The question is: Find a recurrence relation for number of ternary strings of length n that contain two consecutive zeros.
I know for ternary strings with length one, there are 0. For a length of 2, there is just 1 (00), and for a length of 3, there are 5 (000,001,002,100,200).
I did a similar problem, finding a relation for the number of bit strings of length n with two consecutive zeros:
an=an−1+an−2+2n−2
Since you can add "1" to the end of all the an−1
strings, "10" to all the an−2
strings, and "00" any string of size n−2
.
For the ternary string problem, I'm pretty sure you would replace the 2n−2
with 3n−2
, but confused about the other terms of the relation. My guess is that it would have the coefficient 2
in front of the other terms, since you can add either 1
or 2
to the end of an−1
and either 01
or 02
at the end of an−1
.
So I believe the answer for the relation is:
an=2an−1+2an−2+3n−2

The vector equation of system of equations is:

[tex]x_1\begin{bmatrix}a_1\\.\\.\\a_n\end{bmatrix}+x_n\begin{bmatrix}a_{m1}\\.\\.\\a_{mn}\end{bmatrix}=\begin{bmatrix}b_1\\.\\.\\b_n\end{bmatrix}[/tex]

And the matrix equation is,

[tex]\begin{bmatrix} 3 & -1\\10 & 8\\6 & -1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}4\\2\\1\end{bmatrix}[/tex]

The system of eqyuations is:

3x₁- x₂ = 4

10x₁ + 8x₂ = 2

6x₁ - x₂ = 1

Consider the system of equations :

a₁x₁ + . . . + [tex]a_m x_m[/tex]= b₁

.

.

[tex]a_n[/tex]x₁ + . . . + [tex]a_{nm} x_{nm}[/tex] = [tex]b_n[/tex]

We know that the vector form of above system of equations is:

[tex]x_1\begin{bmatrix}a_1\\.\\.\\a_n\end{bmatrix}+x_n\begin{bmatrix}a_{m1}\\.\\.\\a_{mn}\end{bmatrix}=\begin{bmatrix}b_1\\.\\.\\b_n\end{bmatrix}[/tex]

So, required vector equation would be,

[tex]x_1\begin{bmatrix}3\\10\\6\end{bmatrix}+x_2\begin{bmatrix}-1\\8\\-1\end{bmatrix}=\begin{bmatrix}4\\2\\1\end{bmatrix}[/tex]

and the required matrix equation would be,

[tex]\begin{bmatrix} 3 & -1\\10 & 8\\6 & -1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}4\\2\\1\end{bmatrix}[/tex]

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The completed table using this function rule is,

Input    (x)    1    4    5   7

Output (y)    2   11   14   26

Function rule is the rule of writing the relationship between the two variables, one is dependent and another is independent.

The table given in the problem is;

Input    (x)    1    4    5   7

Output (y)

Thus we need to write such a function, which gives the value of (y).

y=3x-1

Complete the table using the above function rule;

At (x) equal to 1,

y=3x-1

y=3(1)-1

y = 2

At (x) equal to 4,

y=3x-1

y=3(4)-1

y = 11

At (x) equal to 5,

y=3x-1

y=3(5)-1

y = 14

At (x) equal to 7,

y=3x-1

y=3(7)-1

y = 26

Hence,  the completed table is,

Input    (x)    1    4    5   7

Output (y)    2   11   14   26

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The solution is, In the binomial expansion (2x + 3)^5 , there are 6 terms.

The formula for expanding the exponential power of a binomial expression is provided by the binomial theorem, sometimes referred to as the binomial expansion. The following is the binomial expansion of (x + y)n using the binomial theorem:

(x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + ... + nCn-1 x1yn-1 + nCn x0yn

here, we have,

According to the question, given that

Binomial expansion  (2x + 3)^5

Number of terms in a binomial expansion of (x + y)^n is

N = n + 1 words in total

In the binomial expansion (2x + 3)^5

n = 5

N = 5 + 1 = 6

Therefore, In Binomial expansion  (2x + 3)^5 there are 6 terms.

The algebraic expression (x + y)n can be expanded according to the binomial theorem, which represents it as a sum of terms using separate exponents of the variables x and y. Each word in a binomial expansion has a coefficient, which is a numerical value.

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The constant of the expression represents the  installation charge, and the coefficient of x represents the number of computer.

Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.

Given that, an equation that the total cost of the computers, including an installation charge for all the computers:

y = $115x+$100

y is the total cost.

Here the constant of the expression $100 represents the installation charge and the coefficient of x represents the number of computer,

For 1 computer, the total cost be:

y = $215

For 2 computers the total cost be:

y = $330

Hence, the constant of the expression represents the  installation charge, and the coefficient of x represents the number of computer.

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You posted a lot of questions. I'll do the first three to get you started.

==================================================

Problem 1

Explanation:

Draw a vertical line to enclose the un-shaded region. Think of it like adding fencing to enclose a paddock or backyard.

What results are two rectangles. The larger rectangle has area = length*width = 8*10 = 80 square inches.

The smaller unshaded rectangle inside has area of 5*5 = 25 square inches.

The difference of those areas is:  80-25 = 55

You have the correct answer. Nice work.

==================================================

Problem 2

Explanation:

Follow the same set of steps as done in the previous problem. Draw a vertical line to form a larger rectangle.

The larger rectangle has area of 18*31.5 = 567 square feet.

The smaller unshaded rectangle has area of 9*9 = 81 square feet.

Subtract those results to get the shaded region only: 567-81 = 486

==================================================

Problem 3

Explanation:

This time we don't have to add any extra lines to enclose the figure.

A = larger area = 6*2.4 = 14.4

B = smaller unshaded area = 2*4.4 = 8.8

C = A-B = 14.4-8.8 = 5.6 square cm

You have the correct answer. Nice work.

Answer:

1)  55 in²

2)  486 ft²

3)  5.6 cm²

Step-by-step explanation:

To calculate the area of each given figure, subtract the area of the cut-out rectangle (marked in blue on the attached diagram) from the area of the larger rectangle.

[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&=8 \cdot 10- 5\cdot 5\\&=80-25\\&=55\; \sf in^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 31.5\cdot 18- 9\cdot 9\\&=567-81\\&=486\; \sf ft^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 6\cdot 2.4- 4.4\cdot2 \\&=14.4-8.8\\&=5.6\; \sf cm^2\end{aligned}[/tex]

The linear equation in the form of y = c is the horizontal line, parallel to the x-axis passing through the point (0,c), because in this equation the x-intercept is 0.

A linear equation is an equation in which the highest power of the variable is always 1.

The line y = c, is always parallel to x-axis because here if we compare to general equation of a line y = mx+c, the m is zero that means the slope is zero and also the x-intercept is zero.

Therefore, the line y = c is the horizontal line, parallel to the x-axis.

Hence, the linear equation in the form of y = c is the horizontal line, parallel to the x-axis passing through the point (0,c), because in this equation the x-intercept is 0.

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The entries of x, 5, 2, and -4, are the coefficients in front of the vectors v1, v2, and v3 respectively.

To find the vectors v1, v2, and v3, we need to perform the matrix-vector multiplication Az = A * x.

A = [3 -2 -4 -5 -2 3]

x = [5 2 -4]

Az = [3 -2 -4 -5 -2 3] * [5 2 -4] = [35 -8 -52 -34 -12 22]

So, the vectors v1, v2, and v3 can be represented as follows:

v1 = [35]

v2 = [-8]

v3 = [-52 -34 -12 22]

These vectors represent the components of the result Az in the same order as they appear in the multiplication. The values of x, 5, 2, and -4, are the coefficients in front of the vectors v1, v2, and v3 respectively.

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____ The given question is incomplete, the complete question s given below:

Consider the matrix A = [ 3 -2 -4 -5 -2 3 ] and the vector x = [5 2 -4] When multiplying A by z on the right, the result Az is a linear combination of three vectors v1, v2, v3, where the entries of x play the role of the coefficients in front of those vectors, as follow (5)v1 + (2)v2 + ( - 4)v3. What are these vectors v1, v2, v3.

The function is defined as g(x) = f (√3x² + 4 ), where "f" is an unknown function.

A function is a mathematical object that assigns a unique output or value for each input. In other words, given an input, a function produces exactly one output.

To understand this function and its properties, we need to first identify and verify the domain and range of the function. The domain of a function is the set of all possible inputs for which the function produces a real output. In this case, the domain of "m" is the set of all real numbers "x" such that the expression √3x² + 4 is real.

Next, we need to understand the composition of functions. In this case, "m" is defined as the composition of two functions, "g" and "f."

The composition of two functions "f" and "g" is defined as "f(g(x))," which means that the output of "g" is used as the input for "f." In other words, we first evaluate "g" for a given value of "x," and then use the output of "g" as the input for "f."

Finally, we can evaluate the function "m" for a specific value of "x" by first evaluating "g" and then "f."

To do this, we simply substitute the value of "x" into the expression for "g" and then evaluate "f" with the resulting output.

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The point satisfies the given system of inequalities is (1, -1)

A system of inequalities is a set of two or more inequalities in one or more variables.

Given is a system of inequalities y < -x+2 and -2x+y < -2,

We know that, the solution set of a system of inequalities is given by studying the graph of the system, The solution set the region covered by lines of the both inequalities.

Plotting the graphs, of the system of inequalities, we get, the set of solution is the area covered with purple shade,

In that area only, (1, -1) lies.

Therefore, the point (1, -1) is the solution of the given system of inequalities

Hence, the point satisfies the given system of inequalities is (1, -1)

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The average rate of change from x = 4 to x = 9 is given as follows:

-1.

The average rate of change of a function is given by the change in the output divided by the change in the input.

The numeric values are given as follows:

f(4) = 25, f(9) = 20.

Hence the change in the output is of:

20 - 25 = -5.

The change in the input is of:

9 - 4 = 5.

Hence the rate is given as follows:

r = -5/5

r = -1.

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Tom's workplace is 0.28 miles from Fort Hood.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

Fort Hood from Tom's house distance = 0.8 miles

Now,

Tom's workplace is 0.35 times as far from Fort Hood as Tom's House.

This means,

Tom's workplace from Fort Hood distance.

= 0.35 x 0.8

= 0.28 miles

Thus,

Tom's workplace is 0.28 miles from Fort Hood.

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A basis for the set of all upper triangular n x n matrices is the set of matrix with all entries below the main diagonal equal to zero. The dimension of this subspace is n(n+1)/2.

A basis for the set of all upper triangular n x n matrices can be found by taking all n x n matrix with all entries below the main diagonal equal to zero. This basis has n(n+1)/2 elements, and so the dimension of this subspace is also n(n+1)/2.The set of all upper triangular n x n matrices is a subspace of mnxn(f), and a basis for this subspace can be found by considering all matrices with all entries below the main diagonal equal to zero. This basis is composed of n(n+1)/2 elements, and so this is also the dimension of the subspace. This means that the subspace is spanned by n(n+1)/2 linearly independent vectors. Each of the matrices in the basis is an upper triangular matrix, and all entries below the main diagonal are equal to zero. As such, each matrix in the basis can be used to represent one of the n(n+1)/2 coordinates of the subspace, and all of the matrices together span the entire space.

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The algebraic structure (℘(S),∪,∩) is a commutative ring.

To prove that the algebraic structure (℘(S),∪,∩) is a commutative ring, we need to show that it satisfies the following properties:

Addition is commutative: A ∪ B = B ∪ A for any A, B ∈ ℘(S)

Addition is associative: (A ∪ B) ∪ C = A ∪ (B ∪ C) for any A, B, C ∈ ℘(S)

There exists an additive identity: there exists a set 0 ∈ ℘(S) such that A ∪ 0 = A for any A ∈ ℘(S)

There exists an additive inverse: for any A ∈ ℘(S), there exists a set -A ∈ ℘(S) such that A ∪ -A = 0

Multiplication is commutative: A ∩ B = B ∩ A for any A, B ∈ ℘(S)

Multiplication is associative: (A ∩ B) ∩ C = A ∩ (B ∩ C) for any A, B, C ∈ ℘(S)

There exists a multiplicative identity: there exists a set 1 ∈ ℘(S) such that A ∩ 1 = A for any A ∈ ℘(S)

Distribution property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) for any A, B, C ∈ ℘(S)

Addition is commutative: This property is true, as the union operation is commutative. Therefore, A ∪ B = B ∪ A for any A, B ∈ ℘(S).

Addition is associative: This property is also true, as the union operation is associative. Therefore, (A ∪ B) ∪ C = A ∪ (B ∪ C) for any A, B, C ∈ ℘(S).

There exists an additive identity: The empty set ∅ is the additive identity, as A ∪ ∅ = A for any A ∈ ℘(S).

There exists an additive inverse: The additive inverse of any set A is its complement -A, as A ∪ -A = ∅ for any A ∈ ℘(S).

Multiplication is commutative: This property is true, as the intersection operation is commutative. Therefore, A ∩ B = B ∩ A for any A, B ∈ ℘(S).

Multiplication is associative: This property is also true, as the intersection operation is associative. Therefore, (A ∩ B) ∩ C = A ∩ (B ∩ C) for any A, B, C ∈ ℘(S).

There exists a multiplicative identity: The set S is the multiplicative identity, as A ∩ S = A for any A ∈ ℘(S).

Distribution property: This property is also true, as the intersection operation distributes over the union operation. Therefore, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) for any A, B, C ∈ ℘(S).

Therefore, we have shown that the algebraic structure (℘(S),∪,∩) is a commutative ring.

Bonus: This structure does not have zero divisors. A zero divisor is an element a of a ring that is not zero and that has a product with some nonzero element b that is equal to zero. However, in

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Answer:

Step-by-step explanation:

a. To sketch the graph of the function F = 9/5 C + 32, we can plot a few points and connect them with a straight line.

For example, when C = 0, F = 32, so we can plot the point (0, 32).  When C = 100, F = 212, so we can plot the point (100, 212). Connecting these two points gives us the following graph

b. The slope of the graph is 9/5. This represents the rate of change of F with respect to C. Specifically, it means that for every 1 degree Celsius increase in temperature, there is a corresponding increase of 9/5 degrees Fahrenheit.

c. The F-intercept is 32. This represents the Fahrenheit temperature when the Celsius temperature is 0. In other words, it is the point where the graph crosses the y-axis.

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9/16 square units is the calculated area of the region r using double integration.

The procedure where two variables, x and y, are involved and you must integrate with each of them is known as a double integral or double integration technique. The area of a given function beneath a curve is calculated using this integration method.

The area of a region, the volume below the surface, and the average value of a function of two variables over a rectangular region may all be determined using double integrals.

A two-variable function, f (x, y), integral over a region R is referred to as a double integral. Iterative integration may be used to get the double integral if R = [a, b] [c, d] (integrate first with respect to y, and then integrate with respect to x). The solution can be seen on the attached image below.

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Question correction:

See on the attached image.

The correct answer is C. $ 3,666.30 is in the savings account.

A percentage is a number or ratio that can be expressed as a fraction of 100. A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

here, we have,

Let's calculate how much money you have in the CD after 5 months, this way:

Gross Income = $ 4,520

Total deductions on taxes = 27.9% (11.75 + 7.65 + 8.50) * 4,520 = $ 1,261.08

Realized income = Gross income - Total deductions on taxes

Realized income = 4,520 - 1,261.08

Realized income = $ 3,258.92

Fixed expenses = 3,258.92 * 0.3 = $ 977.68

Emergency fund = 977.68 * 5 = $ 4,888.40

Certificate of Deposit = 4,888.40 * 0.75 = $ 3,666.30

The correct answer is C. $ 3,666.30

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Answer:

Increasing: (-infinity, 1)  and Decreasing: (1, -infinity)

Step-by-step explanation:

Increasing means as you go from left to right where the x values are increasing, the y values are also increasing.

Decreasing means as you go from left to right where the x values are increasing, the y values are decreasing.

Let f : A → B be a function and the equivalent of stating that f is onto is statements (a), (c), and (e) and they are, (a) 3x EA, Vy E B f(x) y, (c) Vy e B, 3x A f(x) = y, and (e)  Vy e B, 3x E A, 3w E B f(x) = wɅy = w.

(a) 3x EA, Vy E B f(x) y denotes that at least one element x in A must exist for f(x) to equal y for every element function y in B. The meaning of onto is as follows.

(c) According to the formula Vy e B, 3x A f(x) = y, there must be at least one element x in A such that f(x) = y for each element y in B. Simply put, this is the same as assertion (a).

(e) Vy e B, 3x E A, 3w E B f(x) = wy = w indicates that at least one element x in A exists for every element y in B such that f(x) = y. The further requirement that g (x)

It is not equivalent to claim that f is onto based on statements (b) and (d).

(b) y B, Vxe A f(x) y denotes that there is at least one element in A such that f(x) = y for every element y in B. This is different from stating that f is onto because it does not imply that every element in B must actually be mapped to by f.

(d) According to the formula Vae E A, 3y B f(x) = y, there must be at least one element y in B such that f(x) = y for every element x in A. According to this definition, a function is total if it is specified for each element in domain A. The fact that f is assigned to some elements in B does not imply that f is necessarily onto those elements.

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The circumference of the circle is 88in (nearest inches)

The circumference is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure.

The circumference of a circle is given as;

C = πd

d = diameter = 28in

C = 3.14 × 28

C = 88 in ( nearest inches)

therefore the circumference of the circle is 88in

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The equation that represents the third side of the triangle is x + 64 = 104 and its solution is 40 units.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

The perimeter of the triangle is determined by adding together all of its sides.

The perimeter of a triangle is 104 units. The combined length of two of the sides of the triangle is 64 units.

Let the third side be 'x'. Then the equation is given as,

x + 64 = 104

x = 104 - 64

x = 40 units

The equation that represents the third side of the triangle is x + 64 = 104 and its solution is 40 units.

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Answer:

2mn

Step-by-step explanation:

To solve the expression 14m^2n^2/7mn, we need to simplify the fraction.

First, we can simplify the numerator by combining the 14 and m^2:

14m^2 * n^2 / 7mn = (14m^2) * (n^2) / (7mn) = 14mn * mn / 7mn = 14mn / 7

Next, we can simplify the denominator by  7

14mn / 7  = 14 / 7 = 2mn

Answer:

to inform, to persuade, and to entertain

Step-by-step explanation:

Probability of winning, you guess the position of each digit is 0.00833 and Probability of winning when you know the first correct piece is 0.04167.

There are 5 pieces to form a car.

Total number of arrangement of these 5 pieces is 5! = 5×4×3×2×1 = 120 ways

Of these 120 arrangements only 1 arrangement will form a proper car

(a) Probability that each position's guess is correct is = 1/ 120

Thus, the probability of getting all the guesses correct is 0.00833 or 0.833%.

(b) It is given that we know the first correct piece.

That is we need to guess the other 4 from the 4 remaining pieces.

Total number of arrangement of these 5 pieces is 4!

= 4×3×2×1 = 24 ways

Of these 24 arrangements only 1 arrangement will form a correct arrangement with the known first piece.

Probability that each position's guess is correct = 1/24

Thus, the probability of getting all the guesses correct when we know the first correct piece is 0.04167 or 4.17%.

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Answer:

              The true statements to be selected are the following:

"The slope of Function A is greater than the slope of Function B."

"The y-intercept of Function A is greater than the y-intercept of Function B."

Step-by-step explanation:

              Based on the points that were provided for Function A: (-5, -11), (-4, -8), and (2, 10), we can create a linear function to represent the function. The slope of the function is 3, and the y-intercept would be 4, so the linear function of Function A would be y = 3x + 4. And the function of Function B is, as given, y = 2x - 4.

              Now, we can compare the two functions and determine which statements are correct. Function A has a slope of 3 and a y-intercept of (0, 4), whilst Function B has a slope of 2 and a y-intercept of (0, -4). As both the slope and y-intercept of Function A is greater than the slope and y-intercept of Function B, the correct statements to select would be "The slope of Function A is greater than the slope of Function B." and "The y-intercept of Function A is greater than the y-intercept of Function B."

Have a great day! Feel free to let me know if you have any more questions :)

Answer:

y = x + 1

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

x - y = 5 ( subtract x from both sides )

- y = - x + 5 ( multiply through by - 1 )

y = x - 5 ← in slope- intercept form

with slope m = 1

• Parallel lines have equal slopes , then

y = x + c ← is the partial equation

to find c substitute (- 7, - 6 ) into the partial equation

- 6 = - 7 + c ( add 7 to both sides )

1 = c

y = x + 1 ← equation of parallel line

The conversion gives 39 Galloon = 147.63106 Liters

The same attribute is expressed using a unit conversion, but in a different unit of measurement. For instance, time can be expressed in minutes rather than hours, and distance can be expressed in kilometres, feet, or any other measurement unit instead of miles.

Given:

We have to convert 39 galloon to liter

As, 1 Galloon =  3.7854118 Liters

Now, 39 Galloon= 39 x 3.7854118 Liters

                            = 147.63106 Liters

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As per the function f(x) = 2x + 6, the value of f(a) = 2a + 6 and f(a + h) = 2(a + h) + 6 = 2a + 2h + 6.

A function is a mathematical rule that assigns each input (or independent variable) a unique output (or dependent variable).

The function f(x) = 2x + 6 means that for any given value of x, we can compute the corresponding value of f(x) by substituting x into the expression 2x + 6.

The first expression that we need to evaluate is f(a), where a is some arbitrary value. This means that we need to substitute a into the function f(x) and simplify the expression.

Therefore, f(a) = 2a + 6.

The second expression that we need to evaluate is f(a + h), where h is also an arbitrary value. This means that we need to substitute a + h into the function f(x) and simplify the expression.

Therefore, f(a + h) = 2(a + h) + 6 = 2a + 2h + 6.

To simplify the expressions, we can collect like terms, which means combining the terms that have the same variable and exponent 2a + 2h can be written as 2(a + h).

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Answer:

[tex]y=0.11x+1[/tex]

Step-by-step explanation:

y = 0.11x + 1

(y is the monthly cost for x text messages)

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The four roots form the vertices of a square in the complex plane. The side length of the square is √2. The vertices of the square complex number are: A = (-√2, -√2), B = (0, -2√2), C = (√2, -√2), D = (0, 0). The principal root is the one with the smallest argument, which is (-√2 - √2 i).

To find the fourth root of the complex number (-8-8√3), we can use the following steps:

We start by finding the modulus and argument of the number:

|(-8-8√3)| = 16

arg(-8-8√3) = arctan(-√3) = -π/3

Therefore, (-8-8√3) = 16(cos(-π/3) + i sin(-π/3)).

To find the fourth root, we take the square root twice. We can use De Moivre's theorem to simplify the calculation:

(-8-8√3)^(1/4) = (16(cos(-π/3) + i sin(-π/3)))^(1/4)

= 2(cos(-π/12) + i sin(-π/12))

= 2(cos(11π/12) + i sin(11π/12))

= 2(-√2/2 - √2/2 i)

We have expressed the roots in rectangular coordinates as (-√2, -√2), (0, -2√2), (√2, -√2), and (0, 0).

The four roots form the vertices of a square in the complex plane. The side length of the square is |(-√2) - (0)| = √2.

The vertices of the square are:

A = (-√2, -√2)

B = (0, -2√2)

C = (√2, -√2)

D = (0, 0)

The principal root is the one with the smallest argument, which is (-√2 - √2 i). This is the root corresponding to the first quadrant of the complex plane.

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