Determine if each root is a rational or irrational number. explain your reasoning. √ 20 3 √ 96

The 1,000 feet and 200 feet distances of the top and the base of the skyscraper from the point on the ground, indicates, using Pythagorean Theorem that the height of the skyscraper is 400·√6 feet

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the other two sides.

The distance of the top of the skyscraper from a point on the ground = 1000 feet

The distance of the base of the skyscraper from the same point = 200 feet

Therefore, according to the Pythagorean Theorem, in the right triangle formed by the ray from the top of the skyscraper to the point on the ground, the height, h, of the skyscraper, and the distance of the point on the ground from the skyscraper, we get;

1000² = h² + 200²

h² + 200² = 1000²

h² = 1000² - 200² = 960,000

h = √(960,000) = 400·√6

The height of the skyscraper is 400·√6 feet

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The area covered by a sprinkler that sprays water in a circle of an 8-foot radius is 200.96 square feet.

Circle is a 2-Dimensional shape. It has no vertex and edges. It has a center point which equidistant from any point of boundary or circumference of a circle.

Radius refers to the distance between the center and any point on the boundary or circumference of the circle.

The area of a circle is given as the product of a constant pi and the square of the radius.

A =  π[tex]r^2[/tex]

A is the area

r is the radius

r = 8 feet

A = π * 8 * 8

= 3.14 * 8 * 8

= 200.96 square feet

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Answer:

Length of a car is 5m

Length of bus is 19m

Velocity of bus is 25ms-¹

Step-by-step explanation:

Assuming that overtaking start from front most part to back most part

Distance travel by car is 19m+5m=24m

Let velocity of bus be Vc and of car be Vb

Now

Velocity of car is 32ms-¹

The measures of the angles A = 81.2, B = 65.2, and C = 33.6 to the nearest tenth using the cosine and sine rule.

In trigonometry, the cosines rule relates the lengths of the sides of a triangle to the cosine of one of its angles. While sine rule is a relationship between the size of an angle in a triangle and the opposing side.

Considering the given triangle, angle C is calculated with cosine rule as follows;

c² = a² + b² - 2(b)(c)cosC

14² = 25² + 23² - 2(25)(23)cosC

196 = 1154 - 1150cosC

C = cos⁻¹(958/1150)

C = 33.6

by sine rule;

14/sin33.6 = 25/sinA

sinA = (25 × sin33.6)/14 {cross multiplication}

A = sin⁻¹(0.9882)

A = 81.2

B = 180 - (33.6 + 81.2) {sum of interior angles of a triangle}

B = 65.2

With proper application of the cosine and sine rule, we have the measures of the angles A = 81.2, B = 65.2, and C = 33.6 to the nearest tenth.

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Sylvia can fit 9 large cubic boxes and 240 small rectangular boxes into her car trunk.

We first need to convert all measurements to the same unit. Since the large cubic boxes are given in inches, we need to convert the dimensions of the trunk to inches as well.

The trunk measures 51 inches wide (4.25 ft x 12 in/ft), 48 inches deep (4 ft x 12 in/ft), and 53.33 inches high (4*1/3 ft x 12 in/ft).

Next, we need to determine how many large boxes can fit in the trunk. Since each large box is 16 inches wide,

we can fit 3 boxes across the width of the trunk:

(51 inches ÷ 16 inches = 3.19)

Similarly, we can fit 3 boxes deep and 3 boxes high.

Therefore, the maximum number of large boxes Sylvia can fit in her trunk is:

27 (3 x 3 x 3 = 27)

After fitting as many large boxes as possible, we can calculate the remaining volume of the trunk and use this to determine how many small boxes can fit.

The remaining volume is approximately 59,712 cubic inches (51 in x 48 in x 20 in).

Dividing this by the volume of each small box:

(3 in x 6 in x 4 in = 72 cubic inches)

We can fit 829 small boxes in the remaining space.

Therefore, Sylvia can fit a total of 27 large boxes and 829 small boxes in her trunk.

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Answer:

36/49.

Step-by-step explanation:

Probability if rains on one day = (6/7)

Probabiilty it rains the next day  also

= 6/7 * 6/7

= 36/49.

The surface area be 243 square feet.

Hence option (d) is correct.

In the given regular pyramid

Slant height = l = 6 ft

Edge of base = s = 9ft

Then,

Area of base = a = 9x9 = 81 square ft

Perimeter of base = p = 4x9 = 36 ft

Since surface area of regular pyramid = A  = a + (1/2)ps

                                                                       = 81 + (36x9)/2

                                                                       = 81 + 162

                                                                       =  243

Hence, A = 243 square ft.

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Answer:

The surface area be 243 square feet.

Step-by-step explanation:

The surface area  of the given volume of the rectangular prism is equal to 13.86 square centimeters.

Volume of the rectangular prism is 6 cubic centimeters.

Let the dimensions of the rectangular prism be length (l), width (w), and height (h).

Volume of the rectangular prism = l x w x h

⇒ l x w x h = 6

Use the given volume to find one of the dimensions .

Then use that information to find the surface area.

To calculate the surface area at least two of the dimensions known.

Let us assume that the height (h) is 1 centimeter.

⇒l x w x 1 = 6

⇒ l x w = 6

Use this equation to solve for one of the dimensions.

⇒ l = 6/w

Substituting this value of l into the surface area formula, we get,

Surface area = 2lw + 2wh + 2lh

⇒Surface area = 2(6/w)w + 2w(1) + 2(6/w)(1)

⇒Surface area = 12/w + 2w + 12/w

⇒Surface area = 2w + 24/w

Value of w that gives the minimum surface area,

Take the derivative of the surface area formula with respect to w and set it equal to 0,

d/dw (2w + 24/w) = 0

⇒ 2 - 24/w^2 = 0

Solving for w, we get,

⇒w = √(12)

Substituting this value of w back into the surface area formula, we get,

⇒ Surface area = 2(√(12)) + 24/√(12)

⇒Surface area = 4√3 + 4√3

⇒Surface area = 8√3

⇒ Surface area = 13.86 square centimeters

Therefore, the surface area of the rectangular prism is approximately 13.86 square centimeters.

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The approximate height of the tall oak tree is 60.0 feet.

To find the height of the tree, follow these steps:

1. Convert the distance between you and the mirror from inches to feet: 36 inches = 3 feet.
2. Create a proportion using similar triangles, where the height of the tree (h) divided by the distance from the tree to the mirror (24 feet) equals your eye height (5 feet) divided by the distance from your eyes to the mirror (3 feet).
3. Set up the proportion: h / 24 = 5 / 3.
4. Solve for h: h = (5 / 3) * 24.
5. Calculate h: h = 40 feet (height of tree above your eye level).
6. Add your eye height (5 feet) to the height of the tree above your eye level: 40 + 5 = 60 feet.
7. Round the answer to the nearest tenth: 60.0 feet.

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The exact solutions to the equation 2x² - 9x + 1 = 0 are (9 + √73) / 4 and (9 - √73) / 4.

To solve the quadratic equation 2x² - 9x + 1 = 0 using the quadratic formula, we first need to identify the values of a, b, and c. From the given equation, we can see that a = 2, b = -9, and c = 1.

The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values of a, b, and c, we get:

x = (-(-9) ± √((-9)² - 4(2)(1))) / 2(2)
x = (9 ± √(81 - 8)) / 4
x = (9 ± √73) / 4

So, the exact solutions to the equation 2x² - 9x + 1 = 0 are (9 + √73) / 4 and (9 - √73) / 4.

As for the answer choices, it's possible that the answer key has simplified the solution further. However, your answer is correct and provides the exact solutions to the equation.

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Mrs. Ross should expect to have about 26,417 books with no damage in the entire collection. Rounded to the nearest whole number, the answer is 26,417.

Mrs. Ross found that 902 out of the sample of (902 + 80 + 43) = 1025 books had no damage. This means that the proportion of books with no damage in the sample is 902/1025. We can use this proportion to estimate the number of books with no damage in the entire collection.

Let X be the number of books with no damage in the collection of 30,000 books. Then we can write:

902/1025 = X/30000

To solve for X, we can cross-multiply and simplify:

902 × 30000 = 1025 × X

X = 902 × 30000 / 1025

X ≈ 26,417.07

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The dimensions that will minimize the cost of the metal to make the can are approximately r = 2.82 inches and h = 7.10 inches.

We can use the following steps:

Step 1: Write the volume formula for the cylinder.
The volume (V) of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. Since the volume is given as 181 in³, we have:

181 = πr²h

Step 2: Solve for h in terms of r.
Divide both sides by πr²:

h = 181 / (πr²)

Step 3: Write the surface area formula for the cylinder without a top.
The surface area (S) of a cylinder without a top is given by the formula S = 2πrh + πr², where r is the radius and h is the height.

Step 4: Substitute h from step 2 into the surface area formula.
Replace h with 181 / (πr²) in the surface area formula:

S = 2πr(181 / (πr²)) + πr²

Step 5: Simplify the surface area formula.
After simplifying the surface area formula, we get:

S = (362 / r) + πr²

Step 6: Minimize the surface area.
To minimize the surface area, differentiate S with respect to r and set the derivative equal to 0:

dS/dr = -362/r² + 2πr = 0

Step 7: Solve for r.
To find the value of r that minimizes the surface area, solve the equation for r:

r³ = 181/π

r = (181/π)¹/³

Step 8: Find the height h.
Substitute the value of r back into the equation for h from step 2:

h = 181 / (π((181/π)¹/³)²)

Step 9: Calculate the dimensions.
Calculate the dimensions r and h using the values obtained in step 7 and step 8:

r ≈ 2.82 inches
h ≈ 7.10 inches

So, the dimensions that will minimize the cost of the metal to make the can are approximately r = 2.82 inches and h = 7.10 inches.

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The expected value of a ticket in this raffle is $12.50.

We have to given that;

1000 raffle ticket are sold for $1 each and there are ten $20, five $50, and one $100 prize.

Now, For the expected value of a ticket, we need to multiply the probability of winning each prize by the amount of each prize, and then add up those values.

Hence, In this case, there are a total of 16 prizes as,

⇒ (10 $20 prizes, 5 $50 prizes, and 1 $100 prize),

Thus, the probability of winning any one prize is,

⇒ 1/16

So, the expected value of a ticket would be:

= (10/16) $20 + (5/16) $50 + (1/16) x $100

= $12.50

Therefore, the expected value of a ticket in this raffle is $12.50.

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Answer:

168

Step-by-step explanation:

Volume is the ''area'' of a 3d shape

to find the volume, multiply the length, width, and height of the shape.

6x7x4

=42x4

=168 cubic yards

hope it helps

pls mark brainliest!!!

Answer:

168 cubic yards

Step-by-step explanation:

The formula for a rectangular prism volume is:

[tex]V=lwh[/tex]

Since we have all 3, the length, the width, and the height, we can plug in the numbers to substitute:

V=6·7·4

=168

So, the volume of this rectangular prism is 168 cubic yards.

Hope this helps :)

To make a profit of $281, the number of tickets that need to be sold is 150.

The profit from selling local ballet tickets can be determined by using the given function: [tex]p = -0.25x^2 + 0.30x + 6[/tex].

To find the number of tickets required to achieve a profit of $281, we can set p equal to 281 and solve for x. This results in a quadratic equation that can be solved using the quadratic formula.

Once we obtain the two possible values of x, we can select the positive value which represents the ticket price. Using this ticket price, we can then calculate the number of tickets required to achieve a profit of $281, which is 150.

In order to increase the profit, we can try adjusting the ticket price or increasing the number of tickets sold. However, it is important to keep in mind that there may be practical limits to both of these options.

For example, increasing the ticket price too much may deter customers from purchasing tickets, while increasing the number of tickets sold may require additional marketing efforts or larger venues.

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It should be noted that to craft a triangle with an edge measuring 5 units of length, another side 7 units in magnitude, we can get underway by sketching an uninterrupted line of 7-units duration.

Subsequently, draw an additional segment arriving at a peak of 5-units from the starting point of the former line. From the end-point of the line calculated as 7 units, draw a joined line towards the conclusion of the 5 unit-length section. This enables two distinctive triangles:

Also, to guarantee that just a single triangle can be drawn, it is imperative to recognize the angle resting between the two assigned sides. In case we are endowed with the inclination between both varying lengths of five and seven units, then only an original composition of triangle can be constructed.

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The correct option is D, The best measure of variability to use to compare the data sets is the standard deviation is the best measure because both data distributions are symmetric.

Let's examine each of the possibilities we've been provided so we can select the best one.

A. Since the data sets have the same minimum weight, the mean is the best indicator.

Option A is untrue about the measure of variability since the mean measures the central tendency of a data collection.

B. The distribution of zucchini weights is tilted to the left, making the range the most accurate measurement.

Given that both of our box plots are symmetric, option B cannot be true, as can be seen.

C. Because the medians of the data sets differ, the median is the best indicator.

Since the median is a measure of a data set's central tendency rather than variability, option C is the correct answer. not true.

D. Because both data distributions are symmetric, the standard deviation is the most accurate measurement.

Option D is the best option since standard deviation is the best measurement for symmetric data sets and both of our provided box plots are symmetric.

Distributions can take on many different forms, depending on the type of random variable being described. Some common distributions include the normal distribution, the uniform distribution, and the binomial distribution. In statistics, a distribution is a function that describes the probabilities of different outcomes in a random variable. A random variable is a variable whose value is determined by chance, such as the outcome of a coin toss or the height of a randomly selected person.

The normal distribution is perhaps the most well-known and is often used to model real-world phenomena, such as heights or weights of people, IQ scores, or measurements of physical characteristics. Understanding distributions is important in statistics because they allow us to make predictions and draw conclusions about populations based on a sample of data.

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Complete Question:-

The box plots show the distributions of weights of cucumbers and zucchini collected from a garden.

(see attachment)

Which statement explains the best measure of variability to use to compare the data sets?

A.The mean is the best measure because the data sets have the same minimum weight.

B. The range is the best measure because the distribution of zucchini weights is skewed left.

C. The median is the best measure because the data sets have different medians.

D. The standard deviation is the best measure because both data distributions are symmetric.

The task requires estimating the rate of change of atmospheric pressure with respect to altitude using the given data.

First, a table needs to be created to estimate the rate of change of pressure with respect to altitude. The rate of change will be the difference in pressure divided by the difference in altitude between two consecutive data points.

Second, two plots should be created: one illustrating the pressure depending on altitude, and another illustrating the estimated rate of change depending on altitude.

Third, a function should be constructed to model the pressure-altitude data. The function should be selected based on how well it fits the data points.

Fourth, the model function should be used to estimate the rate of change of atmospheric pressure with respect to altitude at different heights over the range from sea level to 10,000 ft.

Fifth, a comparison should be made between the rate of change information computed from the model function and the rate of change information estimated directly from the data. This comparison will be used to assess the accuracy of the model function.

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The width should be 12 feet and the height should be 6 feet.

Let's assume that the rectangular part of the window has a width of x feet. Since the semi-circle at the top is half the width of the rectangle, its radius will also be x/2 feet. Therefore, the height of the rectangle can be expressed as 12 - x/2, since we have a total of 12 feet of frame material available.

To find the area of the rectangle, we can multiply its length and width together: A = x(12 - x/2) = 12x - x²/2. To maximize this area, we can take its derivative with respect to x and set it equal to 0:

dA/dx = 12 - x = 0

x = 12

So the width of the rectangle should be 12 feet, and its height would be 12 - (12/2) = 6 feet. This would maximize the amount of light entering the rectangular part of the window, given the 12 feet of frame material available.

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This system represents the purchase made by Muriel (2 boxes of pencils and 4 erasers for $11) and Ramon (1 box of pencils and 3 erasers for $7).

The correct system of equations to find the price of one box of pencils (p) and one eraser (e) is:

2p + 4e = 11

p + 3e = 7

This system represents the purchase made by Muriel (2 boxes of pencils and 4 erasers for $11) and Ramon (1 box of pencils and 3 erasers for $7).

By solving this system of equations, you can determine the individual prices for a box of pencils and an eraser.

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The velocity of the bowling ball is approximately 20.104 meters per second.

The pace at which an object's position changes in relation to a frame of reference and time is what is meant by velocity.

The formula given is used to calculate the velocity of a moving object in meters per second, given the object's mass in kilograms and its kinetic energy in joules. The formula is:

V = √(2E/m)

where V is the velocity, E is the kinetic energy, and m is the mass of the object.

To use this formula to find the velocity of the bowling ball, we need to substitute the given values into the formula. The mass of the bowling ball is 5.5 kilograms, and the kinetic energy is 2223 joules. Substituting these values, we get:

V = √(2 × 2223 J / 5.5 kg)

Now, we can simplify the equation:

V = √(404.1818)

Using a calculator, we can find the square root of 404.1818:

V = 20.104 m/s (rounded to three decimal places)

Therefore, the velocity of the bowling ball is approximately 20.104 meters per second.

This formula is useful for calculating the velocity of a moving object when the mass and kinetic energy of the object are known. It can be used in a variety of situations, such as in physics experiments, engineering design, or in understanding the motion of objects in sports.

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Kira planted 9/35 acres with corn.

To find out how many acres planted with corn, we first need to understand that the total amount of land she had was 3/5 acres. Then, we know that she planted 3/7 of her land with corn.

To calculate the area planted with corn, we multiply the total area of her land (3/5 acres) by the fraction that represents the portion she planted with corn (3/7). This gives us:

(3/5) x (3/7) = 9/35 acres

Therefore, Kira planted 9/35 acres of her land with corn.

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 3 in this case. This gives us:

(9/3) / (35/3) = 3/11 acres

So, Kira planted approximately 0.26 acres with corn.

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The area of the spill is increasing at a rate of approximately 94.25 square meters per minute at the end of an hour.

Using the formula for the rate of change of the area, we can calculate the rate at which the area is increasing at the end of an hour:

dA/dt = 2πr(dr/dt) = 2π(30 meters)(0.5 meters per minute) ≈ 94.25 square meters per minute

Area is a mathematical concept that refers to the amount of space contained within a two-dimensional shape or surface. It is typically measured in square units, such as square meters or square feet. The formula for calculating the area of a shape depends on its geometry. For example, the area of a rectangle is calculated by multiplying the length and width, while the area of a circle is calculated by multiplying pi (approximately equal to 3.14) by the square of its radius.

Area is used in a variety of fields, including mathematics, physics, engineering, and architecture. In mathematics, the study of area is a fundamental part of geometry, and is used to solve problems related to shapes and their properties. In physics, the concept of area is used to calculate quantities such as force, pressure, and energy.

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We reflect the point (-5, -3) over the x-axis to get (-5, 3), and then translate this point down by 2 units to get the final coordinates of (-5, 1).

To reflect a point over the x-axis, we keep the x-coordinate the same, but negate the y-coordinate. Thus, reflecting the point (-5, -3) over the x-axis gives us the point (-5, 3).

Next, we need to translate the result 2 units down. To do this, we subtract 2 from the y-coordinate of the reflected point. So, subtracting 2 from the y-coordinate of (-5, 3) gives us the final coordinates:

(-5, 3) translated 2 units down is (-5, 1).

In summary, we can reflect a point over the x-axis by negating the y-coordinate, and translate a point down by subtracting a value from its y-coordinate. So, we reflect the point (-5, -3) over the x-axis to get (-5, 3), and then translate this point down by 2 units to get the final coordinates of (-5, 1).

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The lighthouse needs to be at least 270.7 feet tall to allow guests to see 20 miles out with binoculars.

Assuming the Earth is a perfect sphere, the distance a person can see to the horizon is given by: d = 1.22 * sqrt(h)

Where d is the distance in miles, h is the height of the observer in feet, and 1.22 is a constant based on the radius of the Earth.

Using this formula, we can solve for the required height of the lighthouse: 20 = 1.22 * sqrt(h), 20/1.22 = sqrt(h), h = (20/1.22)^2, h ≈ 270.7 feet

Therefore, the lighthouse needs to be at least 270.7 feet tall to allow guests to see 20 miles out with binoculars.

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The quotient of the expression (7¹⁸/7⁻⁹) expressed as a power of 7 is 7 ²⁷.

The quotient of the expression is calculated as follows;

When you divide two numbers with the same base, you subtract the exponents of the base. Using this rule, we can simplify the expression as follows;

7¹⁸ / 7⁻⁹

= 7 ⁽¹⁸ ⁻ ⁻⁹⁾

= 7 ⁽¹⁸ ⁺ ⁹⁾

= 7 ²⁷

Therefore, the quotient of the expression (7¹⁸/7⁻⁹) is 7 ²⁷.

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Answer:

I'm assuming that A and B are two different items, and you want me to work with the same data set for both items. Here are the calculations:

1. Mean price of milkshake:

To find the mean price of a milkshake, you need to add up all the prices and divide by the total number of prices:

(4 + 2 + 9 + 14 + 6) / 5 = 7

Therefore, the mean price of a milkshake is $7.

2. Median price of milkshake:

To find the median price of a milkshake, you need to put the prices in order from lowest to highest:

2, 4, 6, 9, 14

The median is the middle value. Since there are five values, the middle value is the third value, which is 6.

Therefore, the median price of a milkshake is $6.

3. Mode of price of milkshake:

To find the mode of the price of a milkshake, you need to find the price that appears most frequently in the data set. In this case, there is no price that appears more than once, so there is no mode.

Therefore, there is no mode for the price of a milkshake.

I hope that helps! Let me know if you have any further questions.

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K Co Ltd's capital structure, stock dividend, and the arising number of new ordinary shares.
1) To compute the number of new ordinary shares arising out of the 10% stock dividend on K Co Ltd's ordinary shares, follow these steps:

Step 1: Identify the current number of ordinary shares outstanding in K Co Ltd's capital structure.
Step 2: Calculate the 10% stock dividend by multiplying the current number of ordinary shares by 0.1 (10%).
Step 3: The result from Step 2 will give you the number of new ordinary shares arising from this stock dividend issue.

2) To show the new capital structure after this stock dividend issue, follow these steps:
Step 1: Add the number of new ordinary shares (calculated in step 3 of the previous part) to the existing number of ordinary shares.

Step 2: Update the capital structure to reflect the new total number of ordinary shares.
Step 3: If there are any changes in other parts of the capital structure, such as preferred shares or debt, update those as well to reflect the new capital structure.

In summary, K Co Ltd intends to declare a 10% stock dividend on its ordinary shares, which will result in an increase in the number of ordinary shares arising from this issue. By calculating the number of new shares and updating the capital structure, you will be able to see the new capital structure after this stock dividend issue.

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option D is correct ,the triangle are not similar because only angle c and z are congruent .

what is  congruent ?

In mathematics, congruent means having the same size and shape. Two objects are said to be congruent if they have the same dimensions and angles, and can be transformed into each other by a combination of translations, rotations, and reflections.

In the given question,

congruent means having the same size and shape. Two objects are said to be congruent if they have the same dimensions and angles, and can be transformed into each other by a combination of translations, rotations, and reflections.

so we can conclude that option D is correct answer as other 2 angle are not equal

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The required answer is a possible scenario where BC (c) is less than AC * sin(angle A)

To justify if it is possible to have BC < AC sin(angle A) for an acute triangle ABC, let's consider the sine formula for a triangle.

The sine formula for a triangle states that:

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

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite to those sides, respectively.

Now let's isolate side BC (b) in the equation:

b = c * sin(B) / sin(C)

Since triangle ABC is acute, all angles A, B, and C are less than 90°. Therefore, sin(B) and sin(C) will be positive values between 0 and 1.

Let's now compare BC (b) to ACsin(angle A):

b < AC * sin(A)

c * sin(B) / sin(C) < AC * sin(A)

We can rewrite the inequality in terms of angle C:

sin(B) / sin(C) < (AC * sin(A)) / c

Now let's recall that angle C is the angle opposite to side AC (c), and angle B is the angle opposite to side BC (b). Since sine is a positive increasing function for acute angles (0° to 90°), it follows that the sine of a larger angle will result in a larger value.

As angle C is opposite to the longer side (AC), angle C > angle B. Therefore, sin(C) > sin(B), and their reciprocals will have the opposite relationship:

1 / sin(C) < 1 / sin(B)

Now, let's multiply both sides of the inequality by c * sin(B):

c < AC * sin(A)

This inequality represents a possible scenario where BC (c) is less than AC * sin(angle A), justifying the initial claim. So, yes, it is possible to have BC < AC sin(angle A) for an acute triangle ABC.

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