Which irrational number can be multiplied by -~ 41 to get a product that equals 1?

There are 2,646 different ways to form the committee with 5 teachers and 5 students.

There are 7 teachers to choose from, and we need to select 5 of them. This can be done in C(7,5) ways:

C(7,5) = 7! / (5!2!) = 76 / 21 = 21

There are 9 students to choose from, and we need to select 5 of them. This can be done in C(9,5) ways:

C(9,5) = 9! / (5!4!) = 9876 / 4321 = 126

To find the total number of ways to form the committee, we multiply the number of ways to select 5 teachers by the number of ways to select 5 students:

21 × 126 = 2,646

Therefore, there are 2,646 different ways to form the committee with 5 teachers and 5 students.

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Answer:

0, -2 for the slope and y intercept

hooe this helps

Answer:

The slope is -5/6 and the y intercept is -2.

Step-by-step explanation:

One way to solve is rewriting the standard form ax+by=c to slope intercept form y=mx+b

5x+6y=-12

6y=-12-5x

Divide both sides by 6.

6y/6 = (-12-5x)/6

y=-2-(5/6)x

The slope is -5/6 and the y intercept is -2.

The sample proportion (360 out of 1200) can be used to estimate the population proportion, and statistical tests such as a one-sample proportion test or a Z-test can be applied to evaluate the hypotheses and determine the significance of the results.

The null and alternative hypotheses that should be used in this scenario are as follows:

Null Hypothesis (H₀): The proportion of families interested in the program is still 20% or less.

Alternative Hypothesis (H₁): The proportion of families interested in the program has increased, i.e., it is greater than 20%.

Symbolically, we can represent the hypotheses as:

H₀: p ≤ 0.20 (proportion is 20% or less)

H₁: p > 0.20 (proportion is greater than 20%)

In this case, "p" represents the population proportion of families in New Jersey interested in the renewable energy program.

To determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, a hypothesis test can be conducted using appropriate statistical methods.

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The estimate of the horizontal distance traveled by a ball hit at an angle of 55º is given as follows:

348.9 feet.

To find the regression equation, which is also called called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator.

From the table, the points in this problem are given as follows:

(15, 157.2), (20, 189.2), (24, 220.8), (30, 253.8), (34, 269.2), (40, 284.8), (45, 285).

Inserting these points into a calculator, the line of best fit is given as follows:

y = 4.42x + 105.8.

Hence when x = 55 the output is given as follows:

y = 4.42(55) + 105.8

y = 348.9 feet.

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Answer:

7,5 hours

Step-by-step explanation:

1/10+1/30=2/15

The missing sides and angles are listed below:

Triangle A: x = 27.061

Triangle B: θ = 70.562°

Triangle C: θ = 51.756

In this problem we find three triangles, a right triangle (Triangle a)) and two non-right triangles (Triangles b) and c)), whose missing angles and sides.

Triangle A

The missing side is determined by following trigonometric function:

x = 43 · sin 39°

x = 27.061

Triangle B

The missing angle is determined by sine law:

2.2 / sin 42° = 3.1 / sin θ

sin θ = 0.943

θ = 70.562°

Triangle C

The missing side is determine by cosine law:

4.1² = 5.2² + 3.6² - 2 · 5.2 · 3.6 · cos θ

16.81 = 40 - 37.44 · cos θ

37.44 · cos θ = 23.19

cos θ = 0.619

θ = 51.756

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There is no significance difference in the variation in the listening times for men and women.

In this study conducted by Arbitron Media Research Incorporated, the iPod listening habits of men and women were examined. One aspect of the study focused on the mean listening time for both genders. The objective is to determine if there is a significant difference in the variation of listening times between men and women. To do so, statistical analysis will be conducted using the given data.

To determine if there is a significant difference in the variation of listening times between men and women, we can perform a hypothesis test. Let's denote the mean listening time for men as μ₁, the mean listening time for women as μ₂, the standard deviation for men as σ1, and the standard deviation for women as σ₂.

Null Hypothesis (H₀): There is no difference in the variation of listening times between men and women. In mathematical terms, H₀ : σ₁ = σ₂.

Alternative Hypothesis (H₁): There is a difference in the variation of listening times between men and women. In mathematical terms, H₁ : σ₁ ≠ σ₂.

We will use a significance level of 0.10, which indicates that we are willing to accept a 10% chance of rejecting the null hypothesis when it is true.

To test this hypothesis, we can use the F-test for comparing variances. The F-test compares the ratio of the variances between two independent samples. In this case, the samples are the listening times of men and women. The F-test statistic is calculated as follows:

F = (s₁²) / (s₂²)

where s₁ and s₂ are the sample variances. In this case, the sample variances are the square of the standard deviations provided.

Let's calculate the F-test statistic using the given data:

For men:

Sample size (n₁) = 8

Sample standard deviation (s₁) = 12 minutes/day

Sample variance (s₁²) = (12²) = 144 minutes²/day²

For women:

Sample size (n₂) = 10

Sample standard deviation (s₂) = 16 minutes/day

Sample variance (s₂²) = (16²) = 256 minutes²/day²

Calculating the F-test statistic:

F = (s₁²) / (s₂²) = 144 / 256 ≈ 0.5625

Next, we need to determine the critical value for the F-test statistic at a significance level of 0.10. This critical value can be found in the F-distribution table. Since the sample size is small, we will use the F-distribution table.

With a sample size of 8 for men and 10 for women, the degrees of freedom for the numerator (df₁) is 7 (n₁ - 1) and the degrees of freedom for the denominator (df₂) is 9 (n₂ - 1).

Looking up the critical value from the F-distribution table with df₁ = 7 and df₂ = 9, at a significance level of 0.10, we find the critical value to be approximately 2.76.

Now, we compare the calculated F-test statistic with the critical value. If the calculated F-test statistic is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

In this case, the calculated F-test statistic (0.5625) is less than the critical value (2.76). Therefore, we fail to reject the null hypothesis. This means that there is no enough evidence to conclude that there is a significant difference in the variation of listening times between men and women at the 0.10 significance level.

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The correct answer is B. The derivative is 2x - 49.

To find the derivative of the function (x - 50 + (x + 1)), we can apply the product rule, which states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.

Let's break down the given function into its components:

f(x) = (x - 50) + (x + 1)

To differentiate this function, we can consider each term separately:

Term 1: (x - 50)

Term 2: (x + 1)

Now, applying the product rule, we differentiate each term and combine the results:

Derivative of Term 1 = 1  (since the derivative of x is 1)

Derivative of Term 2 = 1  (since the derivative of x is 1)

Using the product rule, the derivative of the function is:

f'(x) = (x - 50) * 1 + (x + 1) * 1

f'(x) = x - 50 + x + 1

f'(x) = 2x - 49

Therefore, the correct answer is B. The derivative is 2x - 49.

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The Volume of the candle is approximately 147.66 cubic inches.

To find the volume of the candle, we need to use the formula for the volume of a cylinder, which is:

V = πr^2h

where r is the radius of the circular base, h is the height of the cylinder, and π is approximately equal to 3.14.

In this case, we are given that the height of the candle (h) is 7.5 inches and the diameter (d) is 5 inches. To find the radius (r), we need to divide the diameter by 2:

r = d/2 = 5/2 = 2.5 inches

Now we can substitute these values into the formula and simplify:

V = πr^2h

V = π(2.5)^2(7.5)

V = π(6.25)(7.5)

V = 147.66 cubic inches (rounded to two decimal places)

Therefore, the volume of the candle is approximately 147.66 cubic inches.

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The largest integer that is a divisor of (n 1)(n 3)(n 57)(n 9 ) for all positive even integers $n$ is the largest prime factor of $n_9$.

The given expression consists of five consecutive odd numbers: $(n_1), (n_3), (n_5), (n_7), (n_9)$. When we multiply these consecutive odd numbers, all their prime factors cancel out except for the largest prime factor, which is the prime factor of the largest odd number.

Since we are considering even numbers, the largest odd number in this case is $n_9$. Therefore, the largest prime factor of the expression is the largest prime factor of $n_9$.

To find the largest prime factor of a number, we can start dividing it by the smallest prime numbers (2, 3, 5, etc.) and keep dividing until we can't divide anymore. The result will be the largest prime factor.

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The distribution of the number of coin tosses required until a tail is observed is geometric, not binomial. Its mean is 2 and variance is 2. so none of the answer options is correct and option is D).

The distribution of x, the number of tosses required until a tail is observed, is a geometric distribution.

It is not a binomial distribution because the trials are not independent (the probability of getting a tail increases with each additional toss). The mean of a geometric distribution with probability of success p is 1/p, so in this case the mean is 1/0.5 = 2.

The variance of a geometric distribution with probability of success p is (1-p)/p², so in this case the variance is (1-0.5)/0.5² = 2. Therefore, none of the answer choices is correct. So, the correct option is D).

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Answer: g(f(3)) = 21.5

Step-by-step explanation:

     g(f(3)) is asking us to substitute 3 into f(x) and then substitute that into g(x).

Given:

   f(x) = x +[tex]\frac{1}{2}[/tex]  

Substiute 3 into f(x) for x:

   f(3) = 3 +[tex]\frac{1}{2}[/tex]  

     f(3) = 3.5

Given:

     g(x) = 2x² –  3

Substiute f(3) = 3.5 into g(x) for x:

      g(f(3)) = 2(3.5)² –  3

     g(f(3)) = 24.5 –  3

     g(f(3)) = 21.5

The recommended time for cleaning and sanitizing a cutting board depends on various factors, such as the type of food being prepared and the risk of contamination.

However, a general guideline suggests that a cutting board should be cleaned and sanitized if it has been used for the same task for more than 4 hours.

Food safety guidelines emphasize the importance of preventing cross-contamination in the kitchen. Cutting boards, especially those used for cutting raw meats, poultry, and seafood, can harbor harmful bacteria that can contaminate other foods if not properly cleaned and sanitized.

The 4-hour guideline is based on the understanding that bacteria can multiply rapidly at room temperature and can reach levels that may pose a health risk. To minimize this risk, it is recommended to clean and sanitize cutting boards after approximately 4 hours of continuous use for the same task.

It's important to note that the 4-hour guideline is a general recommendation and may vary depending on the specific circumstances. Factors such as the type of food being prepared, the level of contamination, and the sanitation practices in place should be considered. Additionally, cutting boards should be cleaned and sanitized immediately if visible signs of contamination or soiling are present, regardless of the time duration.

To ensure food safety, it is always advisable to follow specific guidelines and regulations provided by local health authorities or food safety organizations. These guidelines take into account the specific risks associated with different types of food and provide detailed recommendations for cleaning and sanitizing cutting boards.

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Answer:

the graph plotting changes, the numbers decreases

We need to first determine the equation of the portion of the plane that lies in the first octant. The area of the surface is 1262.5 square units.

The first octant is the region of space where all three coordinates (x, y, and z) are positive. Therefore, we need to find the values of x, y, and z that satisfy the equation 5x + 13y + z = 65 and are all positive.

To do this, we can use the fact that the plane intersects each of the coordinate axes (x, y, and z) when the other two coordinates are equal to 0.

When z = 0 and y = 0, we get:
5x = 65
x = 13

When z = 0 and x = 0, we get:
13y = 65
y = 5

When x = 0 and y = 0, we get:
z = 65

Therefore, the portion of the plane that lies in the first octant is the triangle with vertices at (13, 0, 0), (0, 5, 0), and (0, 0, 65).

To find the area of this triangle, we can use the formula:
Area = 1/2 * base * height

The base of the triangle is the distance between the points (13, 0, 0) and (0, 5, 0), which is the same as the length of the projection of this line segment onto the xy-plane. We can find this length using the Pythagorean theorem:
base = sqrt((13-0)^2 + (0-5)^2) = sqrt(169 + 25) = sqrt(194)

The height of the triangle is the distance between the point (0, 0, 0) and the plane, which is given by the equation z = (65 - 5x - 13y)/1. Plugging in the coordinates of any of the three vertices of the triangle gives:
z = (65 - 0 - 0)/1 = 65

Therefore, the height of the triangle is 65.

Plugging these values into the formula for the area, we get:
Area = 1/2 * sqrt(194) * 65 = 1262.5

So the area of the surface is 1262.5 square units.

To find the area of the surface, we need to consider the part of the plane 5x + 13y + z = 65 that lies in the first octant. In the first octant, all values of x, y, and z are non-negative.

We can find the points where the plane intersects each of the three axes by setting two variables to 0 and solving for the third variable:

1. x-axis: Set y = 0 and z = 0:
  5x = 65
  x = 13

2. y-axis: Set x = 0 and z = 0:
  13y = 65
  y = 5

3. z-axis: Set x = 0 and y = 0:
  z = 65

Now we have three points on the plane in the first octant: (13, 0, 0), (0, 5, 0), and (0, 0, 65). These points form a triangle. To find the area of this triangle, we can use the formula:

Area = (1/2) * base * height

Since we have a right triangle, we can use the distance between the points on the x and y axes as the base and height:

Base = 13 (x-axis distance)
Height = 5 (y-axis distance)

Area = (1/2) * 13 * 5 = 32.5 square units.

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The total surface area of the prism is 684.15 m²

We can decompose the prism into 3 rectangles and two triangles.

The area of a rectangle is equal to the product between its dimensions, now let's find the areas of the rectangles from top to bottom.

a₁ = 15m*13m = 195m²

a₂ = 15m*9m = 135m²

a₃ = 15m*15.81m = 237.15 m²

And the area of a triangle of base B and height H is:

A = B*H/2

Here we have two triangles of height of 13m and a base of 9m, then the area of each triangle is:

A = 13m*9m/2 = 58.5m²

Then the total area of the prism is:

area =195m² +  135m² +  237.15 m² + 2* 58.5m²

area = 684.15 m²

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To find how many miles Skylar runs, we need to use the information given that Everett runs 1.5 fewer miles than Skylar and that Everett runs 7.64 miles. Let x be the number of miles Skylar runs. Then, we know that x - 1.5 is the number of miles Everett runs.

  We can set up an equation:

x - 1.5 = 7.64

To solve for x, we add 1.5 to both sides:

x = 9.14

Therefore, Skylar runs 9.14 miles.

In summary, to find the number of miles Skylar runs, we can use the fact that Everett runs 1.5 fewer miles than Skylar and that Everett runs 7.64 miles. We can set up an equation and solve for x, which represents the number of miles Skylar runs. The answer is 9.14 miles.

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To find how many miles Skylar runs, we need to use the information given that Everett runs 1.5 fewer miles than Skylar and that Everett runs 7.64 miles. Let x be the number of miles Skylar runs. Then, we know that x - 1.5 is the number of miles Everett runs.

 We can set up an equation:

x - 1.5 = 7.64

To solve for x, we add 1.5 to both sides:

x = 9.14

Therefore, Skylar runs 9.14 miles.

In summary, to find the number of miles Skylar runs, we can use the fact that Everett runs 1.5 fewer miles than Skylar and that Everett runs 7.64 miles. We can set up an equation and solve for x, which represents the number of miles Skylar runs. The answer is 9.14 miles.

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Answer:

To compute the average speed, multiply the total distance travelled by the whole time spent. There's a formula for speed, time and distance

Travel distance aboard the pterodactyl = 10 mph x 3 hours = 30 miles

Distance travelled on the spinosaurus = 2 miles x 1 hour = 2 miles

30 miles + 2 miles = 32 miles total distance travelled

Total time required = 3 hours + 1 hour = 4 hours

Average speed = total distance travelled divided by the total time taken = 32 miles divided by 4 hours = 8 mph

As a result, our average speed is 8 mph.

The quotient of the polynomial long division is determined as 4x²  +  11x  +  15. The remainder = -13

The quotient of the polynomial long division is calculated as follows;

                 2x²  +  5.5x + 7.5

               -------------------------------------

( 2x - 6) √ (2x³ - x² - 18x + 32)

               - (2x³  - 12x²)

              ------------------------------------

                         11x² - 18x + 32

                     -  (11x² - 33x )

             -----------------------------------------

                                 15x  + 32

                              - (15x + 45)

                   ----------------------------------------

                                       -13

The quotient of the polynomial long division is determined as;

2x²  +  5.5x + 7.5 = 20x² +  55x²  + 75 = 4x²  +  11x  +  15

The remainder = -13

So (2x³ - x² - 18x + 32) / ( 2x - 6)  =  4x²  +  11x  +  15 [ -13 /  ( 2x - 6) ]

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The complete question is below:

(2x³ - x² - 18x + 32) / ( 2x - 6) . Find the quotient and remainder of this polynomial division.

It represents exponential growth. As x increases, the value of f(x) will increase at an accelerating rate, reflecting the steady growth rate of 8.5% each year in the US credit card debt.

The given equation, f(x) = 86(1.085)^x, represents the US credit card debt in billions of dollars over time. To determine whether this equation shows exponential growth or exponential decay, we need to analyze the properties of the function.

Exponential growth is characterized by an increasing value over time. In this case, if the value of f(x) is increasing as x increases, then it indicates exponential growth.

Let's examine the equation: f(x) = 86(1.085)^x.

The base of the exponential term, 1.085, is greater than 1. This implies that the value of f(x) will grow exponentially as x increases. Each time x increases by 1, the term (1.085)^x will be multiplied by 1.085, causing the function to grow even further. This multiplication by a constant greater than 1 ensures exponential growth.

Additionally, the initial value, 86 billion, further supports the idea of growth. As x increases, the exponential term will cause the function to increase rapidly, reflecting the compounding effect of the growth rate.

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The value of a is of a = -1/3 and the equation of f(x) in factored form is f(x) = -1/3(x + 5)(x - 9)

We are given the roots for each function, hence the factor theorem is used to define the functions.

The function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.

The roots for this problem are given as follows:

Hence the function in factored form is defined as follows:

f(x) = a(x + 5)(x - 9)

When x = -2, f(x) = 11, hence the leading coefficient a is given as follows:

11 = a(3)(-11)

-33a = 11

33a = -11

a = -1/3.

Hence the equation is:

f(x) = -1/3(x + 5)(x - 9).

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Answer:

It would take 6 workers 16 days to complete the job.

This is because the number of workers and the number of days are inversely proportional. This means that if the number of workers increases, the number of days will decrease, and vice versa.

In this case, the number of workers has decreased from 8 to 6. This means that the number of days will increase from 12 to 16.

Here is the formula for calculating the number of days it would take 6 workers to complete the job:

Number of days = (Number of workers x Number of days) / (New number of workers)

Plugging in the values, we get the following:

Number of days = (8 x 12) / 6 = 16

Therefore, it would take 6 workers 16 days to complete the job.

Step-by-step explanation:

Answer:

It will take 6 workers 16 days

Step-by-step explanation:

8 workers at 12 days = 96 work days

We have 6 workers

96/6 = 16 days

It will take 6 workers 16 days

Hello !

Answer:

[tex]\boxed{\sf V{cone} \approx 2408.55 m^3}[/tex]

Step-by-step explanation:

To find the volume of a cone with the radius of its base and its height, we will apply the following formula:

[tex] \sf V{cone} = \dfrac{\pi \times r^2 \times h}{3} [/tex]

Where r is the radius of its base and h is its height.

Given:

Let's substitute our values into the formula:

[tex]\sf V{cone} = \dfrac{\pi (10)^2(23)}{3} = \dfrac{2300\pi}{3} \ \ \\\boxed{\sf V{cone} \approx 2408.55 m^3}[/tex]

Have a nice day ;)

The store keeper used 125 pounds of flour worth $2.40 per pound and 175 pounds of flour worth $3 per pound to get 300 pounds of mixed flour.

The assertion that two expressions are equal when they are connected by the equals sign ("="), is the mathematical definition of an equation. As an example, 2x - 5 = 13. The equations in this example are 2x - 5 and 13. The symbol "=" links these two expressions together.

Let x be the number of pounds of the flour worth $2.40 per pound, and y be the number of pounds of the flour worth $3 per pound. Since the store keeper wants to end up with 300 pounds of flour, we have:

x + y = 300

We also know that the store keeper wants to sell the mixed flour for $2.05 per pound. This means that the total cost of the flour (x pounds at $2.40 per pound, plus y pounds at $3 per pound) must be equal to:

300 pounds × $2.05 per pound = $615

In other words:

2.40x + 3y = 615

Two equations with two unknowns are now present:

x + y = 300

2.40x + 3y = 615

We can find x and y using algebra. One way to do this is to solve the first equation for x:

x = 300 - y

We can then substitute this expression for x into the second equation:

2.40(300 - y) + 3y = 615

Expanding the parentheses and simplifying, we get:

720 - 2.4y + 3y = 615

0.6y = 105

y = 175

So, the store keeper used 175 pounds of flour worth $3 per pound. To find the amount of flour worth $2.40 per pound, we can substitute y = 175 into the equation x + y = 300:

x + 175 = 300

x = 125

Therefore, the store keeper used 125 pounds of flour worth $2.40 per pound and 175 pounds of flour worth $3 per pound to get 300 pounds of mixed flour.

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Answer:

x = 10 and y = 11

Step-by-step explanation:

2x + 3y = 53      (call this equation '1')

3x - y = 19      (call this '2')

multiply '2' by 3:

9x - 3y = 57          (call this '3')

add '1' and '3':

(2 + 9)x  + (3 + -3)y = 53 + 57

11x = 110

x = 10.

now sub that value of x into '1':

2(10) + 3y = 53

20 + 3y = 53

3y = 53 -20

3y = 33

y = 11

no sub both x = 10 and y =11 into '2' to see if everything adds up:

3x - y = 19

3(10) - 11 = 30 - 11 = 19.

so x = 10 and y =11

Answer:

(a) yes

(b) yes

(c) yes

(d) yes

(e) no

(f) no--2x - 1 > 2x, so -1 > 0 (not true)

The z-score of 1.6 corresponds to a probability of 0.0548 or 5.48%. Therefore, the probability that the force acting on the column is greater than 17 kips is 5.48%.

To answer this question, we need to use the concept of deviation. The deviation measures the distance of a data point from the mean. In this case, the mean force acting on the column is 15.0 kips, and the standard deviation is 1.25 kips.
We can use the normal distribution formula to calculate the probability that the force will be within a certain range. Since we are interested in the probability that the force is greater than a certain value, we need to calculate the z-score corresponding to that value. The z-score is the number of standard deviations that the value is from the mean.
To calculate the z-score, we use the formula:
z = (x - mu) / sigma
where x is the value we are interested in, mu is the mean, and sigma is the standard deviation. In this case, we want to find the probability that the force is greater than a certain value, say 17 kips.
z = (17 - 15) / 1.25

= 1.6
In summary, the deviation of a data point from the mean is an important concept when calculating probabilities using the normal distribution. By calculating the z-score, we can determine the probability that a data point falls within a certain range. In this case, the probability that the force acting on the column is greater than 17 kips is 5.48%.

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Answer:

Answer

Step-by-step explanation:

To solve the inequality 1/6(2-x) - 1/3(2-x) > 2, we can simplify it step by step as follows:

1/6(2-x) - 1/3(2-x) > 2

(2-x)/6 - 2(2-x)/6 > 2

[2(2-x) - (2-x)]/6 > 2

(4-2x)/6 > 2

4-2x > 12

-2x > 8

x < -4

Therefore, the solution for x is x < -4.

The sum of the first ten terms of the sequence is approximately -22.17.

Let the first term of the sequence be a, and let the common ratio be r. Then the third term is ar², the fourth term is ar³, and the fifth term is ar⁴.

We are given:

ar² + ar³ = -12

ar² + ar⁴ = 60

ar³ + ar⁴ = 24

Subtracting the first equation from the second gives:

ar⁴ - ar³ = 72

Subtracting the first equation from the third gives:

ar⁴ -ar²= 36

Dividing these two equations gives:

r - 1/r = 2

Multiplying both sides by r gives:

r² - 1 = 2r

r² - 2r - 1 = 0

Using the quadratic formula, we find:

r = 1 + √2 or r= 1- √2

Since the terms of the sequence must be real numbers, we can discard the second solution. Therefore, the common ratio is:

r = 1 + √(2)

To find the first term a, we can use the equation:

ar² + ar³ = -12

Substituting r and simplifying, we get:

a = -12 / (3 + 2√(2))

So, The sum of the first ten terms of the sequence is:

a(1 - r¹⁰) / (1 - r)

=-22.1666... (rounded to the nearest hundredth)

Therefore, the sum of the first ten terms of the sequence is approximately -22.17.

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The expression tan²θ-Sin²θ = tan²θsin²θ is proven that the left hand side is equal to the right hand side.

To prove the expression, tan²θ-Sin²θ = tan²θsin²θ, start from the left-hand side and use the trigonometric identity tan²θ = sin²θ/cos²θ and the Pythagorean identity sin²θ + cos²θ = 1.

Starting from the left-hand side:

tan²θ - sin²θ = (sin²θ/cos²θ) - sin²θ

= sin²θ (1/cos²θ - 1)

= sin²θ ((1-cos²θ)/cos²θ)

= sin²θ (sin²θ/cos²θ)

= tan²θ sin²θ

Thus, it is shown that the left-hand side is equal to the right-hand side: tan²θ - sin²θ = tan²θ sin²θ.

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