At the beginning of a population study, a city had 300,000 people. Each year since, the population has grown by 2.5%.
Let t be the number of years since start of the study. Let y be the city's population.
Write an exponential function showing the relationship between y and t.

Therefore, the equation of the circle in standard form is: (x - 2)² + (y - 3)² = 40.

(xh)2+(yk)2=r2 is the equation of a circle in standard form. The radius is r units, and the centre is at (h,k). Mark points r units up, down, left, and right from the centre of the circle to graph it. Through these four points, draw a circle.

We must first determine the circle's centre and radius before we can determine its equation.

The circle's centre is defined as the point on a line segment that connects the diameter's two endpoints. Taking the average of the x-coordinates and the average of the y-coordinates will get the midpoint:

Midpoint: ((-3+7)/2, (-1+7)/2) = (2, 3)

The distance formula can be used to determine that the circle's radius is equal to half of its diameter:

diameter = √((7-(-3))²+ (7-(-1))²) = √(160) = 4√(10)

radius = (1/2)diameter = 2sqrt(10)

So the center of the circle is (2, 3) and the radius is 2sqrt(10).

Consequently, the circle's standard form equation is as follows:

(x - 2)²+ (y-3)²= (2√(10))²

Simplifying:

(x - 2)²+(y-3)²=40

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The difference is accounted for by d. all of these

The percentage of the nation is 46.8 percent

The economic condition most likely to be caused by a war affecting oil production in the Mid East would be cost-push inflation

The accurate response is (d) all of these. An overflow in population can influence Gross Domestic Product (GDP) by enlarging the labor force, potentially promoting productivity and efficiency as well as heightening output through more accessible assets and advanced technology. Inflation might also affect GDP by escalating prices and reducing buying capability.

Therefore, the precise answer is (d) 46.8 percent. The Lorenz Curve is a pictorial rendering of income inequality and the Gini coefficient estimates the severity of disparity. According to fresh records, the most affluent 20 percent of households in the United States keep virtually 46.8 percent of the national income.

The economic state maybe instigated by a conflict impacting oil production in the Middle East would be (b) cost-push inflation. A disruption in the offering of petroleum may consequence in elevated construction costs for businesses, which could then pass on such augmented charges to customers via higher rates. This manner of inflation is referred to as cost-push inflation.

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Answer:

8) ]-infinity ,1]

9) ]-infinity,-2[

Answer:

8) x ≤ 1     9) x < -2

Hope this helps.

Applying PEMDAS the correct expression showing where to insert brackets to make it correct are:

i. 3 × (5 + 2) + 2 = 23;      ii. (12 ÷ 4) + 2 = 5

i. 3 × (5 + 2) + 2 = 23

In this expression, we need to prioritize addition over multiplication, as per the order of operations (PEMDAS or BODMAS).

Therefore, we should add 5 and 2 first and then multiply the result by 3. Adding the brackets around 5 + 2 forces the addition to be done before multiplication, resulting in the correct answer.

ii. (12 ÷ 4) + 2 = 5

In this expression, we need to prioritize division over addition. Therefore, we should divide 12 by 4 first and then add the result to 2. Adding brackets around 12 ÷ 4 forces the division to be done before addition, resulting in the correct answer.

To understand where to insert brackets, you need to follow the order of operations, which is a set of rules that dictates the order in which arithmetic operations should be performed. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)

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The coordinates of P when written as a fraction and as a decimal, is:

Point P has the coordinates of ( 3/4 , 5/6 ) which means that it is already in fraction form as it has both a numerator and a denominator for the x and ya values.

We can then convert these fractions to decimal form as shown :

x - value : 3 ÷ 4 = 0.75

y - value : 5 ÷ 6 = 0.83

In decimals, it is:

(0.75, 0.83)

In conclusion, as a fraction, point P is ( 3 / 4, 5 / 6 ), and as a decimal, point P is ( 0.75, 0.83 ).

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Answer: 130

Step-by-step explanation:

The problem is asking for an expression that is equivalent to the addition of 91 and 39. An expression is a combination of numbers, variables, and mathematical operations that does not have an equal sign. In this case, the expression we are looking for should evaluate to the same value as the sum of 91 and 39, which is 130.

To obtain this expression, we simply add 91 and 39, since the sum of these two numbers is equivalent to the expression we are looking for. Adding 91 and 39, we get:

91 + 39 = 130

Therefore, the expression that is equivalent to 91 plus 39 is simply 130.

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The mean of the new data set is 15.12. and the median of the new data set is 14.4.

To increase each data value by 20%, we can multiply each value by 1.20. So the new data set becomes:

12 x 1.20 = 14.4

9 x 1.20 = 10.8

17 x 1.20 = 20.4

15 x 1.20 = 18

10 x 1.20 = 12

New data set: 14.4, 10.8, 20.4, 18, 12

To find the mean, we add up all the values and divide by the total number of values:

Mean = (14.4 + 10.8 + 20.4 + 18 + 12) / 5 = 15.12

Therefore, the mean of the new data set is 15.12.

To find the median, we first need to order the data set from smallest to largest:

10.8, 12, 14.4, 18, 20.4

The median is the middle value in the ordered set, which is 14.4.

Therefore, the median of the new data set is 14.4.

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The surface area of the hexagonal pyramid is  740 metres².

The pyramid above is an hexagonal pyramid. The surface area of the pyramid can be found as follows:

Therefore,

surface area of the hexagonal pyramid = A + 1 / 2 ps

where

Therefore,

A = base area = 1 / 2 p a

where

Hence,

p = 10(6) = 60 metres

a = 5√3 metres

A = base area = 1 / 2 × 60 × 5√3 = 150√3 metres

Therefore,

surface area of the hexagonal pyramid = 150√3 + 1 / 2 × 60 × 16

surface area of the hexagonal pyramid = 150√3 + 480

surface area of the hexagonal pyramid = 739.807621135

surface area of the hexagonal pyramid = 740 metres²

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The surface area of the hexagonal pyramid is  740 metres².

The pyramid above is an hexagonal pyramid. The surface area of the pyramid can be found as follows:

Therefore,

surface area of the hexagonal pyramid = A + 1 / 2 ps

where

Therefore,

A = base area = 1 / 2 p a

where

Hence,

p = 10(6) = 60 metres

a = 5√3 metres

A = base area = 1 / 2 × 60 × 5√3 = 150√3 metres

Therefore,

surface area of the hexagonal pyramid = 150√3 + 1 / 2 × 60 × 16

surface area of the hexagonal pyramid = 150√3 + 480

surface area of the hexagonal pyramid = 739.807621135

surface area of the hexagonal pyramid = 740 metres²

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The rate of change of the function given by the table is 1.

The rate of change represents the ratio of one quantity compared to another.

The rate of change is also known as the slope (the Rise/the Run), the gradient, unit rate, or constant rate of proportionality.

The rate of change is computed as the quotient between the Change in the Rise and the Change in the Run.

x     y   Rate of Change

5    3  

6    4  1 (1/1)

7    5  1 (1/1)

8    6 1 (1/1)

Thus, for the function represented by the table, the rate of change or unit rate is 1.

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Answer:

We can use the continuous compounding formula to solve this problem:

A = Pe^(rt)

where A is the final amount, P is the initial amount, r is the interest rate, and t is the time in years.

In this problem, we know the final amount (A = $1255.66), the interest rate (r = 4% = 0.04), and the time (t = 4 years). We want to find the initial amount (P).

Substituting the known values into the formula, we get:

1255.66 = Pe^(0.04*4)

Simplifying the exponent:

1255.66 = Pe^0.16

Dividing both sides by e^0.16:

1255.66 / e^0.16 = P

Using a calculator to evaluate e^0.16, we get:

1255.66 / 1.17351087099 = P

Simplifying:

P = $1069.44

Therefore, the initial amount placed in the account was $1069.44 (rounded to the nearest cent).

Step-by-step explanation:

compounded continuously, given that there is $1255.66 in the account after four years, we can use the formula:

A = Pe^(rt)

where A is the final amount, P is the initial amount, r is the interest rate, and t is the time in years.

Substituting the known values into the formula, we get:

1255.66 = Pe^(0.04*4)

Simplifying the exponent:

1255.66 = Pe^0.16

Dividing both sides by e^0.16:

P = 1255.66 / e^0.16

Using a calculator to evaluate e^0.16, we get:

P = 1069.44

Therefore, the initial amount placed in the account was $1069.44 (rounded to the nearest cent).

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Answer:

We can use the continuous compounding formula to solve this problem:

A = Pe^(rt)

where A is the final amount, P is the initial amount, r is the interest rate, and t is the time in years.

In this problem, we know the final amount (A = $1255.66), the interest rate (r = 4% = 0.04), and the time (t = 4 years). We want to find the initial amount (P).

Substituting the known values into the formula, we get:

1255.66 = Pe^(0.04*4)

Simplifying the exponent:

1255.66 = Pe^0.16

Dividing both sides by e^0.16:

1255.66 / e^0.16 = P

Using a calculator to evaluate e^0.16, we get:

1255.66 / 1.17351087099 = P

Simplifying:

P = $1069.44

Therefore, the initial amount placed in the account was $1069.44 (rounded to the nearest cent).

Step-by-step explanation:

compounded continuously, given that there is $1255.66 in the account after four years, we can use the formula:

A = Pe^(rt)

where A is the final amount, P is the initial amount, r is the interest rate, and t is the time in years.

Substituting the known values into the formula, we get:

1255.66 = Pe^(0.04*4)

Simplifying the exponent:

1255.66 = Pe^0.16

Dividing both sides by e^0.16:

P = 1255.66 / e^0.16

Using a calculator to evaluate e^0.16, we get:

P = 1069.44

Therefore, the initial amount placed in the account was $1069.44 (rounded to the nearest cent).

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The simple interest owed for the use of the money is $56.25.

The following formula can be used to determine the simple interest due for the usage of the money:

Simple Interest = P x r x t ÷ 360

Where P is the principal amount borrowed, r is the annual interest rate as a decimal, and t is the time period in days.

In this case, the principal amount P is $3000, the interest rate r is 2.5% or 0.025 as a decimal, and the time period t is 9 months or 270 days (assuming 30 days per month).

Simple Interest (SI) = $3000 x 0.025 x 270 ÷ 360

SI = $56.25

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Answer:

5 3/8 ft

Step-by-step explanation:

5 7/8 - 1/4 = 5 7/8 - 4/8 = 5 3/8

The five-number summary for the given data set is 18, 22, 26, 30, 31.

The five-number summary is a set of descriptive statistics that provides information about a dataset by summarizing the data's five most important features. It includes the minimum data point, the first quartile, the median, the third quartile, and the maximum data point.

In order to calculate the five-number summary for the given data set (20, 26, 18, 31, 22, 28, 30), we first need to arrange the data in ascending order: 18, 20, 22, 26, 28, 30, 31.

The minimum data point is 18, the first quartile is 22, the median is 26, the third quartile is 30, and the maximum data point is 31.

So, the five-number summary for the given data set is 18, 22, 26, 30, 31.

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The probability that a bus chosen at random has a working radio given that it is ready for service is approximately 85.3%

How to find the probability of a bus having a working radio given that it is ready for service?

We can use Bayes' theorem to find the probability of a bus having a working radio given that it is ready for service,

P(radio | ready) = P(ready | radio) * P(radio) / P(ready)

where

P(radio | ready) is the probability that a bus has a working radio given that it is ready for service.

P(ready | radio) is the probability that a bus is ready for service given that it has a working radio.

P(radio) is the probability that a bus has a working radio.

P(ready) is the probability that a bus is ready for service.

From the given information, we know that:

P(ready) = 0.84

P(radio | ready) = 0.67

To find P(ready | radio), we can use the formula:

P(ready | radio) = P(ready and radio) / P(radio)

From the given information, we know that:

P(ready and radio) = P(radio | ready) × P(ready) = 0.67 × 0.84 = 0.5628

To find P(radio), we can use the law of total probability:

P(radio) = P(radio | ready) × P(ready) + P(radio | not ready) × P(not ready)

We can assume that if a bus is not ready for service, it doesn't matter if it has a working radio or not. So we can simplify the equation to:

P(radio) = P(radio | ready) × P(ready) + P(radio | not ready) × (1 - P(ready))

From the given information, we know that:

P(radio | ready) = 0.67

P(ready) = 0.84

We don't have information about P(radio | not ready), but we can assume that it is lower than P(radio | ready) since a bus that is not ready for service is more likely to have a broken radio. Let's assume P(radio | not ready) = 0.3.

Then, we can calculate,

P(radio) = 0.67 × 0.84 + 0.3 × (1 - 0.84) = 0.6596

Now, we want to find P(radio | ready),

P(radio | ready) = P(ready | radio) × P(radio) / P(ready)

P(ready | radio) = P(ready and radio) / P(radio) = 0.5628 / 0.6596 = 0.853

Therefore, the probability that a bus chosen at random has a working radio given that it is ready for service is approximately 85.3% (rounded to the nearest tenth of a percent).

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The probability that a bus chosen at random has a working radio given that it is ready for service is approximately 85.3%

How to find the probability of a bus having a working radio given that it is ready for service?

We can use Bayes' theorem to find the probability of a bus having a working radio given that it is ready for service,

P(radio | ready) = P(ready | radio) * P(radio) / P(ready)

where

P(radio | ready) is the probability that a bus has a working radio given that it is ready for service.

P(ready | radio) is the probability that a bus is ready for service given that it has a working radio.

P(radio) is the probability that a bus has a working radio.

P(ready) is the probability that a bus is ready for service.

From the given information, we know that:

P(ready) = 0.84

P(radio | ready) = 0.67

To find P(ready | radio), we can use the formula:

P(ready | radio) = P(ready and radio) / P(radio)

From the given information, we know that:

P(ready and radio) = P(radio | ready) × P(ready) = 0.67 × 0.84 = 0.5628

To find P(radio), we can use the law of total probability:

P(radio) = P(radio | ready) × P(ready) + P(radio | not ready) × P(not ready)

We can assume that if a bus is not ready for service, it doesn't matter if it has a working radio or not. So we can simplify the equation to:

P(radio) = P(radio | ready) × P(ready) + P(radio | not ready) × (1 - P(ready))

From the given information, we know that:

P(radio | ready) = 0.67

P(ready) = 0.84

We don't have information about P(radio | not ready), but we can assume that it is lower than P(radio | ready) since a bus that is not ready for service is more likely to have a broken radio. Let's assume P(radio | not ready) = 0.3.

Then, we can calculate,

P(radio) = 0.67 × 0.84 + 0.3 × (1 - 0.84) = 0.6596

Now, we want to find P(radio | ready),

P(radio | ready) = P(ready | radio) × P(radio) / P(ready)

P(ready | radio) = P(ready and radio) / P(radio) = 0.5628 / 0.6596 = 0.853

Therefore, the probability that a bus chosen at random has a working radio given that it is ready for service is approximately 85.3% (rounded to the nearest tenth of a percent).

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The rectangle's height is 2 in, and the rectangle's base is 5 x 2+ 3 = 30 in, whose area is given as 68in².

It has two pairs of equal-length sides that are at right angles to each other. It is the most common type of quadrilateral, and it can be found in everyday objects like windows, doors, and tabletops.

To solve this problem, we need to use the formula for the area of a rectangle (A = l x w), where A is the area, l is the length, and w is the width. We also know that the base is 3 inches shorter than 5 times the height. Therefore, we can set up the equation to solve for the height and the base of the rectangle as follows:

A = l x w

A = (h + 3) x (5h)

A = 5h² + 3h

5h²  + 3h = 68

h= -2 (as height can not be in negative)

h= 2

By using the quadratic formula, we can solve for the height, h, which is equal to 2.

Therefore, the rectangle's height is 2 in, and the rectangle's base is 5 x 2+ 3 = 30 in.

This problem was solved by using the quadratic equation to solve for the height of the rectangle, which was then used to solve for the base.

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The height of the actual pyramid is 98 feet.

How to find the height of the actual pyramid?

To find the height of the actual pyramid, we need to use the two scales given to us and convert the height on the drawing to the height of the actual pyramid. Here are the steps to follow,

Use the scale for the drawing to the replica to convert the height on the drawing to the height of the replica,

[tex]3 \frac{1}{2} \: inches \: \times \frac{2 \: feet}{ 1 \: inch}= 7 \: feet[/tex]

Use the scale for the replica to the actual pyramid to convert the height of the replica to the height of the actual pyramid,

[tex]7 \: feet \times \frac{ 14 \: feet }{1 \: foot}= 98 \: feet[/tex]

Therefore, the height of the actual pyramid is 98 feet. The answer is A. 98 feet.

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Given that lines a and b are parallel, lines e and f are parallel and the angles formed by their intersection is x at the intersection of lines b and e. The value of x is 82 degrees. So, the correct answer is B).

We are given that Lines a and b are parallel and Lines e and f are parallel. Horizontal lines e and f are intersected by lines a and b.

At the intersection of lines a and e, the upper left angle is 82 degrees, and the bottom left angle is 98 degrees.

We need to find the value of x, which is the bottom right angle at the intersection of lines b and e. Here are the steps to solve the problem:

Since lines a and b are parallel, the angle at the top left of the intersection between b and f is also 98 degrees. This is because alternate interior angles are congruent.

We know that the angle at the top left of the intersection between a and e is 82 degrees.

Therefore, the angle at the bottom left of the intersection between f and b is

angle at the bottom left of the intersection between f and b = 180 - (82 + 98) = 0

This means that line f and line b are collinear, and therefore, they do not intersect.

Since line e intersects both lines a and b, the angle at the bottom right of the intersection between b and e is 180 degrees (a straight line).

Therefore, x = 180 - the angle at the bottom left of the intersection between b and e, which is 98 degrees.

x= 180 - 98 = 82

Hence, x = 82 degrees.

So, the correct option is B).

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--The given question is incomplete, the complete question is given

" Lines a and b are parallel and lines e and f are parallel.

Horizontal lines e and f are intersected by lines a and b. At the intersection of lines a and e, the uppercase left angle is 82 degrees, and the bottom left angle is 98 degrees. At the intersection of lines b and e, the bottom right angle is x degrees.

What is the value of x?

8

82

98

172 "--

statement B and D  are true. to determine which of the statements is true, we need to perform the indicated operations and simplify the expressions for the volumes of the boxes.

what is expressions ?

In computer programming, an expression is a combination of values, variables, and operators that produces a result or value. Expressions can be simple or complex, and they can be used in a variety of contexts, such as assigning values to variables, performing calculations, and making logical decisions.

In the given question,

To determine which of the statements is true, we need to perform the indicated operations and simplify the expressions for the volumes of the boxes.

The volume of the oak box is 2x³ + 5x² - 3xc*m³.

The volume of the maple box is 2x³ + 9x² - 2x - 24c*m³.

A. The oak box has a volume 4x² + x 24c*m³ greater than the maple box.

To compare the volumes, we need to subtract the volume of the maple box from the volume of the oak box:

(2x³ + 5x² - 3xcm³) - (2x³ + 9x² - 2x - 24cm³)

= 2x³ - 4x² + 2x + 24c*m³

Therefore, statement A is not true.

B. The maple box has a volume 4x² + x - 24c*m³ greater than the oak box.

To compare the volumes, we need to subtract the volume of the oak box from the volume of the maple box:

(2x³ + 9x² - 2x - 24cm³) - (2x³ + 5x² - 3xcm³)

= 4x² + x - 24c*m³

Therefore, statement B is true.

C. The total volume of 1 oak box and 1 maple box is 4x³ + 14x² + 5x + 24c*m³.

To find the total volume, we need to add the volumes of the oak and maple boxes:

(2x³ + 5x² - 3xcm³) + (2x³ + 9x² - 2x - 24cm³)

= 4x³ + 14x² - x - 27c*m³

Therefore, statement C is not true.

D. The total volume of 2 maple boxes is 4x³ + 18x² + 4x + 48c*m³.

To find the total volume of 2 maple boxes, we need to multiply the volume of 1 maple box by 2:

2(2x³ + 9x² - 2x - 24cm³) = 4x³ + 18x² - 4x - 48cm³

Therefore, statement D is true.

In summary, the correct statements are:

B. The maple box has a volume 4x² + x - 24c*m³ greater than the oak box.

D. The total volume of 2 maple boxes is 4x³ + 18x² + 4x + 48c*m³.

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The  expression is equivalent to the area of metal sheet required to make this square-shaped traffic sign is A)  [tex]x^2+2x+1[/tex].

What is area?

The region that an object's shape defines as its area. The area of a figure or any other two-dimensional geometric shape in a plane is how much space it occupies.

Here the given square shaped traffic sign side length a = x+1

We know that, area of square = [tex]a^2[/tex] square unit.

Then, area of the square shaped traffic sign is

=> A = [tex](x+1)^2[/tex]

=> A = [tex]x^2+2x+1[/tex]

Hence the  expression is equivalent to the area of metal sheet required to make this square-shaped traffic sign is A)  [tex]x^2+2x+1[/tex].

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Answer:

12 units.

Step-by-step explanation:

Convert the expression to vertex form:

d(t)=3t^2-12t+0

= 3(t^2 - 4t)

= 3[(t - 2)^2 - 4]

= 3(t-2)^2 - 12.

Minimum value = 12.

Its value is always one greater than the sample size.

In statistics, the degree of freedom represents the number of values in the final calculation of a statistic that is open to varying. In other terms, it is the number of distinct data points used to calculate a statistic.

The degree of freedom (df) for a sample data set is equal to the sample size minus one (df = n - 1), where 'n' represents the sample size. This means that the sample size is always one less than the degree of freedom.

The answer choices do not accurately depict the concept of the degree of freedom. The sum of all differences between the data values and the sample mean may not equal zero. The value of the degree of freedom does not always equal the sample size; rather, it is always one less than the sample size. Therefore, the correct statement is always that its value exceeds the sample size by one.

Although part of your question is missing, you might be referring to the full question:
Which of the following is true with regard to the degree of freedom?
The sum of all the differences between the data value and the sample mean can be any number
The sum of all the differences between the data value and the sample mean is always zero
Its value is always one more than the sample size
Its value is the same as the sample size

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If the boat travel north at 30 km per hour .  the equivalent speed in still water required to achieve an actual speed of 30 km/h is 30.1 km/h.

Finding the boat's true speed using Pythagoras' theorem using this  formula

Speed = √((30 km/h)^2 + (4/√(2) km/h)^2)

Let plug in the formula

Speed = √(900 + 8) km/h

Speed = √(908) km/h

Speed = 30.1 km/h

Speed = 30 km/h ( Approximately)

Therefore  the speed is 30km/h.

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Answer: x = 9 units

Step-by-step explanation:

Answer:

  5.35 grams

Step-by-step explanation:

Given that a 12 grams of a substance will decay exponentially to 7 grams in 2 hours, you want to know the amount remaining after 3 hours.

The amount remaining can be described by ...

  remaining = (initial amount)·(decay factor)^(t/(decay interval))

  remaining = 12(7/12)^(t/2) . . . . where t is in hours

After 3 hours, the amount remaining is ...

  remaining = 12(7/12)^(3/2) ≈ 5.35 . . . . grams

About 5.35 grams will remain after 3 hours.

__

Additional comment

The half-life is about 2.572 hours.

The kite is approximately 18.57 yards high above the edge of the pond.

The Pythagorean theorem would be used to solve the prob;lem that we have here

The theorye tell us that the squatre of the hypotenuse is the same as the sum of the square of the other two sides

The formula is written as

c^2 = a^2 + b^2

In this case:

c = 37 yards (length of the string)

a = 32 yards (distance from where the kite is tied to the ground to the edge of the pond)

b = height of the kite above the edge of the pond (which we need to find)

Now, plug the values into the Pythagorean theorem:

37^2 = 32^2 + b^2

1369 = 1024 + b^2

Subtract 1024 from both sides:

b^2 = 345

Now, take the square root of both sides to solve for b:

b = √345

= 18.57 yards

The kite is approximately 18.57 yards high above the edge of the pond.

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Answer:

(- 8, 2 )

Step-by-step explanation:

y = [tex]\frac{1}{2}[/tex] x + 6 → (1)

y = - [tex]\frac{3}{4}[/tex] x - 4 → (2)

substitute y = [tex]\frac{1}{2}[/tex] x + 6 into (2)

[tex]\frac{1}{2}[/tex] x + 6 = - [tex]\frac{3}{4}[/tex]x - 4

multiply through by 4 ( the LCM of 2 and 4 ) to clear the fractions

2x + 24 = - 3x - 16 ( add 3x to both sides )

5x + 24 = - 16 ( subtract 24 from both sides )

5x = - 40 ( divide both sides by 5 )

x = - 8

substitute x = - 8 into either of the 2 equations and evaluate for y

substituting into (1)

y = [tex]\frac{1}{2}[/tex] (- 8) + 6 = - 4 + 6 = 2

solution is (- 8, 2 )

[tex]f(x)=\displaystyle\sum_{n=1}^{\infty}c_n\cdot x^n \\\\[-0.35em] ~\dotfill\\\\ \displaystyle\sum_{n=1}^{\infty}c_n\cdot n\cdot x^n\implies n\sum_{n=1}^{\infty}c_n\cdot x^n\implies n\cdot f(x)\quad \textit{assuming "n" is an constant}[/tex]

The possible number of combinations are 9,720.

In mathematics, a combination is a way of selecting a subset of items from a larger set, where the order of selection does not matter. The number of combinations of size k that can be chosen from a set of n items is denoted by the symbol C(n,k) or sometimes by ⁿCk, and is given by the formula:

C(n,k) = n! / (k! ×(n - k)!)

A number is a mathematical concept used to represent quantity or value. It can be a whole number, such as 1, 2, 3, or a decimal, such as 1.5 or 3.14. Numbers are used in various mathematical operations, such as addition, subtraction, multiplication, and division. They are also used in everyday life for counting and measuring.

There are a total of 9,720 possible combinations for a 4-number computer cable lock where each space can be any number 0-9, except for the restriction that all four numbers cannot be the same. To arrive at this number, we can first calculate the total number of possible combinations without the restriction, which is 10⁴ = 10,000. Then we subtract the number of combinations where all four numbers are the same, which is simply 10 (i.e. 0000, 1111, 2222, etc.), to get 9,990. Finally, we subtract the number of combinations where all four numbers are the same and also subtract the number of combinations where three out of four numbers are the same (which is simply 10×4 = 40 since there are 10 possible values for the repeated number and 4 possible positions for it), which gives us a final count of 9,720.

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