Simplify and state the restrictions on the domain:

The 10th term of the geometric sequence whose common ratio is 1/3 and whose first term is 7 is [tex]\frac{7}{19683}[/tex].

What is  geometric sequence?

A unique kind of sequence is a geometric sequence. Every term in the series (apart from the first term) is multiplied by a fixed amount to determine the following term. In other words, we multiply the current phrase in the geometric sequence by a constant term (called the common ratio), and then divide the current term in the geometric sequence by the same common ratio to discover the previous term in the geometric sequence.

nth term of geometric series

a(n) = a(1) * r^(n-1)

You are given a(1) = 7, r = 1/3 and n = 10.

a(10) = 7 * 1/3^(10-1)

[tex]7\left(\frac{1}{3}\right)^{10-1}$$[/tex]

Apply exponent rule: [tex]$\left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$[/tex]

[tex]$$\begin{aligned}& \left(\frac{1}{3}\right)^{10-1}=\frac{1^{10-1}}{3^{10-1}} \\& =7 \times \frac{1^{10-1}}{3^{10-1}}\end{aligned}$$[/tex]

[tex]$$\begin{aligned}& 1^{10-1}=1 \\& =7 \times \frac{1}{3^{10-1}} \\& 3^{10-1}=19683 \\& =7 \times \frac{1}{19683}\end{aligned}$$[/tex]

Convert element to fraction: [tex]$\quad 7=\frac{7}{1}$[/tex]

[tex]=\frac{7}{1} \times \frac{1}{19683}$$[/tex]

Apply the fraction rule: [tex]$\frac{a}{b} \times \frac{c}{d}=\frac{a \times c}{b \times d}$[/tex]

[tex]=\frac{7 \times 1}{1 \times 19683}$$[/tex]

Multiply the numbers: [tex]$7 \times 1=7$[/tex]

[tex]=\frac{7}{1 \times 19683}$$[/tex]

Multiply the numbers: [tex]$1 \times 19683=19683$[/tex]

[tex]=\frac{7}{19683}[/tex]

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The possible amounts Susan will spend on candy is $25 and above.

Expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both.

In this situation, since Susan will buy more than 6 pounds of candy, the same expression will be illustrated by x.

x > ($4 × 6)

x > $24

In this case, she'll have to spend $25 or more.

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A standard normal distribution's characteristics are its mean is equal to 0 and the standard deviation is 1. Thus, the variance is equal to 1.

A normal distribution that has a mean value of 0 and a standard deviation or a variance of 1 is said to be a standard normal distribution.

I.e., μ = 0 and σ² = 1

This is also called z-distribution.

A standard normal distribution has μ = 0 and σ² = 1.

So, the probability of a normal random variable is calculated by using a z-score.

The z-score value is calculated by the formula

z = (X - μ)/σ

Where mean (μ) and standard deviation (σ) for the variable X

By using the standard normal distribution tables with the z-score and probabilities, the required values are calculated.

So, we can conclude that the major characteristics of a standard normal distribution are mean = 0 and variance = 1.

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The mean of the dataset increases and the standard deviation of the data set increases when the 7 is changed to a 70

From the question, we have the following parameters that can be used in our computation:

Dataset: 1 2 3 3 4 4 4 4 5 5 6 7

As a general rule:

To prove the above statements, the mean and the standard deviations are calculated using online calculators

So, we have

Original dataset

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 7

Mean = 4

Standard deviation = 1.58

New dataset

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 70

Mean = 9.25

Standard deviation = 18.36

This means that changing 7 to 70 would increase the mean and the standard deviation

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The golden ratio can be found using the following formula if we take x as the width, and y as the length of the rectangle:

[tex]\frac{y}{x} =\frac{x+y}{y} =1.618[/tex]

What is the golden ratio?

The golden ratio is a unique proportion between two values where the ratio of the two values equals the ratio of their sum to the bigger of the two values.

If we take a square of sides A,B,C,D.

Locate the midpoint of any one side of the square by bisecting it.

Connecting the midpoint (say) P to a corner of the opposite side.

Placing the compass on point P, and the width set to match the distance of P to one of the opposite sides, we draw an arc.

By extending the line where P sits, we see that the arc intersects at a point. say Q.

Extending the opposite line to P, we see that the point Q drawing parallel to the side, we see that it intersects at another point R.

Now, the rectangle AQRD is a golden ratio rectangle.

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Answer:

9.

[tex] \frac{1}{6} [/tex]

10.

[tex] \frac{1}{3} [/tex]

The reason (B) Corresponding Angles Theorem can be used to fill in the blanks in the given question.

A parallelogram is a straightforward quadrilateral with two sets of parallel sides in Euclidean geometry.

A parallelogram's facing or opposing sides are of equal length, and its opposing angles are of similar size.

So, extend segments JM and JK beyond points M and J, respectively, and draw points P and Q.

It is assumed that a parallelogram with segments JM parallel to segments KL and JK parallel to segments ML is presented.

Draw point P and extend segment JM past point M through construction.

Through construction also Continue segment JK draws point Q after point J. 

Consequently, the Alternate Interior Angles Theorem ∠MLK≅∠PML and ∠JML≅∠QJM (1)

Then, the corresponding angles theorem ∠PML≅∠KJM and ∠QJM≅∠LKJ is applied (2)

Equations (1) and (2) and the transitive property of equality are used to create:

∠MLK≅∠KJM and ∠JML≅∠LKJ

As a result, the supplied parallelogram JKLM's opposite angles are congruent. So it was proved.

Therefore, the reason (B) Corresponding Angles Theorem can be used to fill in the blanks in the given question.

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Correct question:
The following is an incomplete flowchart proving that the opposite angles of parallelogram JKLM are congruent:

Parallelogram JKLM is shown where segment JM is parallel to segment KL and segment JK is parallel to segment ML. Extend segment JM beyond point M and draw point P, by Construction. An arrow is drawn from this statement to angle MLK is congruent to angle PML, Alternate Interior Angles Theorem. An arrow is drawn from this statement to angle PML is congruent to angle KJM, numbered blank 1. An arrow is drawn from this statement to angle MLK is congruent to angle KJM, Transitive Property of Equality. Extend segment JK beyond point J and draw point Q. An arrow is drawn from this statement to angle JML is congruent to angle QJM, Alternate Interior Angles Theorem. An arrow is drawn from this statement to angle QJM is congruent to angle LKJ, numbered blank 2. An arrow is drawn from this statement to angle JML is congruent to angle LKJ, Transitive Property of Equality. Two arrows are drawn from this previous statement and the statement angle MLK is congruent to angle KJM, Transitive Property of Equality to opposite angles of parallelogram JKLM are congruent.

Which reasons can be used to fill in the numbered blank spaces?

a. Alternate Interior Angles Theorem

b. Corresponding Angles Theorem

c. Same-Side Interior Angles Theorem

d. Same-Side Interior Angles Theorem

A)the interval on which f is increasing is x ∈ (π/2, 3π/2), The interval on which f is decreasing x∈ ( 0, π/2), (3π/2,2π)

B) local minimum = -6

local  maximum = 6

C) Inflection points are (π/6,-3/4), (5π/6,-3/4).

Local maxima are points in an interval where the values of the function at those points are never greater than the values of the function nearby. Local minima, on the other hand, are locations where the values of the function nearby are higher than the values of the function itself.

A point of inflection is the location where a curve changes from sloping up or down to sloping down or up; also known as concave upward or concave downward. Points of inflection are studied in calculus and geometry. In business, the point of inflection is the turning point of a business due to a significant change.

f(x) = 3cos^2 -6sin(x)

f'(x) = 6cosx d(cosx)/dx -6cosx

f'(x) = 6cosx( -sinx) - 6cosx

f'(x) = -6cosx ( sin x +1)

f"(x) = -6d (cosx)/dx (sinx +1) + -6cosx d ( sin x +1)/dx

f"(x) = -6 ([tex]cos^{2}x - sin^{2}x[/tex]) + 6sinx

a).the interval on which f is increasing is

f'(x)>0

-6cosx ( sin x +1) >0

6cosx ( sin x +1) <0

x ∈ (π/2, 3π/2)

The interval on which f is decreasing

f'(x)<0

-6cosx ( sin x +1) <0

6cosx ( sin x +1) >0

x∈ ( 0, π/2), (3π/2,2π)

since the function is decreasing till x = π/2

so x = π/2 is local minimum (x,y) = ( π/2, -6)

it increasing till 3π/2 and then decreasing

x = 3π/2 is local  maximum values (x,y) = ( 3π/2, 6)

local minimum = -6

local  maximum = 6

c). inflection points. f"(x) = 0

f"(x) = -6 ([tex]cos^{2}x - sin^{2}x[/tex]) + 6sinx = 0

-6 (1-2sin^{2}x) + 6sinx = 0

2[tex]sin^{2}x[/tex] + 2sinx -sinx -1 = 0

sinx = 1/2,-1

x = (π/6,5π/6)

concave up f"(x) >0

-6 (1-2sin^{2}x) + 6sinx > 0

x ∈ (π/6,5π/6)

concave down f"(x) <0

(2sinx -1)(sinx+1) <0

x ∈ (0,π/6) (5π/6,3π/2),(3π/2, 2π)

Inflection points are (π/6,-3/4), (5π/6,-3/4).

A)the interval on which f is increasing is x ∈ (π/2, 3π/2), The interval on which f is decreasing x∈ ( 0, π/2), (3π/2,2π)

B) local minimum = -6

local  maximum = 6

C) Inflection points are (π/6,-3/4), (5π/6,-3/4).

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Answer + Step-by-step explanation:

In the attach pdf

(a) 794g of the initial sample will be left in the sample after 25 years. (b) Time taken to decay to half of its original amount is 3.39 years.

Isotopes(200g) are atoms that have the same number of protons in their nucleus, but a different number of neutrons. This means that they have the same atomic number, but a different atomic mass. Because of this, isotopes have different physical and chemical properties. Isotopes can be stable, meaning that they do not undergo radioactive decay, or they can be unstable, meaning that they will undergo radioactive decay over time.

(a) Substituting 25 for t in the expression, we get:

[tex]A(25) = 200e^{0.0541 \times 25}[/tex]

         [tex]= 200e^{1.3525}[/tex]

Thus, after 25 years, there will be

[tex]200e^{1.3525} = 2003.97[/tex]

794 g of the initial sample left in the sample.

(b) We want to find t such that

[tex]A(t) = 200e^{0.0541}[/tex]                                          

       = 100.

Solving for t, we get:

[tex]= 200e^{0.0541} \times t=100[/tex]

Dividing both sides by 200 and applying the natural logarithm to both sides, we get:

0.0541 × t = ln(0.5)

Therefore,

[tex]t = \frac{ln(0.5)}{0.0541}[/tex]

  = 3.39 years.

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Answer:

No answer

Step-by-step explanation:

Answer:

x3+y3+z3=k

Let x=a, y=b, and z=c.

Then, a3+b3+c3=k

This is a Diophantine equation, which has no general solution. To find particular solutions, we must use trial and error.

Step-by-step explanation:

The required solution of the given inequality is z ≤ 9.

Given that,
To determine the solution of the inequality −2(2z−3)≥2z+24.

Inequality can be defined as the relation of the equation containing the symbol of ( ≤, ≥, <, >) instead of the equal sign in an equation.

Here,
−2(2z−3)≥2z+24
-4z + 6 ≥ 2z + 24
-2z ≥ 18
z ≤ 9

Thus, the required solution of the given inequality is z ≤ 9.

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The value of x using Exterior Angle Theorem is 9.

What is the exterior angle of theorem?

Interior 2 angle = exterior angle

32° + (6x -2)° = 9x + 3

32° + 6x -2° = 9x + 3

30° - 3° = 9x - 6x

3x = 27°

x = 9

Hence, the value of x using Exterior Angle Theorem is 9.

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Answer:

18 miles

Step-by-step explanation:

To determine how many miles Brent can travel, we need to first determine the cost of the base fare and then subtract that from the total amount of money Brent has. The base fare is $1.23, so we can subtract that from the total amount of money Brent has to find out how much money he has left for the per-mile charge:

14.91 - 1.23 = 13.68

Now that we know how much money Brent has left for the per-mile charge, we can divide that amount by the per-mile charge to find out how many miles Brent can travel:

13.68 / 0.76 = 18 miles

Therefore, Brent can travel a maximum of 18 miles given the amount of money he has available to spend on the ride.

Stephen's current average weekly net pay is $354.76. The van he wants to purchase has monthly payments of $325.00. the inequality that correctly shows the comparison between 15% of Stephen's average monthly net pay and the monthly van payment is  $212.86 ≤ $325.00

Information given in the question

Stephen's current average weekly net pay is $354.76

The van he wants to purchase has monthly payments of $325.00.

the inequality that compares 15% of Stephen's average monthly net pay and the monthly van payment = ?

A weekly pay of  $354.76 getting the monthly pay is

= 4 * $354.76

= $1419.04

15% of the monthly pay

= 0.15 * $1419.04

= 212.856

= $212.86

This amount is less than $325.00 which is monthly payment for the van hence  $212.86 ≤ $325.00

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The linear equation written in the slope-intercept form is:

y = (-3/5)*x

The correct option is C.

A general linear equation is written as:

y = a*x + b

Where a is the slope and b is the y-intercept.

Here we can see that the line crosses through the poin (0, 0), so the y-intercept is 0.

b = 0

y = a*x + 0

y = a*x

To find the value of a, we can use another point on the graph.

We can see that the linear equation passes through (5, - 3), replacing these values:

-3 = a*5

-3/5 = a

Then the linear equation is just:

y = (-3/5)*x

The correct option is C.

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On solving the provided question, we got that 18 no. of glasses of punch was Lucas able to fill

a mathematical procedure. The most frequent operations are add, subtract, multiply, and divide . However, there are numerous others, including squaring, taking the square root, logarithms, etc. If it's not a number, it's probably an operation.

9/2= quarts of iced tea

9/4= quarts of lemonade

After mixing,

punch for a party = 9/2 + 9/4 = 27/4

He fills 3/8 =  quart of punch

No. of glasses of punch was Lucas able to fill = 27/4 X  8/3  = 18

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The image by rotation of the triangle ABC with vertices A(x, y) = (8, - 9), B(x, y) = (5, - 4), C(x, y) = (7, - 1) is represented by vertices A'(x, y) = (- 9, - 8), B'(x, y) = (- 4, - 5) and C'(x, y) = (- 1, - 7). The transformation rule is R'(x, y) = (x · cos θ - y · sin θ, x · sin θ + y · cos θ).

In this problem we have the case of triangle set on a Cartesian plane and that must be rotated 90° clockwise, the image can be found by using the following transformation rule:

R'(x, y) = (x · cos θ - y · sin θ, x · sin θ + y · cos θ)

Where:

x, y - Coordinates of the original point.

θ - Angle of rotation, in degrees.

If we know that A(x, y) = (8, - 9), B(x, y) = (5, - 4), C(x, y) = (7, - 1) and θ = - 90°, then the image of the triangle is:

A'(x, y) = (8 · cos (- 90°) - (- 9) · sin (- 90°), 8 · sin (- 90°) + (- 9) · cos (- 90°))

A'(x, y) = (- 9, - 8)

B'(x, y) = (5 · cos (- 90°) - (- 4) · sin (- 90°), 5 · sin (- 90°) + (- 4) · cos (- 90°))

B'(x, y) = (- 4, - 5)

C'(x, y) = (7 · cos (- 90°) - (- 1) · sin (- 90°), 7 · sin (- 90°) + (- 1) · cos (- 90°))

C'(x, y) = (- 1, - 7)

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All of the assertions are false. A statistically significant finding does not necessarily imply that the null hypothesis has been proven incorrect, nor does it indicate how likely it is to be true.

Generally speaking, the word "norm" describes something that is customary, typical, anticipated, or standard. Norms are established definitions of beneficial attitudes and actions that ought to be commonplace, or "the norm," whenever a group is working together. They apply to cooperation and collaboration.

When the variances are known and the sample size is large, a z-test is a statistical test that is used to assess whether two population means differ from one another.

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The other five function values are sin(Ф)= -4/5, tan(Ф)= 4/3, cot(Ф)= 3/4, sec(Ф)= -5/3 and csc(Ф)= -5/4.

Since x is in IIIrd quadrant sin and cos will be negative but tan will be positive

given cos x= -3/5

We know that

sin²x + cos²x=1

sin²x+(-3/5)²=1

sin²x+9/25=1

sin²x=1-9/25

sin²x=16/25

sinx= ±√16/25

sinx= ±4/5

since, x is in IIIrd quadrant.

As sinx is negative IIIrd quadrant,

∴sinx= -4/5

tanx=sinx/cosx

      =-4/5÷-3/5

      =4/3

cotx=1/tanx

      =1/4/3

      =3/4

cosecx=1/sinx

           =1/-4/5

           =-5/4

secx=1/cosx

       =1/-3/5

       =-5/3

Hence, The other five function values are sin(Ф)= -4/5, tan(Ф)= 4/3, cot(Ф)= 3/4, sec(Ф)= -5/3 and csc(Ф)= -5/4.

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Answer:y=2.50x+5

Step-by-step explanation:

y=2.50x+5

that's all i can do with the information provided

The width of the rectangle that maximizes the area is C/2 and the height of the rectangle that maximizes the area is C/2. The maximal area of the rectangle is C2/4.

The gap between a rectangle's length and width must be as small as possible for its area to be as large as possible. Therefore, the length in this scenario must be ceiling (perimeter / 4) and the width will be floor (perimeter /4). The greatest size of a rectangle with a given perimeter is therefore equal to ceiling(perimeter/4) * floor(perimeter/4)

The average of the maximum areas of land planted to a specific crop over the Relevant Period is referred to as the "maximum area for a particular crop."

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The expressions are given to solve and reduce the answer to their simplest form.The x is 1/8.

We move all terms to the left:

7/8x-8-(1/8x-2)=0

Domain of the equation: 8x!=0

x!=0/8

x!=0

x∈R

Domain of the equation: 8x-2)!=0

x∈R

We get rid of parentheses

7/8x-1/8x+2-8=0

We multiply all the terms by the denominator

2*8x-8*8x+7-1=0

We add all the numbers together, and all the variables

2*8x-8*8x+6=0

By multiply elements

16x-64x+6=0

We add all the numbers together, and all the variables

-48x+6=0

We move all terms containing x to the left, all other terms to the right

-48x=-6

x=-6/-48

x    =   1/8

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Answer:

x - 2 [tex]\frac{3}{4}[/tex] = 13 [tex]\frac{1}{4}[/tex]

x = 16

Step-by-step explanation:

x - 2 [tex]\frac{3}{4}[/tex] = 13 [tex]\frac{1}{4}[/tex]

x - [tex]\frac{11}{4}[/tex] = [tex]\frac{53}{4}[/tex]  Add [tex]\frac{11}{4}[/tex] to both sides

x = [tex]\frac{53}{4}[/tex] + [tex]\frac{11}{4}[/tex]

x = [tex]\frac{64}{4}[/tex]

x = 16

Answer:

[tex]x-2 \frac{3}{4}=13 \frac{1}{4}[/tex]

(where x is the original length of the pipe).

Original length of the pipe = 16 feet

Step-by-step explanation:

Let x be the original length of the pipe.

Given a plumber cuts 2 3/4 feet from a pipe and now has a pipe that is 13 1/4 feet long, the equation that models this is:

[tex]\boxed{ x-2 \frac{3}{4}=13 \frac{1}{4}}[/tex]

To determine the length of the original pipe, solve the equation for x.

Add 2 3/4 to both sides of the equation:

[tex]\implies x-2 \frac{3}{4}+2 \frac{3}{4}=13 \frac{1}{4}}+2 \frac{3}{4}[/tex]

[tex]\implies x=13 \frac{1}{4}}+2 \frac{3}{4}[/tex]

When adding mixed numbers, partition the mixed numbers into fractions and whole numbers, and add them separately:

[tex]\implies x=13 +\dfrac{1}{4}}+2 +\dfrac{3}{4}[/tex]

[tex]\implies x=13 +2 +\dfrac{1}{4}}+\dfrac{3}{4}[/tex]

[tex]\implies x=15 +\dfrac{1+3}{4}[/tex]

[tex]\implies x=15 +\dfrac{4}{4}[/tex]

[tex]\implies x=15 +1[/tex]

[tex]\implies x=16[/tex]

Therefore, the original length of the pipe was 16 feet.

a. The inverse variation relationship between a and b can be expressed as a = k/b^2, where k is a constant. We are given that a is 1 when b is 3, so we can substitute these values into the equation to find k: 1 = k/(3^2). Solving for k, we get k = 9.

Substituting this value of k back into the equation a = k/b^2, we can find a when b is 5: a = 9/(5^2) = 9/25 = 0.36. Therefore, a is 0.36 when b is 5.

b. To find the distance it takes for a car traveling at 65 miles per hour to stop, we can use the equation for direct variation: d = kv^2, where d is the distance, v is the speed, and k is a constant. We are given that it takes 112 feet for a car traveling at 40 miles per hour to stop, so we can substitute these values into the equation to find k: 112 = k(40^2). Solving for k, we get k = 0.0056.

Substituting this value of k back into the equation d = kv^2, we can find the distance it takes for a car traveling at 65 miles per hour to stop: d = 0.0056(65^2) = 2256 feet. Therefore, it takes 2256 feet for a car traveling at 65 miles per hour to stop.

The  simplest radical from BC = 2/3.

AC = 26 and BC = 10.

Therefore, BC = 26 - 10 = 14.

we need to find the simplest radical from BC = 14. A simple radical is a fraction with no common factors other than 1 and itself. The simplest radical from BC = 14 is 2/3

Simply put, simplifying a radical eliminates the need to find any further square roots, cube roots, fourth roots, etc. Additionally, it entails eliminating any radicals from a fraction's denominator.

The radical consists of three separate parts:

The Complete Question is :

AC = 26 and BC = 10   find the simplest radical from BC =

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Answer:

[tex](-3, 4)[/tex]

Step-by-step explanation:

The horizontal component is [tex]-3[/tex] and the vertical component is [tex]4[/tex]

The component (-3, 4) from the given vectors are shown ​in the graph.

Let A = v 1, v 2,..., v r be a collection of vectors from Rn. If r > 2 and at least one of the vectors in A can be expressed as a linear combination of the others, A is said to be linearly dependent. The rationale for this description is straightforward: at least one of the vectors is dependent (linearly) on the others. If no vector exists in A, the set is said to be linearly independent. It is also usual to state "the vectors are linearly dependent (or independent)" rather than "the set comprising these vectors is linearly dependent (or independent)."

We have been given that the graph in the question

Here v = -3i + 4j

This means the component (-3, 4) of the provided vectors.

Therefore, the required component (-3, 4) from the provided vectors are depicted ​in the graph.

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