Using a 2-year moving average, the forecast for year 6=3825 miles (round your response to the nearest whole number). b) If a 2-year moving average is used to make the forecast, the MAD based on this = miles (round your response to one decimal place). (Hint: You will have only 3 years of matched data.) c) The forecast for year 6 using a weighted 2-year moving average with weights of 0.45 and 0.55 (the weight of 0.55 is for the most recent period) = whole number).

The direction of the balloon's resultant vector is approximately 53.13°. Therefore, the angle is θ ≈ 53.13°

To determine the direction of the balloon's resultant vector, we can use trigonometry to find the angle between the resultant vector and the east direction.

First, let's draw a vector diagram to represent the displacement of the balloon. Start with a reference point, and from there, draw a line 18.5 kilometers east and then a line 24.6 kilometers north. Connect the starting point to the endpoint of the northward displacement.

Now, we have a right triangle formed by the eastward displacement, northward displacement, and the resultant vector. The angle between the east direction and the resultant vector is the angle we need to find.

Applying trigonometry, we can use the inverse tangent function to find this angle. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle.

Let's denote the angle we want to find as θ. We can use the tangent of θ:

tan(θ) = (opposite side) / (adjacent side)

In this case, the opposite side is the northward displacement of 24.6 kilometers, and the adjacent side is the eastward displacement of 18.5 kilometers.

tan(θ) = 24.6 / 18.5

Using a calculator, we can find the approximate value of θ:

θ ≈ 53.13°

Rounding to the nearest hundredth, the direction of the balloon's resultant vector is approximately 53.13°.

Therefore, the angle is θ ≈ 53.13°.

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1) The radical form is,

[tex]w^{- 5/8} =\sqrt[8] (\frac{1}{w^{5} })[/tex]

2) The radical form is , [tex]w^{1/5} = \sqrt[5]{w}[/tex]

We have,

The expressions are w⁻⁵/⁸ and w⁰.² .

Now, To write an expression with a negative exponent as a radical, we can use the following rule:

a⁻ⁿ = 1/aⁿ

So, we can rewrite w⁻⁵/⁸ as:

[tex]w^{- 5/8} = \frac{1}{w^{5/8} }[/tex]

To write this in radical form, we can convert the exponent to a root:

[tex]w^{- 5/8} = (\frac{1}{w^{5} })^{1/8}[/tex]

Therefore, It can be written as:

[tex]w^{- 5/8} =\sqrt[8] (\frac{1}{w^{5} })[/tex]

So, The radical form is,

[tex]w^{- 5/8} =\sqrt[8] (\frac{1}{w^{5} })[/tex]

Now let's move on to the expression w⁰.²:

To write an expression with a fractional exponent as a radical, we can use the following rule:

[tex]a^{m/n} = (nth root of a )^m[/tex]

So, we can rewrite as:

[tex]w^{1/5} = \sqrt[5]{w}[/tex]

Therefore, the radical form is , [tex]w^{1/5} = \sqrt[5]{w}[/tex]

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Answer:

Step-by-step explanation:

To determine the number of integers 'n' between 11 and 5050, inclusive, for which the expression (n^2 - 1)! / (n!^n) is an integer, we can analyze the prime factors of the given expression.

Let's consider the prime factorization of the expression:

(n^2 - 1)! = (n - 1)! * n! * (n + 1)! * ... * (n^2 - 1)!

Since we have n! in the denominator, we need to make sure that all the prime factors in n! are canceled out by the prime factors in (n^2 - 1)!. This will ensure that the expression is an integer.

For any integer 'n' greater than or equal to 4, the prime factorization of n! will contain at least one instance of a prime number greater than n. This means that the prime factors in n! cannot be fully canceled out by the prime factors in (n^2 - 1)!, resulting in a non-integer value for the expression.

Therefore, we need to check the values of 'n' from 11 to 5050 individually to find the integers for which the expression is an integer.

Upon checking the values, we find that the integers for which the expression is an integer are n = 11, 12, 13, ..., 5050. There are a total of 5040 integers in this range that satisfy the given condition.

Hence, there are 5040 integers 'n' between 11 and 5050, inclusive, for which the expression (n^2 - 1)! / (n!^n) is an integer.

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The given function f(x) = 6x + |−8x| is neither even nor odd.

To determine whether the given function f(x) = 6x + |−8x| is even, odd, or neither, we need to analyze its algebraic properties.

Even function:

A function f(x) is even if f(x) = f(-x) for all x in the domain of f.

Let's check if f(x) = f(-x) for the given function:

f(-x) = 6(-x) + |−8(-x)| = -6x + |8x|

Since f(x) = 6x + |−8x| and f(-x) = -6x + |8x|, we can see that the function is not equal to its reflection across the y-axis.

Odd function:

A function f(x) is odd if f(x) = -f(-x) for all x in the domain of f.

Let's check if f(x) = -f(-x) for the given function:

-f(-x) = -(6(-x) + |−8(-x)|) = -(-6x + |8x|) = 6x - |8x|

Since f(x) = 6x + |−8x| and -f(-x) = 6x - |8x|, we can see that the function is not equal to the negation of its reflection across the y-axis.

Therefore, the given function f(x) = 6x + |−8x| is neither even nor odd.

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The cone has a diameter of 14 inch and a volume and height of 196[tex]\pi[/tex]cubic inches and 12 inches, respectively.

The formula for a cone's volume can be used to get its diameter which is as follows:

[tex]V = (1/3)\pi r^2h[/tex]

V is the volume, r is the radius, and h is the height.

In this particular case, we are informed that the height is 12 inches and the capacity is 196 cubic inches. These values can be substituted in the formula:

[tex]196\pi = (1/3)\pi r^2(12)[/tex]

To simplify the problem, we can multiply both sides by 3 and divide both sides by π:

[tex]588 = r^2(12).[/tex]

Next, we can isolate [tex]r^2[/tex] by dividing both sides by 12: 

[tex]49 = r^2[/tex]

By taking the square root of both, we can get the radius.
[tex]r = \sqrt{49[/tex]
r = 7

We know that,

The diameter is twice the radius, So the diameter is:
d = 2r = 2(7) = 14 inches

Therefore, the diameter of the cone is 14 inches.

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The ellipse's equation with the center at the origin, given the vertex (-6,0) and co-vertex (0,5), is $\frac{x^2}{36} + \frac{y^2}{25} = 1$.

The center of the ellipse is at the origin (0,0) since it has a center at the origin.

The distance from the center to the vertex along the x-axis is denoted as "a," which is 6 units in this case (-6 to 0). The distance from the center to the co-vertex along the y-axis is denoted as "b,"

which is 5 units in this case (0 to 5). These values are used to determine the coefficients in the equation.

Since the center is at the origin, the equation simplifies to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Plugging in the given values, we get $\frac{x^2}{36} + \frac{y^2}{25} = 1$, which represents the ellipse.

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(a)the body temperature of the 1601 lb man will increase by approximately 3.0 °C.(b)the body temperature of the 160 lb man will increase by approximately 2.4 °C.

The specific heat of a human is given as 3.47 J/°C. Using this information, we can calculate the increase in body temperature when a certain number of calories are converted into heat energy. In the first scenario, a 1601 lb man consumes a candy bar containing 287 Cal. In the second scenario, a 160 lb man consumes a roll of candy containing 41.9 Cal. We will calculate the increase in body temperature for each case.

(a) To calculate the increase in body temperature for a 1601 lb man who consumes a candy bar containing 287 Cal, we need to convert calories to joules. Since 1 Calorie (Cal) is equal to 4184 joules, we have:
Energy = 287 Cal × 4184 J/Cal = 1.2 × [tex]10^6[/tex] J
Now, using the specific heat formula Q = mcΔT, where Q is the energy, m is the mass, c is the specific heat, and ΔT is the change in temperature, we can rearrange the formula to solve for ΔT:
ΔT = Q / (mc)
Assuming the mass of the man is converted to kilograms, we have:
ΔT = (1.2 × [tex]10^6[/tex] J) / (1601 lb × 0.4536 kg/lb × 3.47 J/°C) ≈ 3.0 °C
Therefore, the body temperature of the 1601 lb man will increase by approximately 3.0 °C.
(b) For a 160 lb man who consumes a roll of candy containing 41.9 Cal, we repeat the same calculation:
Energy = 41.9 Cal × 4184 J/Cal = 1.75 × [tex]10^5[/tex] J
ΔT = (1.75 × [tex]10^5[/tex] J) / (160 lb × 0.4536 kg/lb × 3.47 J/°C) ≈ 2.4 °C
Thus, the body temperature of the 160 lb man will increase by approximately 2.4 °C.

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You will have saved approximately $192,739 by the end of 17 years. The investment is worth approximately $72,123 today. You will earn approximately $136,000 in interest over the 17-year period.

1. To calculate the savings accumulated over 17 years, we can use the formula for the future value of an annuity:

FV = PMT * [(1 + r)^n - 1] / r

Where:

FV = Future value (unknown)

PMT = Annual savings ($9,000)

r = Annual interest rate (2.0% or 0.02)

n = Number of years (17)

Substituting the given values into the formula:

FV ≈ $9,000 * [(1 + 0.02)^17 - 1] / 0.02

FV ≈ $192,739

Therefore, you will have saved approximately $192,739 by the end of 17 years.

3. To calculate the present value of the investment, we can use the formula for the present value of an annuity:

PV = PMT * [(1 - (1 + r)^(-n)) / r]

Where:

PV = Present value (unknown)

PMT = Quarterly payment ($3,000)

r = Quarterly discount rate (7% or 0.07/4)

n = Number of quarters (15 * 4)

Substituting the given values into the formula:

PV ≈ $3,000 * [(1 - (1 + 0.07/4)^(-60)) / (0.07/4)]

PV ≈ $72,123

Therefore, the investment is worth approximately $72,123 today.

4. To calculate the total interest earned over 17 years, we can multiply the annual deposit by the number of years and subtract the total amount deposited:

Total interest = (Annual deposit * Number of years) - Total amount deposited

Total interest = ($8,000 * 17) - ($8,000 * 17)

Total interest = $136,000

Therefore, you will earn approximately $136,000 in interest over the 17-year period.

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The expression that represents sin (1/60)° is (c) sin (30°/60°). Sine is a periodic function, which means that it repeats itself every 360°. So, sin (1/60)° is the same as sin (360°/60°) = sin 6°.

We can also write sin 6° as sin (30°/60°). This is because sin 6° is the sine of an angle that is 6° less than 30°. In other words, the terminal side of the angle that measures sin 6° is the same as the terminal side of the angle that measures 30°, but rotated 6° counterclockwise.

Therefore, the expression that represents sin (1/60)° is (c) sin (30°/60°).

Angle A measures 30°.

Angle B measures 6°.

The terminal sides of Angle A and Angle B are the same.

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In year 1, the GDP price deflator is calculated to be 0.66 (or 66%), and in year 2, it is 0.77 (or 77%). The rate of inflation between years 1 and 2, as measured by the GDP price deflator, is approximately 16.67%. In contrast, the CPI rate of inflation is calculated to be 20%. The differences in these results can be attributed to the differences in the composition and weighting of the goods included in the GDP price deflator and the Consumer Price Index (CPI).

a) The GDP price deflator measures the average price change of all goods and services produced in an economy. To calculate the GDP price deflator in year 1, we use the formula: (Nominal GDP / Real GDP) * 100. Given the quantities and prices of broccoli and cauliflower in year 1, the nominal GDP is (100 million * $0.50) + (30 million * $0.80) = $65 million, and the real GDP is (100 million * $0.50) + (30 million * $0.50) = $55 million. Thus, the GDP price deflator in year 1 is (65/55) * 100 = 118.18%. In year 2, the nominal GDP is (80 million * $0.60) + (60 million * $0.85) = $88 million, and the real GDP is (80 million * $0.50) + (60 million * $0.50) = $70 million. Therefore, the GDP price deflator in year 2 is (88/70) * 100 = 125.71%. The rate of inflation between years 1 and 2, as measured by the GDP price deflator, is ((125.71 - 118.18) / 118.18) * 100 = 6.36%.

b) The Consumer Price Index (CPI) measures the average price change of a basket of goods and services typically consumed by households. To calculate the CPI in year 1, we assign weights to the prices of broccoli and cauliflower based on their consumption quantities. The CPI in year 1 is (100 million * $0.50) + (30 million * $0.80) = $65 million. In year 2, the CPI is (80 million * $0.60) + (60 million * $0.85) = $81 million. The CPI rate of inflation between years 1 and 2 is ((81 - 65) / 65) * 100 = 24.62%.

c) The differences in the results between parts (a) and (b) can be attributed to the differences in the composition and weighting of goods included in the GDP price deflator and the CPI. The GDP price deflator considers the prices of all goods and services produced in the economy, reflecting changes in production patterns and the overall price level. On the other hand, the CPI focuses on a fixed basket of goods and services consumed by households, reflecting changes in the cost of living. The differences in the weighting and composition of goods between the two measures result in variations in the calculated inflation rates.

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Mean = 15

Median = 15

Mode = 18

To find the mean, median, and mode of the given set of values: 8, 9, 11, 12, 13, 15, 16, 18, 18, 18, 27.

Mean:

The mean is calculated by summing up all the values in the set and dividing by the total number of values.

Sum of the values = 8 + 9 + 11 + 12 + 13 + 15 + 16 + 18 + 18 + 18 + 27 = 165

Total number of values = 11

Mean = Sum of values / Total number of values = 165 / 11 = 15

Therefore, the mean of the given set is 15.

Median:

The median is the middle value in a sorted list of numbers. To find the median, we need to arrange the values in ascending order first.

Arranged in ascending order: 8, 9, 11, 12, 13, 15, 16, 18, 18, 18, 27

Since there are 11 values, the middle value is at position (n + 1) / 2 = (11 + 1) / 2 = 6th position.

Thus, the median of the given set is 15.

Mode:

The mode is the value that appears most frequently in the set.

In the given set, the value 18 appears three times, more than any other value. Therefore, the mode of the set is 18.

To summarize:

Mean = 15

Median = 15

Mode = 18

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AD is parallel to BC and AB is parallel to DC.

AD is parallel to BC and AB is parallel to DC, we need to demonstrate that the slopes of the corresponding sides are equal.

Given the coordinates of the square's vertices, A(2, -4) and C(10, 4), we can determine the slope of the line passing through these points using the slope formula:

slope = (change in y) / (change in x)

For the line passing through A and C, the slope is:

slopeAC = (4 - (-4)) / (10 - 2) = 8 / 8 = 1

Similarly, we can find the slopes for the other sides of the square:

For the line passing through A and B:

slopeAB = (-4 - (-4)) / (2 - 10) = 0 / (-8) = 0

For the line passing through D and C:

slopeDC = (4 - 4) / (10 - 2) = 0 / 8 = 0

We can see that the slope of AD (0) is equal to the slope of BC (0), and the slope of AB (0) is equal to the slope of DC (0). When two lines have equal slopes, they are parallel.

Therefore, we have shown that AD is parallel to BC and AB is parallel to DC in the square ABCD.

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Each side's length of the polygon will change by √3 times.

Here we do not know whether the polygon is a regular or an irregular one.

Hence we get the formula for the area of a polygon to be

Area = a² X n X cot(180/n)/4

where a = length of each side

n = no. of sides

Here Area is given by 144 m²

Hence we get

a²ncot(180/n)/4 = 144

or, a²ncot(180/n) = 144 X 4 = 576

[tex]or, a^2 = \frac{576}{ncot(180/n)}[/tex]

Now if area is tripled we get the polygon with the new side A to be

A²ncot(180/n) = 576 X 3

[tex]or, A^2 = 3 \frac{576}{ncot(180/n)}[/tex]

or, A² = 3a²

or A = √3 a

Hence each side's length will change by √3 times.

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The part of the marbles which are blue is  41.7%.

We are given that;

The number of red marbles=3

The number of white marbles=4

The number of blue marbles=5

Now,

To find the part of the marbles that are blue,

we need to find the total number of marbles and the number of blue marbles.

The total number of marbles is:

3 + 4 + 5 = 12

The number of blue marbles is:

5

So the part of the marbles that are blue is:

5/12

Therefore, by probability the answer will be  41.7%

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(a) The OLS (Ordinary Least Squares) objective function is given by:

minimize: Σ(i=1 to N) ϵ

i

²

where ϵ

i

represents the residuals or errors, and N is the number of observations. The objective of the OLS estimator is to minimize the sum of squared residuals.

b) E(a) = α

c) The OLS estimator is said to be BLUE, which stands for Best Linear Unbiased Estimator.

d) When we say that shocks (ϵ

i

) are heteroskedastic, it means that the error terms or residuals in the linear model have a non-constant variance across the observations

e) In the case of heteroskedasticity, the OLS standard errors, which are used to estimate the precision of the coefficient estimates, become inefficient

(b) When we say that the estimator a is unbiased for α, it means that on average, the estimate of a will be equal to the true value of α. In other words, the expected value of the estimator a is equal to the true value of α. Mathematically, it can be represented as:

E(a) = α

(c) The OLS estimator is said to be BLUE, which stands for Best Linear Unbiased Estimator. This means that among all the linear unbiased estimators, the OLS estimator has the smallest variance. In other words, the OLS estimator is efficient and provides the minimum variance among all unbiased estimators in the class of linear estimators.

(d) When we say that shocks (ϵ

i

) are heteroskedastic, it means that the error terms or residuals in the linear model have a non-constant variance across the observations. In other words, the variability of the errors is not the same for all values of the independent variable(s). This violates the assumption of homoscedasticity.

In the case of heteroskedasticity, the OLS standard errors, which are used to estimate the precision of the coefficient estimates, become inefficient. The standard errors estimated under the assumption of homoscedasticity will be biased and inconsistent. Therefore, to obtain valid standard errors in the presence of heteroskedasticity, it is necessary to employ robust standard errors or use other estimation techniques that account for heteroskedasticity, such as weighted least squares or heteroskedasticity-consistent standard errors.

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The expression in factored form will be (x + 4)(x + 10) .

Given,

x²+14 x+40

Now,

To obtain the factored form of the quadratic equation .

Factorize the quadratic expression ,

x²+14 x+40 = 0

Factorizing,

x² + 10x + 4x + 40 = 0

x(x + 10) + 4(x + 10) = 0

Factored form :

(x + 4)(x + 10) = 0

Thus the values of x ,

x+4 = 0

x = -4

x+ 10 = 0

x = -10

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The triangle inequality is not satisfied. It is not possible for the third side of the reflective sticker to be 2 cm long.

(a) To determine if the third side of the reflective sticker could be 12 cm long, we can apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check if the triangle inequality holds for the given side lengths:

6 cm + 8 cm > 12 cm

14 cm > 12 cm

Since the sum of the two given side lengths (6 cm and 8 cm) is greater than the potential third side length (12 cm), the triangle inequality is satisfied. Therefore, it is possible for the third side of the reflective sticker to be 12 cm long.

To draw the triangle, start by drawing a line segment of length 6 cm. From one endpoint of the 6 cm segment, draw another line segment of length 8 cm. Finally, connect the other endpoints of the two line segments with a line segment of length 12 cm. This will form the triangle with side lengths of 6 cm, 8 cm, and 12 cm.

(b) To determine if the third side of the reflective sticker could be 2 cm long, we again apply the triangle inequality theorem.

Let's check if the triangle inequality holds for the given side lengths:

6 cm + 8 cm > 2 cm

14 cm > 2 cm

In this case, the sum of the two given side lengths (6 cm and 8 cm) is not greater than the potential third side length (2 cm).

Hence, we do not need to draw a triangle for the case where the third side is 2 cm long, as it does not form a valid triangle.

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The values are approximately:

cos(40°) ≈ 0.766

cos(400°) ≈ -0.766

cos(-320°) ≈ -0.766

Certainly! In trigonometry, the cosine function (cos) calculates the ratio of the adjacent side to the hypotenuse of a right triangle. The values obtained from the calculator represent the cosine values for the given angles.

For the angle 40°, the cosine value is approximately 0.7660444431. This means that the adjacent side of a right triangle is approximately 0.766 times the length of the hypotenuse.

For the angle 400°, we can use the concept of periodicity in trigonometric functions. Since the cosine function repeats every 360°, an angle of 400° is equivalent to an angle of 40°. Therefore, the cosine value is approximately the same, -0.7660444431, as it was for 40°.

For the angle -320°, negative angles are obtained by rotating clockwise instead of counterclockwise. In this case, we can use the fact that the cosine function is an even function, which means that cos(-θ) = cos(θ). So the cosine value for -320° is the same as the cosine value for 320°, which is approximately -0.7660444431.

To summarize, the cosine values for the given angles are approximately 0.766 for both 40° and -320°, and approximately -0.766 for 400°.

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The solutions to the equation, 2x²-1=5x are approximately x ≈2.68 and x ≈ -0.18

The given quadratic equation is,

2x²- 1 =  5x

To solve this equation bring all the terms to one side, so we get:

2x² - 5x - 1 = 0

Now we can use the quadratic formula to find the solutions for x:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 2, b = -5, and c = -1, so we get:

x = (-(-5) ± √((-5)² - 4(2)(-1))) / 2(2)

x = (5 ± √(33)) / 4

x = (5 ± 5.74) / 4

Rounding to the nearest hundredth, we get:

x ≈2.68 and x ≈ -0.18

Hence,

The solutions to the equation are approximately x ≈2.68 and x ≈ -0.18

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The formula for the surface area of a right circular cylinder is S = 2πr + 2πr². To solve for h, we can divide both sides of the equation by 2πr, which gives us h = S/2πr².

The surface area of a right circular cylinder is the total area of the top and the two bases, plus the lateral surface area. The lateral surface area is the curved surface area, and it is equal to 2πrh, where r is the radius of the base and h is the height of the cylinder.

The total surface area of the cylinder is therefore S = 2πr² + 2πrh. We can solve for h by dividing both sides of this equation by 2πr, which gives us h = S/2πr².

Here is a step-by-step solution:

Start with the formula for the surface area of a right circular cylinder: S = 2πr + 2πr².

Divide both sides of the equation by 2πr: h = S/2πr².

The answer is (B).

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Let's consider a two-digit number where the value of the red digit is ten times greater than the value of the green digit. We can represent this number as "10r + g," where r represents the value of the red digit and g represents the value of the green digit.

To find the specific number that satisfies this condition, we need to find a pair of digits that meets the given criteria. Since the value of the red digit is ten times greater than the green digit, we have the equation r = 10g By substituting this equation into our representation of the number, we get the number as 10(10g) + g, which simplifies to 100g + g. Therefore, the number can be written as 101g. From this, we can conclude that any two-digit number where the red digit is ten times greater than the green digit can be represented as 101g, where g can be any digit from 0 to 9.

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The domain of the composite function f(g(x)) is (-∞, -3) ∪ (-3, ∞).

To find the domain of f(g(x)), we need to consider two conditions: the domain of g(x) and the restriction imposed by f(x).

First, let's determine the domain of g(x). The expression 2x + 6 appears in the denominator of g(x), so we must ensure that it is not equal to zero. Solving 2x + 6 ≠ 0, we find x ≠ -3. Therefore, the domain of g(x) is (-∞, -3) ∪ (-3, ∞), excluding x = -3.

Next, we consider the restriction imposed by f(x). The expression 4 - x appears in the denominator of f(x). To avoid division by zero, we need to ensure that 4 - x ≠ 0. Solving 4 - x ≠ 0, we find x ≠ 4. Hence, f(x) is defined for all real numbers except x = 4.

Now, for the composition f(g(x)), we need to find the values of x for which both conditions are satisfied. Combining the domains of g(x) and f(x), we take the intersection of (-∞, -3) ∪ (-3, ∞) and (-∞, 4) ∪ (4, ∞). The overlapping interval is (-∞, -3) ∪ (-3, 4) ∪ (4, ∞).

In interval notation, the domain of f(g(x)) is (-∞, -3) ∪ (-3, 4) ∪ (4, ∞). This means that any real number x within this interval will yield a valid output for the composite function.

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The simplified expression is Cos² θ.

Given that is a trigonometric expression, sinθ·cosθ/tanθ, we need to simplify it,

So,

sinθ·cosθ/tanθ

We know tanθ = Sin θ / Cos θ, put the value in the expression,

= [Sin θ · Cos θ] / [Sin θ / Cos θ]

= [Sin θ · Cos θ] × [Cos θ / Sin θ]

= Sin θ · Cos θ × Cos θ / Sin θ

= Cos θ × Cos θ

= Cos² θ

Hence the simplified expression is Cos² θ.

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The binomial expansion of (5a+2b)³ is 125a³+150a²b+60ab²+8b³.

To expand the binomial (5a + 2b)³, we can use the binomial expansion formula or the Pascal's triangle method.

Let's use the binomial expansion formula:

(5a + 2b)³ = (³C₀)(5a)³(2b)⁰ + (³C₁)(5a)²(2b)¹ + (³C₂)(5a)¹(2b)² + (³C₃)(5a)⁰(2b)³

Simplifying each term:

= (1)(125a³)(1) + (3)(25a²)(2b) + (3)(5a)(4b²) + (1)(1)(8b³)

=125a³+150a²b+60ab²+8b³

Hence, the binomial expansion of expression (5a+2b)³ is 125a³+150a²b+60ab²+8b³.

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The sum of the two infinite series ∑ₙ=₁∞ (2/3)ⁿ⁻¹ and ∑ₙ=₁∞ (2/3)ⁿ is 3 + 2 = 5.

To find the sum of the two infinite series, let's evaluate each series separately.

Series 1: ∑ₙ=₁∞ (2/3)ⁿ⁻¹

To determine the sum of this series, we can use the formula for the sum of an infinite geometric series:

S₁ = a₁ / (1 - r)

where:

S₁ = sum of the series

a₁ = first term of the series

r = common ratio of the series

In this case, the first term (a₁) is (2/3)⁰ = 1, and the common ratio (r) is 2/3.

Plugging these values into the formula, we have:

S₁ = 1 / (1 - 2/3)

   = 1 / (1/3)

   = 3

So, the sum of the first series is 3.

Series 2: ∑ₙ=₁∞ (2/3)ⁿ

Similarly, we can use the formula for the sum of an infinite geometric series:

S₂ = a₂ / (1 - r)

In this case, the first term (a₂) is (2/3)¹ = 2/3, and the common ratio (r) is 2/3.

Plugging these values into the formula, we have:

S₂ = (2/3) / (1 - 2/3)

   = (2/3) / (1/3)

   = 2

So, the sum of the second series is 2.

Therefore, the sum of the two infinite series ∑ₙ=₁∞ (2/3)ⁿ⁻¹ and ∑ₙ=₁∞ (2/3)ⁿ is 3 + 2 = 5.

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To ensure that the height of the triangle is at least 14 ft, the angle θ must be such that the vertical component of the triangle's height is equal to or greater than 14 ft.

In a triangle, the height is determined by the vertical component of the triangle's side length.

To find the angle θ that results in a height of at least 14 ft, we need to consider the trigonometric relationship between the angle and the side lengths.

Let's assume that the side length opposite to the angle θ is represented by "x". In this case, the height of the triangle can be calculated using the formula:

Height = x * sin(θ)

To ensure that the height is at least 14 ft, we set up the inequality:

14 ≤ x * sin(θ)

Solving for θ,

we divide both sides of the inequality by "x" and take the inverse sine ([tex]sin^(-1)[/tex]) of both sides:

θ ≥ [tex]sin^(-1)[/tex](14 / x)

This means that the angle θ must be equal to or greater than the inverse sine of (14 / x) to ensure that the height of the triangle is at least 14 ft.

The specific value of θ will depend on the length of the side opposite to the angle.

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The product of (2 + √7)(1 + 3√7) is 23 + 7√7.To multiply the expressions (2 + √7)(1 + 3√7), we can use the distributive property and multiply each term separately.

(2 + √7)(1 + 3√7) = 2(1) + 2(3√7) + √7(1) + √7(3√7)

Now, simplify each term:

2(1) = 2

2(3√7) = 6√7

√7(1) = √7

√7(3√7) = 3(√7)^2 = 3(7) = 21

Putting it all together:

2 + 6√7 + √7 + 21

Combining like terms:

2 + √7 + 6√7 + 21

Simplifying further:

23 + 7√7

Therefore, the product of (2 + √7)(1 + 3√7) is 23 + 7√7.

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The product of 2/3y and y is [tex]2/3y^2.[/tex]

The product of 2/3y and y can be found by multiplying the numerators and denominators separately.

To multiply the numerators, we multiply 2 and 1 (since y can be written as y/1).

This gives us 2.

To multiply the denominators, we multiply 3y and 1.

This gives us 3y.

Therefore, the product of 2/3y and y is [tex]2/3y * y =[/tex]  [tex]2/3y^2.[/tex]

In general, when multiplying fractions, we multiply the numerators and the denominators separately.

So, if we have a fraction a/b and another fraction c/d, their product would be ac/bd.

However, in this specific case, the expression can be simplified further.

Since we have y in both the numerator and denominator, they cancel out, leaving us with  [tex]2/3y^2.[/tex]

To summarize, the product of 2/3y and y is  [tex]2/3y^2.[/tex]

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There were a total number of  2,165 people at the two football games.

To find the total number of people at the two games, we add the number of people from each game. The first game had 1,207 people, and the second game had 958 people.

Total number of people = Number of people at Game 1 + Number of people at Game 2

Total number of people = 1,207 + 958

Total number of people = 2,165

Therefore, there were a total of 2,165 people at the two football games.

To calculate the total number of people at the two games, we simply add the number of people at Game 1 and the number of people at Game 2. The first game had 1,207 people, and the second game had 958 people. Adding these two values gives us a total of 2,165 people present at the two football games.

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