Write one-hundred-two-and-eighty-one-hundredths in standard form.​

1. The  probability of landing on red is 7/20.

2. I will predict 25 to be white.

3. I will predict 48 to be blue.

4. It will most likely land on red.

Probability deals with the study of random events. It involves determining the likelihood or chance of an event occurring, based on the information available.

The basic formula for probability is:

P(event) = number of favorable outcomes / number of possible outcomes

number of possible outcomes = 7 + 5 + 4 + 4 = 20

1) P(red) = number of red / number of possible outcomes

P(red) = 7/20

2) when spun 100 times:

number of white = P(white) * 100

number of white = 5/20 * 100 = 25

3) when spun 240 times:

number of blue = P(blue) * 100

number of blue = 4/20 * 240 = 48

4) That will be the color with highest probability. That is color red with probability of 7/20.

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2x + y = -7

y= -7-2x

put this value in 2nd equation

3x=6+4(-7-2x)

3x=6-28-8x

11x= -22

x= -2

y= -7-2(-2)

y= -7+4

y= -3

If triangles 1 and 2 are similar, the ratio relates a side length of triangle 1 to the corresponding side length of triangle 2 is 2:3.

Since Triangles 1 and 2 are similar, their corresponding sides are proportional. This means that the ratio of the length of any corresponding pair of sides in the two triangles will be equal.

To determine the ratio of the side lengths of triangle 1 to the corresponding side lengths of triangle 2, we can choose any two corresponding sides and divide their lengths.

Let's choose side AC of triangle 1 and side DF of triangle 2. We know that these two sides are corresponding because they are both opposite angles that have the same measure.

The length of side AC in triangle 1 is given as 17, and the length of side DF in triangle 2 is given as 25.5. Therefore, the ratio of the length of side AC to side DF is:

AC/DF = 17/25.5 = 0.6667 or 2/3

So the ratio of the length of a side in triangle 1 to the corresponding side in triangle 2 is 2:3. This means that if we multiply any side length in triangle 1 by 2, we will get the length of the corresponding side in triangle 2.

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The given chain which has a negligible mass is draped over the sprocket which has a mass of 2 kg and a radius of gyration of k0=50 mm.

If the 4-kg block A is released from rest in the position s=2 m. Using energy methods, determine the angular velocity of the sprocket at the instant s=4 m.First of all, draw a FBD of the system at s=2 m and at s=4 m.At s=2 m Let the velocity of the block A be v2m/s.At s=4 m Let the velocity of the block A be v4m/s.Let the angular velocity of the sprocket be ω rad/s. Draw FBD at both states and clearly label datum.For the block A at s=2 m Velocity of the block A, v=0 (Initial velocity)

Kinetic energy of the block A,KE1=12mv²=12×4×(0)²=0Potential energy of the block A,PE1=mgh=4×9.81×2=78.48JFor the block A at s=4 mLet h be the height by which the [tex],KE1=12mv²=12×4×(0)²=0[/tex]block A falls. Velocity of the block A, [tex]v=√2gh=√2×9.81×h[/tex]Kinetic energy of the block A,KE2=[tex]12mv²=12×4×[√2gh]²=8gh[/tex] Joule (J)Total potential energy of the system at s=2 m,PE1=2×g×k0 Joule (J)When the block falls from s=2 m to s=4 m, the change in the potential energy of the system is,ΔPE=PE2-PE1Where,PE2=2×g×(2k0) Joule (J)Let the change in the potential energy of the system be ΔPE1 Joule (J).

So,ΔPE1=PE2-PE1=2gk0 Joule (J)The total energy of the system is conserved.

So,ΔPE1=KE2ω²+Iω²Where,ΔPE1 is the change in the potential energy of the systemKE2 is the kinetic energy of the block A when it is at s=4 mI is the moment of inertia of the sprocket about its axisω is the angular velocity of the sprocket at s=4 mSo,ω²=ΔPE1KE2+Iω²Substitute the values of ΔPE1, KE2 and I in the above equation.

[tex]ω²=(2gk0)8gh+mr²ω²=(2×9.81×50×10⁻³)8×h×h+2×1×(50×10⁻³)²ω²=0.7848h²+2.5×10⁻³[/tex]

Therefore,ω=√(0.7848h²+2.5×10⁻³) rad/s.

Substitute the value of h=2 m in the above equation.ω=2.808 rad/s.So, the angular velocity of the sprocket at the instant s=4 m is 2.808 rad/s.

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The graph for the linear equation y=24x, will be a straight line having coordinate (1,24) i.e. B.

What is a linear equation, exactly?

A linear equation is a first-degree algebraic equation in which each term is either a constant or the product of a constant and a single variable (degree 1). A linear equation is stated as y = mx + b, where y is the dependent variable, x is the independent variable, m is the line's slope, and b= y-intercept .

A linear equation's graph is a straight line. The line's slope decides how steep it is, and the y-intercept indicates where the line crosses the y-axis. Linear equations are used to model relationships between variables that are directly proportional to each other, such as distance and time, or cost and quantity.

Now,

Given equation is y=24x

then for  x,    y is

              1,    24

              2,   48

              3,    72

              4,     96

Hence, the graph will be as represented in option 2.

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a) Approximately 29.50% of brake pads will contain less than 32.5% copper. b)  The approximate distribution of the sample mean copper percentage is a normal distribution with mean 34 and standard error 0.7.

a)To solve this problem, we can use the z-score formula to standardize the value of 32.5 using the mean and standard deviation given:

z = (x - μ) / σ

where x is the value we are interested in (32.5), μ is the mean (34), and σ is the standard deviation (2.8).

Substituting these values, we get:

z = (32.5 - 34) / 2.8 = -0.536

Using a standard normal distribution table or calculator, we can find the probability of a z-score less than -0.536, which is approximately 0.2950 or 29.50%. Therefore, approximately 29.50% of brake pads will contain less than 32.5% copper.

b) The sample mean copper percentage of 16 pads will also follow an approximate normal distribution with mean μ = 34 and standard deviation σ/√n = 2.8/√16 = 0.7, where n is the sample size.

Therefore, the approximate distribution of the sample mean copper percentage can be expressed as:

X ~ N(34, 0.7)

where X is the sample mean copper percentage.

The mean of this distribution is equal to the population mean μ, which is 34. The standard error of the mean (also known as the standard deviation of the sampling distribution) is given by σ/√n, which is 0.7 in this case.

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Your question is incomplete, but probably the complete question is :

Assume the distribution of copper content (%) in sintered brake pads follows an approximate normal distribution with a mean of 34 and a standard deviation of 2.8

a.) What percentage of brake pads will contain less than 32.5% copper?

b.) What is the approximate distribution of the sample mean copper percentage if we sample 16 pads? Include the mean and standard error of this distribution

I Hope it is correct

Step-by-step explanation:

The total number of sodas in the fridge is:

6 + 4 + 5 + 9 + 7 + 3 + 6 = 40

The number of NuGrapes or ginger ales is:

7 (NuGrapes) + 6 (Canada Dry ginger ales) = 13

The probability of selecting a NuGrape or a ginger ale is the number of favorable outcomes (selecting a NuGrape or a ginger ale) divided by the total number of possible outcomes (selecting any soda):

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 13 / 40

Probability = 0.325

So, the probability of selecting a NuGrape or a ginger ale is 0.325 or 32.5%.

The probability that 5 or less students will withdraw from a math course if 30 students are enrolled is 0.6826.

This can be calculated using the binomial probability formula, which states that the probability of a certain number of successes in a certain number of trials is equal to the number of combinations of successes times the probability of each success. In this case, the probability of success (withdrawing) is 0.10 and the number of trials (students enrolled) is 30.

For this example, the probability of 5 or less successes (withdrawals) is calculated by adding together the probabilities of 0 successes, 1 success, 2 successes, 3 successes, 4 successes, and 5 successes. That gives us a probability of 0.6826.

This is equivalent to saying that 68.26% of students enrolled in a math course will not withdraw. It is also equivalent to saying that 31.74% of students enrolled in a math course will withdraw.

To illustrate this, if we assume that 30 students are enrolled in a math course, we would expect that approximately 19 students will not withdraw (68.26%) and approximately 11 students will withdraw (31.74%).

In conclusion, the probability that 5 or less students will withdraw from a math course if 30 students are enrolled is 0.6826.

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The coordinates of point C are (4, 2), which is located in quadrant 1.

In a 30-60-90 degree triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3/2 times the length of the hypotenuse. Since the side opposite the 30-degree angle is the shortest side, it must be the distance between points A and B, which is 8 units.

Let's call point C (x, y). Since angle C is 90 degrees, side AC is perpendicular to side AB, which means it is a vertical line that passes through point A. Similarly, side BC is perpendicular to side AB, which means it is a horizontal line that passes through point B.

Therefore, point C must lie on both the vertical line passing through A and the horizontal line passing through B. The equation of the vertical line passing through A is x = -4, and the equation of the horizontal line passing through B is y = -2. So we have the system of equations

x = -4

y = -2

Solving this system gives us the coordinates of point C: (-4, -2). However, this point is not in quadrant 1, as we desired.

To find a point C in quadrant 1, we need to flip the signs of the x and y coordinates of point C. So the coordinates of point C are (4, 2).

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A correlation coefficient of 21.34 is an impossible value.

In statistics, correlation is defined as the degree of association between two or more variables. Correlation research helps to establish whether a relationship exists between two variables, and if so, to what extent. The correlation coefficient is a statistical measure used to quantify the degree of association between two variables.

The correlation coefficient, which ranges from -1 to +1, indicates the strength of the relationship between the two variables. When the correlation coefficient is 0, it indicates that there is no correlation between the two variables. When the correlation coefficient is positive, it indicates that the two variables are directly related, whereas when the correlation coefficient is negative, it indicates that the two variables are inversely related.

In this case, a correlation study reported a linear correlation coefficient of 21.34, which is not possible. Correlation coefficients range from -1 to +1. The value of the correlation coefficient ranges from -1 to +1, where a value of 0 indicates no correlation, a value of 1 indicates perfect positive correlation, and a value of -1 indicates perfect negative correlation. Therefore, a correlation coefficient of 21.34 is an impossible value.

In this case, it is likely that the correlation coefficient was misreported or that a different statistical measure was used. It is important to carefully examine and verify statistical findings before drawing any conclusions or making decisions based on the results.

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Answer:

6 and 10

Step-by-step explanation:

first find what 1 min is in meters

1/3 is 8 so 8x3 = 1m which is 24

now divide based on the next two numbers

1/4 of 24 = 6

6x 1 = 6

now the next

24/12 = 2

2x5 = 10

these two results, 6 and 10, are the answers to the missing meter values

The equation of the line that passes through the given point (-1, 5) is y = -2x + 7.

The given point (–1, 5) is the solution to a system of linear equations.

One of the equations is y=−2x+3.

Let's discuss linear equations.

A linear equation is an algebraic equation that represents a straight line on the coordinate plane.

In the form y = mx + b,

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

the general form of a linear equation is y = mx + b.

In the equation y = mx + b, m represents the slope of the line, and b represents the y-intercept of the line.

Now, let's find another equation of the given system of linear equations using the given point (-1, 5).

Given equation is y = -2x + 3.

Let's find the slope of the given equation.

Slope (m) = -2Therefore, the slope-intercept form of the equation is y = -2x + b.

To find b, substitute x = -1 and y = 5 in the above equation.

5 = -2(-1) + b Simplifying the above equation.

5 = 2 + b Adding 2 to both sides of the equation.

5 + 2 = b7 = b Therefore, the equation of the line that passes through the given point (-1, 5) is y = -2x + 7.

The system of linear equations is y = -2x + 3y = -2x + 7

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The solution for this question is (10x-4) + (-7+ x) = (10x + x) + (-4 - 7) = 11x - 11

Why it is and what is expression in math?

(10x-4) + (-7+ x) = (10x + x) + (-4 - 7) = 11x - 11

Therefore, the missing values are:

The first missing value is "x", as we have combined the "10x" and "x" terms to get "11x".

The second missing value is "-11", as we have combined the "-4" and "-7" terms to get "-11".

Therefore, the final expression is: 11x - 11

In math, an expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) that can be evaluated to produce a single value. An expression can contain constants (fixed numbers), variables (which represent unknown values), or a combination of both.

For example, "2 + 3" is an expression that represents the sum of 2 and 3, which can be evaluated to produce the value 5. Similarly, "4x - 2" is an expression that contains a variable (x) and represents the result of multiplying x by 4 and then subtracting 2.

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The probability of selecting 46 students out of a college population where 65% still need to take another math class is 0.05433 or 5.433%.

The probability of selecting 46 students out of a college population of which 65% still need to take another math class can be calculated using the binomial probability formula. To calculate the probability, the following information is needed:

Using the binomial probability formula, we can calculate the probability of selecting 46 students out of a college population where 65% still need to take another math class as follows:

[tex]P(X=46) = (N!/((N-r)! * r!)) * (p^r) * (1-p)^(N-r)[/tex]

[tex]= (100!/((100-46)! * 46!)) * (0.65^46) * (1-0.65)^(100-46)[/tex]
[tex]= 0.05433[/tex]

Therefore, the probability of selecting 46 students out of a college population where 65% still need to take another math class is 0.05433 or 5.433%.

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The coordinates for the dilated triangle is obtained as A'(-1, 8), B'(-7, 6), and C'(-7, -4).

What is dilation?

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. The initial shape should be stretched or contracted during a dilatation.

The coordinates point of the triangle is given as -

A (-1,6)

B (-4,5)

C (-4,0)

To dilate triangle ABC by a scale factor of 2 about the point (-1, 4), we can follow these steps -

Translate the triangle so that the center of dilation is at the origin.

We can do this by subtracting (-1, 4) from each vertex -

A' = (-1, 6) - (-1, 4) = (0, 2)

B' = (-4, 5) - (-1, 4) = (-3, 1)

C' = (-4, 0) - (-1, 4) = (-3, -4)

Dilate the translated triangle by multiplying the coordinates of each vertex by the scale factor of 2 -

A'' = 2(0, 2) = (0, 4)

B'' = 2(-3, 1) = (-6, 2)

C'' = 2(-3, -4) = (-6, -8)

Translate the dilated triangle back to its original position by adding (-1, 4) to each vertex -

A'B'C' = A'' + (-1, 4) = (-1, 8)

B'' + (-1, 4) = (-7, 6)

C'' + (-1, 4) = (-7, -4)

Therefore, the coordinates of the dilated triangle A'B'C' are (-1, 8), (-7, 6), and (-7, -4).

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The least amount of money he could pay for the conditions that he will for the given information is = £ 3053.4

What about least amount?

In mathematics, the term "least amount" is not commonly used. However, a similar term that is often used is "minimum".

The minimum value of a set of numbers or a function is the smallest value within that set or function. For example, if we have the set of numbers {2, 5, 7, 8, 10}, the minimum value is 2.

Similarly, in the context of optimization problems, we often seek to find the minimum value of a function to achieve the best possible outcome. This can involve finding the minimum value of a cost function in order to minimize the cost of a process, or finding the minimum value of a profit function to maximize the profit of a business.

According to the given information:

Set the number of party bags pack as x , toy pack as y

Since, 105 party bags in a pack , £1.32

84 toys in a pack £4.15.

Hence, 105x = 84y , cost = 105x × 1.32x + 84y × 4.15 = 763.35x

Find the least common multiple

Hence, 84 × 5 = 420 , 105 × 4 = 420

x  =4

y =5

least cost = 105 × 1.32 × 4 + 84×4.15 ×5

= £3053.4

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Thus, option (d) is the correct answer.

Option (d) as x approaches infinity, the value of g(x) eventually exceeds the values of both f(x) and h(x) is the correct option. Since the functions are:f(x) =[tex]8x² + x + 3g(x) = 4x² – 1h(x) = 3x + 6[/tex]

a) over the interval [3, 5], the average rate of change of g and h is more than the average rate of change of f.It is not possible to determine which of these functions has a higher average rate of change since they all have different derivatives.b) over the interval [0, 2], the average rate of change of f and h is less than the average rate of change of g.The average rate of change of f(x) = [tex]8x² + x + 3[/tex]over the interval [0, 2] is given by:f'(x) = 16x + 1The average rate of change of f(x) = 8x² + x + 3 over the interval [0, 2] is:

[tex]f(2) - f(0)/2 - 0= f(2) - f(0)/2 = [8(2)² + 2 + 3 - (8(0)² + 0 + 3)]/2 = [32 + 2 + 3 - 3]/2 = 34/2 = 17[/tex]

The average rate of change of h(x) = 3x + 6 over the interval [0, 2] is given by:h'(x) = 3The average rate of change of h(x) = 3x + 6 over the interval [0, 2] is:[tex]h(2) - h(0)/2 - 0= h(2) - h(0)/2 = [3(2) + 6 - (3(0) + 6)]/2 = 12/2 = 6[/tex]The average rate of change of g(x) = 4x² - 1 over the interval [0, 2] is given by:g'(x) = 8xThe average rate of change of g(x) = 4x² - 1 over the interval [0, 2] is:[tex]g(2) - g(0)/2 - 0= g(2) - g(0)/2 = [4(2)² - 1 - (4(0)² - 1)]/2 = 16 - 1/2 = 15/2[/tex]

Thus, it can be concluded that the average rate of change of g(x) > f(x) > h(x) over the interval [0, 2]c) as x approaches infinity, the values of g(x) and h(x) eventually exceed the value of f(x).It is not true since g(x) and h(x) both have leading coefficients of 4 and 3, respectively, and will eventually grow faster than f(x) with a leading coefficient of 8.d) as x approaches infinity, the value of g(x) eventually exceeds the values of both f(x) and h(x)The leading coefficient of g(x) = 4x² - 1 is 4, and as x approaches infinity, it will continue to grow faster than both f(x) = 8x² + x + 3 and h(x) = 3x + 6, which have leading coefficients of 8 and 3, respectively.

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Answer:

What is the probability of rolling an odd product? 1/4

What is the probability of rolling a product that is greater than or equal to 20? 2/9

What is the probability of rolling a product that is less than 10? 17/36

What is the probability of rolling a product that is a multiple of 10? 1/6

Answer:

We can use the formula:

time = distance ÷ speed

where distance is in kilometers (km) and speed is in kilometers per hour (km/h).

Substituting the given values, we get:

time = 340 km ÷ 12 km/h

time = 28.33 hours (rounded to two decimal places)

Therefore, it takes approximately 28.33 hours for the car to travel a distance of 340 km at an average speed of 12 km/h.

Answer:28hrs and 20min

Step-by-step explanation:

rgrghwrt

Consequently, x is 70 degrees as a consequence. The supplied diagram labels the angles of the triangle ABC as 40°, 70°, and x.

Three straight lines that meet at three different locations to create a triangle, a two-dimensional geometric shape. Triangles have three sides and three vertices, which are the three places where those three lines intersect. Triangles can be categorised based on the dimensions of their angles and side lengths. . In contrast, a triangle with three equal sides and three equal angles of 60 degrees is called an equilateral triangle. The sides and angles of a scalene triangle are not identical.

given

The angles of the triangle ABC are labelled in the provided figure as 40°, 70°, and x. We can use the knowledge that the sum of a triangle's angles is 180° to find x.

As a result, we have:

x + 40° + 70° = 180°

The left half of the equation is simplified as follows:

x + 110° = 180°

110° from both edges subtracted:

x = 70°

Consequently, x is 70 degrees as a consequence. The supplied diagram labels the angles of the triangle ABC as 40°, 70°, and x.

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The area of given polygon is 44 square units from the given graph.

What is Polygon ?

A polygon is a 2-dimensional geometric shape that is formed by joining a finite number of straight line segments to form a closed shape.

The area of each triangle can be found by using the formula:

Area = (base * height)/2

Triangle 1: Base = 4 units, Height = 5 units

Area of Triangle 1 = (4*5)/2 = 10 square units

Triangle 2: Base = 8 units, Height = 3 units

Area of Triangle 2 = (8*3)/2 = 12 square units

The area of the trapezoid can be found by using the formula:

Area = (Sum of parallel sides * Height)/2

Trapezoid: Height = 4 units, Parallel side 1 = 3 units, Parallel side 2 = 7 units

Area of Trapezoid = ((3+7)*4)/2 = 20 square units

Therefore, the total area of the polygon is:

Total Area = Area of Triangle 1 + Area of Triangle 2 + Area of Trapezoid

Total Area = 10 + 14 + 20 = 44 square units

Hence, The area of given polygon is 44 square units from the given graph.

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A requirement of a random sample is that every individual has an equal chance of being selected.

What is a random sample? A random sample is a process of selecting members from a population, that each member has an equal chance of being chosen. In a random sample, each member of the population has an equal chance of being selected.

Therefore, every individual has an equal chance of being selected is a requirement of a random sample. So, option A is the correct answer. It is the most essential part of random sampling.

There are various types of random sampling techniques, including simple random sampling, stratified sampling, cluster sampling, and systematic sampling.

The objective of random sampling is to acquire an unbiased sample that represents the entire population fairly. The primary purpose of random sampling is to minimize bias while selecting the members of a population.

Hence, most essential part of Random Sampling is that every individual has an equal chance of being selected.

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The functions when ordered from narrow to wide is g(x) = 4x², h(x) = 2x² and f(x) = (1/3)x²

The width of the graph of a quadratic function is determined by the coefficient of the squared term.

The greater the value of this coefficient, the narrower the graph, and vice versa.

So, in order of increasing width, we have:

g(x) = 4x²

h(x) = 2x²

f(x) = (1/3)x²

f(x) has the widest graph because its coefficient is the smallest, followed by h(x), and g(x) has the narrowest graph because its coefficient is the largest.

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The rate of decrease of the side of the cube with a side of 6 cm is 0.25 cm per second.

The rate of decrease of the side of an ice cube can be calculated by considering the volume of the cube. The volume of the cube is decreasing at a rate of 1.5 cm3 per second. Therefore, the rate of decrease of the side of the cube is proportional to the cube's volume.

To calculate the rate of decrease of the side of the cube, we need to use the formula for the volume of a cube, which is: V = s3, where V is the volume and s is the side of the cube.

Since we know that the volume of the cube is decreasing at a rate of 1.5 cm3 per second, we can calculate the rate of decrease of the side of the cube by substituting the value of the rate of decrease of the volume and the current side of the cube (6 cm) into the formula.

Therefore, the rate of decrease of the side of the cube is equal to 1.5 cm3 per second divided by 6 cm3, which is equal to 0.25 cm per second.

In conclusion, the rate of decrease of the side of the cube with a side of 6 cm is 0.25 cm per second.

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If the price of a new car is $24,000, the total owed for tax is $1,518.50.

Tax is the money that the government collects from people and organizations for providing public services and infrastructure.

The total owed for tax on a $24,000 car purchase is calculated by multiplying the sales tax rate of 6.25% by the price of the car.

The result of this calculation is the sales tax portion of the total amount owed.

The title fee of $18.50 is added to this amount to get the total amount due for tax on the purchase of the car.

Mathematically, the equation for the total owed for tax is:

Total Tax = (6.25% x $24,000) + $18.50

Total Tax = ($1,500 + $18.50)

Total Tax = $1,518.50

Therefore, if the price of a new car is $24,000, the total owed for tax is $1,518.50.

This amount includes the sales tax portion of 6.25% of the car's price and a title fee of $18.50. This tax amount must be paid before the buyer can take possession of the car.

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The Recursive expression f(n), that shows the population at the beginning of the year n, as a function of its population the preceding year is

The population at the beginning of the year is given as 10 000

Let P(n) be the population at the beginning of the year n,

where

n is a positive integer.

We can express P(n) in terms of P(n-1) as follows:

P(n) = P(n - 1) + 0.03P(n - 1) - 0.02P(n - 1) + 0.05P(n - 1)

= (1 + 0.03 - 0.02 + 0.05)P(n - 1)

 = 1.06P(n - 1)

Therefore, the recursive expression that shows the population at the beginning of the year n as a function of its population the preceding year is:

f(n) = 1.06f(n - 1)

This recursive formula is derived from the fact that each year,

Resulting in a total increase of 6% per year.

The formula expresses this increase as a multiplication of the previous year's population by a factor of 1.06, resulting in the population at the beginning of the current year.

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Answer:

If the larger bottle of shampoo gives 25% more for the same price, then it must contain 1.25 times the amount of shampoo in the smaller bottle. Let's represent the size of the smaller bottle as x ounces.

Thus, we can set up the following equation:

x * 1.25 = 6.7

Solving for x, we get:

x = 6.7 / 1.25

x ≈ 5.36

Therefore, the size of the original smaller bottle of shampoo is approximately 5.4 ounces (rounded to the nearest tenth).

The different types of outputs that are possible when performing intersect and union include points, lines, and polygons.

Intersect: It is a binary operation that compares two shapes and returns the geometries that they have in common. The result of an intersect operation is either a point, line, or polygon, depending on the shapes being compared. If two points intersect, the result is a point. If two lines intersect, the result is a point or a line. If two polygons intersect, the result is a polygon that contains the intersecting area.

Union: It is a binary operation that combines two shapes into one. The result of a union operation is a polygon, line, or point, depending on the shapes being combined. If two points are combined, the result is a point. If two lines are combined, the result is a line or a polygon. If two polygons are combined, the result is a larger polygon that contains both shapes. In some cases, the result of a union operation may not be a single shape, but rather a set of shapes.

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The work required to pump all the water over the top edge of the cylindrical tank is approximately 83,500 kJ.

To calculate the work required to pump all the water over the top edge of the cylindrical tank, we need to determine the volume of the water in the tank and the force required to lift it to the top edge.

The volume of water in the tank can be found by multiplying the cross-sectional area of the tank by the depth of the water. Since the tank is cylindrical, the cross-sectional area is given by πr^2, where r is the radius of the tank. Therefore, the volume of water in the tank is:

V = πr^2h = π(6m)^2(5m) = 540π m^3

To lift the water to the top edge of the tank, we need to overcome the force of gravity acting on the water. The force required to lift an object of mass m against gravity is given by F = mg, where g is the acceleration due to gravity (9.81 m/s^2).

The mass of the water in the tank can be found by multiplying its volume by its density. The density of water is approximately 1000 kg/m^3. Therefore, the mass of the water in the tank is:

m = ρV = (1000 kg/m^3)(540π m^3) = 1.7 x 10^6 kg

The force required to lift the water to the top edge of the tank is:

F = mg = (1.7 x 10^6 kg)(9.81 m/s^2) = 16.7 x 10^6 N

To find the work required to lift the water, we need to multiply the force by the distance over which it acts. The distance is equal to the height of the water in the tank, which is 5 m.

Therefore, the work required to pump all the water over the top edge of the tank is:

W = Fd = (16.7 x 10^6 N)(5 m) = 83.5 x 10^6 J

Rounding to the nearest kilojoule, the work required is approximately 83,500 kJ.


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According to the given exponential equations it is proved that b = (ac) / (a+b) where a+b ≠ 0.

We are given the first equation as [tex]3^{a} = 21^{b}[/tex] ----(1) i.e 3 to the power a is equal to 21 to the power b.

The next exponential equation is [tex]7^{c} = 21^{b}[/tex] ----(2)

We know, 21 can be written as 3*7 because they are the factors of 21. Therefore, [tex]21^{b} = (3*7)^{b} = 3^{b}7^{b}[/tex]

Substituting equations (1) and (2), we get:

[tex]3^{a}= 3^{b} *7^{b}[/tex]

Taking logarithm base 3 on both sides:

a = blog3(37)

a = b*(log3(3) + log3(7))

a = b*(1 + log3(7))

b = a / (1 + log3(7)) ------ (3)

Similarly, taking logarithm base 7 on both sides of equation (2):

c = b*log7(21)

c = b*(log7(3) + log7(7))

c = b*(1 + log7(3/7))

b = c / (1 + log7(3/7)) ------ (4)

Substituting (3) and (4) in the given equation, we get:

a / (1 + log3(7)) = (ac) / (a+b) * (1 + log7(3/7))

After cross-multiplying and simplifying the equations,

b = (ac) / (a+b)

Hence,b = (ac) / (a+b)

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