Solve 6 cos(4x) = 2 for the smallest three positive solutions. Give your answers accurate to at least two decimal places, as a list separated by commas.

The polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.

To find a polynomial function of degree 3 with real coefficients that has the given zeros, we can use the fact that if a polynomial has a zero of x = a, then (x - a) is a factor of the polynomial. Therefore, if the polynomial has zeros of x = -3, x = 4, and x = -5, then the polynomial can be written in factored form as:

f(x) = (x + 3)(x - 4)(x + 5)

To find the polynomial function in standard form, we can multiply the factors:

f(x) = (x + 3)(x - 4)(x + 5)
f(x) = (x^2 - x - 12)(x + 5)
f(x) = x^3 + 5x^2 - x^2 - 5x - 12x - 60
f(x) = x^3 + 4x^2 - 17x - 60

Therefore, the polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.

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There are 70 ways to assign 4 runners from a group of 8 to run the legs of the race.

Permutation and combination are two fundamental concepts in mathematics that deal with counting and arranging objects.

This is a combination problem. The formula for calculating the number of combinations of r items from a set of n items is:

nCr = n! / (r! * (n-r)!)

where n! represents the factorial of n, or the product of all positive integers up to and including n.

In this problem, we want to choose 4 runners from a group of 8, without regard to order. So we can use the combination formula as follows:

8C4 = 8! / (4! * (8-4)!)

= (8765) / (4321)

= 70

Therefore, there are 70 ways to assign 4 runners from a group of 8 to run the legs of the race.

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The probabilities are given as follows:

The probabilities are different, as the theoretical probability is higher.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.

Out of 20 tosses, 4 resulted in heads, hence the experimental probability is given as follows:

p = 4/20

p = 1/5.

A coin is equally as likely to be heads or tails, hence the theoretical probability is given as follows:

p = 1/2.

Over a large number of trials, it is expected that the two probabilities assume close values, but for a small number of trials such as 20 it is expected for there to be differences.

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The triangles are similar and the value of x is 2 that makes AABC~ADEF congruent.

what is congruent ?

In geometry, two figures are said to be congruent if they have the same shape and size. In other words, if all corresponding angles are congruent and all corresponding sides are of equal length, then the two figures are congruent.

When two figures are congruent, we can superimpose one on top of the other and they will match up exactly. This means that all parts of the two figures will coincide, including angles, sides, and diagonals.

According to the question:
To determine the value of x that makes AABC~ADEF, we need to find a scaling factor that relates the corresponding sides of the two triangles.

Since AABC has side lengths of 10, 6x, and 20, its perimeter is 10 + 6x + 20 = 30 + 6x. Similarly, the perimeter of ADEF is 25 + 30 + 50 = 105.

Since the two triangles are similar, their corresponding sides are proportional. This means that:

10/25 = (6x)/30 = 20/50

Simplifying each of these ratios, we get:

2/5 = x/5 = 2/5

This tells us that x/5 = 2/5, or x = 2. Therefore, the value of x that makes AABC~ADEF is x = 2.

To check that the triangles are indeed similar, we can also check that their corresponding angles are congruent. In AABC, the ratio of the side lengths is 1:6x/10:2, which simplifies to 1:3x/5:1. Since the sum of the angles in a triangle is 180 degrees, we know that:

angle A + angle B + angle C = 180 degrees

Using the Law of Cosines, we can find the measure of angle B:

[tex]cos(B) = (10^2 + (6x)^2 - 20^2)/(210(6x)) = (100 + 36x^2 - 400)/(120x) = (36x^2 - 300)/(120x)[/tex]

[tex]B = cos^-1((36x^2 - 300)/(120x))[/tex]

Using this expression, we can express the measures of the angles in AABC in terms of x:

[tex]angle A = sin^{-1(1/(2x))} = 30 degrees[/tex]

[tex]angle B = cos^{-1((36x^2 - 300)/(120x))}[/tex]

angle C = 180 - 30 - B = 150 - B

Similarly, in ADEF, the ratio of the side lengths is 5:6:10. Using the Law of Cosines, we can find the measures of the angles:

[tex]angle D = cos^{-1((25^2 + 30^2 - 50^2)/(22530))} = 36.87 degrees[/tex]

[tex]angle E = cos^{-1((25^2 + 50^2 - 30^2)/(22550))} = 53.13 degrees[/tex]

angle F = 180 - 36.87 - 53.13 = 90 degrees

Comparing the angles in the two triangles, we can see that angle A is congruent to angle F (both are 30 degrees) and angle C is congruent to angle D (both are 180 - B - 30). Therefore, the triangles are similar.

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Earl works 5 days a week, which is 50percentage of Monday through Friday. He works in the Tutoring center one day, cleans bathrooms one day, and has another day for unspecified work.

Earl works 5 days a week. He works in the Tutoring center one day, cleans bathrooms one day, and has another day for unspecified work. This means that he is working 50% of Monday through Friday. On Saturday and Sunday, he has no work. His work schedule is a great example of how one can balance work with free time. It also shows how one can make the most out of their working hours by doing different activities each day. It provides variety and prevents boredom from setting in. The days off also give Earl time to rest and relax, allowing him to be productive the other days. His weekly schedule is a great example of how one can efficiently manage their time.

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Alice therefore requires 72 feet of netting.

An example of a quadrilateral with equal and aligned opposing edges is a rectangular. It is a rectangle with four sides and four edges that are each 90 degrees. A rectangular is a form with only two dimensions.

If the longer dimension of the rectangle is 25 feet, then we can write:

x + 14 = 25

Solving for x, we get:

x = 25 - 14 = 11

Therefore, the height of the rectangle is 11 feet.

To find the total amount of fencing Alice needs, we need to add up the length of all four sides of the rectangle. Since opposite sides of a rectangle are equal in length, we can write:

Total fencing = 2(length + height)

Substituting the values of length and height, we get:

Total fencing = 2(25 + 11)

Total fencing = 72 feet

Therefore, Alice needs 72 feet of fencing.

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The equation of the line is x = 3.

To find the equation of a line passing through two points, we need to use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.

First, let's find the slope of the line using the formula: m = (y2 - y1) / (x2 - x1)

Plugging in the given points, we get:

m = (-8 - 9) / (3 - 3) = -17 / 0

Since the denominator is zero, this means that the slope of the line is undefined. This means that the line is vertical and has an equation of the form x = c, where c is a constant.

Since both of the given points have an x-coordinate of 3, the equation of the line is:

x = 3

So, the equation of the line passing through the given points is x = 3. This is the final answer in slope-intercept form.

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Answer:

sin 45°= x/110 ft

sin45° * 110 = x

Step-by-step explanation:

try it i dont have calculator here

The amount of element left after the decay is found as 43 grams.

Decay rate is given as:

N = N₀(1 - r)ˣ

x - is the time in minutes

N₀ - initial mass (570 grams)

r - rate of decay (18%)

Put the values:

N = 570(1 - 0.18)¹³

N = 570*0.0757

N = 43

Thus, the amount of element left after the decay is found as 43 grams.

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The finance charge on a $15,820 car loan if monthly payments were $325 for 60 months is: $3,680.

The total amount paid over 60 months is:

Total amount paid = $325/month x 60 months

Total amount paid = $19,500

The amount of interest paid is:

Amount of interest = $19,500 - $15,820

Amount of interest = $3,680

Therefore, the finance charge on the car loan is $3,680.

Therefore the finance charge is $3,680.

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Answer:

Slope = -250

Interpretation: Every month the value of the car goes down $250.

Step-by-step explanation:

The slope of this linear function is -250 because every 4 months, the value goes down by $1000 so...

-1000/4 = -250

Interpretation: Every month the value of the car goes down $250.

Hope this helps!

So, by resolving the given, we obtain the result: The final simplified expressions phrase is as follows: -4x + 38/3

Mathematical operations include doubling, dividing, adding, and subtracting. A phrase is constructed as follows: Expression, monetary value, and mathematical operation Numbers, parameters, and functions make up a mathematical expression. It is possible to use words and terms in contrast. Every mathematical statement including variables, numbers, and a mathematical operation between them is called an expression, sometimes referred to as an algebraic expression. For example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

Let's start by condensing the first expression:

(2/3) (3x - 1) = 2x - 2/3

Now, we may reformat the phrase as follows:

(2x - 2/3) + 4x + 9 - 10x + 5

Mixing related concepts gives us:

-4x + 38/3

The final simplified phrase is as follows:

-4x + 38/3

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Answer:

The total commission earned by the company is 10% of the selling price, which is:

0.10 x $180,000 = $18,000

Marge receives 1/4 of the company's commission, which is:

(1/4) x $18,000 = $4,500

Therefore, Marge receives $4,500 in commission.

For a=7 the quadratic equation have no real roots. (4)

For a quadratic equation y=ax^(2)-6x+3 to have no real zeroes, the discriminant of the equation must be less than zero. The discriminant of a quadratic equation is given by the formula b^(2)-4ac, where a, b, and c are the coefficients of the equation. In this case, a=a, b=-6, and c=3.

Plugging these values into the formula for the discriminant, we get:

(-6)^(2)-4(a)(3)<0

36-12a<0

12a>36

a>3

Therefore, the values of a that will make the quadratic equation have no real zeroes are those that are greater than 3. This means that the correct answer is (4) a=7, since this is the only value of a that is greater than 3.

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Answer:

13

an example:In words: 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

Trust me its 13

10 + 3³/⁹ answer is  11.4422495703074083.

An algebraic expression is a combination of variables, constants, and mathematical operations. It represents a mathematical relationship between quantities and can be used to model real-world situations or solve problems.

Algebraic expressions can contain variables, which are symbols that represent unknown values, constants, which are fixed values, and mathematical operations such as addition, subtraction, multiplication, division, and exponents.

We can simplify the expression step by step as follows:

10 + 3³/⁹ = 10 + 3^(1/3) (since 3^(3/9) = (3^3)^(1/9) = 27^(1/9) = (3^3)^(1/3) = 3^(3/3) = 3^1 = 3)

= 10 + 1.4422495703074083 (using a calculator to evaluate 3^(1/3) to about 15 decimal places)

= 11.4422495703074083

Therefore, 10 + 3^(3/9) = 11.4422495703074083.

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Answer:

tgv7cutc

Step-by-step explanation:

txr. thh5d tdhg6643gchctc

A sinusoidal function is a type of periodic function that can be written in the form y = a sin(bx + c) + d, where a is the amplitude, b is the frequency, c is the phase shift, and d is the vertical shift.

Given the parameters in the question, we can plug them into the general form of a sinusoidal function to find the specific function.

Amplitude = 4
Period = 3π
Vertical shift = -8

The period of a sinusoidal function is related to the frequency by the equation period = (2π)/b. Therefore, we can solve for b to find the frequency:

3π = (2π)/b

b = (2π)/(3π)

b = 2/3

Plugging in the given values for amplitude, frequency, and vertical shift, we get the sinusoidal function:

y = 4 sin((2/3)x) - 8

This is the sinusoidal function that satisfies the given conditions. It has an amplitude of 4, a period of 3π, and a vertical shift down 8 units.

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Answer: To find the nth term of the sequence -5, -2, 3, 10, 19, we need to identify the pattern in the sequence.

Starting with the first term, we can see that we are adding 3 to get to the second term, adding 5 to get to the third term, adding 7 to get to the fourth term, and adding 9 to get to the fifth term.

So, the pattern is that each term is obtained by adding (n-1) x 2, where n is the position of the term in the sequence (starting with n=1 for the first term).

Therefore, the nth term of the sequence is:

-5 + (n-1) x 2

Simplifying this expression, we get:

-5 + 2n - 2 = 2n - 7

So, the nth term of the sequence is 2n - 7.

Step-by-step explanation:

a)  The demand (price) function in terms of x : P(x) = 12 - 0.025(x - 1500)

b) the marginal revenue function is [tex]MR(x) = dR(x)/dx = 12 - 0.05(x - 1500)[/tex] The marginal revenue from the sales of 1800 T-shirts is -$3. This means that for each additional T-shirt sold after the 1800th, the revenue will decrease by $3 per T-shirt

To determine the demand (price) function, we need to find a relationship between the price of the T-shirts (P) and the number of T-shirts sold (x). We are given that for every $0.50 decrease in price, 20 more T-shirts will be sold.

Let's first find the decrease in price per T-shirt sold. If 20 more T-shirts are sold for every $0.50 decrease, then the decrease per T-shirt is $0.50/20 = $0.025. Now, we can set up the demand function. The base price is $12, and we decrease it by $0.025 for each additional T-shirt sold. So the demand function is:

P(x) = 12 - 0.025(x - 1500) To determine the marginal revenue function, we first need to find the total revenue function. Total revenue (R) is the product of the price (P) and the number of T-shirts sold (x):

R(x) = P(x) * x = (12 - 0.025(x - 1500)) * x Now we need to find the derivative of the total revenue function with respect to x, which will give us the marginal revenue function: MR(x) = dR(x)/dx = 12 - 0.05(x - 1500)

To determine the marginal revenue from the sales of 1800 T-shirts, we simply plug in x = 1800 into the marginal revenue function: MR(1800) = 12 - 0.05(1800 - 1500) = 12 - 0.05(300) = 12 - 15 = -$3

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The dataset's median, which has a value of 82 and a total of 11 values, is the sixth figure.

When a dataset is ordered from smallest to largest, the median, a measure of central tendency, shows the middle value of the dataset. It is frequently used as a more reliable substitute for the mean, which can be vulnerable to anomalies in the data with extremely high or low values. We must first sort the numbers in a dataset from smallest to largest before we can determine the median. The median is just the middle value when the dataset has an odd number of numbers. For instance, the median, or middle number, in the dataset "2, 5, 7, 10, 12" is 7.

given

Values that are noticeably higher or lower than the remainder of the dataset are found using the fences.

Step 1: Sort the information by size, starting from the smallest:

[tex]54, 72, 75, 81, 81, 82, 83, 83, 83, 85, 100[/tex]

Step 2: Calculate the middle

The dataset's median represents its middle number. The median is the sixth number in the dataset, which contains 11 values total:

Average: 82

Step 3: Determine the first and third quartiles (Q1 and Q3, respectively):

The medians for the dataset's lower and upper halves are Q1 for the lower part and Q3 for the upper. We don't use the median value to determine Q1 and Q3 because the dataset has an odd amount of values.

54, 72, 75, 81, 81 in the bottom half.

Top half: 83, 83, 83, 83, 85, and 100

Q1 = 75 for the bottom half's median.

Q3 = 83 for the top half's median.

The dataset's median, which has a value of 82 and a total of 11 values, is the sixth figure.

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Answer:

We can use the given information to find the ant's crawling speed and then use that to calculate how far it can crawl in a given amount of time.

The ant crawls 12 feet in 10 minutes, so its crawling speed is:

12 feet / 10 minutes = 1.2 feet/minute

To find how far the ant can crawl in 20 minutes, we can multiply its crawling speed by the time:

Distance in 20 minutes = crawling speed x time

Distance in 20 minutes = 1.2 feet/minute x 20 minutes

Distance in 20 minutes = 24 feet

Therefore, the ant can crawl 24 feet in 20 minutes.

To find how far the ant can crawl in 30 minutes, we can use the same formula:

Distance in 30 minutes = crawling speed x time

Distance in 30 minutes = 1.2 feet/minute x 30 minutes

Distance in 30 minutes = 36 feet

Therefore, the ant can crawl 36 feet in 30 minutes.

Answer:

20 mins=24 feet

30 mins=36 feet

Step-by-step explanation:

if 12 feet = 10 mins and 10 times 2 = 20 then 12 times 2 =24

same for 30 mins.

10 times 3= 30 then 12 times 3= 36

Answer:6

Step-by-step explanation:If jayden made 5% of his free throws and he shot 120 free throws you have to find 5% of 120.

5% of 120 = 6  

So Jayden shot a total of 6 free throws during the season

The meaning of point J(2,6) on the graph is given as follows:

The cost of 2 ounces of orange juice is $6.00.

The general format of an ordered pair is given by the rule presented as follows:

(x,y).

In which:

The input and output variables for this problem are given as follows:

Hence the point (2,6) represents that the cost of 2 ounces of orange juice is $6.00, meaning that the second option is the correct option in the context of this problem.

The coordinates of the point are (2,6).

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There are 59,049 different types of bags that can be made containing 10 chocolate bars and 54 x 729 = 39,366 different types of bags that can be made containing 10 chocolate bars with at least 1 Kit Kat, 2 Reese's cups, and 1 Almond Joy in the bag.

(A) To calculate the number of different types of bags that can be made with 10 chocolate bars, we can use the concept of combinations. Since each bag can contain Kit Kats, Reese's cups, and Almond Joys in different quantities, we can think of this as selecting 10 items from a group of 3 different types of items with replacement (since we can have more than one of each type of chocolate bar in a bag).

The formula for the number of combinations with replacement is: n^r, where n is the number of items to choose from and r is the number of items to choose

In this case, n = 3 (since there are 3 different types of chocolate bars) and r = 10 (since we are choosing 10 chocolate bars for each bag). Therefore, the number of different types of bags that can be made is: 3^10 = 59,049

So there are 59,049 different types of bags that can be made containing 10 chocolate bars.

(B) To calculate the number of different types of bags that can be made containing at least 1 Kit Kat, 2 Reese's cups, and 1 Almond Joy, we can use a combination of permutations and combinations. We need to choose 4 chocolate bars (1 Kit Kat, 2 Reese's cups, and 1 Almond Joy) out of the 10, and then choose the remaining 6 chocolate bars from the 3 types of chocolate bars.

First, we choose the 4 chocolate bars with the required distribution:

We can choose 1 Kit Kat in 3 ways (since there are 3 Kit Kats to choose from).

We can choose 2 Reese's cups in 6 ways (since there are 6 ways to choose 2 out of 4 Reese's cups).

We can choose 1 Almond Joy in 3 ways (since there are 3 Almond Joys to choose from).

Therefore, the number of ways to choose the 4 required chocolate bars is: 3 x 6 x 3 = 54

Next, we choose the remaining 6 chocolate bars from the 3 types of chocolate bars. This can be done using the formula for combinations with replacement, as in part (A): n^r, where n is the number of items to choose from and r is the number of items to choose

In this case, n = 3 (since there are 3 types of chocolate bars) and r = 6 (since we are choosing 6 chocolate bars for each bag). Therefore, the number of different types of bags that can be made with the required distribution of chocolate bars is: 3^6 = 729

So there are 54 x 729 = 39,366 different types of bags that can be made containing 10 chocolate bars with at least 1 Kit Kat, 2 Reese's cups, and 1 Almond Joy in the bag.

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The actual interest rate for this loan is 20.26%.

The actual interest rate for this loan can be calculated using the formula for annual percentage rate (APR). Since the loan will be paid off in four weeks, which is equivalent to 28 days, the APR can be calculated as follows:

[tex]APR = ((Fee / Loan Amount) * (365 / Time Period)) * 100[/tex]

[tex]= (($40 / $530) * (365 / 28)) * 100[/tex]

= 20.26%

Therefore, the actual interest rate for this loan is 20.26%.

The APR is the annual rate charged for borrowing money, expressed as a percentage of the loan amount. In this case, the loan amount is $530, and the fee charged by the tax preparer is $40. The time period for the loan is 28 days, which is the time it takes for the refund check to arrive. Since the loan is being paid off with the refund check, it is a no-interest loan. However, the fee charged by the tax preparer effectively increases the cost of the loan, resulting in an actual interest rate of 20.26%. This means that the borrower is effectively paying 20.26% in interest for the loan over the course of a year.

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Check the picture below.

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ A(\stackrel{x_1}{2}~,~\stackrel{y_1}{1})\qquad B(\stackrel{x_2}{5}~,~\stackrel{y_2}{5})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ AB=\sqrt{(~~5 - 2~~)^2 + (~~5 - 1~~)^2} \implies AB=\sqrt{( 3 )^2 + ( 4 )^2} \\\\\\ AB=\sqrt{ 9 + 16 } \implies AB=\sqrt{ 25 }\implies AB=5[/tex]

Therefore , the solution of the given problem of unitary method comes out to be  6.10 is the standard deviation.

The task can be completed by integrating what was learned and putting this variable technique into practise, which also incorporates all supplemental data from two individuals who used a specific tactic. Or, to put it another way, if the desired expression outcome happens, either the entity stated in the algorithm will also be found, or both essential processes will actually bypass the colour. For forty pens, a refundable charge of Rupees ($1.01) might be required.

Here,

Using the ideas of mean and standard deviation, we can interpret the three grades in terms of achievement.

The three categories' mean (average) is:

=> Mean = (75 + 85 + 90) / 3 = 83.33

This indicates that the achievement as a whole is above average (since the mean of 72, 82, and 88 is 80).

Which is:

=> Variance = [(75 - 83.33)² + (85 - 83.33)² + (90 - 83.33)²] / 3

=>  [67.11 + 1.36 + 44.11] / 3

=>  37.19

The variance's square root equals the standard deviation.

=> SD= √(37.19)

=> 6.10 is the standard deviation.

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Yes, f(x) and g(x) are inverse functions.

An inverse function is a function that "undoes" the original function. In other words, if we plug the output of the original function into the inverse function, we will get the original input back.

To determine if two functions are inverses of each other, we can use the following steps:

1. Plug the first function into the second function, replacing x with f(x). In this case, we will plug f(x) = 3x - 4 into g(x) = 4 - 3x:

g(f(x)) = 4 - 3(3x - 4)

2. Simplify the expression:

g(f(x)) = 4 - 9x + 12

g(f(x)) = 16 - 9x

3. Now plug the second function into the first function, replacing x with g(x):

f(g(x)) = 3(4 - 3x) - 4

4. Simplify the expression:

f(g(x)) = 12 - 9x - 4

f(g(x)) = 8 - 9x

5. If the two simplified expressions are equal to each other, then the functions are inverses of each other. In this case, g(f(x)) = 16 - 9x and f(g(x)) = 8 - 9x are not equal to each other, so the functions are not inverses of each other.

However, if we rearrange the equation for g(x) to solve for x, we get:

g(x) = 4 - 3x

3x = 4 - g(x)

x = (4 - g(x))/3

This is the same as the equation for f(x), but with g(x) in place of x. Therefore, f(x) and g(x) are inverse functions.

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