Find the slope and y-intercept from the following graph of a linear equation.

To find the formula for the nth term of a geometric sequence, we can use the formula:

a_n = a_1 * r^(n-1)

where a_1 is the first term of the sequence, r is the common ratio, and n is the position of the term.

In this case, the first term of the sequence is a_1 = 7 and the common ratio is r = (-21)/7 = -3. Plugging these values into the formula, we get:

a_n = 7 * (-3)^(n-1)

Therefore, the formula for the nth term of the geometric sequence is:

a_n = 7 * (-3)^(n-1)

Answer:

[tex]a_n=7\left(-3\right)^{n-1}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given geometric sequence:

To find the common ratio, divide a term by the previous term:

[tex]\implies r=\dfrac{a_3}{a_2}=\dfrac{63}{-21}=-3[/tex]

Substitute the found common ratio and given first term into the formula to create an equation for the nth term:

[tex]a_n=7\left(-3\right)^{n-1}[/tex]

An equation in the form x = a , where a is a constant can never be a function.

What is a function?

A function is defined as a relation between a set of inputs having one output each.

In simple words, a function is a relationship between inputs where each input is related to exactly one output.

Every function has a domain and codomain or range.

A function is generally denoted by f(x) where x is the input.

The general representation of a function is y = f(x).

According to the given question:

An equation is in the form x = a , where a is a constant.

This equation is a vertical line parallel to y axis. For a function a relation of y = f(x) is missing in the given equation since a is a constant.

Hence, x = a can never be a function.

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g is continuous for all x in the interval [0,7] is correct statement that would be sufficient to conclude that there exists a number c in the interval [0,7] such that g (c) = 50.

Here the given function g(x) gives values for selected values of x.

Now we are to find a condition which will conclude that there exist a number 'c' in the interval [0,7] such that g(c)=50

If a function f : a, b [tex]\rightarrow[/tex] R be continuous on R with [tex]$\mathrm{f}(\mathrm{a}) \neq \mathrm{f}(\mathrm{b})$[/tex]  then the function f(x) attains every value between f(a) and f(b) at least once in the interval [a, b]

I. The option is false.

Because if the function is defined in the interval [0,7] then it is not necessary that there exist a point in this interval, where the function will attain the value 50.

II. The given option is false.

Because if the function is increasing in the interval [0,7] then it is not necessary that there exist a point in this interval, where the function will attain the value 50.

III. This option is correct.

Since g(0) = 24 and g(7) =  68 and 'g' is continuous on the interval [0,7] , so the function g(x) attains every value between 24 and 68 at least once in the interval [0,7].

That is there must be a point 'c' in the interval [0,7] such that g(c)=50.

Therefore option (B) is correct.

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The time that is left before the fossil fuel reserves run out would be= 2000 years.

A fossil fuel is defined as the type of fuel that is gotten from dead and decayed organic matter that has been buried for years underneath the earth surface.

The quantity of usable fossil fuel existing on earth = 2.1 x 10⁸metric tonnes.

The rate of fossil fuel used per year = 1 x 10⁵

Mathematically,

If 1 year = 1 x 10⁵

X years = 2.1 x 108

make X years the subject of formula;

X years = 2.1× 10⁸/1× 10⁵

X years = 2.1 × 10³ or 2100

X year = 2000( to one significant figure)

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Function notation, such as f(x) = 3x + 5, can be interpreted by plugging in different values for x and evaluating the resulting expression.

For example, if we plug in 2 for x, we get f(2) = 3(2) + 5 = 11. This means that the value of the function at x = 2 is 11. By plugging in different values for x, we can evaluate the function at any point on its domain.

A connection between two variables can be expressed using function notation. We are accustomed to writing straight-line equations in the form y = m x + c .

We can express this in function notation as f (x) = m x + c by substituting y with f (x).

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Sebastian uses the method of similar triangles to find the distance across the river, and the distance across the river is 372 foot.

A triangle is a polygon with three sides and three vertices.

The triangle's total number of angles comes to 180°.

The distances between the formed the sight-lines are;

RB = 210 feet

OC = 275 feet

The distance between the point close to the river and the next point further from the river = 115 feet

In triangles ΔPRB and ΔPOC,

we have;

∠PRE = ∠POC = 90°

Given;

∠PER ≅ ∠PCO

By corresponding angle formed between two parallel lines and a common transversal.

Using angle-angle similarity theorem;

∴ ΔPRE is similar to ΔPOC Which gives;

PR / PO = RE / OC

Let x represent the distance across the river,

we have;

PR = x

PO = 115 + x

Which gives;

x / (115+x) = 210 / 275

275x = 210 × (115 + x)

275x = 24150 + 210x

275x - 210x = 24150

65x = 24150

x = 371. 54

Therefore, the distance across the river is 372 foot.

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to estimate the average coffee intake of Coursera students, measured in cups of coffee. C) μ=0.55, σ=1 is not necessarily true.

Given that survey is of 1000 students so n = 1000, average is 0.55 cups per day  i.e. mean (x bar) = 0.55 and standard deviation is of 1 cup per day ,i.e. S.D (s) = 1 .

First, determine the mean. Step 2: Calculate the square of each data point's variance from the mean. Add the values from Step 2 in Step 3. Divide by the total number of data points in step 4.

Just because the sample statistics are these values doesn't mean the population values will be exactly equal to them, therefore it's not necessarily true μ=0.55, σ=1 .

C) μ=0.55, σ=1 is not necessarily true.

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Part 1) The rule of the translation is    (x,y) ------> (x+4,y+5) ,Part 2) B'(4,8) Part 3) C'(3,3)

Find out the rule of the translation

we know that

The transformation of the point A to A' is equal to

A(-4,1) ------> A'(0,6)

so

The rule of the translation is equal to

(x,y) ------> (x + a ,y + b)

(-4,1) ------> (-4 + a, 1 + b)

Find the value of a

-4 + a = 0 -----> a = 4

Find the value of b

1 + b = 6 -----> b = 5

substitute the values of a and b

(x,y) ------> (x+4,y+5)

That means -----> The translation is 4 units at right and 5 units up

Find out the coordinates of B'

Applying the rule of the translation

(x,y) ------> (x+4,y+5)

so

B(0,3) ------> B'(0+4,3+5)

B(0,3) ------> B'(4,8)

Find out the coordinates of C'

Applying the rule of the translation

(x,y) ------> (x+4,y+5)

so

C(-1,-2) ------> C'(-1+4,-2+5)

C(-1,-2) ------> C'(3,3)

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Answer:

It can be any number.

Step-by-step explanation:

The x xalue is an independent variable, which means its value does not depend on that of another variable (y)

Answer: 224 pens more sold in 2nd shop

Step-by-step explanation:

Given,    

1st shop, 6 pens are sold more than previous day,

      1st day : 60

      2nd day : 60+6 =66

      3rd day : 66+6 =72

      4th day : 72+6 =78

      5th day : 78+6 =84

      6th day : 84+6 =90

      7th  day : 90+6 =96  

Then,

2nd shop, each day doubles the previous sold,

      1st day: 5

      2nd day:5*2 =10

      3rd day:10*2 =20

      4th day:20*2 =40

      5th day:40*2 =80

      6th day:80*2 =160

      7th day:160*2 =320

Therefore, second shop sells more in a week which is calculated by 320-96=224 more sold than the 1st shop.

The least to greatest of the ratio is 18 : 32, 5 : 8, and 11 : 16

Least to greatest arrangement can also be referred to as ascending order. Ascending order is the order such that each element is greater than or equal to the previous element.

5 : 8

= 5/8

= 0.625

11 : 16

= 11/16

= 0.6875

18 : 32

= 18/32

= 0.5625

Therefore, the ratio can be arranged as 18 : 32, 5 : 8, and 11 : 16 in ascending order.

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The rent per unit will be $8,354.22 to have a Net Operating Income (NOI) of $100,000.

The net operating income (NOI) is the difference between the gross operating income and the operating expenses.

The net operating income excludes some fixed or period expenses.

                                                                                            Total

Rent per unit (per month) = $696.19 ($8,354.22/12)   $150,376

Vacancy Rate (5% of Potential Gross Income)                   7,519

Effective Gross Income                                                $142,857

Operating Expenses (30% of Effective Gross Income) 42,857

Net Operating Income (NOI)                                      $100,000

Number of Units 18

Gross operating income = net operating income + operating expenses

Operating expenses = 30% of effective gross income

Net operating income = 70% of effective gross income (100 - 30%)

= $100,000

Effective Gross Income (100%) = $142,857 ($100,000 ÷ 70%)

Vacancy rate = 5% of potential gross income

Effective Gross Income = 95% (100 - 5)

Potential gross income = $150,376 ($142,857 ÷ 95%)

Number of units = 18

Rent per unit = $8,354.22 ($150,376/18)

Rent per month = $696.19 ($8,354.22 ÷ 12)

Thus, a rent of $8,354.22 per unit can generate a Net operating income of $100,000.

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Answer:

The correct answers are (x+3) . 2 and 2x+6.

In the first expression, (x+3) . 2, the parentheses indicate that x+3 should be treated as a single term and multiplied by 2. This is equivalent to 2(x+3).

In the second expression, 2x+6, the 2 is multiplied by x and the 3 from x+3, giving 2x+6, which is also equivalent to 2(x+3).

The other expressions are not equivalent because they do not properly distribute the multiplication. For example, in 2 . x+3, the 2 is only being multiplied by x, not x+3, and in 2x+5, the 2 is only being multiplied by x and not 3.

The perimeter of the rectangle is equal to 2(√90 + √40).

A coordinate plane is a 2D plane which is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.

The perimeter is defined as the sum of all the sides of the rectangle.

Given coordinates of the rectangle are (4, −3), (2, 3), (11, 6), and (13, 0). The length and width of the rectangle are calculated by the distance formula.

D = √ [ ( y₂ - y₁ )² + ( x₂ - x₁ )² ]

Here, D is the distance, and the  y₂ and y₁ are the coordinates.

L = √(13-4)² + 3² = √90

W = √ ( 4-2)² + (3 + 3)² = √40

The perimeter is,

P = 2(L + W)

Here, P is the perimeter and L is the length, and W is the width.

P = 2 (√90 +√ 40)

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Answer:

  a.  k = -8/3 = -2 2/3

  b.  k = 32/5 = 6.4

Step-by-step explanation:

You want to find the values of k that place the point (k, 1) on the line through the point (4, -3) when that line is (a) parallel to 3x +5y = 10, and (b) perpendicular to 3x +5y = 10.

The equation of the parallel line will have the same x- and y-coefficients, but will have a constant that make the equation true at the point (4, -3).

  3x +5y = 3(4) +5(-3) = 12 -15 = -3

The equation of the parallel line is

  3x +5y = -3

When y=1, the value of k is ...

  3k +5(1) = -3

  3k = -8

  k = -8/3 = -2 2/3 . . . . . . on line parallel to 3x+5y=10

The equation of the perpendicular line will have swapped x- and y-coefficients, with one of them negated. The constant will be chosen to make the equation true at the point (4, -3).

  5x -3y = 5(4) -3(-3) = 20 +9 = 29

The equation of the perpendicular line is

  5x -3y = 29

When y=1, the value of k is ...

  5k -3(1) = 29

  5k = 32

  k = 32/5 = 6.4 . . . . . . on th eline perpendicular to 3x+5y=10

We can write an equation to represent the relationship between the three integers n - 1, n, and n + 1 by multiplying these three numbers together:

We can then solve this equation to find the value of n.

To solve the equation, we can first factor the left-hand side to get (n - 1) * (n + 1) * n = 210. This expression can be further simplified to n^2 - 1 = 210. We can then solve this equation by adding 1 to both sides to get n^2 = 211, and then taking the square root of both sides to get n = sqrt(211).

The value of n must be an integer, so the only possible value for n is 14. This means that the three consecutive integers are 13, 14, and 15.

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The product of 0.123(overline) and 9 is equal to 149/142.

To multiply 0.123(overline) and 9, we can first rewrite 0.123(overline) as a fraction. To do this, we can express 0.123(overline) as a fraction with a denominator of 1000, which represents the number of thousandths that 0.123(overline) represents:

To find the product of 123/1000 and 9, we can use the distributive property:

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common factor, which is 7:

Therefore, the product of 0.123(overline) and 9 is equal to 149/142.

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The rate per student for the gym is given as $5.

The arithmetic operations are the fundamentals of all mathematical operations. The example of these operators are addition, subtraction, multiplication and division.

The rent of gym for 30 students is $150.

And, for 70 students it is $350.

The rate of rent in both the cases can be found by taking the ratio as follows,

The unit cost in the first case = Total rent ÷ Number of students

                                                 =  150 ÷ 30 = $5  

And, the unit cost in the second case = 350 ÷ 70 = $5

Hence, the required rate is obtained as $5 per student for both the cases.

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The relationship's slope and common difference are both 70 and constant.

The slope of the line is a tangent angle made by line with horizontal. i.e. m =tanx where x in degrees.

here,
As the  relationship given in the table is linear,
The slope of the relationship is given as,
M = (y₂ - y₁) /  (x₂ - x₁)
Now, putting values from the table,
m = 140 - 70 / 2 - 1
m = 70 jumps per minute

Now,
The common difference between the consecutive minutes of jumping,
d = 140 - 70 = 210 - 140
d = 70 = 70

From the above evaluation, it can be said that the common difference and rate are constant.

Thus, the slope of the relationship and the common difference is 70, as well as constant.

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Answer:

7

Step-by-step explanation:

54 - 3 = 51

51 divided by 7 is 7 with 2 left over

Answer:

Step-by-step explanation:

a. To calculate the incremental income after taxes, we need to determine the incremental costs and revenues associated with extending credit to the new customers. The additional credit sales will generate $180,000 in revenue, but we need to subtract the expected uncollectible amount of 12% * $180,000 = $21,600. The net additional credit sales will be $180,000 - $21,600 = $158,400.

The production and marketing costs will be 72% * $180,000 = $129,600. The total incremental costs will be $129,600 + $16,200 (collection expense) = $145,800.

The incremental income before taxes will be $158,400 - $145,800 = $12,600. The income tax will be 34% * $12,600 = $4,284. The incremental income after taxes will be $12,600 - $4,284 = $8,316.

To calculate the return on incremental investment, we need to determine the incremental investment required to extend credit to the new customers. Since no additional investment in assets is required, the incremental investment will be equal to the incremental costs of $145,800. The return on incremental investment will be $8,316 / $145,800 = 5.7%.

Based on the return on incremental investment of 5.7%, the company should not extend credit to the new customers, as it is below the desired return of 10%.

b. If an additional investment in inventory is required to extend credit to the new customers, we will need to recalculate the incremental income after taking into account the additional investment. If the additional investment in inventory is $x, the incremental investment will be $145,800 + $x. The incremental income before taxes will be $158,400 - ($129,600 + $16,200 + $x) = $12,600 - $x. The income tax will be 34% * ($12,600 - $x) = $4,284 - $x * 0.34. The incremental income after taxes will be ($12,600 - $x) - ($4,284 - $x * 0.34) = $8,316 - $x * 0.34.

To determine whether credit should be extended, we need to compare the return on incremental investment to the desired return of 10%. The return on incremental investment will be ($8,316 - $x * 0.34) / ($145,800 + $x). Setting this equal to the desired return of 10% and solving for x, we find that the additional investment in inventory cannot exceed $24,764 for the company to meet its desired return. If the additional investment in inventory is less than or equal to $24,764, the company should extend credit to the new customers. If it is greater than $24,764, the company should not extend credit to the new customers.

To find the original price of the T-shirt before the discount, we need to divide the discount by the percentage to find the discount amount, then add the discount amount to the sale price to find the original price. In this case, the discount was $10.50 * 30% = $<<10.5030.01=3.15>>3.15. So the original price of the T-shirt before the discount was $10.50 + $3.15 = $<<10.50+3.15=13.65>>13.65. Answer: \boxed{13.65}.

Answer:

$15.00

Step-by-step explanation:

The original price is x, an unknown.

x is 100% of the original price.

The discount was 30%.

100% - 30% = 70%

The discounted price was 70% of the original price.

70% of x is $10.50

0.7x = 10.5

x = 15

Answer: $15.00

Correlation Coefficient between the given variables is 0.0023

A statistical concept known as the correlation coefficient aids in establishing a relationship between expected and actual values gained through statistical experimentation. The estimated correlation coefficient's value explains how well the expected and actual values match.

Correlation Value of the coefficient is always between -1 and +1. If the correlation coefficient value is positive, the two variables have a similar and same relationship. Otherwise, it shows how the two variables are different.

The Pearson correlation coefficient is the result of dividing the covariance of two variables by the sum of their standard deviations. It is typically displayed as ρ(rho).

The correlation coefficient can be determined using the formula if the two variables being discussed are x and y.

Formula for Correlation Coefficient:

r×{[n∑x∧2-(∑x)∧2][n∑y∧2-(∑y)∧2]}=n(∑xy)-∑x×∑y.

From given data:

∑x=3.95+4.18+7.5+6.19+6.35+7.23+7.98+8.15=51.53

∑y=16.12+15.75+21.45+20.08+22.60+21.95+26.42+28.38=172.75

∑x²=[tex]3.95^{2}+4.18^{2}+7.5^{2}+6.19^{2}+6.35^{2}+7.23^{2}+7.98^{2}+8.15^{2}=350.3393[/tex]

∑y²=[tex]16.12^{2}+15.75^{2}+21.45^{2}+20.08^{2}+22.60^{2}+21.95^{2}+26.42^{2}+28.38^{2}=3867.2291[/tex]

∑xy=[tex]3.95\cdot16.12+4.18\cdot15.75+7.5\cdot21.45+6.19\cdot20.08+6.35\cdot22.6+7.23\cdot21.95+7.98\cdot26.42+8.15\cdot28.38=1159.0163[/tex]

By substituting in formula:

[tex]r=\frac{8*1159.0163-51.53*172.75}{(8*350.3393-51.53^2)(8*3867.2291-172.75^2)}=\frac{370.3229}{147.3735*1095.2703}[/tex]

r=0.00229=0.0023

Correlation Coefficient between the given variables is 0.0023

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The correlation coefficient between the following variables is  0.76.

A statistical measure called correlation shows how much two or more variables fluctuate in connection to one another. When two variables rise or decrease simultaneously, there is a positive correlation; when there is a negative correlation, one variable increases as the other falls.

We know the coefficient of correlation is [tex]r = \frac{\sum{XY}}{\sum{X^{2}Y^2}}[/tex].

Now, [tex]\sum{XY}[/tex] = (3.95)×(16.12) + (4.18)×(15.75) + (7.50)×(21.45) + (6.19)×(20.08)

+ (6.35)×(22.60) + (7.23)×(21.95) + (7.98)×(26.42) + (8.15)×(28.38).

[tex]\sum{XY} =[/tex] 1159.02.

[tex]\sum{X} =[/tex] 51.53.

[tex]\sum{X^2} =[/tex] 463.35.

[tex]\sum{Y} =[/tex] 172.75.

[tex]\sum{Y^2} =[/tex] 5047.015.

∴ [tex]r = \frac{\sum{XY}}{\sum{X^{2}Y^2}}[/tex].

= [tex]\frac{1159.02}{\sqrt{463.35\times 5057.015}}[/tex].

= 1159.02/1530.74.

= 0.76.

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The solution for the given inequality is -6 < x < 9.

What is linear equality?

In mathematics a linear inequality is an inequality that involves a linear function. A linear inequality contains one of the symbols of inequality. It shows the data which is not equal in graph form.

The given inequality is:

         -7 < x - 1 < 8

       −7 + 1 < x − 1 + 1 < 8 + 1 -------- (Add 1 to all parts)

             -6 < x < 9

Hence, the solution for the given inequality is -6 < x < 9.

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The correct hypotheses for this test are:

Null hypothesis (H0): The standard deviation of the mutual fund's monthly rate of return is greater than or equal to 3%.

Alternative hypothesis (H1): The standard deviation of the mutual fund's monthly rate of return is less than 3%.

To determine if there is sufficient evidence to reject the null hypothesis and conclude that the fund has moderate risk, you would need to perform a hypothesis test. In this case, since you have a sample of 27 monthly rates of return and the normal probability plot indicates that the data is normally distributed, you can use a z-test for the population standard deviation.

To perform the test, you would need to calculate the test statistic and the p-value. The test statistic is calculated as follows:

test statistic = (sample standard deviation - population standard deviation) / (standard error)

where the sample standard deviation is 2.55%, the population standard deviation is 3%, and the standard error is calculated as:

standard error = sample standard deviation / sqrt(sample size)

Plugging in the values, the test statistic is:

test statistic = (2.55 - 3) / (2.55 / sqrt(27)) = -0.44

The p-value is the probability of observing a test statistic at least as extreme as the one calculated, given that the null hypothesis is true. To calculate the p-value, you can use a z-table or a statistical software package.

If the p-value is less than the chosen level of significance (a=0.05 in this case), you can reject the null hypothesis and conclude that the fund has moderate risk. If the p-value is greater than the level of significance, you cannot reject the null hypothesis and cannot conclude that the fund has moderate risk.

Answer:

r = 2/ r = -2/3

Step-by-step explanation:

so you move the terms to the left side

6r^2 - 8r = 8

6r^2 -8r - 8 = 0

the common factor

6r^2 - 8r -8 = 0

2(3r^2 - 4r - 4) = 0

then you divideboth side by the same factor

2(3r^2 - 4r - 4) = 0

3r^2 - 4r - 4 = 0

use the quadratic formula

you would get 2.3

then you simplify

r = 4+8 over 6

seperate the equations

r = 4+8 over 6 change the plus into a minus

after that rearrange and issolate variable

r = 2

r = -2/3

therfore your answer is -2/3