement is 16 percent.

1. What amount of excess reserves will Goldstar Bank now have as a result of this deposit?

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Therefore, it will take John 4 hours to catch up with Karabo.

In mathematics, an equation is a statement that shows the equality of two expressions. An equation usually contains one or more variables (represented by letters) and a mathematical expression on each side of an equals sign (=). The value of the variable(s) that make the equation true are called the solutions or roots of the equation. Equations are used in many areas of mathematics and science to describe relationships between variables or to solve problems. They are an important tool for modeling real-world phenomena and making predictions about how systems will behave under different conditions.

Here,

Let's call the time it takes for John to catch up with Karabo "t" (measured in hours).

In the hour that Karabo has been driving, she has covered a distance of:

distance = speed x time = 80 km/h x 1 h = 80 km

When John starts driving, he is behind Karabo by this distance. However, he is driving faster, so he will catch up to her eventually.

We can set up an equation to represent this situation:

distance covered by John = distance covered by Karabo

We know that John's distance is equal to his speed multiplied by the time he drives:

distance covered by John = 100 km/h x t

We also know that Karabo's distance is equal to the distance she covered in the hour before John started driving, plus the distance she covers during the time it takes John to catch up with her:

distance covered by Karabo = 80 km + 80 km/h x t

Now we can set these two expressions equal to each other and solve for t:

100 km/h x t = 80 km + 80 km/h x t

100 km/h x t - 80 km/h x t = 80 km

20 km/h x t = 80 km

t = 80 km / 20 km/h

t = 4 hours

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The simplified form of the rational expression is given as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)] = 1/[(x + 4)(x - 3)]

The rational expression for this problem is defined as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)].

To simplify the rational expression, first we must factor the quadratic equations at the denominator of each of the fractions, hence:

The simplification of the first quadratic function, x² - 3x - 10 = (x - 5)(x + 2), means that the numerators of each of the fractions can be simplified.

As the numerators of each of the fractions are simplified along with x² - 3x - 10, the remaining part is x² + x - 12, hence the simplified expression is given as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)] = 1/[(x + 4)(x - 3)]

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In the molecule of phosphorus pentachloride:

In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. It can be thought of as a completely flat and level surface, like a tabletop or a sheet of paper.

Collinear points, on the other hand, are points that lie on the same straight line. If three or more points are collinear, they can be connected by a straight line, and the line they lie on is sometimes called a "collinear line".

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The coins take up approximately 35.259 cubic inches of space inside the treasure chest.

Volume refers to the amount of space occupied by a three-dimensional object, measured in cubic units.

Solution:

The volume of each coin is a cylinder with radius 1.4 inches and height 0.0625 inches. The volume V of one coin is given by:

V = πr²h

= 3.14 x 1.4² x 0.0625

= 0.1533 cubic inches (rounded to four decimal places)

Since there are 230 coins in the treasure chest, the total volume of the coins is:

Total volume = 230 x V = 230 x 0.1533 = 35.259 cubic inches (rounded to three decimal places)

Therefore, the coins take up approximately 35.259 cubic inches of space inside the treasure chest.

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Answer: 41/4 or 10.25 or 10 1/4

Step-by-step explanation:Exact Form:

x

=

41

4

Decimal Form:

x

=

10.25

Mixed Number Form:

x

=

10

1

4

Answer: A

Step-by-step explanation:
To form a triangle, the two smaller lengths must be bigger than the largest side(Hopotonuse or sum, cant spell)

8 + 7 = 15
15>14

The possible lengths to complete a triangle with side lengths of 14 in. and 8 in. are  6 < x < 22.

The triangle inequality theorem states that for any given triangle, the total of the two sides is always greater than the sum of the three sides. The Triangle is a polygon with three line segments as its boundaries. That is the tiniest polygon imaginable.

We have side lengths of 14 in. and 8 in.

Using Triangle inequality we know that the sum of two is greater than the third side.

Also, the range for third side is

14 + 8 = 22 inch

14 - 8 =  6 inch

So, the third side lies between as 6 < x < 22.

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Step-by-step explanation:

To find the area of the rose garden, we need to find the sum of the areas of the rectangle and the semicircle.

Area of rectangle = length x width = 22 ft x 14 ft = 308 sq ft

Area of semicircle = (1/2) x pi x radius^2, where radius = diameter/2

The diameter of the semicircle is the width of the rectangle, which is 14 ft. So the radius is 7 ft.

Area of semicircle = (1/2) x 3.14 x 7^2 = 3.14 x 24.5 = 77.03 sq ft

Total area of rose garden = area of rectangle + area of semicircle

= 308 sq ft + 77.03 sq ft

= 385.03 sq ft

Therefore, the area of the rose garden is 385.03 square feet.

A(n) = 3 + 7(n-1)

Arithmetic sequences have common differences and change by the same amount between each term.

Arithmetic Sequences

Arithmetic sequences change by the same amount each term. In the sequence above, each term increases by 7. This means that 7 is added to the previous term to make the new term. Using this information we can write a function to represent this sequence.

Explicit Rule

The function that describes an arithmetic sequence is known as an explicit rule. Explicit rules are written as A(n) = A(1) + d(n-1). In this equation, A(1) represents the first term in a sequence and d represents the common difference. As you can see, the first term is 3, so A(1) = 3. The common difference is the change between terms. The previous paragraph shows that the common difference for this sequence is 7.

This means the explicit rule for this sequence is A(n) = 3 + 7(n-1).

Answer:

[tex]20x^{50}[/tex]

Step-by-step explanation:

This is because you are simplifying the radical by breaking the radicand up into a product of known factors, assuming positive real numbers.

There is your answer!

The expression [tex]i^{8}+i^{9}+i^{10}+i^{11}[/tex] is equivalent to 0 + i - 1 - i, which simplifies to -1.

Using the property of imaginary unit i, we know that [tex]i^2=-1[/tex]. Therefore, i^8 can be written as [tex](i^2)^4[/tex] which is equal to[tex]1^4 = 1[/tex]. Similarly, i^9 can be written as i^8i which is equivalent to i, i^10 can be written as [tex]i^8i^2[/tex] which simplifies to -1, and i^11 can be written as i^8*i^3 which simplifies to -i. Thus, [tex]i^{8}+i^{9}+i^{10}+i^{11}[/tex] can be simplified to 1 + i - 1 - i, which equals 0 - 0i. Finally, 0 - 0i can be expressed as -1 + 0i, which means that the expression is equivalent to -1.

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The solution is, after paying the agent commission, Brian will get 155.605265 pounds.

And, the percentage increase in the price 7.9%

the car travel in 2 hours 45 minutes is 291.5 km.

Currency conversion can be stated as the process of converting one currency into another currency. This is usually done in order to make transactions or investments in another country or to keep track of the value of assets held in different currencies.

here, we have,

we know.

The value of one currency is generally expressed in terms of another currency using exchange rates, which means  the value of one currency in terms of another currency at a specific point in time.

We know that, exchange rate can fluctuate over time, and currency conversion can be done manually or through various financial services or tools.

we have,

To calculate that how many pounds Brian will get for R2 100,

At first we need to calculate the amount of commission he will have to pay to the agent.

Commission = 1.5% of R2 100

Commission = 0.015 x R2 100

Commission = R31.50

Now we can calculate how much money Brian will have left after paying the commission:

Amount left after commission = R2 100 - R31.50

Amount left after commission = R2 068.50

Next, we can use the exchange rate given to convert Rands to Pounds:

1 Rand = 0.075199 Pounds

Therefore, to find out how many Pounds Brian will get for R2 068.50, we can multiply R2 068.50 by the exchange rate:

2 068.50 x 0.075199 = 155.605265 pounds

Next Part:

The price of petrol is increased from R12, 58 per litre to R13, 28 per litre.

i.e. increase of price = 100

so, the percentage increase in the price = 100 * 100/1258

                                                                   =7.9%

Next Part:

A motor car drives at an average speed of 106 km/h.

the car travel in 2 hours 45 minutes = 106* 165 / 60

                                                           =291.5 km

So The solution is, after paying the agent commission, Brian will get 155.605265 pounds.

And, the percentage increase in the price 7.9%

the car travel in 2 hours 45 minutes is 291.5 km.

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The number 100-x has a decreasing worth as x rises.

A mathematical statement is referred to as an expression or an algebraic expression if it contains numbers, factors, and a numerical operation between them..For example, the arithmetic sign + separates the terms 4m and 5 from of the variable m in the expression 4m + 5..

The number 100-x has a decreasing worth as x rises. This is due to the fact that the phrase 100-x denotes a linear equation with a -1 slope. This indicates that the expression's value drops by one for each unit rise in x. As a result, the expression's value decreases as x increases. For instance, if x is 50, the expression's value is 50; however, if x is 70, the expression's value is 30. The meaning of the equation decreases in proportion to how much x increases.

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The possible range of y-values that form a solution to the inequality along with x=-4 is y<-12

The possible range of y-values that, along with x=-4, form a solution to the inequality 10x-3y>-4 can be found by isolating y on one side of the inequality. Here are the steps to do so:

1. Start with the given inequality: 10x-3y>-4
2. Plug in x=-4: 10(-4)-3y>-4
3. Simplify the left side of the inequality: -40-3y>-4
4. Add 40 to both sides of the inequality: -3y>36
5. Divide both sides of the inequality by -3 and reverse the inequality sign: y<-12

Therefore, the possible range of y-values that form a solution to the inequality along with x=-4 is y<-12. This means that any value of y that is less than -12 will satisfy the inequality when x=-4.

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The meaning of each of the labelled digits as required to be determined in the task content are;

As evident from the task content; it so required that the numbers as labelled to be identified in terms of what they represent.

Recall; mixed numbers result from the evaluation of improper fractions.

Hence, given the mixed number; 2 ⅓.

The number 2 which is whole is; the quotient.

The number 3 which is the denominator is the divisor.

The number 1 which is the numerator is the remainder.

Complete question; The number is 2⅓.

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Therefore, the original price of the shirts was $18.75 each.

An equality on both sides of the equal to sign signifies a mathematical equation, which is a relationship between two expressions. Here is an example of an equation: 3y = 16.

If the selling price of the shirts was 80% of their regular price, then we can use the following equation to find the original price:

Original price * 0.8 = Selling price

Let's substitute the given values into the equation:

Original price * 0.8 = 15

To solve for the original price, we need to isolate it on one side of the equation. We can do this by dividing both sides by 0.8:

Original price = 15 ÷ 0.8

Original price = 18.75

Therefore, the original price of the shirts was $18.75 each.

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The total money Eva would have to pay for parking if she left her car in the lot for 5 hours is $32.

Eva would have to pay (12 + 4t) if she left her car in the lot for t hours.

Automatic fee = $12

Additional fee per hour = $4

Total fee paid = y

Number of hours = t

y = 12 + 4t

When t = 5 hours

y = 12 + 4t

= 12 + 4(5)

= 12 + 20

y = 32

Therefore, the total money Eva would have to pay for 5 and t hours is $32 and (12 + 4t) respectively.

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65x + 17y = -108  is the result of completing the multiplication 7x+3y=-8 4x-y=-13 of two equations together.

To find the result of multiplying these two equations, we need to use the distributive property. The distributive property states that a(b + c) = ab + ac. In this case, we can distribute the 7 and the 4 across the equations:

7(7x + 3y) = 7(-8)
4(4x - y) = 4(-13)

Simplifying these equations gives us:

49x + 21y = -56
16x - 4y = -52

Now, we can combine like terms:

65x + 17y = -108

This is the result of multiplying the two equations together. We can check our work by plugging in values for x and y and seeing if the equation holds true. For example, if x = 1 and y = 2, then the equation becomes:

65(1) + 17(2) = -108
65 + 34 = -108
99 = -108

This is not true, so we know that our answer is correct.

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a) This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].

b) The null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.

c) The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).

a). (1.5pts) This statement is possibly false. It is not always true that if the linear system Ax = b has infinitely many solutions, then 6 is a linear combination of the columns of A. A concrete counterexample would be the linear system:
2x + 3y = 6
4x + 6y = 12
This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].

b). (1.5pts) This statement is always true. If a 3x3 matrix A has rank 2, then Null(A) is a plane through the origin. This is because the rank of a matrix is the number of linearly independent columns in the matrix, and if the rank is 2, then there are 2 linearly independent columns in the matrix. This means that the null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.

c). (1.5pts) This statement is possibly false. It is not always true that if {ū, v} is linearly dependent then so is {ü, v, w}. A concrete counterexample would be the vectors ū = [1, 0], v = [0, 1], and w = [1, 1]. The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).

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The equation is now in the form y=mx+b, where m is 2 and b is -4. The second equation is now in the form y=mx+b, where m is -0.5 and b is -2.

An equation is an expression that shows the relationship between two or more variables. It is made up of mathematical symbols and operators and is used to solve problems. Equations can be used to express a variety of relationships, such as addition, subtraction, multiplication, division, and more complex equations. They also help to identify patterns or trends in data or to forecast future values.

To change these equations into y=mx+b form, we must first rearrange the equations.

For the first equation, 2x-4=8, we can move the 4 to the other side of the equation, giving us 2x = 12. We can then divide both sides by 2, giving us x = 6. So the equation is now in the form y=mx+b, where m is 2 and b is -4.

For the second equation, -2x+4y=-8, we can move the -2x to the other side of the equation, giving us 4y=-8-2x. We can divide both sides by 4, giving us y=-2-0.5x. So the equation is now in the form y=mx+b, where m is -0.5 and b is -2.

Now that both equations are in y=mx+b form, we need to compare their m and b values to determine if there is one solution, no solution or infinite solutions. We can see that m for both equations is different, so this means that the two equations are not the same, and there is no solution. This means that the two equations have no solution.

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Answer:

A. The quantities that change in this problem situation are the number of books Ben orders and the total cost. The quantities that remain constant are the cost per book ($12) and the shipping and handling charge per order ($5).

The independent variable is the number of books Ben orders, as this is the variable that Ben has control over and chooses to change. The dependent variable is the total cost, as it depends on the number of books Ben orders.

B.
(imagine this as a chart)

Number of books                                                                            Total cost

1                                                                                                               $17

2                                                                                                              $29

3                                                                                                               $41

5                                                                                                              $65

10                                                                                                             $125

----------------------------------------------------------------------------------------------------------

To create this table, we used the formula:

Total cost = (Cost per book x Number of books) + Shipping and handling charge

For example, when Ben orders 3 books, the total cost is:

Total cost = ($12 x 3) + $5 = $41

Similarly, when Ben spends $65, the number of books he can order is:

Number of books = (Total cost - Shipping and handling charge) / Cost per book

Number of books = ($65 - $5) / $12 = 5

And so on for the other values in the table.

Answer:

See below

Step-by-step explanation:

Let x be he cost per book and y be the total cost including shipping and handling.

The relevant equation is

y = 12x + 5

A. Variables are number of books ordered(x) and total cost(y)
The constant is the shipping and handling cost

Since the total cost depends on the number of books ordered, the independent variable x = number of books

The total cost y is the dependent variable

--------------------------------------------------------------------------------------

B. Cost of 1 or 2 books can be found by plugging in x = 1 and x =2 into the equations and solving for y
Total Cost of 1 book = 12(1) + 5 = $17
Total cost of 2 books = 12(2) + 5 = 24 + 5 = $29

To compute the number of books that can be ordered for different total cost amounts is obtained by substituting for y and solving for x

For $41:
41 = 12x + 5
41 - 5 = 12x

36 = 12z

x = 36/12 = 3 books

For $65:
65 = 12x + 5
65 - 5 = 12x
60 = 12x
x = 60/12 = 5 books

For $125:

125 = 12x + 5
125 - 5 = 12x
120 = 12x
x = 120/12 = 10 books

Here is the table

Number                            Total Cost(y)
of books (x)                    

          1                                  $17

          2                                 $29

          3                                  $41

          5                                  $65

          10                                 $125

Answer:

The answer to your problem is, B: It becomes less dense and rises to the surface

Step-by-step explanation:

Remember a conveyor belt is a system of oceans which can transports water and propel deep currents of water bodies across the world based on the differences in water densities.

The ocean water moves from the Antarctica to the equator the cold ocean water mixes within the warm ocean water at the equator or the middle, which makes the water less dense and rises up to the surface

Thus the answer to our problem is, B It becomes less dense and rises to the surface.

Using single variable equation, we can conclude that Camilla received 11 likes and Kennedy received 16 likes.

Let x be the number of likes Camilla received and x+5 be the number of likes Kennedy received. We can set up an equation to solve for x:

x + (x+5) = 27

Simplifying the equation:

2x + 5 = 27

Subtracting 5 from both sides:

2x = 22

Dividing both sides by 2:

x = 11

So Camilla received 11 likes and Kennedy received 11+5=16 likes.

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The value of the given trigonometric expression is -1.45.

The sides and angles of a right-angled triangle are dealt with in Trigonometry. The ratios of acute angles are called trigonometric ratios of angles. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).

The given trigonometric expression is tan(-2π/3)/sin(7π/4) -sec(-π)

We know that, π=180°

tan(-2×180°/3)/sin(7×180°/4) -sec(-180°)

= tan(-2×60°)/sin(7×45°)-sec(-180°)

= tan(-120°)/sin315°-(-1)

= √3/(-√2/2)+1

= -2√3/√2 +1

= -√6+1

= -2.45+1

= -1.45

Therefore, the value of the given trigonometric expression is -1.45.

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Answer:

Least acceptable weight= 15.6 ounces

and Greatest acceptable weight=16.4 ounces

Step-by-step explanation:

A cereal company fills boxes with 16 ounces of cereal.

The acceptable percent error in filling a box is 2.5%.

i.e. the error in decimal is: 2.5/100 x 16 = 25/1000 x 16 = 0.4

Hence,

the least acceptable weight is obtained by subtracting the actual amount with the error.

16-0.4=15.6 ounces.

and the greatest acceptable weight is obtained by adding the error to the actual weight.

16+0.4=16.4 ounces

Hence,

Least acceptable weight= 15.6 ounces

and Greatest acceptable weight=16.4 ounces.

The frequency of each interval is:

81-100: 2

61-80: 2

1-20: 1

21-40: 1

41-60: 2

What is frequency interval?

Frequency refers to the number of times a particular value or event occurs within a given dataset or sample. In statistics, frequency is used to describe the distribution of values in a dataset, and it is often represented as a frequency table or histogram.

Frequency is often used in conjunction with other statistical measures such as mean, median, and mode to describe the central tendency and variability of a dataset. By analyzing the frequency of values within a dataset, we can identify patterns and trends, and make inferences about the population from which the sample was drawn.

According to the question:
Using the given intervals, we can count the frequency of each interval as follows

81-100: 2 (81, 97)

61-80: 2 (65, 44)

1-20: 1 (2)

21-40: 1 (24)

41-60: 2 (25, 38)

Total: 8

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The coordinates of v with respect to the basis β are (cos(1), sin(1)).

The coordinates of a vector with respect to a given basis can be found by expressing the vector as a linear combination of the basis vectors and finding the coefficients of the linear combination. These coefficients are the coordinates of the vector with respect to the given basis.

(i) In the first case, we have V=R2, β={(1,1),(1,−1)}, and v=(1,2). We can express v as a linear combination of the basis vectors as follows:
v = a(1,1) + b(1,-1) = (a+b, a-b)
Equating the coordinates, we get:
a+b = 1
a-b = 2
Solving for a and b, we get a=3/2 and b=-1/2. Therefore, the coordinates of v with respect to the basis β are (3/2, -1/2).

(ii) In the second case, we have V=P2(R), β={1,1+x,1+x+x2}, and v=(x+1)2. We can express v as a linear combination of the basis vectors as follows:
v = a(1) + b(1+x) + c(1+x+x2) = a + (b+c)x + cx2
Equating the coefficients, we get:
a = 1
b+c = 2
c = 1
Solving for a, b, and c, we get a=1, b=1, and c=1. Therefore, the coordinates of v with respect to the basis β are (1, 1, 1).

(iii) In the third case, we have V=span{sin(x),cos(x)}, β={sin(x),cos(x)}, and v=sin(x+1). Using the trigonometric identity sin(x+1) = sin(x)cos(1) + cos(x)sin(1), we can express v as a linear combination of the basis vectors as follows:
v = a(sin(x)) + b(cos(x)) = (a)sin(x) + (b)cos(x)
Equating the coefficients, we get:
a = cos(1)
b = sin(1)
Therefore, the coordinates of v with respect to the basis β are (cos(1), sin(1)).

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