6x-5y=5
3x+5y=4
The x-coordinate of the solution to this system of equation is

Answer:

4.DH

Step-by-step explanation:

let's assume the point P to be the point that X will be after it is  dilated.

we know that after dilation the length of PM should be two two times the length of XY

PM = 2.XY ===>[tex]\frac{PM}{XY}[/tex] = 2

and from proportionality theorem we now that:

[tex]\frac{PM}{XY}[/tex] = [tex]\frac{MX}{MP}[/tex] = 2

So we know XY should be half the size of MP  and we can see the only line matching is DH thus the answer is DH

Rectangles and squares always have congruent diagonals due to their geometrical properties, making option c (rectangle, square) the correct answer.

The question asks which of the given sets of quadrilaterals must have congruent diagonals. The key to answering this question is understanding the properties of each kind of quadrilateral mentioned. Rectangles and squares always have congruent diagonals because their diagonals bisect each other and are of equal length due to the right angles at the corners of the shapes.

A rhombus, while having diagonals that bisect each other at right angles, does not necessarily have congruent diagonals because the angles between adjacent sides do not guarantee equal lengths of diagonals. As such, the correct answer is c.rectangle, square, where both of these shapes must have diagonals of equal length due to their geometrical properties.

Answer:

Step-by-step explanation:

[tex]\left\{\begin{array}{ccc}y=-2x+12&(1)\\3y-x+6=0&(2)\end{array}\right\qquad\text{substitute (1) to (2):}\\\\3(-2x+12)-x+6=0\qquad\text{use the distributive property}\\(3)(-2x)+(3)(12)-x+6=0\\-6x+36-x+6=0\qquad\text{combine like terms}\\(-6x-x)+(36+6)=0\\-7x+42=0\qquad\text{subtract 42 from both sides}\\-7x=-42\qquad\text{divide both sides by (-7)}\\\boxed{x=6}\qquad\text{put it to (1)}\\\\y=-2(6)+12\\y=-12+12\\\boxed{y=0}[/tex]

Answer:

[tex]\text{\fbox{(6,~0)}}[/tex]

Step-by-step explanation:

[tex]\left \{ {{\text{y~=~-2x~+~12}} \atop {\text{3y~-~x~+~6~=~0}} \right. \\ \\ \text{We~already~have~the ~value~of ~y ~so~ substitute~ this~ value~~ of ~y ~into~ the ~second ~equation.} \\ \\ \text{3(-2x~+~12)~-~x~+~6~=~0} \\ \\ \text{Distribute~ 3 ~inside~ the~ parentheses.} \\ \\ \text{-6x~+~36~-~x~+~6~=~0} \\ \\ \text{Combine~ like~ terms. ~You ~can~ subtract~ -6x ~and ~x ~and ~add ~36 ~and ~6.} \\ \\ \text{-7x~+~42~=~0} \\ \\ \text{Subtract~ 42 ~from~ both~ sides ~of~ the ~equation.} \\ \\ \text{-7x~=~-42} \\ \\ \text{Now ~solve~ for ~x ~by ~dividing~ both~ sides ~by~ -7.} \\ \\ \text{\fbox{x~=~6}} \\ \\ \text{To~ find~ y, ~substitute~ 6 ~for~x~ into~ the~first~ equation.} \\ \\ \text{y~=~-2(6)~+~12} \\ \\ \text{Multiply ~-2~ and~ 6.} \\ \\ \text{y~=~-12~+~12} \\ \\ \text{Combine~ like ~terms~ to ~complete~ solving~ for ~y.} \\ \\ \text{\fbox {y~=~0}} \\ \\ \text{The~ solution~ to ~this ~system ~of ~equations ~is ~\fbox{(6~,~ 0)}~.}[/tex]

Answer:

AB=20

Step-by-step explanation:

Given:

r= 14.5

AB cuts r=14.5 in two parts one parts length=4

remaining length, x = 14.5 - 4 =10.5

draw a line from center of circle to point A making right angled triangle

Now hypotenuse=r=14.5

and one side of triangle=10.5

Using pythagoras theorem to find the third side:

c^2=a^2+b^2

14.5^2=10.5^2+b^2

14.5^2-10.5^2=b^2

b^2=100

b=10

AB=2b

     =2(10)

     =20

Hence length of cord AB=20!

Answer:

6

Step-by-step explanation:

imputing 5 into g which makes the equation: 2(5) - 4

2(5) - 4

10 - 4

6

Answer:

6

Step-by-step explanation:

2(5)-4

2*5 is 10

10 subtracted by 4 is 6

Hope this helps ^-^

The first step in finding the equation of the line would be to find the slope of the points.

The slope-intercept form of a line is given by:

[tex]\[ y = mx + b \][/tex]

where:

- [tex]\( m \)[/tex] is the slope of the line, and

- [tex]\( b \)[/tex] is the y-intercept.

To find the equation of the line that passes through the points (5, -4) and (-1, 8) in slope-intercept form, you need to follow these steps:

1. Find the slope [tex](\( m \))[/tex]:

  The slope [tex](\( m \))[/tex] is given by the formula:

  [tex]\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \][/tex]

  Pick two points, let's say (x1, y1) = (5, -4) and (x2, y2) = (-1, 8), and substitute them into the formula to find [tex]\( m \)[/tex].

 [tex]\[ m = \frac{{8 - (-4)}}{{-1 - 5}} \][/tex]

  Simplify the expression to find [tex]\( m \)[/tex].

2. Use the slope and one of the points to find the y-intercept [tex](\( b \))[/tex]:

  Substitute the slope [tex](\( m \))[/tex] and one of the points (let's use (5, -4)) into the slope-intercept form equation and solve for [tex]\( b \)[/tex].

  [tex]\[ -4 = m \cdot 5 + b \][/tex]

  Substitute the value of [tex]\( m \)[/tex] that you found in step 1 and solve for [tex]\( b \)[/tex].

3. Write the equation in slope-intercept form:

  Once you have the values of [tex]\( m \) and \( b \)[/tex], substitute them into the slope-intercept form equation.

  [tex]\[ y = mx + b \][/tex]

  Write the final equation.

By following these steps, you can find the equation of the line passing through the given points in slope-intercept form.

To find the equation of the line passing through (5,-4) and (-1,8), calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1), which results in a slope of -2.

The first step in finding the equation of the line that passes through the points (5,-4) and (-1,8) in slope-intercept form is to calculate the slope (m) of the line. The slope of a line is determined by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, we would use the following calculation:

With the slope, you can then use the point-slope form or directly the slope-intercept form (y = mx + b) to find the equation, using either of the two points to solve for b, the y-intercept.

Answer:

○C. x + 2

Step-by-step explanation:

x^4 + 2x^3 -x-2

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

x^3 -1

Factor the numerator by grouping.  Take an x^3 from the first 2 terms and -1 from the last 2 terms

x^3( x + 2) -1(x+2)

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

x^3 -1

now lets factor out the x+2

( x + 2)(x^3 -1)

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

x^3 -1

Canceling out the x^3-1, we are left with

x+2

Answer:

C. x+2

Step-by-step explanation:

The given expressions are two polynomials which have to be divided in order to find the quotient. The long division method will be used to find the quotient of the two terms.

The long division is done and the picture is attached for detail.

From the picture, we can see that the correct answer is:

C. x+2 ..

Answer:

Option C 3 + 4 = 9 or 5 · 2 = 11 is correct answer

Step-by-step explanation:

Disjunction states that if we have p or q then the disjunction is true if either p is true or q is true or p and q are true.

The disjunction is false when both p and q both are false.

1. 2 · 3 = 6 or 4 + 5 = 10

Disjunction is true because 2.3 =6 is true while 4+5=10 is false

2.  5 - 3 = 2 or 3 · 4 = 12

Disjunction is true because 5-3 =2 is true and 3.4=12 is also true.

3. 3 + 4 = 9 or 5 · 2 = 11

Disjunction is false because 3+4 =7 and not 9 and 5.2 =10 and not 11. Since both are false so, this disjunction is false.

4. 6 · 2 = 11 or 3 + 5 = 8

Disjunction is true because 6.2=12 and not 11 is false but 3+5 = 8 is true.

So, Option C 3 + 4 = 9 or 5 · 2 = 11 is correct answer.

Answer: cos(C)

Step-by-step explanation: Use SohCahToa. Sin(A) is opposite over hypotenuse. The opposite is line BC. Use angle C. To get line BC, you will need the adjacent, and also the hypotenuse. Cos will get you this. Therefore, your answer is Cos(C).

The one which is equivalent to [tex]sin(A)[/tex] will be [tex]Cos(C)[/tex] .

Trigonometric ratios are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

There are six trigonometric ratios [tex]Sin\theta,Cos\theta,Tan\theta,Cot\theta,Sec\theta,Cosec\theta[/tex] .

We have,

[tex]\triangle ABC[/tex] Right angled at  [tex]B[/tex] .

So,

[tex]sin(A)=\frac{Perpendicular}{Height}[/tex]

So, in  [tex]\triangle ABC[/tex] ,

[tex]sin(A)=\frac{BC}{AC}[/tex]

Now,

To find its equivalent we will look from angle [tex]C[/tex] ,

So, in options we two  trigonometric ratios from angle [tex]C[/tex],

So, take  [tex]Cos(C)[/tex] ,

[tex]Cos(C)=\frac{Base}{Hypotenuse}[/tex]

[tex]Cos(C)=\frac{BC}{AC}[/tex]

i.e. Equivalent to [tex]sin(A)[/tex]  is [tex]Cos(C)[/tex]

Hence, we can say that the one which is equivalent to [tex]sin(A)[/tex] will be [tex]Cos(C)[/tex] which is in option [tex](c)[/tex] .

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Answer:

A''(1,-1) B''(2,2) C''(5,2)

Step-by-step explanation:

Points A(1,−5) B(2,−2) C(5,−2)

reflection over y=-1

Perpendicular distance between points y-coordinates of points (A, B and C) and y=-1 are 4,1 and 1

after reflections, the perpendicular distance will be 8,2,2 and the points will be at

A'(1,3) B'(2,0) C'(5,0)

again  reflection over y=1

Perpendicular distance between points y-coordinates of points (A', B' and C') and y=1 are 2,1 and 1

after reflections, the perpendicular distance will be 4,2,2 and the points will be at

A''(1,-1) B''(2,2) C''(5,2)!

[tex]\huge{\boxed{y=-\frac{5}{6} x+4}}[/tex]

Parallel lines share the same slope, so the slope of the parallel line in this case must be [tex]-\frac{5}{6}[/tex].

Point-slope form is [tex]y-y_1=m(x-x_1)[/tex], where [tex]m[/tex] is the slope and [tex](x_1, y_1)[/tex] is any known point on the line.

Plug in the values. [tex]y-(-1)=-\frac{5}{6} (x-6)[/tex]

Simplify and distribute. [tex]y+1=-\frac{5}{6} x+5[/tex]

Subtract 1 from both sides. [tex]\boxed{y=-\frac{5}{6} x+4}[/tex]

Answer:

y = -5/6x   +   4        (slope - intercept form)

OR

5x + 6y -24 = 0          (standard form)

Step-by-step explanation:

What is the equation of the line that goes through the point (6,-1) and is parallel to the line represented by the equation below?

y=-5/6 x+3

To solve this;

We need to first find the slope of the the equation given

Comparing the equation given with y=mx + c, the slope (m) = -5/6, any equation parallel to this equation will have the same slope as this equation.

Since our new equation is said to be parallel to this equation the slope(m) of our new equation is also -5/6.

Now we will proceed to find the intercept of our new equation, to find the intercept, we will simply plug in the value of the points given and the slope into the formula y=mx + c and then simplify

The value of the points given are; (6, -1) which implies x=6 and y=-1  slope(m)= -5/6

y = mx + c

-1 = -5/6 (6)  +  c

-1 = -5  +  c

Add 5 to both-side of the equation to get the value of c

-1+5 = -5+5 + c

4 = c

c=4

Therefore the intercept(c) of our new equation is 4

We can now proceed to form our new equation. To form the equation, all we need to do is to simply insert the value of our slope (m) and intercept (c) into y = mx +  c

y = -5/6x   +   4

This above equation is in slope-intercept form, we can further simplify it to be in the standard form.

6y = -5x + 24

5x + 6y -24 = 0

Answer:

y=4

Step-by-step explanation:

-9x - 3y = 6

Let x =-2

-9(-2) -3y =6

18 -3y =6

Subtract 18 from each side

18-18-3y = 6-18

-3y = -12

Divide each side by -3

-3y/-3 =-12/-3

y=4

Answer:

[tex]\large\boxed{a_n=6\left(\dfrac{1}{10}\right)^{n-1}}[/tex]

Step-by-step explanation:

Check:

[tex]n=1\\\\a_1=6\left(\dfrac{1}{10}\right)^{1-1}=6\left(\dfrac{1}{10}\right)^0=6(1)=6\qquad\bold{CORRECT}\ (1,\ 6)\\\\n=2\\\\a_2=6\left(\dfrac{1}{10}\right)^{2-1}=6\left(\dfrac{1}{10}\right)^1=6\left(\dfrac{1}{10}\right)=\dfrac{6}{10}=0.6\qquad\bold{CORRECT}\ (2,\ 0.6)\\\\n=3\\\\a_3=6\left(\dfrac{1}{10}\right)^{3-1}=6\left(\dfrac{1}{10}\right)^2=6\left(\dfrac{1}{100}\right)=\dfrac{6}{100}=0.06\qquad\bold{CORRECT}\ (3,\ 0.06)[/tex]

Answer:

[tex]a_n=6\left(\frac{1}{10}\right)^{n-1}[/tex]

Option 3 is correct

Step-by-step explanation:

The coordinates are (1,6) (2,0.6) and (3,0.06)

If we make table of given coordinate:

x   :    1          2         3  

y   :   6        0.6     0.06

[tex]a_1=6,a_2=0.6,a_3=0.06[/tex]

Ratio of the sequence:

[tex]r=\dfrac{a_2}{a_1}=\dfrac{0.6}{6}=0.1[/tex]

Formula of geometric sequence:

[tex]a_n=ar^{n-1}[/tex]

[tex]a_n=6\cdot 0.1^{n-1}[/tex]

[tex]a_n=6\left(\frac{1}{10}\right)^{n-1}[/tex]

Hence, The sequence model by [tex]a_n=6\left(\frac{1}{10}\right)^{n-1}[/tex]

Answer:

The smaller angle= 68°

The larger angle=112°

Step-by-step explanation:

Supplementary angles add up to 180°

Let the smaller of the angles to x then the larger angle will be x+44.

Adding the two then equating to 180°:

x+(x+44)=180

2x+44=180

2x=180-44

2x=136

x=68

The smaller angle= 68°

The larger angle=68+44=112°

Answer:

The angles are 68° , 112°

Step-by-step explanation:

Let the smaller angle be x

so the larger angle = x + 44

x , x + 44 are supplementary.

so,    x  + (x + 44) = 180

         x  + x + 44  = 180

                      2x  = 180 - 44 = 136

                          x = 136/2 = 68    

 the larger angle = x + 44 = 68 + 44 = 112

Answer:

1 4/21

Step-by-step explanation:

(5/9*3/5)+6/7

The 5's cancel

(3/9)+6/7

Cancel a 3 in the numerator and a 3 in the denominator

1/3 + 6/7

We need to get a common denominator of 21

1/3 *7/7  + 6/7 *3/3

7/21 + 18/21

25/21

This is an improper fraction

21 goes into 25 1 time with 4 left over

1 4/21

Answer: [tex]\frac{25}{21}[/tex]

Step-by-step explanation:

The first step is to make the multiplication of the fractions inside the parentheses. To do this, you must multiply the numerator of the first fraction by de numerator of the second fraction and the denominator of the first fraction by the denominator of the second fraction:

[tex](\frac{5}{9}*\frac{3}{5})+\frac{6}{7}=\frac{15}{45}+\frac{6}{7}[/tex]

Now you can reduce the fraction  [tex]\frac{15}{45}[/tex]:

[tex]=\frac{1}{3}+\frac{6}{7}[/tex]

 And make the corresponding addition: in this case the Least Common Denominator (LCD) will be the multiplication of the denominators.   Divide each denominator by the LCD and multiply this quotient by the corresponding numerator and then add the products. Therefore you get:

[tex]=\frac{7+18}{21}=\frac{25}{21}[/tex]

Answer:

the Answer is 60 miles.

Step-by-step explanation:

The current of the river is towards city B; hence, the reason the ferry moves faster from town A to B than when coming back from town B to A. The distance between both towns is 60m.

Let

[tex]s \to[/tex] speed in still water

[tex]d \to[/tex] distance between both cities

The given parameters are:

[tex]t_{AB} =2[/tex] -- from A to B

[tex]t_{BA} =1.5[/tex] -- from B to A

[tex]s_c = 5mph[/tex] -- speed of the current

Speed is calculated as:

[tex]Speed = \frac{Distance}{Time}[/tex]

The speed (s) from town A to town B is:

[tex]s = s_c + \frac{d}{t_{AB}}[/tex] --- i.e. speed of the current + speed in still water from A to B

[tex]s = 5 + \frac{d}{2}[/tex]

Multiply through by 2

[tex]2s = 10 + d[/tex]

Make d the subject

[tex]d = 2s - 10[/tex]

The speed (s) from town B to town Ais:

[tex]s = -s_c + \frac{d}{t_{BA}}[/tex] --- i.e speed in still water from B to A - . speed of the current

[tex]s = -5 + \frac{d}{1.5}[/tex]

Multiply through by 1.5

[tex]1.5s = -7.5 + d[/tex]

Make d the subject

[tex]d = 1.5s + 7.5[/tex]

So, we have:

[tex]d = 1.5s + 7.5[/tex] and [tex]d = 2s - 10[/tex]

Equate both values of d

[tex]2s - 10 = 1.5s + 7.5[/tex]

Collect like terms

[tex]2s - 1.5s= 10 + 7.5[/tex]

[tex]0.5s= 17.5[/tex]

Divide both sides by 0.5

[tex]s = 35[/tex]

Substitute [tex]s = 35[/tex] in [tex]d = 1.5s + 7.5[/tex] to calculate distance (d)

[tex]d =1.5 *35 + 7.5[/tex]

[tex]d =52.5 + 7.5[/tex]

[tex]d =60[/tex]

The distance between town A and B is 60m.

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Given

106 + (147x + 92)

Combine like terms

106 + 92 = 198

Simplify

147x + 198

Answer

147x + 198

106 + (147x + 92) = what

To solve this expression, you need to apply the distributive property, which states that a(b + c) = ab + ac.

Therefore, 106 + (147x + 92) = 106 + 147x + 92, which simplifies to 239 + 147x.

Answer:

The correct answer option is B. 2.1 inches.

Step-by-step explanation:

We are given that the cylinder is to hold 42 in³ of liquid while the height of this container is 3 inches tall.

We are to find the best approximation for the radius of this container.

We know that the formula for the volume of a cylinder is given by:

Volume of cylinder = [tex]\pi r^2h[/tex]

Substituting the given values in the above formula to get:

[tex]42=\pi \times r^2 \times 3[/tex]

[tex]r^2 = \frac{42}{\pi \times 3 }[/tex]

[tex]\sqrt{r^2} = \sqrt{4.46}[/tex]

r = 2.1 inches

Answer:

A

Step-by-step explanation:

since you know the measure of the angle that isn't a is 50, because 180 - 130 = 50, and since the sum of all the angles in the triangle has to be 180, y = 130/2 which equals 65

Answer: C

Step-by-step explanation:

First, you should see that the bottom right corner is an angle on the other side. So subtract 180 from 95 to get 85. Since all angles in a triangle add up to 180, You do 85 + 60 + x = 180. You simplify further to get 145 + x = 180.

Subtracting 145 from both sides leaves you with x = 35, which gives you C.

So C is the correct answer.

Answer:

1st piece = 15 inch

2nd piece = 15 inch

3rd piece = 30 inch

Step-by-step explanation:

2.

Let length of first piece be x

since 2nd piece is same, so 2nd piece's length is also x

third piece is TWICE, so its length is 2x

Total length of all the 3 pieces is 60, so we setup an equation and solve for x:

x + x + 2x = 60

4x = 60

x = 60/4 = 15

Hence, first piece is 15, second piece is 15, third piece is 2(15) = 30

Answer:

if the length, width and height of a cube quadruple, then the volume of the cube will be multiplied by 64.

Step-by-step explanation:

Given a cube of width W, height H and length L, the volume of the cube will be: Volume = W*H*L

If all the three parameters quadruple, then:

New Width = 4W

New Height = 4H

New Length = 4L

New Volume = 4W*4H*4L = 64WHL

Therefore, if the length, width and height of a cube quadruple, then the volume of the cube will be multiplied by 64.

Answer:

Volume will be multiplied by 64.

Step-by-step explanation:

We are to find the effect of quadrupling the length, width and height of a cube on its volume.

Since the length, width and height of a cube are all of the same dimension, let us represent it was a variable [tex] S [/tex].

[tex] V o l u m e = S ^ 3 [/tex]

[tex]New Volume = (4S)^3 =64S^3[/tex]

Therefore, if the length width and height of a cube all are quadrupled, the volume will be 64 times.

Answer:

3 is the correct option.

Step-by-step explanation:

The given expression is:

3k+6/(k-2)+(2-k)

Break the numerators:

3k/(k-2) + 6/(2-k)

Now Re-arrange the term (2-k) in the denominator as (-k+2)

3k/(k-2) + 6/(-k+2)

Now takeout -1 as a common factor from (-k+2)

3k/(k-2) + 6/-1(k-2)

Now  move a negative (-1)from the denominator of 6/-1(k-2) to the numerator

3k/(k-2) + -1*6/(k-2)

Now take the L.C.M of the denominator which is k-2 and solve the numerator

3k - 6/ (k-2)

Take 3 as a common factor from the numerator:

3(k-2)/(k-2)

k-2 will be cancelled out by each other:

Thus the answer will be 3.

The correct option is 3....

Answer:

f(- 5) = 13

Step-by-step explanation:

To evaluate f(- 5) substitute x = - 5 into f(x)

f(- 5) = - 3 × - 5 - 2 = 15 - 2 = 13

Answer:

c. 1; rational

Step-by-step explanation:

k² − 10k + 25 = 0

The discriminant of ax² + bx + c is b² − 4ac.

If the discriminant is negative, there are no real roots.

If the discriminant is zero, there is 1 real root.

If the discriminant is positive, there are 2 real roots.

If the discriminant is a perfect square, the root(s) are rational.

If the discriminant isn't a perfect square, the root(s) are irrational.

Finding the discriminant:

a = 1, b = -10, c = 25

(-10)² − 4(1)(25) = 0

The discriminant is zero, so there is 1 rational root.

Answer:

f(-5) = 1/ 243

Step-by-step explanation:

f(x) = 3^x

Let x=-5

f(-5) = 3^-5

Since the exponent is negative, it will move to the denominator

f(-5) = 1/3^5

f(-5) = 1/ 243

For this case we have the following function:

[tex]f (x) = 3 ^ x[/tex]

We must evaluate the function for[tex]x = -5[/tex]

So, we have:

[tex]f (-5) = 3 ^ {-5}[/tex]

By definition of power properties it is fulfilled that:

[tex]a ^ {- 1} = \frac {1} {a ^ 1} = \frac {1} {a}[/tex]

Thus:

[tex]f (-5) = \frac {1} {3 ^ 5} = \frac {1} {3 * 3 * 3 * 3 * 3} = \frac {1} {243}[/tex]

Answer:

[tex]\frac {1} {243}[/tex]

Answer:

it would take roughly 6 and a half minutes i think

Let's consider each of the options and evaluate whether the given number is rational or not. Remember, a rational number is any number that can be expressed as the quotient or fraction of two integers (where the denominator is not zero).
1. **3•π**: This number is not rational because π (pi) is an irrational number. An irrational number is a number that cannot be expressed as a simple fraction - its decimal goes on forever without repeating. When you multiply an irrational number by an integer (in this case 3), the result is still irrational.
2. **2/3 + 9.26**: To determine if this sum is rational, we can evaluate each addend. The fraction 2/3 is clearly rational, as it is already expressed as a quotient of two integers. The decimal 9.26 can be expressed as a fraction because it is a terminating decimal; in fraction form, it is \( \frac{926}{100} \) which simplifies to \( \frac{463}{50} \) when reduced to lowest terms. The sum of two rational numbers is also rational (since both can be written as fractions, and the sum of two fractions is a fraction), so this number is indeed rational.
3. **45 + 36**: Both 45 and 36 are integers and the sum of two integers is also an integer. Integers are a subset of rational numbers, because they can be expressed as a fraction with a denominator of 1 (e.g., \( \frac{45}{1} + \frac{36}{1} \)). Thus, this number is rational.
4. **14.3 + 5.78765239**: The number 14.3 is a terminating decimal and can be represented as a fraction (\( \frac{143}{10} \)). However, 5.78765239 is given without any indication that it is a repeating or terminating decimal. If it is a non-repeating and non-terminating decimal, then it cannot be expressed as a fraction and would be considered irrational. Without further information, we cannot determine if this number is rational or not. Therefore, we can neither confirm nor deny that the sum is rational.
Given these considerations, the rational option from the given choices is **45 + 36**.

Answer:

[tex]a_n=9.5 \cdot (0.2)^{n-1}[/tex]

Step-by-step explanation:

If this is a geometric sequence, it will have a common ratio.

The common ratio can be found by dividing term by previous term.

The explicit form for a geometric sequence is [tex]a_n=a_1 \cdot r^{n-1} \text{ where } a_1 \text{ is the first term and } r \text{ is the common ratio}[/tex]

We are have the first term is [tex]a_1=9.5[/tex].

Now let's see this is indeed a geometric sequence.

Is 0.076/0.38=0.38/1.9=1.9/9.5?

Typing each fraction into calculator and see if you get the same number.

Each fraction equal 0.2 so the common ratio is 0.2.

So the explicit form for our sequence is

[tex]a_n=9.5 \cdot (0.2)^{n-1}[/tex]

Final answer:

A geometric sequence follows a specific pattern where each term is obtained by multiplying the previous term by a constant ratio. The explicit rule for a geometric sequence is defined by the first term, the term number, and the common ratio.

Explanation:

Geometric series are sequences in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The explicit rule for a geometric sequence is of the form an = a₁ * r⁽ⁿ⁻¹⁾, where a₁ is the first term, n is the term number, and r is the common ratio.

[tex]\displaystyle\\\text{We will use numbers from 1 to 9.}\\\\1+2+3+4+5+6+7+8+9=\frac{9(9+1)}{2}=\frac{9\times10}{2}=\frac{90}{2}=\boxed{\bf45}\\\\\text{the sums of the numbers in each row, in each column are }=\frac{45}{3}=\boxed{\bf15}\\\\\text{Solution:}\\\\\boxed{\,2\,}\boxed{\,7\,}\boxed{\,6\,}\\\boxed{\,9\,}\boxed{\,5\,}\boxed{\,1\,}\\ \boxed{\,4\,}\boxed{\,3\,}\boxed{\,8\,}\\\\\text{Convenient rotation of the square gives 8 solutions.}[/tex]