Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Match each expression to make pairs of equivalent expressions.

Answer:

6255 lbs

Step-by-step explanation:

so the questions says 1 gallon of water = 8.34

So, 8.34 * 750 = 6255Lbs

Answer:

Step-by-step explanation:

    1 x 8.34 = 8.34 lb   (one gal multiplied by the weight of 8.34)

750 x 8.34 =  6255 lb   (750 gal multiplied by the weight of 8.34)

6,255 pounds is added.

The values of x representing side length for which triangle is acute is given by 4, 5, and 6 .

For a triangle to be acute,

Sum of the squares of the two shorter sides must be greater than the square of the longest side.

Mathematically,

a^2 + b^2 > c^2

where a, b, and c are the side lengths of the triangle,

With c being the longest side.

The sides are given as 3, 6, and x.

Without loss of generality,

Assume that 3 and 6 are the shorter sides,

3^2 + 6^2 > x^2

Simplifying this inequality, we get,

45 > x^2

Taking the square root of both sides, we get,

6.71 > x

Since x must be an integer, the possible values of x are 4, 5, and 6.

Triangle inequality is satisfied,

Sum of any two sides of a triangle must be greater than the third side.

3 + 6 > x

3 + x > 6

6 + x > 3

Inequalities are all satisfied for x = 4, 5, and 6.

Therefore, there are 3 possible values of x is 4, 5, and 6 for which the triangle is acute.

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Answer:

Step-by-step explanation:

The total amount paid for the house is the sum of the down payment and the total amount paid for the mortgage.

The total amount paid for the mortgage can be calculated as follows:

Number of monthly payments = 15 years x 12 months/year = 180 months

Total amount paid for the mortgage = 180 x $2843.81 = $511,086.80

Therefore, the total amount paid for the house is:

$386,000 + $511,086.80 = $897,086.80

To calculate the amount of interest paid, we need to subtract the principal amount (the original amount borrowed) from the total amount paid for the mortgage.

Principal amount = Total amount borrowed - Down payment = $386,000 - $49,000 = $337,000

Total interest paid = Total amount paid for the mortgage - Principal amount = $511,086.80 - $337,000 = $174,086.80

Therefore, they paid a total of $897,086.80 for the house, and they paid $174,086.80 in interest on their mortgage.

The integral of [sin x / cos x] + [cos x / sin x] is (1/2) x ln |tan x| - (1/2) x ln |sec x| + C

The integral you have provided can be rewritten as:

∫ [sin x / cos x] + [cos x / sin x] dx

Using algebraic manipulation, we can simplify this expression to:

∫ (sin² x + cos² x) / (cos x x sin x) dx

Now, we can use the method of partial fractions to break down the integrand into simpler fractions. To do this, we first need to factor the denominator:

cos x x sin x = (1/2) x (sin 2x)

We can then express the integrand as:

(sin² x + cos² x) / [(1/2) x (sin 2x)]

Using the partial fractions technique, we can express the integrand as:

(sin² x + cos² x) / [(1/2) x (sin 2x)] = A / sin 2x + B / cos 2x

where A and B are constants that we need to determine. To solve for A and B, we can multiply both sides by sin 2x x cos 2x, which gives us:

sin² x + cos² x = A x cos 2x + B x sin 2x

We can then use the trigonometric identities sin² x + cos² x = 1 and cos 2x = 2 x cos² x - 1, and sin 2x = 2 x sin x x cos x, to simplify the equation to:

1 = (2A - B) x cos² x + (2B) x sin x x cos x - A

We now have two equations (for x = 0 and x = π/2) and two unknowns (A and B), which we can solve simultaneously to obtain:

A = 1/2 and B = -1/2

Using these values, we can express the integrand as:

(sin² x + cos² x) / [(1/2) x (sin 2x)] = (1/2) x [1 / sin 2x - 1 / cos 2x]

We can now integrate each term separately:

∫ [sin x / cos x] + [cos x / sin x] dx = ∫ [(1/2) x (1 / sin 2x - 1 / cos 2x)] dx = (1/2) x ln |tan x| - (1/2) x ln |sec x| + C

where C is the constant of integration. Therefore, the final answer to the given integral is:

∫ [sin x / cos x] + [cos x / sin x] dx = (1/2) x ln |tan x| - (1/2) x ln |sec x| + C

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Complete Question:

what is the integral of

[sin x / cos x] + [cos x / sin x]

The mean weight of 16 fruit picked at random, then their mean weight will be greater than 414.72 grams 20% of the time

The mean value of a set of data, is the sum of the values in the data, divided by the number of data in the dataset.

The population mean = 408 grams

The population standard deviation = 32 grams

The mean of 16 fruits such that the probability of the mean is larger than the value is 0.2, can be obtained as follows;

The standard error of the mean = The standard deviation/√(Sample size)

Therefore;

The standard error = 32/√(16) = 8

The mean that corresponds to a probability of 0.2, which is the 80th percentile of the normal distribution, can be found using the z-score of the 80th percentile, which is about 0.84

Therefore, we get;

z = (Sample mean - 408)/8 = 0.84

The sample mean = 8 × 0.84 + 408 =  414.72

The sample mean = 414.72 grams

Therefore;

The mean weight of the 16 fruit such that their weight is greater than the mean weight of the population 20% of the time is 414.72 grams.

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The answer of the given question is (a) TR = p(40 - (4/3)p), or TR = 40p - (4/3)p² , (b)  TC = 9q + (3/10)q² + 30 , (c) π = 40p - (4/3)p² - 9q - (3/10)q² - 30 , (d) TR ≈ 342.67 , (e) the profit when 20 units are produced is approximately RM188.27 , (f)  the profit when 7 units are produced is approximately -RM24.44, indicating a loss , (g) the output required to obtain a profit of RM100 is approximately 8.78 units.

An equation is  mathematical statement that asserts yhe equality of two expressions. It typically consists of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division, among others. Equations are often used to solve problems, to model real-world phenomena, and to describe mathematical relationships.

(a) The total revenue is given by TR = p x q. Substituting the demand equation q = 40 - (4/3)p, we get TR = p(40 - (4/3)p), or TR = 40p - (4/3)p².

(b) The total cost is given by TC = q x AC. Substituting the given average cost function, we get TC = 9q + (3/10)q² + 30.

(c) The profit is given by π = TR - TC. Substituting the expressions we found in parts (a) and (b), we get π = 40p - (4/3)p² - 9q - (3/10)q² - 30.

(d) At the break even point, the firm earns zero profit, so we set π = 0 and solve for q. Substituting the expression we found in part (a) for p, we get:

0 = 40p - (4/3)p² - 9q - (3/10)q² - 30

0 = 40(40/3 - (3/4)q) - (4/3)(40/3 - (3/4)q)² - 9q - (3/10)q² - 30

0 = 533.33 - 51.25q - 0.22q^2

Solving for q using the quadratic formula, we get:

q = (51.25 ± sqrt(51.25² - 4(-0.22)(533.33))) / 2(-0.22)

q ≈ 22.75 or q ≈ 206.58

We reject the solution q ≈ 206.58 because it is outside the relevant range of output, which is between 0 and 40. Therefore, the total output at the break even point is approximately 22.75 units. To find the total revenue at the break even point, we substitute q = 22.75 into the demand equation from part (a) and get:

p = (40/3) - (3/4)q

p ≈ 15.08

TR = p x q

TR ≈ 342.67

(e) To find the profit when 20 units are produced, we substitute q = 20 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ 188.27

Therefore, the profit when 20 units are produced is approximately RM188.27.

(f) To find the profit when 7 units are produced, we substitute q = 7 into the expression for profit we found in part (c) and get:

π = 40p - (4/3)p² - 9q - (3/10)q² - 30

π ≈ -24.44

Therefore, the profit when 7 units are produced is approximately -RM24.44, indicating a loss.

(g) To find the output required to obtain a profit of RM100, we set the profit equation equal to 100 and solve for q:

Profit = TR - TC

100 = pq - ACq

100 = (40-(4/3)p)*q - (9+(3/10)q+30/q)*q

100 = (40-(4/3)p - 9q - 3q²/10)

Multiplying by 10 and rearranging terms, we get a quadratic equation in q:

3q² + 91q - 310 = 0

Solving for q using the quadratic formula, we get:

q = (-91 ± sqrt(91² - 43(-310)))/(2*3)

q ≈ 8.78 or q ≈ -29.44

Since the quantity produced cannot be negative, the output required to obtain a profit of RM100 is approximately 8.78 units.

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The midline equation of the function is y = 1.5.

The gap between a trigonometric function's highest and least values is known as its amplitude. The difference between the greatest and minimum numbers is, in other words, divided by two. For instance, the amplitude is A in the equation y = A sin(Bx) + C. The amplitude, also known as the average value of the function across a period, denotes the "height" of the function above and below the midline. It gauges the magnitude or intensity of the oscillation of the function's representation of. The oscillation is more prominent and subtler depending on the amplitude, which ranges from higher to lower values.

Given that, the function has minimum point at (1, 1.5) and an amplitude of 1.5.

Using the definition of the amplitude the midline is the given amplitude.

Hence, the midline equation of the function is y = 1.5.

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Answer:

3

Step-by-step explanation:

Khan Academy

The equation of the line that is parallel to the given line and passes through the point (-3, 2) is 3x - 4y = -17.

If the line's equation is axe + by + c = 0 and the coordinates are (x1, y1).

To find the equation of a line parallel to a given line, we must first understand that parallel lines have the same slope. As a result, we must first determine the slope of the given line.

3x - 4y = -17 is the given line. To determine its slope, solve for y and write the equation in slope-intercept form:

-3x - 4y = -3x - 17 y = (3/4)x + (17/4)

This line has a 3/4 slope.

Now we want to find the equation of a parallel line that passes through the point (-3, 2). Because the new line is parallel to the given line, it has a slope of 3/4.

We can write the equation of the new line using the point-slope form of the equation of a line as:

y - y1 = m(x - x1)

where m represents the slope and (x1, y1) represents the given point (-3, 2).

When m = 3/4, x1 = -3, and y1 = 2, we get:

y - 2 = (3/4)(x - (-3))

y - 2 = (3/4)(x + 3)

Divide both sides by 4 to get rid of the fraction.

4y - 8 = 3x + 9

3x - 4y = -17

As a result, the equation of the parallel line that passes through the point (-3, 2) is 3x - 4y = -17.

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Answer:

 (c) ∠STU

Step-by-step explanation:

Transversal UT between parallel sides RU and ST creates alternate interior angles RUT and STU. These are congruent.

 ∠STU has the same measure as ∠RUT

_____

The figure shown is a trapezoid, not a parallelogram.

Answer:

  (a, b)  -2

  (c) -1, -1.5, -2, -2.5

Step-by-step explanation:

You want the average and instantaneous rates of change at various times in year 3, given the money in the bank after t years is 180+3t-t².

The amount at the beginning of year 3 is ...

  180 +3t -t² = 180 +t(3 -t)

  180 +2(3 -2) = 182 . . . . . . t=2

The amount at the end of year 3 is ...

  180 +3(3 -3) = 180 . . . . . . t=3

The amount increased by 180 -182 = -2 thousand dollars.

This change occurred in one year, so the average rate of change is ...

  change/years = -2/1 = -2 thousand dollars per year

The derivative of the amount function will give its instantaneous rate of change:

  da/dt = 3 -2t

The values of t at the beginning of the quarters in year 3 are ...

  t = 2: da/dt = 3 -2·2 = -1 thousand per year at start of 1st quarter

  t = 2.25: da/dt = 3 -2(2.25) = -1.5 thousand per year at start of Q2

  t = 2.50: da/dt = 3 -2(2.50) = -2 thousand per year at start of Q3

  t = 2.75: da/dt = 3 -2(2.75) = -2.5 thousand per year at start of Q4

The code segment that assigns bonus correctly for all possible integer values of score is D, which uses nested if statements to implement the game's rules for assigning a value to bonus based on the value of score.

The code segment that assigns bonus correctly for all possible integer values of score is D:

IF(score < 50)

{

   bonus ← Ø

}

ELSE

{

   IF (score > 100)

   {

       bonus ← score (10)

   }

   ELSE

   {

       bonus ← score

   }

}

This code segment correctly implements the rules for assigning a value to bonus based on the value of score. It first checks if score is less than 50, and if so, it assigns 0 to bonus. If score is greater than or equal to 50, it checks if score is greater than 100, and if so, it assigns 10 times score to bonus. Otherwise, it assigns score to bonus. This covers all possible integer values of score and ensures that bonus is assigned correctly according to the game's rules.

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Complete question is in the image attached below

Answer:

1.Neither

2.Perpendicular

3.Perpendicular

4.Neither

5.Perpendicular

6.Perpendicular

7.Neither

8.Neither

9.Perpendicular

10.Neither

11. Perpendicular

12.Perpendicular

13.Neither

14.Neither

15.Neither

16.Neither

17.Parallel

18.Neither

here are the answers in order from top to bottom

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. b. 20! = 20 x 19 x 18 x ... x 2 x 1. c. There are 16 zeros at the end of the decimal representation of (20!)2.

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

b. 20! = 20 x 19 x 18 x ... x 2 x 1. To write this in factored form, we can identify the prime factors of each number and write the product using exponents. For example, 20 = 2² x 5, so we can write 20! as:

20! = (2² x 5) x 19 x (2 x 3²) x 17 x (2² x 7) x 13 x (2 x 2 x 3) x 11 x (2³) x (3) x (2) x 7 x (2) x 5 x (2) x 3 x 2 x 1

Simplifying, we get:

20! = 2¹⁸ x 3⁸ x 5⁴ x 7² x 11 x 13 x 17 x 19

c. The number of zeros at the end of (20!)² in decimal form is determined by the number of factors of 10, which is equivalent to the number of factors of 2 x 5. Since there are more factors of 2 than 5 in the prime factorization of (20!)², we only need to count the number of factors of 5. There are four factors of 5 in the prime factorization of 20!, which contribute four factors of 10 to the square. Therefore, (20!)²ends in 8 zeros when written in decimal form.

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The critical numbers of f(x)  = x^6 - 6x^5 are x = 0 and x = 5 at the given extremum.

To find the derivative of f(x), we use the chain rule and power rule:

f(x) = (x + 4)^(2/3)

f'(x) = (2/3)(x + 4)^(-1/3)(1)

f'(-4) = (2/3)(-4 + 4)^(-1/3)(1) = DNE (the derivative does not exist at x = -4)

To find the critical numbers of f(x), we first find the derivative:

f(x) = x^6 - 6x^5

f'(x) = 6x^5 - 30x^4

Next, we set the derivative equal to zero and solve for x:

6x^5 - 30x^4 = 0

6x^4(x - 5) = 0

So the critical numbers of f(x) are x = 0 and x = 5.

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The distance from the base of the tower to the end of the wire is approximately 16.41 feet, and the height of the tower is approximately 23.41 feet.

What is Distance?

Distance is a numerical measurement of how far apart objects or points are. It is a scalar quantity, which means it only has a magnitude (size) and not a direction.

Let h be the height of the tower, and let d be the distance from the base of the tower to the end of the wire.

We are told that h = d + 7, and we know that the wire is 35 feet long. We can use the Pythagorean theorem to relate d and h to the length of the wire:

[tex]d^2 + h^2 = 35^2[/tex]

Substituting h = d + 7, we get:

[tex]d^2 + (d + 7)^2 = 35^2[/tex]

Expanding the left-hand side, we get:

[tex]2d^2 + 14d - 696 = 0[/tex]

Dividing both sides by 2, we get:

[tex]d^2 + 7d - 348 = 0[/tex]

We can solve this quadratic equation using the quadratic formula:

[tex]d =\frac{ (-b \pm \sqrt{(b^2 - 4ac)}) }{2a}[/tex]

where a = 1, b = 7, and c = -348. Substituting these values, we get:

d ≈ 16.41 or d ≈ -21.41

Since d represents a distance, it must be positive, so we take d ≈ 16.41.

Finally, we can use h = d + 7 to find the height of the tower:

h ≈ 23.41

Therefore, the distance from the base of the tower to the end of the wire is approximately 16.41 feet, and the height of the tower is approximately 23.41 feet.

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Answer:

Since opposite angles in a parallelogram are congruent, we know that:

m∠DAB = m∠BCD = 67°

Since ∠BAC and ∠BCD are adjacent angles, we know that:

m∠BAC + m∠BCD = 180°

So we can solve for m∠BAC:

m∠BAC + 67° = 180°

m∠BAC = 113°

Since opposite angles in a parallelogram are congruent, we know that:

m∠BAD = m∠BC

So we can solve for m∠BC:

m∠BAD = 16

m∠BC = 16°

Finally, we can solve for x by using the fact that the interior angles of a quadrilateral sum to 360 degrees:

m∠DAB + m∠BAC + m∠BCD + m∠CDA = 360°

67° + 113° + m∠BCD + 90° = 360°

m∠BCD = 90°

So we can solve for x:

m∠4B - m∠BCD = 90° - 90° = 0

4x - 59° = 0

4x = 59°

x = 14.75°

Therefore, m/BAC = 113°, m/B = 16°, and x = 14.75°.

Answer:

perimeter of plot=119m

circumference of pond=44m

volume of house=9000 m^3

surface area of room=248 m^2

Step-by-step explanation:

The function that satisfies  F Has An Inverse G That Satisfies G'(2) = 1/6 is f(x) = 2x³ (option a).

More precisely, if f(x) is a function, then its inverse function g(x) satisfies the following two conditions:

g(f(x)) = x for all x in the domain of f

f(g(x)) = x for all x in the domain of g

In other words, if we apply f(x) to an input value x, and then apply g(x) to the resulting output, we get back to the original input value.

Now, let's look at the given condition: G'(2) = 1/6. This means that the derivative of the inverse function at x=2 is 1/6. We can use this condition to eliminate some of the options.

f(x) = 2x³

If we take the derivative of f(x), we get: f'(x) = 6x²

To find the inverse function, we can solve for x in the equation y = 2x³:

x = [tex]y/2^{(1/3)}[/tex]

Now we can express the inverse function g(x) in terms of y:

g(y) = [tex]y/2^{(1/3)}[/tex]

To find the derivative of g(x), we use the chain rule:

g'(x) = f'(g(x))⁻¹

g'(2) = f'(g(2))⁻¹

g'(2) = f'([tex]1/2^{(1/3)}[/tex])⁻¹

g'(2) = 6([tex]1/2^{(1/3)}[/tex])²)⁻¹

g'(2) = 6/36 = 1/6

Since g'(2) = 1/6, option a) is the correct answer.

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10% chance of contamination by a particular: It's not clear what random variable is being referred to here, but if it's the probability of contamination.

In general terms, a distribution refers to the way something is divided or spread out. In the context of statistics and probability theory, a distribution is a mathematical function that describes the likelihood of different possible outcomes or values that a variable can take.

There are various types of distributions, but some of the most commonly used ones include:

Normal distribution: also known as the Gaussian distribution, it is a continuous probability distribution that is symmetrical around the mean, with most of the data falling within one standard deviation of the mean.

Binomial distribution: this is a discrete probability distribution that describes the likelihood of a certain number of successes in a fixed number of trials.

Poisson distribution: another discrete probability distribution that describes the likelihood of a certain number of events occurring in a fixed interval of time or space.

Exponential distribution: a continuous probability distribution that describes the time between events occurring at a constant rate.

Distributions are essential in statistical analysis as they can help to understand and analyze data, make predictions, and draw conclusions about a population based on a sample of data.

Given by the question.

a. The wages of academician and non-academician workers in UPSI: This random variable can be approximated to a continuous distribution as wages can take on any numerical value within a range. However, it's worth noting that in practice, there may be discrete intervals or categories of wages, in which case a discrete distribution may be more appropriate.

b. The time taken to submit online quiz answer's document: This random variable can also be approximated to a continuous distribution as it can take on any numerical value within a range.

c. The prices of SAMSUNG mobile phones displayed at a phone shop: This random variable can be approximated to a continuous distribution as prices can take on any numerical value within a range.

d. The number of pumps at Shell petrol stations in Perak: This random variable can be approximated to a discrete distribution since the number of pumps can only take on integer values.

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To earn an A grade, a student needs to score at least 82.8 , calculated using the inverse normal cumulative distribution function with a mean of 70, a standard deviation of 10, and a 10th percentile of 0.10.

Given that the grades are normally distributed with a mean of 70 and a standard deviation of 10.

We need to find the score which is at the 10th percentile of the distribution.

Using the standard normal distribution table, we can find the z-score that corresponds to the 10th percentile.

From the table, we can see that the z-score is approximately -1.28.

Using the formula for standardizing a normal distribution:

z = (x - μ) / σ

where z is the z-score, x is the raw score, μ is the mean, and σ is the standard deviation.

Substituting the given values, we have:

-1.28 = (x - 70) / 10

Solving for x, we get:

x = (-1.28 * 10) + 70

x = 82.8

Therefore, a student would need to earn a score of approximately 82.8 to receive an A grade.

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So the solution equation is 6x + 3 and the total miles is equal to 6x + 3.

Where x represents the length of Lincoln Park School.

A linear equation is an algebraic equation of the form y=mx+b. m is the slope and b is the y-intercept. The above is sometimes called a "linear equation in two variables" where y and x are variables. 

Mile is a unit of measurement that equals 1760 yards or approximately 1.6 kilometer's. It the mostly used in the continent of North America.

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The value of the triple integral is 7/180.

To evaluate the triple integral, we first need to set up the limits of integration. Since the solid tetrahedron T is defined by the four vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1), we can use these points to set up the limits of integration as follows:

For z, we integrate from the bottom of the solid (z = 0) to the top (z = 1 - x - y).

For y, we integrate from the left side of the solid (y = 0) to the right (y = 1 - x).

For x, we integrate from the back of the solid (x = 0) to the front (x = 1).

Therefore, the triple integral can be written as:

∫∫∫ t 7x^2 dv = ∫∫∫t 7x^2 dV

= ∫[0,1] ∫[0,1-x] ∫[0,1-x-y] 7x^2 dz dy dx

∫[0,1] ∫[0,1-x] ∫[0,1-x-y] 7x^2 dz dy dx

= ∫[0,1] ∫[0,1-x] 7x^2 (1 - x - y) dy dx

= ∫[0,1] [7x^2 (1 - x) (1/2 - x/3)] dx

= 7/180

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Answer:

FIRST ONE "Deb sold vases for two years, neither sold nor bought the next year and then sold bases for two more years"

Step-by-step explanation:

Notice the number of bases in debs collection is DECREASING as the years passes for the first and third period. This is she is selling her vases but in the middle the number is the same (two point in the same horizontal line) this means she neither sold nor bought any vase in that period.

Step-by-step explanation:

Please provide the figure then it will be easy to answer

Considering the secant-tangent theorem, the value of x is given as follows: x = 17º.

The secant-tangent theorem states that when a tangent and a secant line intersect at a point outside a circle, the measure of the angle of intersection is given by half the difference between the arc length of the far arc by the arc length of the near arc.

Hence the equation is given as follows:

y = (far arc - near arc)/2.

The parameters for this problem are given as follows:

Hence the value of x is obtained as follows:

64 = (145 - x)/2

145 - x = 128

x = 17º.

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In response to the stated question, we may state that We know that these two angles are complimentary since their total is 90 degrees.

An angle is a form in Euclidean geometry that is composed of a pair of rays, known as such angle's sides, that meet at a center point known as the angle's vertex. Two rays may merge to generate an angle in the plane in which they are located. An angle is formed when two planes collide. They are known as dihedral angles. In plane geometry, an angle is a potential arrangement of two rays or lines whose share a termination. The English term "angle" is derived from the Latin word "angulus," which means "horn." The apex is the point in which the two rays, often known as the angle's sides, converge.

Angle Sum Theorem formal definition:

The total of the three interior angles of a triangle is always equal to 180 degrees.

The Angle Sum Theorem states:

According to the Angle Sum Theorem, the sum of a triangle's internal angles is always equal to 180 degrees. In other terms, using new numbers:

Consider a triangle having three angles of 70 degrees, 60 degrees, and 50 degrees. The Angle Sum Theorem states that the sum of these angles should be 180 degrees.

[tex]70 + 60 + 50 = 180\\90 + x + y = 180\sx + y = 90[/tex]

We know that these two angles are complimentary since their total is 90 degrees.

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The volume of the solid is given by the equation V = 36π (√2 - 1) using the triple integral.

A three-dimensional object's volume in three-dimensional space may be determined using the triple integral. The three-variable function is represented by the triple integral. If in space is a closed region, the region's entire volume may be expressed as V = Ddv, which is equivalent to V = D d x d y d z.

Cone: z = [tex]\sqrt{x^2+y^2}[/tex]

sphere: x² + y² + z² = 18

Here, we will use cylindrical coordinates to evaluate volume:

x = rcosθ , y = rsinθ, z = z

so, z = [tex]\sqrt{r^2cos^2\theta+r^2sin^2\theta} =r[/tex]

z = [tex]\sqrt{18-(x^2+y^2)} =\sqrt{18-r^2}[/tex]

r = [tex]\sqrt{18-r^2}[/tex]

r = 3

Finding limits,

[tex]Volume = \int\limits^2_0 \int\limits^3_0\int\limits^a_r {rdzdrd\theta} \, \\\\= \int\limits^2_0 \int\limits^3_0rz \ drd\theta\\\\= \int\limits^2_0 \int\limits^3_0 r(\sqrt{18-r^2}-r ) \ drd\theta\\\\[/tex]

Now, we have

[tex]\int\limits^3_0 {r\sqrt{18-r^2} } \, dr = -(9-18\sqrt{2} ) = 18\sqrt{2} -9[/tex]

Now the integral becomes,

Volume = 2π [(18√2-9) - 9]

= 2π x 18√2 - 18

V = 36π (√2 - 1)

Therefore, the volume of the solid is given by V = 36π (√2 - 1).

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To conduct a hypothesis test comparing variances of independent samples from two populations, the test statistic will have an F-distribution.

The F-distribution is a probability distribution that describes the ratio of two independent chi-squared distributions divided by their degrees of freedom. In this case, the numerator and denominator degrees of freedom are based on the sample sizes and variances of the two populations being compared.

The null hypothesis for the F-test is that the variances of the two populations are equal, and the alternative hypothesis is that they are not equal. The F-test allows us to determine if the difference in variances is statistically significant, and if we reject the null hypothesis, we can conclude that the variances are significantly different.

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If 2 books are chosen at random, then the probability that both are statistics books is (a) 1/21.

The number of statistics book in bookcase is = 2;

The number of biology books in bookcase is = 5;

So, the total number of books is = 7;

The Probability of choosing a statistics book on the first draw is 2/7, since there are 2 statistics books out of a total of 7 books.

After the first book is chosen, there will be 6 books left, including 1 statistics book out of a total of 6 books.

So, the probability of choosing another statistics book on the second draw is 1/6.

In order to find the probability of both events happening together (i.e. choosing 2 statistics books in a row), we multiply the probabilities of each event:

So, P(choosing 2 statistics books) = P(1st book is statistics) × P(2nd book is statistics given that the 1st book was statistics);

⇒ (2/7) × (1/6)

⇒ 1/21

Therefore, the required probability is (a) 1/21.

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The given question is incomplete, the complete question is

A bookcase contains 2 statistics books and 5 biology books. If 2 books are chosen at random, the chance that both are statistics books is

(a) 1/21

(b) 10/21

(c) 11

(d) 21/11