How many atoms of your element must be lined up to to make a line 1 inch long? Argon (Explain or show how you did it)

HINT: Your atomic radius in probably given in picometers. Convert those picometers to inches. That gives you the radius (half of the length of an atom) in units of inches.

AR = 188 pm
B. What would the mass of 1 Liter (1000 mL or 1000 cm3) of your element be?

Density = 1.78.10 -3 g.cm -3

The probability distribution can be constructed by dividing each value of p(x) by the sum of all p(x) values. This will give the proportion of each value in the distribution. The completed table is as follows: x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | p(x)| 0.030 | 0.061 | 0.139 | 0.243 | 0.186 | 0.117 | 0.064.

To construct the probability distribution, we need to find the probabilities for each value of x. The p(x) values given in the table represent the frequencies or counts of each value. To convert these counts into probabilities, we need to divide each p(x) value by the sum of all p(x) values.

In this case, the sum of all p(x) values is 7 + 14 + 32 + 56 + 43 + 27 + 15 = 194. To find the probability for each x value, divide each p(x) value by 194.

For example, the probability for x=0 is 7/194 ≈ 0.036. Similarly, the probability for x=1 is 14/194 ≈ 0.072, and so on.

The completed probability distribution table is as follows:
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | p(x)| 0.036 | 0.072 | 0.165 | 0.289 | 0.222 | 0.140 | 0.082.

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The distance between the points A and B is √117 and the midpoint is (-3/2, 5).

To solve the inequality, x+26/x+5 ≥ 5, we can begin by subtracting 5 from both sides to get:

x+26/x+5 - 5 ≥ 0

x+26/x+5 - 5

x+5/x+5 ≥ 0

Common denominator = (x+5)(x+5), which means:

((x+26)(x+5) - 5x(x+5))/((x+5)(x+5)) ≥ 0

Simplifying gives:x²+21x-74 ≥ 0Therefore,x ≤ -7orx ≥ 4

The midpoint formula is [(x1+x2)/2, (y1+y2)/2] where (x1,y1) and (x2,y2) are the coordinates of the points A and B.  

Given two points A(-6, 8) and B(3, 2), the distance and midpoint are found as follows:

Distance between A and B=√(x2−x1)^2+(y2−y1)^2

Distance between A and B=√(3−(−6))^2+(2−8)^2=√81+36=√117

Midpoint of AB=[((x1+x2)/2),((y1+y2)/2)]

Midpoint of AB=[((-6+3)/2),((8+2)/2)]

Midpoint of AB=[(-3/2), (5)]

Therefore, the distance between the points A and B is √117 and the midpoint is (-3/2, 5).

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The cost of a 13-foot tube with a 5-inch diameter is $13000.

To find the cost, we can set up a proportion using the given information. Since the cost varies jointly with the length and diameter, we can write:

C ∝ L × D

where C is the cost, L is the length, and D is the diameter.

Using the given values for the 14-foot tube with a 5-inch diameter (C = $280, L = 14, D = 5), we can set up a proportion:

280 ∝ 14 × 5

To find the cost of the 13-foot tube, we can rearrange the proportion:

C ∝ L × D

C = (280/14) × (13 × 5)

C = 20 × 65

C = $1300

Therefore, the cost of a 13-foot tube with a 5-inch diameter is $1300.

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Answer:

No, North St. and South St. are not parallel. If East St. intersects both North St. and South St., then North St. and South St. must intersect each other at some point. Two lines that intersect cannot be parallel.

To find the height of a triangle using trigonometry, you can use the sine or cosine ratios. The specific ratio to use depends on the information you have about the triangle.

If you have the length of one side of the triangle and the measure of the angle opposite that side, you can use the sine ratio to find the height. The sine ratio is defined as the length of the side opposite the angle divided by the length of the hypotenuse.

Here are the steps to find the height using the sine ratio:

1. Identify the side of the triangle that represents the height.
2. Determine the angle opposite that side.
3. Measure the length of the side adjacent to the angle or obtain that information from the problem.
4. Use the sine ratio: height = length of adjacent side * sin(angle).

For example, let's say you have a right triangle with an angle of 30 degrees and a side adjacent to that angle measuring 6 units. To find the height, you would use the sine ratio:

height = 6 * sin(30)
height ≈ 3 units

If you have the length of two sides of a right triangle and you need to find the height, you can use the cosine ratio. The cosine ratio is defined as the length of the side adjacent to the angle divided by the length of the hypotenuse.

Here are the steps to find the height using the cosine ratio:

1. Identify the side of the triangle that represents the height.
2. Determine one of the acute angles of the triangle.
3. Measure the lengths of the two sides adjacent to that angle or obtain that information from the problem.
4. Use the cosine ratio: height = length of adjacent side * cos(angle).

For example, let's say you have a right triangle with an angle of 45 degrees and two sides adjacent to that angle measuring 4 units and 4√2 units. To find the height, you would use the cosine ratio:

height = 4 * cos(45)
height ≈ 2.828 units

In summary, to find the height of a triangle using trigonometry, you can use the sine or cosine ratios. The sine ratio applies when you have the length of one side and the angle opposite that side, while the cosine ratio applies when you have the lengths of two sides adjacent to an angle.

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The multiplication and division rule with significant figures: Result has the fewest significant figures as the measurement involved.

When performing multiplication or division calculations with measurements, it is important to consider the significant figures in the numbers involved.

The rule states that the result of the calculation should be rounded to the same number of significant figures as the measurement with the fewest significant figures.

For example, let's consider the multiplication of 3.4 cm and 2.16 g. The measurement with the fewest significant figures is 3.4 cm, which has two significant figures. Therefore, the result of the multiplication should also have two significant figures, yielding 7.3 cm².

Similarly, for division, suppose we have 8.25 m divided by 2.1 s. The measurement with the fewest significant figures is 2.1 s, which has two significant figures.

Following the division rule, the result should also be rounded to two significant figures, resulting in 3.9 m/s.

By applying the multiplication and division rule with significant figures, we ensure that the precision of the result aligns with the least precise measurement involved in the calculation.

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Answer:

45 + 2x - 5 = 180

2x + 40 = 180

2x = 140, so x = 70

An angle between 0 and 2π that is coterminal with the given angle is:

1. 19π/5 is 9π/5

2. -14π/3 is 4π/3

3. 57π/2 is 53π/2

4. 19π/11 is 41π/11

To find an angle that is coterminal with a given angle, we need to add or subtract integer multiples of 2π until we obtain an angle between 0 and 2π.

1. 19π/5

To find an angle coterminal with 19π/5, we can subtract 2π repeatedly until we get an angle between 0 and 2π:

19π/5 - 2π = 19π/5 - 10π/5 = 9π/5

The angle coterminal with 19π/5 is 9π/5.

2. -14π/3

To find an angle coterminal with -14π/3, we can add 2π repeatedly until we get an angle between 0 and 2π:

-14π/3 + 6π = -14π/3 + 18π/3 = 4π/3

The angle coterminal with -14π/3 is 4π/3.

3. 57π/2

To find an angle coterminal with 57π/2, we can subtract 2π repeatedly until we get an angle between 0 and 2π:

57π/2 - 2π = 57π/2 - 4π/2 = 53π/2

The angle coterminal with 57π/2 is 53π/2.

4. 19π/11

To find an angle coterminal with 19π/11, we can add 2π repeatedly until we get an angle between 0 and 2π:

19π/11 + 2π = 19π/11 + 22π/11 = 41π/11

The angle coterminal with 19π/11 is 41π/11.

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The compound formed between Ag and Se is Ag₂Se.

To determine the formula of the compound, we need to consider the charges of the individual ions. Ag is the symbol for silver, which commonly forms a 1+ cation (Ag⁺). Se is the symbol for selenium, which commonly forms a 2- anion (Se²⁻).

To combine the two ions in a neutral compound, we need to find the ratio that balances their charges. Since Ag has a 1+ charge and Se has a 2- charge, we need two Ag⁺ ions to balance the charge of one Se²⁻ ion.

Therefore, the formula for the compound is Ag₂Se.

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Freud is a stage theorist. Sigmund Freud, the renowned Austrian neurologist and psychoanalyst, is widely recognized as one of the pioneers in the field of psychoanalysis.

He proposed a developmental theory that included psychosexual stages of development. According to Freud, human development progresses through distinct stages, each characterized by a specific focus on different erogenous zones. These stages include the oral stage, stage, phallic stage, latency stage, and genital stage. Freud believed that the way individuals navigate these stages influences their personality and psychological well-being in adulthood.

Although Freud's stage theory has been critiqued and modified over time, his ideas regarding the importance of early childhood experiences and unconscious processes have had a profound impact on psychology and continue to shape our understanding of human development.

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Remy should expect a profit of RM81,200 from the site for the coming year.

To calculate the expected profit for the coming year, we need to consider the cost structure of the establishment. From the given information, we know that last year the total cost was RM120,000 and the profit was RM26,000. This implies that the major costs of operating the establishment are RM120,000 - RM26,000 = RM94,000.

Now, Remy expects the total revenue for the coming year to increase by 20 percent to RM175,200. To calculate the expected profit, we need to subtract the expected costs from the expected revenue.

Expected Profit = Expected Revenue - Expected Costs

Expected Costs = RM94,000 (major costs of operating the establishment)

Expected Revenue = RM175,200 (20% increase from the previous year)

Expected Profit = RM175,200 - RM94,000

Expected Profit = RM81,200

Therefore, Remy should expect a profit of RM81,200 from the site for the coming year.

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Answer: a

Step-by-step explanation:

The exact value of s in the interval [(3π)/(2), 2π] where sin(s) = -(1)/(2) is s = 11π/6.

To find the exact value of s in the interval [(3π)/(2), 2π] where sin(s) = -(1)/(2), we can use the properties of the unit circle and the trigonometric function sin.

In the interval [(3π)/(2), 2π], the angle s lies in the fourth quadrant of the unit circle. In this quadrant, the sine function is negative.

We know that sin(s) = -(1)/(2). Looking at the unit circle, we can see that there is a special angle in the fourth quadrant where sin is equal to -(1)/(2). That special angle is -π/6.

Since we are working in the interval [(3π)/(2), 2π], we need to find an angle s that is equivalent to -π/6 within this interval.

Adding 2π to -π/6 gives us the equivalent angle within the interval:

-π/6 + 2π = 11π/6

Therefore, the exact value of s in the interval [(3π)/(2), 2π] where sin(s) = -(1)/(2) is s = 11π/6.

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The density of the rectangular prism, you need to divide its mass by its volume.

1. Mass of the prism = 6.161 grams

2. Length of the prism = 1.669 cm

3. Width of the prism = 1.845 cm

4. Height of the prism = 6.907 cm

The volume of a rectangular prism is calculated by multiplying its length, width, and height:

Volume = Length * Width * Height

Volume = 1.669 cm * 1.845 cm * 6.907 cm

Volume ≈ 21.325

Now, divide the mass by the volume to obtain the density:

Density = Mass / Volume

Density = 6.161  / 21.325

Density ≈ 0.289 (rounded to three decimal places)

For the second part of your question, we need to calculate the percent abundance of zinc in the sample with the given densities.

1. Density of the sample = 7.801

2. Density of pure copper = 8.96

3. Density of pure zinc = 7.13

Since the densities are given in different units, we need to convert them to the same unit. We'll convert the density of the sample from g/mL to g/cm^3:

Density of the sample = 7.801 g/mL * (1 mL / 1 cm)

Density of the sample ≈ 7.801

Now, we can calculate the percent abundance of zinc using the densities:Percent abundance of zinc = (Density of sample - Density of copper) / (Density of zinc - Density of copper) * 100

Percent abundance of zinc = (7.801 - 8.96 ) / (7.13  - 8.96 ) * 100

Percent abundance of zinc ≈ -11.48%

The negative value indicates that the sample contains a higher Percentage of copper compared to zinc.

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The value of X in the equation  is approximately 1.982 (rounded to three decimals).

To solve for the value of X in the given equation, let's substitute the given values and solve step by step.

We have:

a = 0.912

b = 0.46

a^2 = b^2 * x^2

Let's start by substituting the given values:

0.912^2 = 0.46^2 * x^2

Squaring 0.912:

0.831744 = 0.2116 * x^2

Dividing both sides by 0.2116:

0.831744 / 0.2116 = x^2

Calculating the left-hand side:

3.928685 = x^2

To find the value of x, we need to take the square root of both sides:

x = sqrt(3.928685)

Using a calculator or software to calculate the square root:

x ≈ 1.982

Therefore, the value of X is approximately 1.982 (rounded to three decimals).

In the given equation, when we substitute the values of a = 0.912 and b = 0.46, we find that X ≈ 1.982 satisfies the equation a^2 = b^2 * X^2.

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The answer is:

Work/explanation:

Our equation is

-10 = x - 7

Flip

x - 7 = -10

Add 7 on each side

x = -10 + 7

Hence, the answer is x = -3.

The piecewise function can be described as follows:

[tex]\[f(x) = \begin{cases} -3 & \text{if } x \in (-\infty, -1) \\3 & \text{if } x \in (-1, \infty) \\\end{cases}\][/tex]

The given piecewise function consists of three different intervals. In the first interval, for all values of \(x\) that belong to the open interval \((-∞, -1)\), the function \(f(x)\) takes a value of -3. This can be denoted as \((-∞, -1) \rightarrow -3\).

In the second interval, for all values of \(x\) that belong to the open interval \((-1, ∞)\), the function \(f(x)\) takes a value of 3. This can be expressed as \((-1, ∞) \rightarrow 3\).

It's important to note that the intervals are represented using interval notation. In interval notation, parentheses indicate that the endpoint is not included, and the arrow pointing to the corresponding function value shows the relationship between the interval and the function value.

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The amount to which $800 will grow under each of the given conditions increases as the compounding period decreases.

The amount to which $800 will grow under each of these conditions is as follows:a) 8% compounded annually for 9 years

When compounded annually for 9 years at 8%, the formula is: Amount = Principal x [(1 + rate/n)^(n*t)]

Where: Principal = $800 Rate = 8% Time = 9 years

Compounded annually = n = 1 Amount = $1,447.91 (rounded to the nearest cent)

b) 8% compounded semiannually for 9 years Compounded semiannually for 9 years at 8%, the formula is:

Amount = Principal x [(1 + rate/n)^(n*t)]

Where: Principal = $800 Rate = 8% Time = 9 years Compounded semiannually = n = 2 Amount = $1,471.16 (rounded to the nearest cent)

c)  8% compounded quarterly for 9 years Compounded quarterly for 9 years at 8%, the formula is:

Amount = Principal x [(1 + rate/n)^(n*t)]

Where: Principal = $800 Rate = 8% Time = 9 years Compounded quarterly = n = 4 Amount = $1,491.03 (rounded to the nearest cent)

d) 8% compounded monthly for 9 years Compounded monthly for 9 years at 8%, the formula is: Amount = Principal x [(1 + rate/n)^(n*t)]

Where: Principal = $800 Rate = 8% Time = 9 years Compounded monthly = n = 12 Amount = $1,505.91 (rounded to the nearest cent)

e) 8% compounded daily for 9 years Compounded daily for 9 years at 8%, the formula is:

Amount = Principal x [(1 + rate/n)^(n*t)]

Where: Principal = $800 Rate = 8% Time = 9 years Compounded daily = n = 365Amount = $1,511.74 (rounded to the nearest cent)

The observed pattern of FVs (future values) occurs due to compounding. Compounding is the process of earning interest not only on the principal amount invested but also on the interest earned from the principal. This results in an increase in the interest earned and the future value of the investment. The more frequent the compounding, the higher the future value of the investment. Hence, the amount to which $800 will grow under each of the given conditions increases as the compounding period decreases.

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The following measurement to be rewritten with a simpler unit if possible, 1.8 m⋅s² kg⋅m²⟹ This unit is in Joules (J).

Using the formula for kinetic energy,K = 1/2 m v²where K is the kinetic energy, m is the mass and v is the velocity of the object. It can be seen that K can be expressed in terms of the mass and velocity squared only.

Now, the given measurement is in m⋅s² kg⋅m² which when simplified, becomes:

K = (1.8 m/s²) (2 kg) (1 m²)

K = 3.6 Joules (J)

Therefore, the simplest unit of measurement is Joules (J).

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The linear equation that passes through the points (2, -2) and (0, -1) is y = -1/2x - 1.

The two points are (2, −2) and (0, −1), we will use the point-slope form to write the equation of a line through these points.

Point-slope form of a linear equation is given asy − y1 = m(x − x1)

where (x1, y1) is any point on the line and m is the slope of the line.

Let us find the slope of the line through the given two points.

The slope m is given asm = (y2 − y1) / (x2 − x1)

Substituting the given values, we getm = (-1 - (-2)) / (0 - 2) = 1 / 2

So, the slope of the line is 1 / 2.

Using the coordinates of the given points (2, -2) and (0, -1):

m = (-1 - (-2)) / (0 - 2)

= (1) / (-2)

= -1/2

Now that we have the slope, let it be one of the points You can find the y-intercept (b) by substituting in the slope-intercept form with Let's use point (2, -2):

-2 = (-1/2)(2) + b

Simplification:

-2 = -1 + b

add 1 to both sides

-2 + 1 = b

b = -1

Now that we know the slope (m = -1/2) and the y-intercept (b = -1) we can write the equation .

y = -1/2x - 1

Let us choose the point (2, −2) to write the equation of the line.

y − y1 = m(x − x1)y − (−2)

= (1 / 2)(x − 2)y + 2

= (1 / 2)x − 1y

= (1 / 2)x − 3

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The equation of the line that passes through point (2, - 2) and (0, - 1) is equal to y = (- 1 / 2) · x - 1.

In this question we must derive the equation of a line that passes through points (2, - 2) and (0, - 1). Lines are defined by equations of the form:

y = m · x + b

m = Δy / Δ x

Where:

First, determine the slope of the line:

m = [- 1 - (- 2)] / (0 - 2)

m = - 1 / 2

Second, find the intercept:

b = y - m · x

b = - 1 - (- 1 / 2) · 0

b = - 1

Third, write the resulting equation of the line:

y = (- 1 / 2) · x - 1

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The line passing through points (5, 8) and (-2, 8) has a slope of 0, indicating that it is a horizontal line parallel to the x-axis.

To find the slope (m) of the line passing through the points (5, 8) and (-2, 8), we can use the slope formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the coordinates:

x₁ = 5, y₁ = 8

x₂ = -2, y₂ = 8

m = (8 - 8) / (-2 - 5)

m = 0 / -7

m = 0

Therefore, the slope (m) of the line passing through the given points is 0.

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The equation sec²(x) - 2tan²(x) = -2 can be simplified to tan(x) = ±√3. The solutions in the interval [0,2π) are x = π/3, 2π/3, 4π/3, and 5π/3.

To solve the equation sec²(x) - 2tan²(x) = -2, we can rewrite it using trigonometric identities. Recall that sec²(x) = 1 + tan²(x).

Replacing sec²(x) with its equivalent expression, we have:

1 + tan²(x) - 2tan²(x) = -2

Combining like terms, we get:

1 - tan²(x) = -2

Rearranging the equation, we have:

tan²(x) = 3

Taking the square root of both sides, we get:

tan(x) = ±√3

To find the solutions in the interval [0,2π), we can use the unit circle or a calculator. The values of x that satisfy tan(x) = √3 are x = π/3 and x = 4π/3. The values of x that satisfy tan(x) = -√3 are x = 2π/3 and x = 5π/3.

Therefore, the solutions in the interval [0,2π) are x = π/3, 2π/3, 4π/3, and 5π/3.

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(a) The total cost of producing 10 units is $10,201.

(b) The value of C(100) is $92,200.

(c) The meaning of C(100) is that it represents the total cost (in dollars) of producing 100 units, which is $92,200.

(a) To find the total cost of producing 10 units, we substitute x = 10 into the cost function C(x) = 900x + 0.1x^2 + 1200:

C(10) = 900(10) + 0.1(10)^2 + 1200

= 9000 + 1 + 1200

= 10201 dollars.

Therefore, the total cost of producing 10 units is $10201.

(b) To find the value of C(100), we substitute x = 100 into the cost function:

C(100) = 900(100) + 0.1(100)^2 + 1200

= 90000 + 1000 + 1200

= 92200 dollars.

Therefore, C(100) = 92200.

(c) The meaning of C(100) is the total cost (in dollars) of producing 100 units. In this case, it costs $92200 to produce 100 units.

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From the following distances, the point closest to (3, -5) is (−2, −4) from the given options.

To determine which of the given points is closest to the point (3, -5), we can calculate the distance between each point and (3, -5) using the distance formula. The point with the smallest distance will be the closest.

Distance formula:

The distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Calculating the distances:

a) Distance between (3, -5) and (0, 0):

d = √((0 - 3)^2 + (0 - (-5))^2)

= √(9 + 25)

= √34

≈ 5.83

b) Distance between (3, -5) and (-2, -4):

d = √((-2 - 3)^2 + (-4 - (-5))^2)

= √(25 + 1)

= √26

≈ 5.10

c) Distance between (3, -5) and (3, 2):

d = √((3 - 3)^2 + (2 - (-5))^2)

= √(0 + 49)

= 7

d) Distance between (3, -5) and (-1, 1):

d = √((-1 - 3)^2 + (1 - (-5))^2)

= √(16 + 36)

= √52

≈ 7.21

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The formula for the cost of a European call option can be derived in five lines as follows:

Start with the formula for the call option value: C = S(0)Φ(ω) - Ke^(-rt)Φ(ω - σ√t)

This formula represents the call option value (C) as the difference between two terms.

Use the notation S(0) to represent the current stock price at time 0.

S(0) is the starting price of the underlying asset (stock) at the beginning of the option contract.

Use Φ(ω) to represent the cumulative standard normal distribution of the random variable ω.

Φ(ω) represents the probability that the underlying asset price will be above the strike price (K) at expiration.

Use K to represent the strike price of the option.

K is the predetermined price at which the option holder can buy the underlying asset.

Use e^(-rt)Φ(ω - σ√t) to represent the present value of the expected payoff at expiration.

e^(-rt) is the present value factor that discounts the future payoff to its present value.

Φ(ω - σ√t) represents the probability that the option will be exercised based on the difference between the expected asset price and the strike price.

In summary, the formula C = S(0)Φ(ω) - Ke^(-rt)Φ(ω - σ√t) is derived to calculate the cost of a European call option. The formula combines the current stock price, strike price, time to expiration, risk-free interest rate, and volatility to estimate the value of the call option at a particular point in time.

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The cylindrical tank holds approximately 108.1 gallons of water.

To find the volume of the cylindrical tank, we can use the formula

Volume of a cylinder (V) = πr^2h.

Given that the diameter of the tank is 18.4 inches, we can find the radius by dividing the diameter by 2: r = 18.4 / 2 = 9.2 inches. The height of the tank is given as 31.9 inches. Plugging these values into the formula, we have

V = 3.14 * 9.2^2 * 31.9. Evaluating this expression, we find,

V ≈ 9727.41 cubic inches.

To convert this volume to gallons, we use the conversion factor 1 gallon = 231 cubic inches. Dividing the volume in cubic inches by the conversion factor, we have 9727.41 / 231 ≈ 42.1 gallons. Rounding to the nearest tenth of a gallon, the tank holds approximately 108.1 gallons of water.

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The model for the country's medical marijuana revenue in billions of dollars, with x as the number of years after 2017, is given by the equation R(x) = 0.037x^2 + 0.06x + 1.688.

To write the model R(x) with x equal to the number of years after 2017, we need to adjust the equation to account for the shift in the starting year. Since the original equation models the revenue with x as the number of years after 2009, we need to convert it to the number of years after 2017.

Given that 2017 is 8 years after 2009, we can substitute (x - 8) with (x - (2017 - 2009)) to align the equation with the number of years after 2017.

The adjusted model R(x) is:

R(x) = 0.037(x - (2017 - 2009))^2 + 0.652(x - (2017 - 2009)) + 4.536

Simplifying further:

R(x) = 0.037(x - 8)^2 + 0.652(x - 8) + 4.536

Expanding the squared term:

R(x) = 0.037(x^2 - 16x + 64) + 0.652(x - 8) + 4.536

Distributing and simplifying:

R(x) = 0.037x^2 - 0.592x + 2.368 + 0.652x - 5.216 + 4.536

Combining like terms:

R(x) = 0.037x^2 + 0.06x + 1.688

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We know that in a spherical triangle, the sides are arcs of great circles, and the angles are angles between these arcs. To prove the given formulas using the formulas of spherical trigonometry, let's start with (a):

(a) sina = sinαsinc

. In triangle ABC, since angle C is a right angle, angle α is opposite side BC and angle a is opposite side AC. Using the Law of Sines, we have sina/sinA = sinα/sinC.

Since angle C is a right angle, sinC = 1. Therefore, sina/sinA = sinα. Rearranging, we get sina = sinαsinc.

Similarly, we can prove (b) tana = tanαsinb,

(c) tana = cosβtanc,

(d) cosc = cosacosb, (e) cosα = sinβcosa,

(f) sinb = sinβsinc, (g) tanb = tanβsina, (h) tanb = cosαtanc,

(i) cosc = cotαcotβ, and (j) cosβ = sinαcosb using the formulas of spherical trigonometry.
The given formulas have been proved using the formulas of spherical trigonometry.

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The difference quotient is a mathematical expression that represents the average rate of change of a function over a given interval. To find the difference quotient, we need to find the value of the function at two different points within the interval and calculate the slope of the secant line connecting these two points.

Let's say we have a function f(x) and we want to find the difference quotient at a point x=a. The formula for the difference quotient is:

[f(a+h) - f(a)] / h

where h is a small change in the x-value.

To simplify the answer, we can expand the numerator and combine like terms. After simplifying, we can cancel out the h in the denominator.

For example, if f(x) = 3x^2, and we want to find the difference quotient at x=2, we substitute the values into the formula:

[f(2+h) - f(2)] / h

= [(3(2+h)^2) - (3(2)^2)] / h

= [(3(4+4h+h^2)) - 12] / h

= [12 + 12h + 3h^2 - 12] / h

= (12h + 3h^2) / h

= 12 + 3h

Therefore, the difference quotient for the given function at x=2 is 12 + 3h.

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Green's Theorem: Green's Theorem states that the line integral around a simple closed curve C is the same as the double integral over the plane region D bounded by C.

Circulation: It is the integral of the tangential component of the vector field around the curve. It gives a measure of the amount of rotation around the curve.

Outward Flux: It is the flux flowing out of a closed curve C. It gives the amount of flow from a vector field through the surface of C. The vector field F=(5x−7y)i+(9y−7x)j.

To use Green's Theorem, we need to first calculate the partial derivatives of the vector field F:∂Q/∂x = -7∂P/∂y = 5∂P/∂x = 5∂Q/∂y = 9

Therefore, the circulation of F around C is equal to the line integral of F around the boundary of the square.

Since C is a square with sides of length 4, we can compute the circulation as follows: Circulation = ∫CF · dr = ∫C (5x - 7y) dx + (9y - 7x) dy= ∫_0^4 (5x-0) dx + ∫_0^4 (9y-4) dy + ∫_4^0 (5x-4) dx + ∫_4^0 (9y-4) dy= 40 + 32 - 40 + 32= 64.

The outward flux of F through C is equal to the double integral of the curl of F over the interior of the square. Since C is a square with sides of length 4, we can compute the outward flux as follows: Outward Flux = ∫∫_R ( ∂Q/∂x - ∂P/∂y ) dA= ∫∫_R (9 - 5) dA= 4 ∫_0^4 ∫_0^4 4dxdy= 64

Therefore, the counterclockwise circulation of F around C is 64 and the outward flux of F through C is also 64.

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