Graph the equation y = − x² + 8 x − 12 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the roots of the equation − x² + 8 x − 12 = 0

The row of zeros at the bottom immediately lets us conclude "infinitely many solutions". This system is consistent and dependent. This is because we have 0x+0y+0z = 0 aka 0 = 0 which is always true for any choice of x, y, and z.

The second row of values lead to the equation 0x+0y+1z = 6, aka z = 6

The first row says: 1x+6y+0z = 1 which is the same as x+6y = 1

Let's say we isolate x

x+6y = 1

x+6y-6y = 1-6y

x = 1-6y

Then we have these three values or expressions

All of the infinitely many solutions are of the form (x,y,z) = (1-6y, y, 6)

A lot of textbooks will use a parameter such as t, so we could write it as (1-6t, t, 6).

The choice of the letter for the parameter does not matter. I think t is most popular because it represents time. Each (x,y,z) point could represent a particle's location at any given time.

and so on.

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

Conclusion:

There are infinitely many solutions of the form (x,y,z) = (1-6t, t, 6) where t is any real number.

This system is consistent and dependent.

Let's assume that the number of Android phones sold is x and the number of iOS phones sold is y. Then we can write two equations based on the given information:

x + y = 110    (equation 1)

400x + 650y = 54000   (equation 2)

To solve for x and y, we can use any method of solving systems of equations. Here, we will use the substitution method:

From equation 1, we can solve for x in terms of y: x = 110 - y

Substitute x = 110 - y into equation 2 and solve for y:

400(110 - y) + 650y = 54000

44000 - 400y + 650y = 54000

250y = 10000

y = 40

Substitute y = 40 into equation 1 and solve for x:

x + 40 = 110

x = 70

Therefore, the phone store sold 70 Android phones and 40 iOS phones.

Answer:

[tex]\frac{10n}{1-30n}:\frac{m}{10(1+3m)}[/tex]

Step-by-step explanation:

Solve for 'm' and 'n':

'm' = [tex]\frac{10n}{1-30n}[/tex]

'n' = [tex]\frac{m}{10(1+3m)}[/tex]

Ratio: [tex]\frac{10n}{1-30n}:\frac{m}{10(1+3m)}[/tex]

The situation that requires a square root operation is calculating the height of a cylinder given the area of the base. third option

It should be noted that the operation that needs square root among the given option lies on the selected option above, this is because from the formular we can see that there is square of r which is r^2

However thegeometry  fiqure of the volume of a cylinder can be calculated using this formular V = πr^2h

height of a cylinder can be calculated using the formular V/ πr2.

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The repeated nearest neighbor algorithm is applied to the given graph to find the circuit of lowest cost starting from vertex A. The vertices are visited in the order A, D, B, C, E, F, and back to A, with a total cost of 38. Therefore, the answer is ADBCEFA.

To apply the repeated nearest neighbor algorithm, we start at vertex A and repeatedly choose the nearest unvisited vertex until we have visited all vertices and return to the starting vertex.

Start at vertex A. Vertex D is the nearest unvisited vertex from A with a distance of 4. Move to vertex D. Vertex C is the nearest unvisited vertex from D with a distance of 5. Move to vertex C. Vertex F is the nearest unvisited vertex from C with a distance of 2.

Move to vertex F. Vertex E is the nearest unvisited vertex from F with a distance of 6. Move to vertex E. Vertex B is the nearest unvisited vertex from E with a distance of 8. Move to vertex B. Return to vertex A to complete the circuit.

Therefore, the circuit starting and ending at vertex A using the repeated nearest neighbor algorithm is ADFCEBA with a total distance of 38.

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Answer: A

Step-by-step explanation:

well i did alll of them and got A

The reason for the statements of the proof to show that ∠A ≅ ∠C are:

A quadrilateral with two sets of parallel sides is referred to as a parallelogram.

In a parallelogram, the opposing or confronting sides are of equal length, and the opposing angles are of equal size.

The ASA congruence theorem states that two triangles are congruent if two angles and the side separating two angles of one triangle are congruent with the corresponding angles and side of another triangle.

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The original price of this car is equal to $6,600.

In Mathematics, a percentage can be defined as any number that is expressed as a fraction of hundred (100). This ultimately implies that, a percentage indicates the hundredth parts of any given number.

Assuming the original price is represented by the variable x, we have the following:

12.5/100 × x = x - 5,775

0.125x = x - 5,775

5,775 = (x - 0.125x)

5,775 = 0.875x

x = 5,775/0.875

x = $6,600.

In conclusion, we can logically deduce that the percentage Pat Bain paid is 87.5%.

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a) The average slope or rate of change between (0, 1) and (2, 4) is 3/2.

b) The average slope or rate of change between (-2, 1/4) and (-1, 1/2) is 1/4.

c) The slope or rate of change is not constant between these two pairs of points, since the average slopes are different.

d) The function connecting these pairs of points is not a linear function.

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.

Any two different points on a line can be used to calculate the slope of any line. The ratio of "vertical change" to "horizontal change" between two different locations on a line is calculated using the slope of a line formula.

The average slope or rate of change between two points (x1, y1) and (x2, y2) on a line is given by:

average slope = (y2 - y1) / (x2 - x1)

For the points (0, 1) and (2, 4), the average slope is:

average slope = (4 - 1) / (2 - 0) = 3/2

For the points (-2, 1/4) and (-1, 1/2), the average slope is:

average slope = (1/2 - 1/4) / (-1 - (-2)) = 1/4

When the average slopes of these two pairs of points differ, the slope or rate of change is not constant. As a result, the function connecting these point pairs is not linear.

Because a linear function has a constant slope, a function with a variable slope cannot be linear.

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Answer:

C

Step-by-step explanation:

if the mean and median are the same, the graph will be symmetrical

If she can give out 12 to each of her friends and have 3 left over. or she can give 9 out to each of her friends and have 12 left over, Malaika has 3 friends who can receive candies.

Let's suppose Malaika has "c" candies, and "f" is the number of friends she has.

According to the problem statement, we have two equations:

c = 12f + 3

c = 9f + 12

To solve for "f", we can set the two equations equal to each other:

12f + 3 = 9f + 12

Simplifying the equation, we get:

3f = 9

Dividing both sides by 3, we get:

f = 3

We can verify our solution by plugging "f=3" back into one of the original equations:

c = 12f + 3

c = 12(3) + 3

c = 39

So, Malaika has 39 candies in total. We can also check the other equation to make sure it is true:

c = 9f + 12

c = 9(3) + 12

c = 39

Both equations are true, so our solution of "f=3" is correct.

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The total units of length for the ice rink away from the origin is 14.42 units.

In Mathematics and Geometry, the distance between two (2) end points that are on a coordinate plane can be calculated by using the following mathematical equation:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

x and y represent the data points (coordinates) on a cartesian coordinate.

Note: Ice rink is located at point (12, 8).

By substituting the given end points into the distance formula, we have the following;

Distance = √[(12 - 0)² + (8 - 0)²]

Distance = √[(12)² + (8)²]

Distance = √[144 + 64]

Distance = √208

Distance = 14.42 units.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Using interval notation, the domain of f.g and f+g is [1/4, ∞).

Given, f(x)= 1/ (5x² + 3) and [tex]g(x) = \sqrt{4x-1}[/tex]

To find f.g, we substitute g(x) into f(x) and simplify:

f.g(x) = f(g(x))

[tex]=f(\sqrt{4x-1})[/tex]

[tex]=\frac{1}{5(\sqrt{4x-1})^2 +3}[/tex]

= 1/(5(4x-1)+3)

= 1 / (20x - 2)

To find f+g, we add the functions f(x) and g(x):

f+g(x) = f(x) + g(x)

= [tex]\frac{1}{5x^2+3}+\sqrt{4x-1}[/tex]

The domain of f.g and f+g is the intersection of the domains of f(x) and g(x).

The domain of f(x) is all real numbers except for the values of x that make the denominator zero, i.e., 5x² + 3 = 0. This equation has no real solutions, so the domain of f(x) is all real numbers.

The domain of g(x) is the set of all real numbers that make the argument of the square root non-negative, i.e., 4x - 1 ≥ 0. Solving this inequality, we get x ≥ 1/4.

Therefore, the domain of f.g and f+g is x ≥ 1/4.

Using interval notation, the domain of f.g and f+g is [1/4, ∞).

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a) The  piecewise function is 80p, if p > 100 b) For p > 100, the graph will be a straight line passing through the point (100, 100 × 80) with a slope of 8

a) We can define a piecewise function f(p) to represent the cost of catering service depending on the number of persons (pax).

f(p) =

120p, if p ≤ 30

105p, if 31 ≤ p ≤ 60

95p, if 61 ≤ p ≤ 100

80p, if p > 100

b) To graph the piecewise function, we can plot the points for each of the four parts of the function, and connect them with line segments. The graph will have four segments, each with a different slope.

For p ≤ 30, the graph will be a straight line passing through the origin with a slope of 120.

For 31 ≤ p ≤ 60, the graph will be a straight line passing through the point (31, 31 × 105) with a slope of 105.

For 61 ≤ p ≤ 100, the graph will be a straight line passing through the point (61, 61 × 95) with a slope of 95.

For p > 100, the graph will be a straight line passing through the point (100, 100 × 80) with a slope of 80.

c) To find how many persons they invite to their wedding, we can use the inverse function of f(p) and plug in the given cost of RM9025.

Let C = RM9025, then

120p, if p ≤ 30

105p, if 31 ≤ p ≤ 60

95p, if 61 ≤ p ≤ 100

80p, if p > 100

Case 1: p ≤ 30

120p = C

p = C/120 = 9025/120 = 75.2083

Since p must be an integer, we round up to the nearest integer. Therefore, if they invite 76 persons, the cost of catering service will be more than RM9025.

Case 2: 31 ≤ p ≤ 60

Let x = p - 30, then

105x + 120 × 30 = C

105x = C - 3600

x = (C - 3600)/105 = (9025 - 3600)/105 = 54.5238

Therefore, if they invite 31 + 54 = 85 persons, the cost of catering service will be RM9025.

Case 3: 61 ≤ p ≤ 100

Let x = p - 60, then

95x + 120 × 30 + 105 × 30 = C

95x = C - 9750

x = (C - 9750)/95 = (9025 - 9750)/95 = 0.7632

Therefore, if they invite 60 + 0.7632 × 100 = 137 persons, the cost of catering service will be RM9025.

Case 4: p > 100

Let x = p - 100, then

80x + 120 × 30 + 105 × 30 + 95 × 40 = C

80x = C - 17700

x = (C - 17700)/80 = (9025 - 17700)/80 = -109.375

Since p must be a positive integer, this case is not possible.

Therefore, Mohammed and Mira should invite 85 persons to their wedding to pay RM9025 for the catering service.

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The value of the summation notation [tex]\sum\limits^{12}_{i=1} {-25x_i}[/tex] for the 12 measurements 69, -5, 47, -10, -80, 13, 64, -76, -95, 45, 9, 7 is 300

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

69, -5, 47, -10, -80, 13, 64, -76, -95, 45, 9, 7

The summation notation is given as

[tex]\sum\limits^{12}_{i=1} {-25x_i}[/tex]

The summation notation  [tex]\sum\limits^{12}_{i=1} {-25x_i}[/tex] means we multiply each value by -25 and add the results

Using the above as a guide, we have the following:

Sum = -25 * (69 - 5 + 47 - 10 - 80 + 13 + 64 - 76 - 95 + 45 + 9 + 7)

Evaluate

Sum = 300

Hence, the value of the summation notation [tex]\sum\limits^{12}_{i=1} {-25x_i}[/tex] for the 12 measurements 69, -5, 47, -10, -80, 13, 64, -76, -95, 45, 9, 7 is 300

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Using the properties of a parallelogram, the values of x and y are:

x = 9; y = 22.

Since the opposite sides are parallel and equal to each other in the parallelogram above, therefore, Opposite angles are equal (or congruent) while the consecutive angles are supplementary to each other.

Therefore, we have:

6y = 180 - 48

6y = 132

Divide both sides by 6:

6y/6 = 132/6

y = 22

(5x + 3) + 6y = 180

5x + 3 + 6(22) = 180

5x + 135 = 180

5x = 180 - 135

5x = 45

x = 9

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The function that has the greatest rate of change is y = x

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

y = -x ,

y = 3/4x +1,

y = 1/2x +2,

y = x

A linear function is represented as

y = mx + c

Where

rate of change = m

So, we have

y = -x , rate = -1

y = 3/4x +1, rate = 3/4

y = 1/2x +2, rate = 1/2

y = x, rate = 1

This means that the function with the highest rate is y = x

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a. the standard deviation is approximately 25.01 and the variance is approximately 625.21.

b.the five-number summary for the data is: 30, 57, 71, 81, 98.

The standard deviation is a measurement of how widely apart a group of data points are from the mean (average) value. You can see how far each data point deviates from the mean value. When the standard deviation is minimal, the data points are densely packed around the mean; when it is great, the data points are more widely spaced from the mean. The standard deviation is a statistical concept that is frequently used to compare the degree of variability across various data sets as well as to infer and make conclusions about a population from a sample.

a. To find the standard deviation and variance for the given data, we can use the following formulas:

Variance: σ² = ∑(x - μ)² / n

Standard deviation: σ = sqrt(σ²)

where x is the data point, μ is the mean of the data, and n is the sample size.

First, we need to find the mean of the data:

μ = (30 + 35 + 43 + 44 + 47 + 48 + 54 + 55 + 56 + 57 + 59 + 62 + 63 + 65 + 66 + 68 + 69 + 69 + 71 + 72 + 72 + 73 + 74 + 76 + 77 + 77 + 78 + 79 + 80 + 81 + 81 + 82 + 83 + 85 + 89 + 92 + 93 + 94 + 97 + 98) / 40

μ = 69.3

Now, we can use the variance formula:

σ² = ∑(x - μ)² / n

σ² = [(30 - 69.3)² + (35 - 69.3)² + ... + (98 - 69.3)²] / 40

σ² = 625.21

Finally, we can find the standard deviation:

σ = sqrt(σ²)

σ = sqrt(625.21)

σ ≈ 25.01

Therefore, the standard deviation is approximately 25.01 and the variance is approximately 625.21.

b. To find the five-number summary, we need to find the minimum, maximum, median, and quartiles (Q1 and Q3) of the data:

Minimum: 30

Q1: 57

Median: 71

Q3: 81

Maximum: 98

Therefore, the five-number summary for the data is: 30, 57, 71, 81, 98.

c. To construct a boxplot for the given data, we can use the five-number summary:

Draw a number line and mark the minimum, Q1, median, Q3, and maximum.

Draw a box from Q1 to Q3.

Draw a vertical line inside the box at the median.

Draw whiskers from the ends of the box to the minimum and maximum, or to the nearest data point within 1.5 times the interquartile range (IQR).

Mark any outliers outside the whiskers.

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As A is the only option that is three units distant from (5, 3), the response is: A) (1, 3) (1, 3) .

The placement of a point in a particular space or on a certain graph is represented by coordinates, which are numbers. Coordinates are normally two figures written in parenthesis and separated by a comma in two-dimensional space, where x denotes the point's horizontal position and y denotes its vertical position. Three integers enclosed in parentheses and separated by commas are used to indicate coordinates in three-dimensional space. The three numbers are generally represented as (x, y, z), where x, y, and z stand for the positions all along x-, y-, and z-axes, respectively. The location of objects, points, and other entities in space is described using coordinates frequently in the domains of mathematics, physics, engineering, and many others.

given

Any other vertex must be three units away from the specified vertex because the square has three units on each side.

The distance between the supplied vertex (5, 3) and each of the possible answers can be calculated using the distance formula:

Option A: Distance between (1 and 3) = sqrt((1 - 5)2 + (3 - 3)2) = sqrt(16) = 4 (not three units away)

Option B: Distance between (8 and 6) = sqrt((8 - 5)2 + (6 - 3)2) = sqrt(27) (not three units away)

Option C: (4, 0)

Distance is equal to sqrt((4 - 5)2 + (0 - 3)2 = sqrt(10) (not three units away)

Option D: Distance = sqrt((2 - 5)2 + (1 - 3)2) = sqrt(10) for the pair (2, 1). (not three units away)

As A is the only option that is three units distant from (5, 3), the response is: A) (1, 3) (1, 3) .

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The area of the triangle that has height of √6 and a base of √24 is calculated as: 6 square units.

The area of a triangle = 1/2 * b * h, where:

h is the height of the triangle, and

b is the base length of the triangle.

Given the following:

Height of triangle = √6

Base length of the triangle = √24

Plug in the values:

Area of triangle = 1/2 * √24 * √6

= (√24 * √6) / 2

= √144 / 2

= 12/2

Area of triangle = 6 square units.

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The logarithm expression In(9/x) can be expanded as In(9) - In(x)

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

In(9/x)

Using the logarithmic identity log a/b = log a - log b, we can expand ln(9/x) as follows:

ln(9/x) = ln(9) - ln(x)

Therefore, ln(9/x) can be expressed as the difference of two natural logarithms: ln(9) and ln(x).

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The ordered pairs of the solutions to the linear expression are В. (-7, 0) C. (0, -7) and F. (3, -4)

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

The linear expression -2x - 2y =14

To determine the ordered pairs of the linear expression, we set x and y to the values in the list of options and check for the true equation

Using the above as a guide, we have the following:

А. (2, 3) В. (-7, 0) C. (0, -7)

-2(2) - 2(3) = 14

-10 = 14 --- false

-2(-7) - 2(0) = 14

14 = 14 --- true

-2(0) - 2(-7) = 14

14 = 14 --- true

D. (-7, 1) E. (-1, 7) F. (3, -4)

-2(-7) - 2(1) = 14

12 = 14 --- false

-2(-1) - 2(7) = 14

-12 = 14 --- false

2(3) - 2(-4) = 14

14 = 14 --- true

Hence, the ordered pairs of the linear expression are В. (-7, 0) C. (0, -7) and F. (3, -4)

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Complete question

Select all the ordered pairs that are solutions

-2x - 2y = 14

А. (2, 3) В. (-7, 0) C. (0, -7)

D. (-7, 1) E. (-1, 7) F. (3, -4)

The "equivalent-form" of equation "x/12 - 7 = x/4" is "x - 84 = 3x", the correct option is (a).

In order to find the equivalent form of the equation "x/12 - 7 = x/4", we first need to solve for "x",

So, we first, simplify left side of equation by finding a common denominator for the two fractions:

⇒ x/12 - 7 = x/4,

⇒ x - 84 = 3x,

⇒ -84 = 2x,

⇒ x = -42,

Now, we check the given options by substituting the value of "x",

Option (a) : x - 84 = 3x,

Substituting x = -42,

We get,

⇒ -42 - 84 = 3(-42),

⇒ -126 = -126

This equation is true, so option (a) is an equivalent form of the original equation.

Option (b) : x - 74 = 12x,

Substituting x = -42,

We get,

⇒ -42 - 74 = 12(-42),

⇒ -116 = -504,

This equation is not true, so option (b) is not an equivalent form of the original equation.

Option (c) : x - 7 = 3x,

Substituting x = -42,

We get,

⇒ -42 - 7 = 3(-42),

⇒ -49 = -126,

This equation is not true, so option (c) is not an equivalent form of the original equation.

Therefore, the correct option is (a) .

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The given question is incomplete, the complete question is

Select an equivalent form of this equation: x/12 - 7 =x/4

(a) x - 84 = 3x

(b) x - 74 = 12x

(c) x - 7 = 3x

A graph of the piecewise function is shown on the coordinate plane in the image attached below.

In Mathematics, a piecewise-defined function can be defined as a type of function that is defined by two (2) or more mathematical expressions over a specific domain.

Generally speaking, the domain of any piecewise-defined function simply refers to the union of all of its sub-domains. By critically observing the graph of the given piecewise-defined function, we can reasonably infer and logically deduce that it is constant over several intervals such as -1 < x ≤ 0.

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Answer:Given: Three lines AD, CF, and BE intersect at point O such that and

To prove: O is the midpoint of all three lines, i.e., AO = BO = CO

Proof:

Since point O lies on line AD, we can write

(1) AO + OD = AD

Similarly, since O lies on line BE, we have

(2) BO + OE = BE

Also, since O lies on line CF, we have

(3) CO + OF = CF

From the given information, we have

(4) AD = BE

Substituting (4) in (1) and (2), we get

(5) AO + OD = BO + OE

Adding (3) to (5), we get

(6) AO + BO + CO + OD + OE + OF = AD + BE + CF

From the given information, we have

(7) CF = AD + BE

Substituting (7) in (6), we get

(8) AO + BO + CO + OD + OE + OF = 2AD + 2BE

Since we know that AD = BE, we can simplify (8) as

(9) AO + BO + CO + OD + OE + OF = 4AD

Dividing both sides of (9) by 4, we get

(10) AO + BO + CO = AD

But from (4), we also know that AD = BE, so we can write

(11) AO + BO + CO = BE

Dividing both sides of (11) by 2, we get

(12) AO = BO = CO

Hence, we have proved that O is the midpoint of all three lines, i.e., AO = BO = CO.

Step-by-step explanation:

Answer:The graph of y = 4f(x-5) is not the same as the graph of y=f(x) because it is a transformation of the function f(x) that involves a horizontal shift to the right and a vertical stretch by a factor of 4.

Answer:

14:10

Step-by-step explanation:

We are given that the total train journey from Reading to Worle is the same for all trains

For the first train the time taken = 13:45 - 12:05

13:45
   -
12:05
--------
1 : 40
--------

So total travel time = 1 hour 40 minutes

For the third train time taken
15:05  -  13:25

15:05 = 14:05 + 0:60 since there are 60 minutes in an hour (we carry 60 from the hours part)

14:05 + 0:60 = 14:65

15:05 - 13:25 =

14:65
   -
13:25
---------
1 : 40 or 1 hour 40 minutes same as train #1

We want to find the arrival time of train #2 at Worle

Departure time of train #2 from Reading = 12:30

Time of travel = 1 hour 40 minutes

Arrival time = 12:30 + 1:40

12:30 time
   +
 1: 40  hours
----------
13:70   time
------------------
Since there are 60 minutes in 1 hour, to convert 13:70 to proper time units
subtract 60 from 70 to get the minute part of time and add 1 to 13 to get the hour part of time

13:70 = 14:10

Therefore the time that goes into the gap = 14:10

Verify

If the second train leaves Reading at 12:30 and arrives at Worle at 14:10, then time taken =

14:10 - 12:30 = 13:70 - 12:30 = 1:40 = 1 hour 40 minutes

Hope that helps and please don't hesitate to ask more questions if you have not fully understood my explanation

The best estimate of 7,625,750,263 is 7x10⁹. So the correct option is B.

The given number is 7,625,750,263.

To express the above number in words, it is "seven billion six hundred twenty-five million seven hundred fifty thousand two hundred sixty-three". So, the given value ultimately has a billion value.

To find the correct answer let's see the given options and eliminate each one to find the correct answer. Option A is 7 x 1010 = 7070 which is not even coming to a close, so option A is eliminated.

Let's see the second option which is 7x10⁹ which is 7000000000, which is 7 billion. So, option B is very near to the answer.

Let's see the third option which is 8x10¹⁰ which is 80 billion, it is not an estimated value. So, it is also eliminated and the final option is 8x10⁹ which is 8 billion which is also high value So, it is also eliminated.

From the above analysis, we can conclude that the best estimated value for 7,625,750,263 is 7x10⁹ which is option B.

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