A landscaping company charges a $12 fee to make a house call plus an hourly rate for labor if the total bill for a job that takes two and a half hours comes to $95.75 write an equation that can be used to solve for the hourly labor rate use R to represent the rate then solve the equation and explain all steps​

The Function  f(g(x)) = g(f(x)) = x + 8.

The functions are: f(x) = x + 4 and g(x) = x + 4. We can find f(g(x)) by substituting g(x) in place of x in f(x).

f(g(x)) = f(x + 4) = (x + 4) + 4 = x + 8

Similarly, we can find g(f(x)) by substituting f(x) in place of x in g(x).g(f(x)) = g(x + 4) = (x + 4) + 4 = x + 8

Thus, we can see that f(g(x)) and g(f(x)) are equal to each other,

which is x + 8.

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The Area of the shaded portion is: 28 in²

The formula for the area of a circle is:

A = πr²

Where:

A is Area

R is radius

Thus:

Area of circle = π * 10² = 314.2 in²

Area of quadrant = 314.2 in²/4 = 78.5 in²

Area of triangle is given by the formula:

A = ¹/₂ * base * height

Thus:

A = ¹/₂ * 10 * 10

A = 50 in²

Area of shaded portion = 78.5 in² - 50 in²

Area = 28 in²

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The total weight of Rice that Mark buys is given as follows:

2.5 pounds.

The total weight of Rice that Mark buys is obtained applying the proportions in the context of the problem.

The weight of each bag is given as follows:

5/8 pounds = 0.625 pounds.

The number of bags is given as follows:

4 bags.

Hence the total weight of Rice that Mark buys is given as follows:

4 x 0.625 = 2.5 pounds.

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The measure of the arc angle WX is 100 degrees.

The arc angle WX can be found as follows:

The measure of an inscribed angle is half of the measure of the intercepted arc and half the measure of the central angle intersecting the same arc.

Therefore,

arc WY = 2(75)

arc WY =  150 degrees

Therefore, let's find the arc  angle  WX as follows:

arc  angle  WX  = 360 - 150 - 110

arc  angle  WX  = 210 - 110

arc  angle  WX  =  100 degrees

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Answer:

Flipping a coin twice and obtaining heads the first time and tails the second time has a probability of (1/2) * (1/2) = 1/4.

   On the initial draw, there is a 4/10 chance that a drink with a lemon-lime flavor will be randomly chosen from the cooler. Because the bottle is put back in the cooler and mixed before the second draw, the likelihood of choosing a lemon-lime beverage at random is also 4/10. The likelihood of winning a lemon-lime drink on both draws is (4/10) * (4/10), which equals 16/100 or 4/25.

Step-by-step explanation:

Answer:

[tex]\textsf{30)} \quad \dfrac{1}{4}=25\%[/tex]

[tex]\textsf{31)} \quad \dfrac{4}{25}=16\%[/tex]

Step-by-step explanation:

Probability is a measure of the likelihood or chance of an event occurring.  The basic formula for probability is:

[tex]\boxed{{\sf Probability\:of\:an\:event\:occurring = \dfrac{Number\:of\:ways\:it\:can\:occur}{Total\:number\:of\:possible\:outcomes}}}[/tex]

A fair coin has two possible outcomes: heads (H) or tails (T).

Each flip of the coin is independent, meaning the outcome of one flip does not affect the outcome of the other flip.

The probability of flipping a head is P(H) = 1/2.

The probability of filliping a tail is P(T) = 1/2.

To find the probability that the first flip lands heads-up and the second flip lands tails-up, we multiply the individual probabilities together:

[tex]\sf P(H)\;and\;P(T)=\dfrac{1}{2} \cdot \dfrac{1}{2}=\dfrac{1 \cdot 1}{2 \cdot 2}=\dfrac{1}{4}[/tex]

So the probability of flipping a coin twice and getting heads on the first flip and tails on the second flip is 1/4 or 25%.

[tex]\hrulefill[/tex]

There are 10 bottles in total, so there are 10 possible outcomes.

There are 4 lemon-lime drinks in the cooler, therefore the probability of selecting a lemon-lime bottle is:

[tex]\sf P(lemon$-$\sf lime)=\dfrac{4}{10}[/tex]

As you are randomly selecting two bottles with replacement, meaning you return the bottle to the cooler before selecting the next one, the probability of selecting a lemon-lime bottle each time is the same.

To find the probability of that both drinks are lemon-lime flavored, multiply the individual probabilities together:

[tex]\begin{aligned}\sf Probability &= \textsf{P(lemon-lime)} \cdot \textsf{P(lemon-lime)}\\\\&= \dfrac{4}{10} \cdot \dfrac{4}{10}\\\\&=\dfrac{4 \cdot 4}{10 \cdot 10}\\\\&=\dfrac{16}{100}\\\\&=\dfrac{4}{25}\end{aligned}[/tex]

Therefore, the probability of randomly selecting two drinks from the cooler and getting a lemon-lime drink both times is 4/25 or 16%.

To find the expected number of times you would wait between 19 and 23 minutes, we need to calculate the z-scores for these values and use them to find the area under the normal distribution curve between those values.

First, we calculate the z-score for 19 minutes:

z = (19 - 22) / 3 = -1

Next, we calculate the z-score for 23 minutes:

z = (23 - 22) / 3 = 0.33

Using a standard normal distribution table or calculator, we can find the area under the normal distribution curve between these z-scores:

P(-1 < z < 0.33) = 0.4082 - 0.3413 = 0.0669

This means that there is a probability of 0.0669 of waiting between 19 and 23 minutes for a single visit to the restaurant.

To find the expected number of times you would wait between 19 and 23 minutes over 37 visits, we multiply the probability for a single visit by the number of visits:

Expected number of times = 0.0669 x 37 ≈ 2.47

Rounding to the nearest whole number, we would expect to wait between 19 and 23 minutes about 2 times over 37 visits to the restaurant.

The perimeter of the figure, rounded to the nearest tenths place, is 41.1 units.

To find the perimeter of the figure composed of a square and four semicircles, we need to determine the lengths of the square's sides and the semicircles' arcs.

Given that the side length of the square is 4 units, the perimeter of the square is simply the sum of all four sides, which is 4 + 4 + 4 + 4 = 16 units.

Now, let's focus on the semicircles. Each semicircle's diameter is equal to the side length of the square, which is 4 units. Therefore, the radius of each semicircle is half of the diameter, or 2 units.

The formula to find the arc length of a semicircle is given by θ/360 * 2πr, where θ is the angle of the arc and r is the radius. In this case, the angle of the arc is 180 degrees since we are dealing with semicircles.

Using the formula, the arc length of each semicircle is 180/360 * 2π * 2 = π * 2 = 2π units.

Since there are four semicircles in the figure, the total length of the arcs is 4 * 2π = 8π units.

Finally, we can calculate the perimeter by adding the length of the square's sides and the length of the semicircles' arcs:

Perimeter = Length of square's sides + Length of semicircles' arcs

= 16 units + 8π units

To round the perimeter to the nearest tenths place, we need to determine the approximate value of π. Taking π as approximately 3.14, we can calculate the approximate perimeter as:

Perimeter ≈ 16 + 8 * 3.14 ≈ 16 + 25.12 ≈ 41.12 units.

Therefore, the perimeter of the figure, rounded to the nearest tenths place, is approximately 41.1 units.

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Answer:kkkkkkkkkkkkkkkkkkkkkkkkkkk mean ok 26 times

Joey's actual speed is 143.59 mph, and his actual direction is slightly west of northwest.

Given that Joey's aircraft speed: 188 mph

Wind speed: 60 mph

Wind direction: 150 degrees (measured clockwise from due north)

We can consider the wind as a vector, which has both magnitude (speed) and direction.

The wind vector can be represented as follows:

Wind vector = 60 mph at 150 degrees

We convert the wind direction from degrees to a compass bearing.

Since 150 degrees is measured clockwise from due north, the compass bearing is 360 degrees - 150 degrees = 210 degrees.

Joey's aircraft speed vector = 188 mph at 0 degrees (due northwest)

Wind vector = 60 mph at 210 degrees

To find the resulting velocity vector, we add these two vectors together. This can be done using vector addition.

Converting the wind vector into its x and y components:

Wind vector (x component) = 60 mph × cos(210 degrees)

= -48.98 mph (negative because it opposes the aircraft's motion)

Wind vector (y component) = 60 mph×sin(210 degrees)

= -31.18 mph (negative because it opposes the aircraft's motion)

Now, we can add the x and y components of the two vectors to find the resulting velocity vector:

Resulting velocity (x component) = 188 mph + (-48.98 mph) = 139.02 mph

Resulting velocity (y component) = 0 mph + (-31.18 mph) = -31.18 mph

Magnitude (speed) = √((139.02 mph)² + (-31.18 mph)²)

= 143.59 mph

Direction = arctan((-31.18 mph) / 139.02 mph)

= -12.80 degrees

The magnitude of the resulting velocity vector represents Joey's actual speed, which is approximately 143.59 mph.

The direction is given as -12.80 degrees, which indicates the deviation from the original northwest direction.

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Answer:

see explanation

Step-by-step explanation:

26

given y varies directly as x then the equation relating them is

y = kx ← k is the constant of variation

to find k use the condition y = [tex]\frac{15}{4}[/tex] when x = 15

[tex]\frac{15}{4}[/tex] = 15k ( divide both sides by 15 )

[tex]\frac{\frac{15}{4} }{15}[/tex] = k , then

k = [tex]\frac{15}{4}[/tex] × [tex]\frac{1}{15}[/tex] = [tex]\frac{1}{4}[/tex]

y = [tex]\frac{1}{4}[/tex] x ← equation of variation

when x = 11 , then

y = [tex]\frac{1}{4}[/tex] × 11 = [tex]\frac{11}{4}[/tex]

27

given y varies inversely as x then the equation relating them is

y = [tex]\frac{k}{x}[/tex] ← k is the constant of variation

to find k use the condition y = 4 when x = 9

4 = [tex]\frac{k}{9}[/tex] ( multiply both sides by 9 )

36 = k

y = [tex]\frac{36}{x}[/tex] ← equation of variation

when x = 7 , then

y = [tex]\frac{36}{7}[/tex]

Answer:

26)  y = 11/4

27)  y = 36/7

Step-by-step explanation:

Direct variation is a mathematical relationship between two variables where a change in one variable directly corresponds to a change in the other variable. It is represented by the equation y = kx, where y and x are the variables and k is the constant of variation.

To find the constant of variation, k, substitute the given values of y = 15/4 when x = 15 into the direct variation equation and solve for k:

[tex]\begin{aligned}y&=kx\\\\\dfrac{15}{4}&=15k\\\\k&=\dfrac{1}{4}\end{aligned}[/tex]

To find the value of y when x = 11, substitute the found value of k and x = 11 into the direct variation equation, and solve for y:

[tex]\begin{aligned}y&=kx\\\\y&=\dfrac{1}{4} \cdot 11\\\\y&=\dfrac{11}{4}\end{aligned}[/tex]

Therefore, if y varies directly as x, then y = 11/4 when x = 11.

[tex]\hrulefill[/tex]

Inverse variation is a mathematical relationship between two variables where an increase in one variable results in a corresponding decrease in the other variable, and vice versa, while their product remains constant. It is represented by the equation y = k/x, where y and x are the variables and k is the constant of variation.

To find the constant of variation, k, substitute the given values of y = 4 when x = 9 into the inverse variation equation and solve for k:

[tex]\begin{aligned}y&=\dfrac{k}{x}\\\\4&=\dfrac{k}{9}\\\\k&=36\end{aligned}[/tex]

To find the value of y when x = 7, substitute the found value of k and x = 7 into the inverse variation equation, and solve for y:

[tex]\begin{aligned}y&=\dfrac{k}{x}\\\\y&=\dfrac{36}{7}\end{aligned}[/tex]

Therefore, if y varies inversely as x, then y = 36/7 when x = 7.

The difference in the addition of both fractions is that they have different lowest common multiples.

We want to add the fraction expression given as:

¹/₂ + ¹/₄

Taking the Lowest common multiple of 8, we have:

(4 + 2)/8 = 6/8

However, for the second fraction expression, we have:

¹/₂ + ¹/₃

Taking the lowest common multiple which is 6, we have:

(3 + 2)/6 = 5/6

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The angles  ZBOC + ZAOC = 180°.

In geometry, angles are two rays that share a common endpoint. The common endpoint is called a vertex, and the rays are known as sides.

In a plane, when two lines intersect, they form four angles at the point of intersection, and when a line segment intersects a line, they form two angles.Geometry: Anglea) Draw a line segment AB.

Mark a point O on AB and draw an angle BOC. Measure ZBOC and ZAOC. Verify that ZBOC + ZAOC = 180°.

To solve this problem, the following steps should be followed:

Step 1: Draw a line segment AB

Step 2: Mark point O on AB and draw an angle BOC

Step 3: Measure angles ZBOC and ZAOC

Step 4: Add angles ZBOC and ZAOCZBOC + ZAOC = 180°

The sum of angles ZBOC and ZAOC is 180°. It is because an angle is the amount of turn between two rays with a common endpoint.

When the rays of two angles form a straight line, the two angles are called supplementary angles.The sum of supplementary angles is always 180°.

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Step-by-step explanation:

First    2x * 2x =   4x^2

Outer   2x * -3 =  -6x

Inner     3 * 2x = 6x

Last    3 * - 3 = -9

4x^2 -6x + 6x - 9   =  4x^2 - 9

Answer

4x²-9

Step-by-step explanation:

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The central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

Given that, a survey was conducted in which 2400 people were asked to name their favorite sport to watch and the table below shows their answers: Sport percentage of people are:

Football 29%

Soccer 9%

Baseball 14%

Basketball 8%

Hockey 8%

Other 32%

We need to find the central angle measure, x, for the Baseball slice in the circle graph.

For this, we need to use the formula that gives the central angle measure in degrees for a sector of a circle:

Central angle measure = (Percentage/100) × 360°

Now, using the above formula, we can calculate the central angle measure for each sport as shown below:

Football: (29/100) × 360° = 104.4°

Soccer: (9/100) × 360° = 32.4°

Baseball: (14/100) × 360° = 50.4°

Basketball: (8/100) × 360° = 28.8°

Hockey: (8/100) × 360° = 28.8°

Other: (32/100) × 360° = 115.2°

The sum of all central angle measures should be 360°, which is the measure of a full circle.

So we can check if the calculations are correct: 104.4° + 32.4° + 50.4° + 28.8° + 28.8° + 115.2° = 360°

We see that the sum is indeed 360°, so the calculations are correct. The central angle measure for the Baseball slice is 50.4°.

Therefore, the central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

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The angle measure of an arc π meters is 45°.

Given that, the diameter of a circle is 8 meters.

Radius of a circle = 8/2 = 4 meters

The formula to find the arc length of a circle is θ/360° ×2πr.

Here, θ/360° ×2πr.

π=θ/360° ×2π×4

1=θ/360° ×8

360°=8θ

θ=360°/8

θ=45°

Therefore, the angle measure of an arc π meters is 45°.

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Answer:

no solution!

Step-by-step explanation:

The absolute value of a quantity is always non-negative, meaning it cannot be negative. However, in this equation, we have the absolute value of x + 4 equaling -8, which is a negative value. Therefore, there is no solution to this equation.

Using law of sine, the value of A is 54°, b is 4.4 units and c is 6.1 units

The sine rule, also known as the law of sines, is a mathematical principle used in trigonometry to relate the sides and angles of a triangle. It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all sides and angles of the triangle.

The formula is given as;

a / sin A = b / sin B = c / sin C

To find the value of angle A

A + B + C = 180°

Reason: The sum of angles in a triangle is equal to 180°

46° + A + 80° = 180°

126° + A = 180°

A = 180° - 126°

A = 54°

Using this, we can apply sine rule;

a / sin A = b / sin B

5/ sin 54 = b / sin 46

b = 5sin46 / sin 54

b = 4.4 units

Using sine rule again;

a/ sin A = c / sin C

5/ sin 54 = c / sin80

c = 5sin80 / sin 54

c = 6.1 units

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Find the common ratio, r, of the sequence

12 000 * r = 9600

r = 9600/12000 =  0.8

12000 ( 0.8)^6   = $ 3145.73  in year 7  ( I used '6' instead of '7' because 12 000 is listed as year '1'   and not year '0' )

The equation of the straight line parallel to 2x + 4y = 1 and passing through the point (1, 1) is y = (-1/2)x + 3/2.

To determine the equation of a straight line that is parallel to the line 2x + 4y = 1 and passes through the point (1, 1), we can use the fact that parallel lines have the same slope.

First, let's rearrange the given equation 2x + 4y = 1 into slope-intercept form, y = mx + b,

where m is the slope and b is the y-intercept.

2x + 4y = 1

4y = -2x + 1

y = (-2/4)x + 1/4

y = (-1/2)x + 1/4

From this equation, we can see that the slope of the given line is -1/2.

Since the parallel line we want to find has the same slope, we can use the point-slope form of a linear equation:

y - y1 = m(x - x1),

where (x1, y1) is the given point.

Plugging in the values (1, 1) and the slope -1/2 into the equation, we have:

y - 1 = (-1/2)(x - 1)

To simplify, we distribute the -1/2:

y - 1 = (-1/2)x + 1/2

Next, we isolate y by adding 1 to both sides of the equation:

y = (-1/2)x + 1/2 + 1

y = (-1/2)x + 3/2.

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Step-by-step explanation:

The formula:

�

�

=

�

2

(

�

1

+

�

�

)

S

n

=

2

n

(a

1

+a

n

)

is used to solve for the sum of the arithmetic sequence given the first term a₁, the number of terms n, and the last term in an.

Example:

3, 6, 9, 12, 15,...,123

The first term, a₁ = 3

The last term an = 123

Common difference, d = 3 (because the sequence are multiples of 3)

Number of terms, n= ?

Find the number of terms, n:

an = a₁ + (n-1) (d)

123 = 3 + (n-1) (3)

123 = 3 - 3 + 3n

123/3 = 3n/3

n = 41

To find the sum of the given sequence without adding 3 + 6 + 9, ... + 123, we use the formula:

S₄₁ = (41/2) (3 + 123)

S₄₁ = (41/2) (126)

S₄₁ = (41)(63)

S₄₁ = 2,583 ⇒ the sum of the given sequence

a. The total impedance, z is 6 Ohms

b. The modulus of the total impedance is 6 Ohms, which represents its magnitude or absolute value. The angle is not provided for the principal argument.

From the information given, we have that;

z₁ = R₁ + Xₐ

z₂ = R₂ - Xₙ

We have that the values are;

Now, substitute the values, we have;

z₁ = 3 + 3

Add the values

z₁ = 6 Ohms

z₂ = 4 - 4

z₂ = 0 Ohms

To determine the total impedance, we have;

1/z = 1/z₁ + 1/z₂

Substitute the values

1/z = 1/6 + 1/0

1/z = 1/6

z = 6 Ohms

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[tex]1\frac{3}{4}[/tex] feet as a multiplication expression using the unit, 1 foot, as a factor is [tex]1\frac{3}{4}[/tex]×1.

The given fraction is [tex]1\frac{3}{4}[/tex].

[tex]1\frac{3}{4}[/tex] feet can be written as a multiplication expression as follows: 1 foot × 1 3/4. This is because [tex]1\frac{3}{4}[/tex] is the same as 1 + 3/4.

Furthermore, 3/4 can be written as 0.75, which is the same as 0.75 × 1 foot.

Therefore, the multiplication expression is 1 foot × [tex]1\frac{3}{4}[/tex] = 1 foot × (1 + 0.75) = 1 foot × 1 + 1 foot × 0.75 = 1 foot + 1.75 feet.

Therefore, [tex]1\frac{3}{4}[/tex] feet as a multiplication expression using the unit, 1 foot, as a factor is [tex]1\frac{3}{4}[/tex]×1.

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Step-by-step explanation:

1ft=12in

1¾ft=x

x=12×7/4=21in

1¾ft=1¾×1 ft

The statements that are true about triangle PQR are QR = (sqrt(3))/2 * PQ and PR = (sqrt(3))/2 * PQ.The correct answer is option B and D.

Let's analyze the statements one by one:

A. PQ = 2PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x.

Therefore, PQ = x, and PR = x√3. Since √3 is not equal to 2, this statement is false.

B. QR = (sqrt(3))/2 * PQ:

This statement is true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x.

Therefore, QR = x√3/2 = (sqrt(3))/2 * x = (sqrt(3))/2 * PQ. This statement holds true.

C. OR = 2PR:

We don't have any information regarding the length of OR, so we cannot determine if this statement is true or false based on the given information.

D. PR = (sqrt(3))/2 * PQ:

This statement is true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, PR = x√3 = (sqrt(3))/2 * 2x = (sqrt(3))/2 * PQ. This statement is correct.

E. QR = sqrt(3) * PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, QR = x√3, and PR = x√3. So, QR = PR, but not QR = sqrt(3) * PR.

F. PQ = sqrt(3) * PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, PQ = x, and PR = x√3. So, PQ = PR/√3, but not PQ = sqrt(3) * PR.

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The probable question may be:

In triangle PQR , m angle P = 60 deg , m angle Q = 30 deg and m angle R = 90 deg Which of the following statements about triangle PQR are true?

Check all that apply

A. PQ = 2PR

B.QR = (sqrt(3))/2 * PQ

C. OR = 2PR

D.PR = (sqrt(3))/2 * PQ

E. QR = sqrt(3) * PR

F. PQ = sqrt(3) * PR

(1) (x + 1)/(x - 2) - (x + 1)/(x^2 + 1)

(2) -2/(2x - 1) + (3/2)/(2x + 1)

To decompose the rational expressions into partial fractions, we first need to factorize the denominators. Let's start with the first expression:

Factorizing the denominator:

x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1)

Decomposing the fraction:

We have a linear factor and a quadratic factor, so the partial fraction decomposition will be of the form:

A/(x - 2) + (Bx + C)/(x^2 + 1)

Finding the values of A, B, and C:

Multiplying both sides of the equation by the common denominator (x - 2)(x^2 + 1) gives:

x^5 - 2x^4 + x^3 + x + 5 = A(x^2 + 1) + (Bx + C)(x - 2)

By equating coefficients of corresponding powers of x, we get:

A = 1

-2A + B = -2

A - 2B + C = 1

Solving this system of equations, we find A = 1, B = -1, and C = 0.

Therefore, the partial fraction decomposition is:

(x + 1)/(x - 2) - (x + 1)/(x^2 + 1)

Now let's move on to the second expression:

Factorizing the denominator:

4x^2 - 1 = (2x - 1)(2x + 1)

Decomposing the fraction:

Since we have two linear factors, the partial fraction decomposition will be of the form:

A/(2x - 1) + B/(2x + 1)

Finding the values of A and B:

Multiplying both sides of the equation by the common denominator (2x - 1)(2x + 1) gives:

4x^2 - 14x + 2 = A(2x + 1) + B(2x - 1)

By equating coefficients of corresponding powers of x, we get:

4A + 4B = 4 (coefficients of x^2)

A - B = -7 (coefficients of x)

A - B = 1 (constant term)

Solving this system of equations, we find A = -2 and B = 3/2.

Therefore, the partial fraction decomposition is:

-2/(2x - 1) + (3/2)/(2x + 1)

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Total mangoes: 15

Leftover mangoes after monkey took two fifth:

2/5 * 15

= 6

So, two-fifths of the mangoes are 6.

Now calculating leftover:

15 - 6

= 9 mangoes(ANSWER)

The two pairs of polar coordinates for the point (3, -3) when x² - y² = 4 in polar coordinates are (3√2, -45°) or (3√2, 315°) and (-3√2, 135°).

Given the Cartesian coordinate (3, -3), and the equation x² - y² = 4.To convert Cartesian coordinates to polar coordinates, we use the formula:r = sqrt(x² + y²)θ = tan⁻¹(y/x)

The first step is to substitute the given Cartesian coordinates into the formula:

r = sqrt(3² + (-3)²)r = sqrt(18)r = 3√2

To determine the angle θ, we first look at the sign of both the x and y coordinates. Since x is positive and y is negative, the angle θ is in the fourth quadrant.

To determine the angle θ, we use the formula:θ = tan⁻¹(y/x)θ = tan⁻¹(-3/3)θ = -45°Alternatively, we can add 360° to get the angle in the fourth quadrant:θ = 315°

Therefore, one pair of polar coordinates is (3√2, -45°) or (3√2, 315°).To determine the second pair of polar coordinates, we can add 180° to the angle and negate the radius,

since this will give us the same point but in the opposite direction.θ = -45° + 180° = 135°r = -3√2

Therefore, the second pair of polar coordinates is (-3√2, 135°).

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The exact value of sin(pi/3) is √3. By definition, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, sin(pi/3) = √3/1 = √3.

The exact value of sin(pi/3) can be determined using trigonometric properties and identities.

First, we know that pi/3 is equivalent to 60 degrees. In a unit circle, the point corresponding to 60 degrees forms an equilateral triangle with the origin and the x-axis. This triangle has side lengths of 1, 1, and √3.

To find the sine of pi/3, we consider the side opposite the angle (pi/3) in the triangle. In this case, the opposite side has a length of √3. The hypotenuse of the triangle is 1, as it is the radius of the unit circle.

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T satisfies both the additive and scalar multiplication properties, it can be concluded that T is a linear transformation.

To prove that T is a linear transformation, we need to show that it satisfies two properties: additive property and scalar multiplication property.

Additive property:

Let z = (x1, x2) be an arbitrary vector in R2. We want to confirm that T(z1 + z2) = T(z1) + T(z2).

Let z1 = (x1, x2) and z2 = (y1, y2) be arbitrary vectors in R2.

T(z1 + z2) = T((x1 + y1, x2 + y2)) = (x1 + y1, x2 + y2)

T(z1) + T(z2) = T(x1, x2) + T(y1, y2) = (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

We can see that T(z1 + z2) = T(z1) + T(z2), thus satisfying the additive property.

Scalar multiplication property:

Let z = (x1, x2) be an arbitrary vector in R2 and k be an arbitrary scalar. We want to confirm that T(kz) = kT(z).

T(kz) = T(kx1, kx2) = (kx1, kx2)

kT(z) = kT(x1, x2) = k(x1, x2) = (kx1, kx2)

We can see that T(kz) = kT(z), thus satisfying the scalar multiplication property.

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The solution of the given algebraic expression is: ⁷/₁₂ + ⁴/₆q

We are given the algebraic expression as:

¹¹/₁₂ - ¹/₆q + ⁵/₆q - ¹/₃

Combining Like terms gives us:

(¹¹/₁₂ - ¹/₃) + (⁵/₆q - ¹/₆q)

Solving both brackets individually gives us:

((11 - 4)/12) + ⁴/₆q

= ⁷/₁₂ + ⁴/₆q

Thus, we conclude that is the solution of the given algebraic expression problem

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