Write a recursive formula for the sequence 3, 9, 15, 21 27,. Then find the next term

The answer of the question based on the angles statements the answer is

always true based on condition.

Range is the difference between highest and lowest values.

Based on the provided image, the statement "The range of a function is always a subset of its codomain" is true.

In mathematics, the codomain of a function is the set of all possible output values, while the range is the set of actual output values produced by the function for a given input. Since the range is a subset of the codomain, the statement is true.

One way to determine the truth of such statements is to understand the definitions of the relevant terms and to reason logically based on those definitions.

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1) m(x) = 0 when x = 3, and when x = 4.

2) The graph of m intercepts the x-axis at x = 3, and x = 4.

3) The zeros of m are 3 and 4.

1) m(x) = 0 when x = 3, and when x = 4.

To find the zeros of m(x), we set the function equal to zero and solve for x:

m(x) = 0

(2x - 6)(x - 4) = 0

This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.

Solving 2x - 6 = 0 gives x = 3, and solving x - 4 = 0 gives x = 4.

2) The graph of m intercepts the x-axis at x = 3, and x = 4.

The x-intercepts of a function are the points where the graph intersects the x-axis, or where y = 0. So, we can find the x-intercepts of m(x) by setting y = m(x) = 0:

m(x) = 0

(2x - 6)(x - 4) = 0

This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.

So, the x-intercepts of m(x) are (3, 0) and (4, 0).

3) The zeros of m are 3 and 4.

The zeros of a function are the values of x for which the function is equal to zero. So, the zeros of m(x) are x = 3 and x = 4.

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The given question is incomplete, the complete question is:

The zeros of a function are the values of x for which the function is equal to zero.

Enter a number in each blank to make true statements about the function m(x)=(2x−6)(x−4).

1)m(x) = 0 when x =__, and when x =___  

2) the graph of m intercept the x axis at x = __, and x =___ .  

3) zeros of m are ___ and ____?

Considering a recipe of the dessert for 16 people, the needed amounts of butter, sugar, flour and cheese are given as follows:

The amounts are obtained applying the proportions in the context of the problem, as we are given the amount needed for 8 people, hence we must obtain the ratio between the number of people and 8, and then multiply the amounts by this ratio.

For 8 people, the amounts of the ingredients are given from the problem as follows:

The ratio between 16 people and 8 people is given as follows:

16/8 = 2.

Hence the amount of each ingredient will double, thus the needed amounts are given as follows:

Gina is preparing a recipe for 8 people, and the amounts are given as follows:

The problem asks for the necessary amounts for 16 people.

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Answer:

8

Step-by-step explanation:

.13 times 8 = 1.04

The amount spent by each friend is 5d + 9.75 = 63.75; d = 10.8 and the mathematics sentence is -10(r + 12.5) = 60.5

In the question, we have

Total = 63.75

Meal = 9.75 each

Dessert = d each

So, the equation is

5d + 9.75 = 63.75

Evaluate the like terms

5d = 54

Divide by 5

d = 10.8

So, the equation and the result are 5d + 9.75 = 63.75; d = 10.8

Here, we have

Negative ten times the sum of a number and 12.5 is 60.5.

Let the number be r

So, we have

-10(r + 12.5) = 60.5

Based on the problem statement, we have

8.94 * 3 = 1/3 * (17.95 + j)

So, we have

80.46 = 17.95 + j

Evaluate

j = 62.51

Hence, the cost of the jalapeno peppers is $62.51

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In the function [tex]V(t) = 230(1.12)^t[/tex], the number 230 represents the initial or starting value of the company in millions of dollars at the beginning of the time period in question, which is 2018.

The function models the growth of the company's value over time as an exponential function, with a base of 1.12. The exponent t represents the number of years that have passed since 2018, and the resulting value V(t) is the projected value of the company t years after 2018, given the assumed growth rate of 12% per year.

So, at the start of 2018, the company was worth 230 million dollars, according to the model.

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Answer:

C

Step-by-step explanation:

Answer:

[tex]16666\frac{2}{3}[/tex]

Step-by-step explanation:

Solving using PEMDAS

[tex]\frac{1000\times 100}{2+8-4}[/tex]

Since this is a fraction, we can work on the top and the bottom. Lets do the top first. Multiply.

[tex]\frac{100000}{2+8-4}[/tex]

Now we can add two, then subtract.

[tex]\frac{100000}{6}[/tex]

Since this yields an irrational number if we divide, we can simplify this fraction.

[tex]16666\frac{2}{3}[/tex]

The pοint οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and  [tex]\mathrm {f^{(-1)}}[/tex](x) are perpendicular is apprοximately (0.71, 0.42).

A graph is a visual representatiοn οf data that shοws the relatiοnship between different variables οr sets οf data. Graphs are used tο display and analyze data in a way that makes it easier tο understand patterns, trends, and relatiοnships.

Tο find the pοints οn the graph οf the inverse functiοn where the tangents οf f(x) and its inverse are perpendicular, we need tο use the fact that the prοduct οf slοpe οf twο perpendicular lines is -1.

Let y = f(x) = 1/(x²+1) + (1-2x[tex])^{(1/3)[/tex], x >= 0

We want tο find the pοints οn the graph οf  [tex]\mathrm {f^{(-1)}}[/tex]  where the tangent οf f(x) and  [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular. Let (a, b) be a pοint οn the graph οf f^(-1) such that [tex]\mathrm {f^{(-1)}}[/tex] (a) = b.

The slοpe οf the tangent tο f(x) at x =  [tex]\mathrm {f^{(-1)}}[/tex] (a) is 1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).

f'(x) = -2x/(x²+1)² - (1-2x[tex])^{(-2/3)[/tex] / (3 * (1-2x[tex])^{(2/3)[/tex])

[tex]\mathrm {f^{(-1)}}[/tex] (a) = b implies a = f(b).

Therefοre, the slοpe οf the tangent tο  [tex]\mathrm {f^{(-1)}}[/tex]  at b is f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).

Sο, we need tο find a pοint (a, b) οn the graph οf  [tex]\mathrm {f^{(-1)}}[/tex]  such that:

1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) * f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) = -1

Simplifying, we get:

-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] a)² + 1)² - (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = -1

Simplifying further, we get:

2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] (a)² + 1)² + (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = 1

Let y =  [tex]\mathrm {f^{(-1)}}[/tex] (x), then x = f(y).

Substituting x = a and y = b, we get:

a = f(b)

2b/(b²+1)² + (1-2b[tex])^{(-2/3)[/tex] / (3 * (1-2b[tex])^{(2/3)[/tex]) = 1

This equatiοn cannοt be sοlved analytically, sο we need tο use numerical methοds tο apprοximate the sοlutiοn.

Using a graphing calculatοr οr sοftware, we can plοt the graphs οf f(x) and  [tex]\mathrm {f^{(-1)}}[/tex] (x) and find the pοints where the tangents are perpendicular. One such pοint is (0.71, 0.42) (rοunded tο twο decimal places).

Therefοre, the pοint οn the graph οf  [tex]\mathrm {f^{(-1)}}[/tex]  where the tangent οf f(x) and  [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular is apprοximately (0.71, 0.42).

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Answer: 7-2 i

Step-by-step explanation:

The solution of the given problem of percentage comes out to be In 2050, there will be about 153,000 African elephants left.

In statistics, a "a%" is a figure or statistic that is expressed as a percentage of 100. The words "pct," "pct," but instead "pc" are also not frequently used. However, the sign "%" is frequently used to represent it. The percentage sum is flat; there are no dimensions. Percentages are truly integers because their numerator almost always equals 100. Either the % symbol (%) or the additional term "fraction" must come before a number to denote that it is a percentage.

Here,

We can apply the exponential decay formula if we presume that this rate of decline stays constant:

=>   [tex]N(t) = N0 * (1/2)^(t/T)[/tex]

The half-life, T, is a problem we want to address. Since we are aware that the population is declining by 8% annually:

=>  [tex](1/2)^{(1/T)} = 0.92[/tex]

Using both sides' natural logarithms:

=> [tex]ln[(1/2)^{(1/T)}] = ln(0.92) (0.92)[/tex]

=>  (1/T) * ln(1/2) Equals ln (0.92)

=>  1/T Equals ln(0.92) / ln(1/2)

=>  8.6 years T

This indicates that the number of African elephants is predicted to decrease by half every 8.6 years.

=>  2050 - 2016 = 34 years

There will be roughly 3.95 half-lives between 2016 and 2050 because the population halves every 8.6 years.

Consequently, the population in 2050 will be roughly:

=> N(2050)=N0*(1/2)*(3.95)=350,000*(0.5)*(3.95)=153,000 elephants

Consequently, if the rate of decrease remains constant, we can calculate that there

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The sum of the first 18 terms of the series -100 + 122 - 148. 84 + 181. 5848–… is option (C) -15840.45

To find the sum of the first 18 terms of the given series, we need to first identify the pattern in the series.

The given series is: -100 + 122 - 148.84 + 181.5848 - ...

We can observe that each term is obtained by multiplying the previous term by -1.22 and then adding a constant. In other words, if the nth term is represented by Tn, then:

Tn = (-1.22) × T(n-1) + C

where C is a constant.

To find the constant C, we can use the first term of the series, which is -100:

-100 = (-1.22) × T(0) + C

where T(0) represents the 0th term of the series, which is not given. However, we can find T(0) by dividing the first term by (-1.22):

T(0) = -100 / (-1.22) = 81.9672

Substituting this value of T(0) in the above equation, we get:

-100 = (-1.22) × 81.9672 + C

C = 100 + 1.22 × 81.9672 = 200.2046

Therefore, the nth term of the series can be represented as:

Tn = (-1.22) × T(n-1) + 200.2046

Using this formula, we can find the sum of the first 18 terms of the series as follows:

S18 = T1 + T2 + T3 + ... + T18

= -100 + 122 - 148.84 + 181.5848 - ... + (-1)^17 × T(17)

= -100 + 122 - 148.84 + 181.5848 - ... + (-1)^17 × (-1.22)^17 × T(0) + (-1)^17 × 200.2046

= -100 + 122 - 148.84 + 181.5848 - ... - 1.3579774 × 10^8 + 200.2046

= -15840.45

Therefore, the correct option is (3) -15840. 45

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The identity numbers of next three selected customers are 0181, 0231, and 0281. The interquartile range and range is 18 and 30. The data is negatively skewed and the mean and range of combined sample is 41.8 and 60 respectively.

(a) Since the sample was selected using systematic sampling, we can determine the sampling interval by dividing the population size by the sample size:

Sampling interval = Population size / Sample size = 1000 / 20 = 50

Since customer 0131 was included in the sample, the next three selected customers are:

0131 + 50 = 0181

0181 + 50 = 0231

0231 + 50 = 0281

(b) To find the interquartile range, we first need to find the median. Since the sample size is even, we take the average of the middle two values:

Median = (75 + 80) / 2 = 77.5

The first quartile (Q₁) is the median of the lower half of the data, and the third quartile (Q₃) is the median of the upper half of the data. We can use the ordered data to find these values:

Ordered data: 60, 62, 63, 64, 65, 70, 75, 80, 85, 90

Lower half: 60, 62, 63, 64, 65, 70

Upper half: 75, 80, 85, 90

Q₁ = median of lower half = (64 + 65) / 2 = 64.5

Q₃ = median of upper half = (80 + 85) / 2 = 82.5

Therefore, the interquartile range is:

IQR = Q₃ - Q₁ = 82.5 - 64.5 = 18

To find the range, we subtract the minimum value from the maximum value:

Range = 90 - 60 = 30

(c) To comment on the skewness of the data, we can compare the mean, median, and mode. If the mean is equal to the median and mode, then the data is symmetrical. If the mean is greater than the median, then the data is positively skewed. If the mean is less than the median, then the data is negatively skewed.

Mean = (60 + 62 + 63 + 64 + 65 + 70 + 75 + 80 + 85 + 90) / 10 = 72.4

Median = 77.5

Mode = there is no mode

Since the mean is less than the median, the data is negatively skewed.

(d) To find the mean of the combined sample, we can use the formula:

Mean = (sum of all data) / (number of data)

The sum of the data in the original sample is:

60 + 62 + 63 + 64 + 65 + 70 + 75 + 80 + 85 + 90 = 694

The sum of the data in the new sample is:

50 + 60 + 70 + 80 + 90 + 100 + 110 = 560

The sum of all the data is:

694 + 560 = 1254

The number of data is 20 + 10 = 30

Therefore, the mean of the combined sample is:

Mean = 1254 / 30 = 41.8

To find the range of the combined sample, we subtract the minimum value from the maximum value:

Range = 110 - 50 = 60

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We have shοwn that the LHS equals the RHS, and hence, we have prοved the identity: sec²x/2) tan(x) = csc²(x).

Trigοnοmetry is a branch οf mathematics that deals with the study οf relatiοnships between the sides and angles οf triangles. It is a fundamental area οf mathematics that has applicatiοns in many fields, including physics, engineering, and astrοnοmy.

Trigοnοmetry invοlves the study οf six trigοnοmetric functiοns: sine (sin), cοsine (cοs), tangent (tan), cοsecant (csc), secant (sec), and cοtangent (cοt). These functiοns describe the relatiοnships between the angles and sides οf a right-angled triangle.

Trigοnοmetry alsο includes the study οf trigοnοmetric identities, which are equatiοns that invοlve trigοnοmetric functiοns and are true fοr all pοssible values οf the variables.

In the given question,

Starting with the left-hand side (LHS) of the given identity:

sec²(x/2) tan(x)

Using the identity, sec²(x) = 1/cos²(x), we can write:

sec²(x/2) = 1/cos²(x/2)

Substituting this into the LHS:

1/cos²(x/2) * tan(x)

Now, using the identity, tan(x) = sin(x)/cos(x), we can write:

1/cos²(x/2) * sin(x)/cos(x)

Rearranging and simplifying:

sin(x) / cos(x) * 1/cos²(x/2)

Using the identity, csc(x) = 1/sin(x), we can write:

1/sin(x) * 1/cos(x) * 1/cos²(x/2)

Now, using the identity, cos(2x) = 1 - 2sin²(x), we can write:

cos(x) =√(1 - sin²(x/2))

Substituting this into the above equation:

1/sin(x) * 1/√(1 - sin²(x/2)) * 1/cos²(x/2)

Simplifying:

1/sin(x) * 1/√(cos²(x/2)) * 1/cos²(x/2)

Using the identity, csc²(x) = 1/sin²(x) and simplifying:

csc²(x) * cos²(x/2) / cos²(x/2)

The cos²(x/2) terms cancel out, leaving:

csc²(x).

Therefore, we have shown that the LHS equals the RHS, and hence, we have proved the identity: sec²x/2) tan(x) = csc²(x).

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Yes, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.

Co-interior angles, also known as consecutive interior angles, are those between two lines that are split by a third line (transversal), and are located on the same face of the transversal.

The majority of the time, it comes from the latin word "com-," which often means "along with." Co-interior angles are located on the same face of a transversal as well as between two lines. The significant correlations angles in each diagram are referred to as co-interior angles. Co-interior angles are supplementary if the two lines remain parallel since they add to 180 degrees.

In the given diagram:

= 75 + 105 (co-interior angles)

= 180° (supplementary angles)

So,

line 1 || line 2

Thus, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.

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Answer:

-3 ≤ 3x ≤ 6

To solve for "x", we can divide each part of the inequality by 3:

-1 ≤ x ≤ 2

Therefore, the number "x" must lie between -1 and 2 in order to satisfy the condition in the sentence.

Step-by-step explanation:

The area included by means of the beam of the Cape Florida Lighthouse light is about 177 square miles whilst rounded to the nearest square mile.

To find the area covered by means of the beam of the Cape Florida Lighthouse light, we need to first find the radius of the circle that the beam sweeps over. We recognise that the most distance the mild can be visible is 15 miles, so the radius of the circle is also 15 miles.

Next, we want to discover the valuable attitude of the circle that the beam sweeps over. We know that the beam sweeps in an arc of 270°, which is three-quarters of a complete circle. therefore, the critical attitude of the circle that the beam sweeps over is also 270°.

Now, we are able to use the formula for the area of a sector of a circle to discover the area covered through the beam:

area of sector = (central angle/360°) x π x radius^2

Substituting the given values, we get:

Thus, the area included by means of the beam of the Cape Florida Lighthouse light is about 177 square miles whilst rounded to the nearest square mile.

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The value of n is 3, n is a positive integer

We know that:

1 + 2 + 3 + 4 + 5 + 6 = 21

Therefore:

(1 + 2 + 3 + 4 + 5 + 6)² = 21² = 441

Now, let's look at the sum of cubes:

1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²

We already know that 1 + 2 + ... + 6 = 21, so we can rewrite the equation as:

1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²

1³ + 2³ + ... + n³ = (1 + 2 + 3 + 4 + 5 + 6 + ... + n)²

We want to find the value of n that makes this equation true. We know that the sum of the first n positive integers is:

1 + 2 + 3 + ... + n = n(n+1)/2

So we can rewrite the equation as:

1³ + 2³ + ... + n³ = [n(n+1)/2]²

Now we substitute the value we know for 1 + 2 + 3 + 4 + 5 + 6:

441 = [6(7)/2]²

441 = 21²

So n(n+1)/2 = 7, which means:

n(n+1) = 14

The only positive integer solution for n in this case is 3, because:

n(n+1) = 14

n² + n - 14 = 0

(n-3)(n+4) = 0

The positive integer solution is n = 3, which means:

1³ + 2³ + 3³ = [3(4)/2]² = 36² = 441

So the value of n is 3.

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Answer:

$0.79

Step-by-step explanation:

$9.48 ÷ 12= 0.79

each pen costs $0.79

The figure's overall area is 8,065.76 square metres (3,136 + 4,929.76 square metres).

To find the area enclosed by the figure, we need to calculate the area of the square and the four semicircles and then add them together. The area of the square is 56 × 56 = 3,136 square meters.

The diameter of each semicircle is equal to the side of the square, which is 56 meters. Therefore, the radius of each semicircle is 28 meters. The area of one semicircle is (1/2) × pi × 28² = 1,232.44 square meters. The area of all four semicircles is 4 × 1,232.44 = 4,929.76 square meters.

Thus, the total area of the figure is 3,136 + 4,929.76 = 8,065.76 square meters.

The error that Frank made is likely in the calculation of the area of the semicircles. He may have used the formula for the area of a circle instead of a semicircle or made a mistake in the calculation. It is also possible that he rounded the area to two decimal places, leading to a small error in the final answer.

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Option A is an example of a function with a domain (-∞, +∞) and a range (-∞, +∞). We can check this by verifying that there are no restrictions on the domain and that the function can output any real number.

The domain of a function in mathematics is the collection of all potential input values (also known as the independent variable) for which the function is specified. It is the collection of all x-values that can be inserted into a function to generate a valid output.

In the given question, for any value of x, the expression [tex](2x)^10[/tex] will result in a real number, since any real number raised to an even power will have a positive result. Therefore, there are no restrictions on the domain.

Similarly, since any real number raised to an even power is positive, multiplying [tex](2x)^10[/tex] by -2 will also result in a real number, which means that the function can output any real number. Therefore, the range is also (-∞, +∞).

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Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.

In mathematics, inequality is a comparison between two values or expressions using an inequality symbol such as >, <, ≥, or ≤. It is used to compare different values to each other and determine whether one is greater than, less than, or equal to the other. Inequality can be used to express relationships between two or more variables, to solve certain equations, and to graph certain data.

In order to answer these questions, we need to compare the given values. In the first question, Brittany's rope was compared to Sam's rope, and in the second question, Alex's rope was compared to Sam's rope.

For the first question, Sam's rope was 1.5 x 4/5 = 6/5. This value was compared to Brittany's rope which was 1 1/4. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.

For the second question, Sam's rope was 1.5 x 3/2 = 9/4. This value was compared to Alex's rope which was 5/4. Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope.

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The greater of the two even integers is 6. The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

The four fundamental operations, often referred to as "arithmetic operations",are said to be able to describe all real numbers. The four mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.

Let the consecutive integers be 'x' and 'x+2'.

We are given that four times the sum of two consecutive even integers is 40.

So,

4 (x + x +2) = 40

On solving this, we get

⇒4 (2x +2) = 40

⇒2x + 2 = 10

⇒2x = 8

⇒x = 4

The next integer will be 6.

Hence, the greater of the two even integers is 6.

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Therefore , the solution of the given problem of equation comes out to be Taylor purchased 3 bags of popcorn and 9 candies.

The use of the same variable word in mathematical formulas frequently ensures agreement between two assertions. Mathematical equations, also referred to as assertions, are used to demonstrate expression the equality of many academic figures. Instead of dividing 12 into 2 parts in this instance, the normalise technique adds b + 6 to use the sample of y + 6 instead.

Here,

Let's describe our variables first:

x is the quantity of popcorn bags bought, and y is the quantity of sweets bought.

The following system of equations can be constructed using the information provided:

The price of the packages of popcorn and candies is $81, or

=>  9x + 4.5y.

=> y = x + 6 (Taylor purchased 6 more candies than bags of popcorn) 

The first step in solving this system of equations numerically is to rewrite the first equation in slope-intercept notation as follows:

=> 4.5y = -9x + 81

=> y = (-2)x + 18

Let's plot the y-intercept at (0,6) and then locate another point by moving up 1 unit and to the right 1 unit to graph the second equation, y = x + 6. This demonstrates our argument (1,7).

The two lines now meet at the number (3,9), indicating that Taylor purchased 3 bags of popcorn and 9 candies.

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In response to the query, we can state that Therefore, the equation of the line that passes through the point (-15,-3) with slope[tex]-3/7 - 7/3 is 12x + 7y = -201.[/tex]

An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.

The point-slope form of the equation of a line is given by:

y - y1 = m(x - x1)

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

[tex]y - (-3) = (-3/7 - 7/3)(x - (-15))\\y + 3 = (-36/21)(x + 15)\\y + 3 = (-12/7)(x + 15)\\7y + 21 = -12x - 180\\12x + 7y = -201[/tex]

Therefore, the equation of the line that passes through the point (-15,-3) with slope[tex]-3/7 - 7/3 is 12x + 7y = -201.[/tex]

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[tex]\textit{volume of a hemisphere}\\\\ V=\cfrac{2\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ V=2712.3 \end{cases}\implies 2712.3=\cfrac{2\pi r^3}{3}\implies (3)(2712.3)=2\pi r^3 \\\\\\ \cfrac{(3)(2712.3)}{2\pi }=r^3\implies \sqrt[3]{\cfrac{(3)(2712.3)}{2\pi }}=r\implies 10.90\approx r[/tex]

Pοlynοmial f(x) has exactly 3 cοmplex rοοts

The fundamental theοrem οf algebra, alsο knοwn as d'Alembert's theοrem, οr the d'Alembert–Gauss theοrem, states that every nοn-cοnstant single-variable pοlynοmial with cοmplex cοefficients has at least οne cοmplex rοοt. This includes pοlynοmials with real cοefficients, since every real number is a cοmplex number with its imaginary part equal tο zerο.

The fundamental theοrem οf algebra states that any nοn-cοnstant pοlynοmial οf degree n has exactly n cοmplex rοοts (cοunting multiplicities). In this case, the pοlynοmial [tex]f(x) = 8x^3[/tex]+ 216 is a nοn-cοnstant pοlynοmial οf degree 3, sο it has exactly 3 cοmplex rοοts (cοunting multiplicities).

We can alsο use the factοr theοrem tο cοnfirm that f(x) has exactly 3 rοοts. The factοr theοrem states that a pοlynοmial f(x) has a factοr (x - a) if and οnly if f(a) = 0. In this case, we can factοr οut 8 frοm the pοlynοmial tο get:

[tex]f(x) = 8(x^3 + 27)[/tex]

Setting f(x) = 0, we get:

[tex]8(x^3 + 27) = 0[/tex]

This equatiοn is satisfied if and οnly if [tex]x^3 + 27 = 0.[/tex] We can factοr this equatiοn as fοllοws:

[tex]x^3 + 27 = (x + 3)(x^2 - 3x + 9)[/tex]

The quadratic factοr [tex]x^2 - 3x + 9[/tex] has nο real rοοts, since its discriminant is negative [tex](b^2 - 4ac = (-3)^2 - 4(1)(9) = -27)[/tex]. Hοwever, it dοes have twο cοmplex rοοts, which are cοnjugates οf each οther. Therefοre, the pοlynοmial f(x) has exactly 3 cοmplex rοοts (cοunting multiplicities).

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The angle m∠P is equal to 50 degrees.

Since triangle OPQ is isosceles with PQ congruent to QO, we know that angle OPQ is congruent to angle OQP. Let's call this angle x. Then, we can set up an equation based on the fact that the angles in a triangle add up to 180 degrees: x + x + 50 = 180

Simplifying the equation, we get

2x + 50 = 180

Subtracting 50 from both sides, we get

2x = 130

Dividing by 2, we get:

x = 65

Therefore, angle OPQ and angle OQP are both equal to 65 degrees. Since angle OPQ and angle P are supplementary (they add up to 180 degrees), we can find angle P as:

m∠P = 180 - m∠OPQ

=> 180 - 2x

=> 180 - 2(65)

=> 50 degrees.

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