"Evaluate the integral using the indicated trigonometric
substitution. Sketch and label the associated right triangle."
∫dx / x^2√4-x^2

To calculate the cost of each tile, we need to first determine the volume of each tile. The length and width of the tile are given as 16 inches, and the thickness is given as 1/4 inches, which can be converted to 0.25 inches. Therefore, the volume of each tile is 16 x 16 x 0.25 = 64 cubic inches.

Next, we need to determine the weight of gold in each tile. Since the density of gold is 11.17 ounces per cubic inch, the weight of gold in each tile is 64 x 11.17 = 715.68 ounces.

Finally, we can calculate the cost of each tile by multiplying the weight of gold by the price of gold per ounce. The price of gold is given as $1,303.80 per ounce, so the cost of each tile is 715.68 x $1,303.80 = $933,526.78. Rounded to the nearest dollar, each tile will cost $933,527.

In summary, each square tile made of solid gold and measuring 16 inches long and 1/4 inches thick will cost approximately $933,527. This cost is based on the density of gold, which is 11.17 ounces per cubic inch, and the price of gold, which is $1,303.80 per ounce.

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The landscaper will need 10.99 feet of fencing to place around the circumference of the pond. Option B is the correct answer.

We need to find how many feet of the fence is needed to decorate the fence around the circumference of the pond. we can determine it by finding the circumference of a pound or circle. The circumference of a circle is calculated using the formula,

C = 2πr

Where:

C = the circumference

r = radius

Given data:

π = 3.14

r =  1. 75 feet.

Substuting the value of the radius in the formula we get

C = 2πr

C = 2π(1.75)

= 10.99

Therefore, the landscaper will need 10.99 feet of fencing to place around the circumference of the pond.

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Based on the information, the three numbers are 14, 34, and 70.

Based on the information, the second number = 3x - 8

The third number is five times the first number, which can be written as:

third number = 5x

The sum of the three numbers is 118, so we can write an equation:

x + (3x - 8) + 5x = 118

9x - 8 = 118

Adding 8 to both sides:

9x = 126

x = 236 / 914

Now we can use this value of x to find the other two numbers:

second number = 3x - 8 = 3(14) - 8 = 34

third number = 5x = 5(14) = 70

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The measure of side OP in quadrilateral OPQR is 0.

To find the measure of side OP in quadrilateral OPQR, which is similar to quadrilateral KLMN, follow these steps:

1. Identify the corresponding sides in both quadrilaterals. In this case, side OP corresponds to side KL, side OQ corresponds to side KM, side PQ corresponds to side LN, and side QR corresponds to side MN.

2. Determine the scale factor between the quadrilaterals by comparing the lengths of corresponding sides. Since we have the lengths of sides KM (13), LN (27), and MN (57), we can use the ratio of KM/LN (13/27) or MN/LN (57/27) as the scale factor.

3. Apply the scale factor to the length of side KL (0) to find the length of side OP. Since the length of side KL is 0, multiplying by the scale factor (either 13/27 or 57/27) will still result in a length of 0 for side OP.

4. Round your answer to the nearest tenth if necessary. In this case, the length of side OP is 0, so rounding is not necessary.

So, the measure of side OP in quadrilateral OPQR is 0.

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So the expression that represents the average distance of a planet from the sun using radicals is: d = k/2√13 * √x

An exponent, also known as a power, is a mathematical notation that indicates the number of times a quantity is multiplied by itself. It is usually written as a small number (the exponent) placed to the right and above a larger number (the base). Exponents are used in many mathematical concepts, including logarithms, roots, and scientific notation.

Here,

The expression ((1)/∛(52)x)²) can be simplified using exponent rules:

((1)/∛(52)x)²) =((1)/∛(52)x)²) * ∛x²)

= 1/(∛52² * ∛x²)

The average distance of a planet from the sun measured in astronomical units can be represented using the formula:

d = k * √T

where d is the distance from the sun, T is the time it takes for the planet to orbit the sun, and k is a constant of proportionality.

We can rewrite this formula in terms of Earth weeks by noting that there are 52 weeks in a year, so T = (1/52)x years. Substituting this into the formula, we get:

d = k * √((1/52)x)

Simplifying this expression using exponent rules, we get:

d = k * √(1/52)* √x

So an equivalent expression using radicals to represent the average distance of a planet from the sun is:

d = k * √(1/(52)) * √x

which simplifies to:

d = k/√(52) * √x

or

d = k/2√13 * √x

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The required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.

A triangle with two sides that are perpendicular to one another is known as a right triangle, right-angled triangle, or orthogonal triangle. It was previously known as a rectangled triangle. Trigonometry is based on the relationship between the right triangle's sides and other angles.

Given data

GH = 14 units and GJ = 16 units

Using Pythagorean theorem;

[tex]16^2 = 14^2 + HJ^2[/tex]

[tex]HJ = \sqrt{16^2-14^2}[/tex]

HJ = √60

HJ = 2√15 units

And

Cos(G) = 14/16 = 7/8

∠G = cos⁻¹(7/8)

∠G = 67.5 Degrees

And

Sin(J) = 14/16 = 7/8

∠J = Sin⁻¹(7/8)

∠J = 61.04 degree

Thus, required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.

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The length of the shorter diagonal is approximately 7.07 units.

A rhombus is a four-sided polygon in which all sides are equal in length. It is also called a diamond shape because it is often used for diamond-shaped figures. In this case, we are given that the perimeter of the rhombus is 88. Since all sides of the rhombus are equal, we can divide the perimeter by 4 to find the length of one side.

88 ÷ 4 = 22

Therefore, each side of the rhombus measures 22 units.

We are also given that two of the angles in the rhombus are acute angles measuring 40° each. Since all angles in a rhombus are equal, we can find the measure of the other two angles by subtracting the sum of the acute angles from 360.

360 - 2(40) = 280

Each of the other two angles measures 140°.

To find the length of the shorter diagonal, we can use the formula:

Shorter diagonal = (2 × Area) / Length of longer diagonal

The area of a rhombus can be found by multiplying the length of the longer diagonal by the length of the shorter diagonal and then dividing by 2.

Area = (diagonal1 × diagonal2) / 2

We know that the longer diagonal is twice the length of the shorter diagonal.

Longer diagonal = 2 × Shorter diagonal

Substituting these values into the formula, we get:

Shorter diagonal = (2 × (22 × Shorter diagonal × sin 40°)) / (2 × Longer diagonal)

Simplifying, we get:

Shorter diagonal = (22 × Shorter diagonal × sin 40°) / Longer diagonal

Plugging in the values we know, we get:

Shorter diagonal = (22 × Shorter diagonal × sin 40°) / (2 × Shorter diagonal)

Shorter diagonal = 11 × sin 40°

Using a calculator, we can find that:

Shorter diagonal ≈ 7.07

Therefore, the length of the shorter diagonal is approximately 7.07 units.

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(a) Identifying the population, parameter, sample, and statistic for a study on the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts before and after a campaign.

(b) Stating the null and alternative hypotheses for a significance test on whether the percentage of male drivers not using seatbelts has decreased after the campaign.

(a) Population: All male drivers between the ages of 19 and 29 in the major urban area.

Parameter: The percentage of male drivers between the ages of 19 and 29 in the major urban area who do not regularly use seatbelts after the radio and television campaign and stricter enforcement by the local police.

Sample: 100 male drivers between the ages of 19 and 29 who were polled.

Statistic: The percentage of male drivers between the ages of 19 and 29 in the sample who did not wear their seatbelts, which is 24%.

(b) The null hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has not decreased after the radio and television campaign and stricter enforcement by the local police.

The alternative hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has decreased after the radio and television campaign and stricter enforcement by the local police.

Mathematically, the hypotheses can be stated as follows:

H0: p >= 0.28

Ha: p < 0.28

where p is the proportion of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area after the radio and television campaign and stricter enforcement by the local police.

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The question is -

In a major urban area, the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts was 28%. After a major radio and television campaign and stricter enforcement by the local police, researchers want to know if the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts has decreased.  They polled a random sample of 100 males between the ages of 19 and 29 and find the percentage who didn’t wear their seatbelts was 24%.

(a) Identify the population, parameter, sample, and statistic.

(b) State appropriate hypotheses for performing a significance test.

Answer:

3

Step-by-step explanation:

We can see that in figure ABC, line segment AB is 5 ft.

We can also see that in figure DEF, line segment DE is 15 ft.

How did we get from 5 to 15?

We multiplied by 3, so the scale factor is 3.

Hope this helps! :)

The calculated value of the circumference of the circle is 31.4 units

From the question, we have the following parameters that can be used in our computation:

Radius, r = 5

Using the above as a guide, we have the following:

Circumference = 2 * π * r

Substitute the known values in the above equation, so, we have the following representation

Circumference = 2 * 5 * 3.14

Evaluate the products

Circumference = 31.4

HEnce, the value of the circumference is 31.4 units

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Answer:

The data in the table represent an exponential function.

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

If Sarah uses 3/4 yard of ribbon to make a hair bow, she will need 6 and 3/4 yards of ribbon to make 9 hair bows.

To find out how many yards of ribbon Sarah will use to make 9 hair bows, we need to multiply the amount of ribbon used for one hair bow (3/4 yard) by the number of hair bows she wants to make (9).

So, the equation we need to use is:

3/4 yard of ribbon per hair bow x 9 hair bows = ? yards of ribbon

To solve for the answer, we can simplify the equation:

3/4 x 9 = 27/4

So Sarah will need 27/4 yards of ribbon to make 9 hair bows.

To convert this fraction to a mixed number, we can divide the numerator (27) by the denominator (4) and write the remainder as a fraction:

27 ÷ 4 = 6 with a remainder of 3

In summary, Sarah will need 6 and 3/4 yards of ribbon to make 9 hair bows, if she uses 3/4 yard of ribbon to make one hair bow.

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The measure of angle x is 77 degrees.

We have two angles given: 65 degrees and 38 degrees. Let's call the measure of angle x as "x".

From the information, we have the measure of the missing angle of the angles in a triangle are 38°, 65° and x. Then we can set up an equation:

x + 65 + 38 = 180 (the sum of the measures of the angles in a triangle is 180)

Simplifying this equation, we get:

x + 103 = 180

x = 77

Therefore, the measure of angle x is 77 degrees.

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Complete question

Find the measure of the missing angle of the angles in a triangle are 38°, 65° and x.

The slope of the line through points (6,3) and (12,7) is 2/3.

To find the slope of a line, we use the formula:

Slope = (y2 - y1) / (x2 - x1)

In this case, we have two points: (6, 3) and (12, 7). We can label them as follows:

x1 = 6
y1 = 3
x2 = 12
y2 = 7

Now we can plug these values into the formula:

Slope = (y2 - y1) / (x2 - x1)
Slope = (7 - 3) / (12 - 6)
Slope = 4 / 6
Slope = 2/3

Therefore, the slope of the line through the points (6, 3) and (12, 7) is 2/3.

The slope of a line tells us how steep it is. A positive slope means the line goes up as you move from left to right, while a negative slope means the line goes down. In this case, since the slope is positive (2/3), we know that the line goes up as we move from left to right.

The slope also tells us how much the y-value changes for every one unit of x-value. In this case, for every one unit we move to the right, the y-value goes up by 2/3.

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(a) To minimize the area of the right triangle formed by the cable, ground, and building, we need to minimize the length of the cable. To do this, we can use the Pythagorean theorem:

c^2 = a^2 + b^2

where c is the length of the cable, a is the distance from the guidepost to the point where the cable is attached to the building, and b is the distance from that point to the ground.

Since we want to minimize c, we can differentiate the equation with respect to a and set the derivative equal to zero:

dc/da = 2a/c = 0

Solving for a, we get a = c/2. This means that the point where the cable is attached to the building should be halfway up the building, or 10 feet high.

(b) To minimize the length of the cable, we can use the principle of least action, which states that the path taken by the cable is the one that minimizes the integral of the tension along the cable.

Assuming that the tension in the cable is constant, we can use the law of sines to find the angle θ:

sin θ / 20 = sin (90° - θ) / c

where c is the length of the cable.

We want to minimize c, so we can differentiate the equation with respect to θ and set the derivative equal to zero:

d(c)/d(θ) = -20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * dc/d(θ) = 0

Solving for dc/d(θ), we get:

dc/d(θ) = 20c * tan(θ)

Substituting this into the original equation, we get:

-20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * 20c * tan(θ) = 0

Simplifying, we get:

cos(θ) / sin(θ) = tan(θ)

Solving for θ, we get:

θ = 45°

Therefore, to minimize the length of the cable, it should make an angle of 45° with the face of the building.

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Let the fraction be x/y.

We know that x + y = 12, and that (x) / (y + 3) = 12.

Multiplying both sides of the second equation by (y + 3), we get:

x = 12(y + 3)

Substituting this into the first equation, we get:

12(y + 3) + y = 12

Expanding and simplifying, we get:

13y + 36 = 12

Subtracting 36 from both sides, we get:

13y = -24

Dividing both sides by 13, we get:

y = -24/13

Substituting this value of y into the equation x + y = 12, we get:

x - 24/13 = 12

Multiplying both sides by 13, we get:

13x - 24 = 156

Adding 24 to both sides, we get:

13x = 180

Dividing both sides by 13, we get:

x = 180/13

Therefore, the fraction is 180/13 divided by -24/13, which simplifies to -15/2.

Ion has a total of $4,378.05 in his savings account at the end of 3 years.

To solve this problem, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the total amount of money in the account at the end of the time period

P = the principal amount (initial deposit)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period (in years)

In this case, we have:

P = $120

r = 0.029 (2.9% expressed as a decimal)

n = 12 (compounded monthly)

t = 3 (years)

We also need to calculate the total number of deposits made during the 3-year period. Since a deposit of $120 is made at the end of every month, and there are 12 months in a year, the total number of deposits made is:

12 deposits/year × 3 years = 36 deposits

Now we can plug in the values into the formula:

A = [tex]120(1 + 0.029/12)^(12 × 3) + 120[(1 + 0.029/12)^(12 × 3) - 1]/(0.029/12)[/tex]

A = $4,378.05

Therefore, Ion has a total of $4,378.05 in his savings account at the end of 3 years.

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The minimum number of people that need to be persuaded is two. When there are 0 contributor, 1 contributor, 2 or more contributor this is a the Nash equilibria.

a. Let's first calculate the benefit to each citizen when there are n contributors. According to the problem, the benefit is n^2 dollars. So when there are 0 contributors, the benefit to each citizen is 0 dollars. When there is 1 contributor, the benefit to each citizen is 1 dollar. When there are 2 contributors, the benefit to each citizen is 4 dollars. And so on, up to 10,000 dollars per citizen when all 100 citizens contribute.

Now let's think about the incentives of each citizen to contribute. If no one contributes, everyone gets 0 dollars of benefit. If one person contributes, that person gets 1 dollar of benefit, and everyone else gets 0 dollars. So each person has an incentive to free-ride, hoping that someone else will contribute.

But if two people contribute, each person gets 4 dollars of benefit, which is more than the 1 dollar cost of contributing. So once there are at least two contributors, it becomes rational for everyone else to contribute as well.

Therefore, the minimum number of people that need to be persuaded is two. Once two people contribute, it becomes rational for everyone else to contribute as well.

b. Let's consider the Nash equilibria of the game where each citizen is deciding whether to contribute. A Nash equilibrium is a situation where no one has an incentive to change their strategy, given the strategies of all the other players.

In this case, each citizen has two strategies: contribute or free-ride. Let's consider the case where n citizens are contributing. If everyone else is contributing, then it is rational to contribute as well, since the benefit of contributing is greater than the cost.

If everyone else is free-riding, then it is rational to free-ride as well, since the cost of contributing is greater than the benefit. However, if some people are contributing and some people are free-riding, then it may be rational to contribute, since the benefit of contributing may outweigh the cost, depending on the number of contributors.

Therefore, there are multiple Nash equilibria in this game, depending on the number of contributors. When there are 0 contributors, everyone is free-riding and this is a Nash equilibrium. When there is 1 contributor, that person is contributing and everyone else is free-riding, and this is a Nash equilibrium. When there are 2 or more contributors, everyone is contributing, and this is a Nash equilibrium.

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The event that has a probability of 0 is selecting a yellow marble

Other probabilities are listed below

The items in the jar are given as

6 black, 4 brown, 8 white, and 2 blue.

These items are the sample space of this event

Hence, the sample space is 6 black, 4 brown, 8 white, and 2 blue.

For the probability Julie will draw a white marble, we have

P(White) = White/Total

So, we have

P(White) = 8/20

Simplify

P(White) = 2/5

For the event that is more likely to happen, we have

P(black marble) = 6/20

P(brown or blue marble) = (4 + 2)/20

P(brown or blue marble) = 6/20

The probabilities are equal

So, both events have equal likelihood

The event that has a probability of 0 could be the probability of selecting a yellow marble

This is because the jar has no yellow marble

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Since 1.1584 > 1, the truck cannot pass under the archway without going into the other lane.

A semi-elliptic archway with a height of 15 feet at the center and a width of 50 feet can be visualized as half of an ellipse.

The major axis of the ellipse corresponds to the width, while the minor axis corresponds to the height. In this case, the major axis (a) is 25 feet, and the minor axis (b) is 15 feet.

To determine if a truck that is 12 feet high and 14 feet wide can drive under the archway without going into the other lane, we can use the equation of an ellipse: (x²/a²) + (y²/b²) = 1.

The truck will occupy half of the road width, which is 25 feet, so its horizontal distance from the center (x) is 25 - 7 = 18 feet, and its height (y) is 12 feet.

Plugging these values into the equation, we get: (18²/25²) + (12²/15²) = (324/625) + (144/225) ≈ 0.5184 + 0.64 = 1.1584.

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The given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is any real number. The base a is typically a number greater than 1, and the function grows or decays rapidly depending on whether a is greater than or less than 1.

Exponential functions are commonly used to model processes that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest. They also arise in various areas of mathematics and science, including calculus, probability theory, and physics.

We are given the exponential function [tex]y=3^x.[/tex]

Let x1 be any value of x, then the corresponding y-value is [tex]y1=3^{(x_1)[/tex]

Let x2=x1+1 be the next value of x, then the corresponding y-value is [tex]y2=3^x2=3^(x1+1)=3*3^x1.[/tex]

So, we can see that y2 is 3 times y1, which means the y-values increase by a factor of 3 between any two points separated by x2−x1=1.

Therefore, the given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

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The critical numbers of f(x) are x = -1, 0, and 1 and The sum of all critical numbers is 0.

To find the critical numbers of f(x) = x²/4(x+1), we need to find the values of x where the derivative of f(x) is equal to zero or does not exist.

The derivative of f(x) is:

f'(x) = [(x+1)(2x) - x²(4)] / [4(x+1)²]

Simplifying, we get:

f'(x) = [2x(x+1) - 4x²] / [4(x+1)²]

f'(x) = [2x(x+1-2x)] / [4(x+1)²]

f'(x) = [2x(1-x)] / [4(x+1)²]

f'(x) = [x(1-x)] / [2(x+1)²]

The critical numbers are the values of x where f'(x) is equal to zero or does not exist.

Setting f'(x) = 0, we get:

x(1-x) = 0

This equation is true when x = 0 or x = 1.

Now, let's check if f'(x) does not exist at x = -1 (which is the only possible point where the derivative may not exist):

f'(x) = [x(1-x)] / [2(x+1)²]

When x = -1, the denominator of f'(x) becomes zero, so the derivative does not exist at x = -1.

Therefore, the critical numbers of f(x) are x = -1, 0, and 1.

The sum of all critical numbers is:

-1 + 0 + 1 = 0

Hence, the answer is 0.

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The volume, in cubic feet, of the right rectangular prism is 9 cubic feet

From the question, we have the following parameters that can be used in our computation:

A right rectangular prism has a length of 2 feet, a width of 3 feet, and a height of 1 2/4 feet

Using the above as a guide, we have the following:

Volume = Length * Width * height

Substitute the known values in the above equation, so, we have the following representation

Volume = 2 * 3 * (1 2/4)

Evaluate

Volume = 9

Hence, the volume is 9 cubic feet

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Given that quadrilateral PQRS is a parallelogram, you can prove that it is also a rectangle by A: Use the distance formula to find the length of both diagonals to see if they are congruent and B: Find the slopes of all sides to determine if the angles are right angles. Therefore, the correct option is C: C. Both A and B are valid.

To prove that PQRS is a rectangle, we need to show that all angles are right angles.

Option A: Using the distance formula, we can find the lengths of both diagonals, PR and QS. If PR and QS are congruent, then we know that opposite sides of the parallelogram are congruent and parallel (since PQRS is a parallelogram). If we can also show that PR and QS intersect at a 90-degree angle, then we have proven that PQRS is a rectangle.

Option B: Finding the slopes of all sides can help us determine if the angles are right angles. If the product of the slopes of adjacent sides is -1, then we know that the sides are perpendicular (since the slope of a line perpendicular to another line is the negative reciprocal of its slope). If we can show that all adjacent sides have slopes that multiply to -1, then we have proven that PQRS is a rectangle.

Both options A and B can be used to prove that PQRS is a rectangle, so the correct answer is C.

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A total number of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.

To calculate the total area that needs to be tiled, we'll start by converting the given dimensions from millimeters to meters:

Bathroom Inner Dimensions:

Length = 6030 mm = 6.03 m

Width = 5130 mm = 5.13 m

Bathroom Outer Dimensions (including bedroom areas):

Length = 12330 mm = 12.33 m

Width = 4680 mm = 4.68 m

Kitchen Dimensions:

Length = 6030 mm = 6.03 m

Width = 5130 mm = 5.13 m

Total area to be tiled in the bathroom (excluding the bath area):

Area = Length x Width = 6.03 m x (5.13 m - 0.86 m) = 6.03 m x 4.27 m = 25.7701 m²

Total area to be tiled in the kitchen:

Area = Length x Width = 6.03 m x 5.13 m = 30.9919 m²

Total area to be tiled (bathroom + kitchen):

Total Area = 25.7701 m² + 30.9919 m² = 56.762 m²

To account for breakages, we need to purchase 5% more tiles. So, the total number of tiles needed is:

Total Number of Tiles = Total Area x 1.05 (to account for 5% extra)

Total Number of Tiles = 56.762 m² x 1.05 = 59.6001 tiles

The building manager stated that 150 tiles are needed. Comparing this with our calculation:

150 tiles < 59.6001 tiles

Therefore, the statement made by the building manager is not valid. According to our calculations, a total of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.

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The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

What are inequalities ?

Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.

here , we have,

The symbol a < b indicates that a is smaller than b.

When a > b is used, it indicates that a is bigger than b.

a is less than or equal to b when a notation like a ≤ b.

a is bigger or equal value of an is indicated by the notation a ≥ b.

Assuming the cost of soccer cleats to be 'C' and the number of subscriptions to be 's'.

∴ The inequality that represents this situation is 3s + 25 ≥ C to real his goal.

Hence, The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

The number of triangles that can be formed using 14 points in a plane such that 4 points are collinear is 360.

If 4 points are collinear, then we can choose any 3 of those points to form a line segment, which cannot be extended to form a triangle.

we need to choose 3 points from the 14 points such that no 3 of them are collinear.

The number of triangles that can be formed from n non-collinear points in a plane is given by the formula:

                  C(n,3) = n(n-1)(n-2)/6

where C(n,3) is the binomial coefficient for choosing 3 items out of n.

So, to find the number of triangles that can be formed using 14 points in a plane such that 4 points are collinear,

we first need to find the number of ways to choose 3 points from 14 points, and then subtract the number of ways to choose 3 points from the 4 collinear points.

That is:

Total number of triangles = C(14,3) - C(4,3)

Total number of triangles = 364 - 4

Total number of triangles  = 360

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The expected value of the sale is $13,000.

The expected value is a statistical measure that represents the average outcome of a probability distribution, weighted by the probabilities of each outcome. In this case, the investor is planning to sell a property and wants to know what the expected value of the sale will be. To determine this value, we must consider the potential outcomes and their probabilities.

According to the real estate consulting firm, there is a 50% chance of making a profit of $50,000, a 30% chance of breaking even, and a 20% chance of suffering a $60,000 loss. To calculate the expected value of the sale, we multiply the potential profit or loss by the probability of each outcome occurring and then sum those products.

To determine the expected value of the sale, we need to multiply the potential profit or loss by the probability of each outcome occurring and then sum those products.

Expected value = (0.5 * $50,000) + (0.3 * $0) + (0.2 * -$60,000)

Expected value = $25,000 - $12,000

Expected value = $13,000

Therefore, the expected value of the sale is $13,000.

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According to the information, and example of a counterexample is: A line can cross the circle only at one point.

To find a counterexample to this statement we must read it carefully and identify the main idea of it. In this case you are stating that a line always crosses a circle at two points.

Later we must analyze this statement and evaluate if it is true or false. In this case it is false because we can make a line that crosses a circle only once. So, the counter example sentence would be.

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