The arithmetic mean of two numbers is half their sum, and the geometric mean is the square root of their product. using log properties, show how we can think of the log of the geometric mean of v and w as the arithmetic mean of two related numbers, p and q. expand and simplify

Answer:

To apply the first derivative test to the function f(x) = 2x^3 + 3x^2 - 12x + 5, we need to find its derivative:

f'(x) = 6x^2 + 6x - 12

Then, we need to find the critical points by solving for f'(x) = 0:

6x^2 + 6x - 12 = 0

Dividing both sides by 6, we get:

x^2 + x - 2 = 0

Factoring the left side, we get:

(x + 2)(x - 1) = 0

So the critical points are x = -2 and x = 1.

To determine the intervals where f(x) is increasing and decreasing, we need to evaluate f'(x) on each side of the critical points. We can use a sign chart to do this:

x     |  -2  |  1  |

------|-----|----|

f'(x) | -12 |  0 |

Since f'(x) is negative to the left of x = -2 and positive to the right of x = -2, the function is decreasing to the left of x = -2 and increasing to the right of x = -2. Since f'(x) changes sign at x = 1, there is a local minimum at x = 1.

Therefore, using the first derivative test, we can conclude that the function has a local minimum at x = 1.

Step-by-step explanation:

Step-by-step explanation:

the ratio is 30/21600 = 3/2160 = 1/720

that means for every degree the hour hand moves, the second hand does 2 full rotations (2×360°).

h = s×30/21600 = s/720

FYI :

as there are 12 hours on a clock, every hour means 360/12 = 30° rotation for the hour hand.

so, for each hour the second hand moves 21600°.

that means 21600/360 = 60 full rotations.

that means one rotation of the second hand is 1 minute.

so, the second hand is truly a second hand (counting the seconds).

60 seconds in a minute (a full rotation by the second hand), that means each second corresponds to 360/60 = 6° rotation of the second hand.

After comparing the image and the figure, we found that the direction and angle of rotation are 180° counterclockwise rotation.

A transformation known as a rotation involves turning the figure either clockwise or counterclockwise. The centre of rotation is the fixed location where the rotation occurs. The term "angle of rotation" refers to the amount of rotation. A figure can be rotated 90 degrees, or a quarter turn, in either a clockwise or counterclockwise direction. You have rotated the figure 180 degrees when you have spun it exactly halfway. The figurine may be rotated 360° by turning it all the way around. A counterclockwise turn has a positive magnitude because a clockwise rotation indicates a negative magnitude. Rotational symmetry exists in every regular polygon. When an object is rotated around its centre, it retains its original appearance. The object is then referred regarded as having rotational symmetry.

Let's write the coordinates of the original figure,

A = (1, -5)

B = (6, -2)

C = (6, -5)

D = (1, -8)

Now, if we rotate it 90 degrees counterclockwise, the coordinates are:

A' = ( 5, 1)

B' = (2, 6)

C' = (5, 6)

D' = (8, 1)

Now if we rotate it again 90 degrees counterclockwise, the coordinates are:

A'' = ( -1, 5)

B'' = (-6, 2)

C'' = (-6, 5)

D'' = (-1, 8)

Now from the figure, let's write the coordinates.

A' = ( -1, 5)

B' =  (-6, 2)

C' = (-6, 5)

D' =  (-1, 8)

This is the same as the coordinates of the second 90 degrees counterclockwise rotation.

Since we did two 90 degrees rotations, we can say we did a total of 180° counterclockwise rotation.

Therefore after comparing the image and the figure, we found that the direction and angle of rotation are 180° counterclockwise rotation.

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Answer: 180 degrees counterclockwise rotation.

Answer: First we have to check the information they are giving in the problem. do we can see that the photocopier can copy 4 pages every 2 seconds.

so with this info we can calculate the ratio, or the number of copies that can be done in one second:

So we made a rule of 3 to solve that

So, if the photocopier can print 2 copys every second, the we can calculate the the time that takes to print 120 pages:

solving this rule of 3:

so is going to take 60 seconds to copy 120 pages.

This question does not require a diagram to be drawn, so based on the scale factor or ratio of 1 inch : 4 yards and since the actual width of the neighborhood park is 60 yards, the width of the drawing is 15 inches.

The scale factor is the ratio of a scale drawing or model to the actual dimension of the object.

The scale factor is also measured as the dimension of the new shape ÷ dimension of the original shape.

The Ratio of the scale drawing = 1 inch : 4 yards

The actual width of the park = 60 yards

The scaled-down width of the drawing = 15 inches (60/4).

Thus, we can conclude that the scale drawing's width that Darrell made was 15 inches.

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The Addition Property of Equality, If a=b, then a+c=b+c depicts the notion that "If equals are added to equals, then the wholes are equal". The correct answer is option e.

Addition Property of Equality is a law that describes the possibility of adding the same quantity on both sides of an equation to maintain equality. If a=b, then a+c=b+c represents the Addition Property of Equality. It denotes that if you add the same quantity on both sides of an equation, the result will remain equal.

The Addition Property of Equality also mentions that if two numbers are equal to each other, then adding the same number to both of them will also result in two equal numbers. This means that if a=b, then a+c=b+c. Therefore, the correct answer is option "e.  

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[tex]\stackrel{a}{-117}+\stackrel{b}{44}i \\\\[-0.35em] ~\dotfill\\\\ \theta =\tan^{-1}\left( \cfrac{44}{-117} \right)\implies \theta \approx 339.39^o~\hfill r=\sqrt{(-117)^2 + 44^2}\implies r=125 \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\sqrt[n]{z}=\sqrt[n]{r}\left[ \cos\left( \cfrac{\theta+2\pi k}{n} \right) +i\sin\left( \cfrac{\theta+2\pi k}{n} \right)\right]\quad \begin{array}{llll} k\ roots\\ 0,1,2,3,... \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \boxed{k=0}\hspace{3em} \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 339.39^o + 360^o( 0 )}{3} \right) +i \sin\left( \cfrac{ 339.39^o + 360^o( 0 )}{3} \right)\right] \\\\\\ \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 339.39^o }{3} \right) +i \sin\left( \cfrac{ 339.39^o }{3} \right)\right][/tex]

[tex]5[ \cos(113.13^o) +i \sin(113.13^o)]\approx \boxed{-1.96~~ + ~~4.60i} \\\\[-0.35em] ~\dotfill\\\\ \boxed{k=1}\hspace{3em} \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 339.39^o + 360^o( 1 )}{3} \right) +i \sin\left( \cfrac{ 339.39^o + 360^o( 1 )}{3} \right)\right] \\\\\\ \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 699.39^o }{3} \right) +i \sin\left( \cfrac{ 699.39^o }{3} \right)\right] \\\\\\ 5[\cos(233.13^o)+i\sin(233.13^o)]\approx \boxed{-3.00~~ - ~~4.00i} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\boxed{k=2}\hspace{3em} \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 339.39^o + 360^o( 2 )}{3} \right) +i \sin\left( \cfrac{ 339.39^o + 360^o( 2 )}{3} \right)\right] \\\\\\ \sqrt[ 3 ]{125} \left[ \cos\left( \cfrac{ 1059.39^o }{3} \right) +i \sin\left( \cfrac{ 1059.39^o }{3} \right)\right] \\\\\\ 5[\cos(353.13^o)+i\sin(353.13^o)]\approx \boxed{4.96~~ - ~~0.60i}[/tex]

Cube roots of complex number - 117 + 14i are,

- 1.96 + 4.60i

- 3.00 - 4.00i

4.96 - 0.60i

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

Complex number is,

⇒ - 117 + 14i

Let a = - 117

b = 44

Then, tan ⁻¹ (44/117) = 339.39°

⇒ θ = 339.39°

And, r = √117² + 44² = 125

Hence, We have;

[tex]\sqrt[n]{z} = \sqrt[n]{r} [cos (x + 2\pi k/n) + i sin (x + 2\pi k/n)][/tex]

Where, k = 0,1,2, 3, ..

Now, At k = 0

[tex]\sqrt[3]{z} = \sqrt[3]{125} [cos (339.39 + 0/3) + i sin (339.39 + 0/n)][/tex]

    = - 1.96 + 4.60i

At k = 1;

[tex]\sqrt[3]{z} = \sqrt[3]{125} [cos (339.39 + 360/3) + i sin (339.39 + 360/n)][/tex]

     = - 3.00 - 4.00i

At k = 2;

[tex]\sqrt[3]{z} = \sqrt[3]{125} [cos (339.39 + 360*2/3) + i sin (339.39 + 360*2/3)][/tex]

     = 4.96 - 0.60i

Thus, All Cube roots of complex number - 117 + 14i are,

- 1.96 + 4.60i

- 3.00 - 4.00i

4.96 - 0.60i

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The graph of the rational function y = f(x) is represented by the blue line on the graph above. The x-intercept is the point at which the function crosses the x-axis and is labeled (0, 0).

The vertical asymptote at x = 3 is the red line. The local approximation for the vertical asymptote at x = 3 is represented by the black line and is given by y = 568 - 332(x - 3). Similarly, the vertical asymptote at x = -2 is the purple line and the associated local approximation is given by y = -2(x + 2). Finally, the horizontal asymptote at y = 2 is the orange line.

To illustrate the graph, assume the function f(x) is the following: f(x) = (x + 2)(x - 3) / (x + 5). This function is a rational function, as it is the ratio of two polynomials, and it has the given constraints.

To graph the rational function, we begin by finding the x-intercept, which is (0, 0). Then, we look at the vertical asymptote. At x = 3, the local approximation is given by y = 568 - 332(x - 3). We solve the equation y = 568 - 332(x - 3) for x, and get x = 3, thus we mark this point on the graph as the vertical asymptote. We do the same for the vertical asymptote at x = -2 and mark this point on the graph. Finally, we draw the line for the horizontal asymptote at y = 2. This gives us the complete graph of the rational function y = f(x).

In conclusion, the graph of the rational function y = f(x) that displays the four constraints can be sketched and labeled as shown in the figure above.

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The expression -a^(2)+b^(4)-7a^(3) is a polynomial.

A polynomial is a mathematical expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The given expression satisfies all these conditions, and therefore, it is a polynomial.

The type of the polynomial is determined by the number of terms it has. In this case, the polynomial has 3 terms, so it is a trinomial.The degree of a polynomial is the highest exponent of the variable. In this case, the highest exponent is 4 (in the term b^(4)), so the degree of the polynomial is 4. Therefore, the expression -a^(2)+b^(4)-7a^(3) is a trinomial of degree 4.

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The number in the bottom box is 9 yd. The number in the top is 9 yd. The area of the table is 90 yd².

A trapezoid is a four-sided polygon that has at least one pair of parallel sides. The legs of a trapezoid can be of different lengths, and the angles between the legs and the bases can also vary.

The formula for finding the area of a trapezoid is:

Area = ((Base 1 + Base 2) x Height) / 2

where Base 1 and Base 2 are the lengths of the parallel sides of the trapezoid, and Height is the distance between the parallel sides.

There are several types of trapezoids, including:

Isosceles Trapezoid: An isosceles trapezoid has two parallel sides of equal length and two non-parallel sides of equal length. The angles between the legs and the bases are also equal.

Right Trapezoid: A right trapezoid has one right angle between the leg and the base.

Scalene Trapezoid: A scalene trapezoid has two non-parallel sides of different lengths, and the angles between the legs and the bases are also different.

Let the number in the bottom box=

7.5*2=15 yd

Because the longest side of the table is twice as long as the table's width.

So the number in top box is :

15-3-3=9 yd

Area=( 15 + 9 )* 7.5/2 = 12* 7.5 = 90 yd²

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The complete question is:

The time it will take for Alex to earn enough money to purchase his shoes is given as follows:

13.6 weeks.

The time needed to obtain enough money to purchase the shoes is obtained applying the proportions in the context of the problem.

Alex saves $10 every 4 weeks. He earned an extra $12 for cleaning his neighbor’s garage, hence the amount he must save is of:

45.99 - 12 = $33.99.

Considering that he saves $10 each week, and he must save $33.99 to purchase the shoes, the number of periods of four weeks that he needs is of:

33.99/10 = 3.399 periods of four weeks.

As each period is composed by four weeks, the number of weeks that he needs to save enough money is obtained as follows:

4 x 3.399 = 13.6 weeks.

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Answer:

141 inch^2

Step-by-step explanation:

13×7= 91 for the rectangle

13-5=8

diameter of circle=8

radius=4

area if circle=4^2 times π

=16×π

=16π

16π+91=141.2654....

to the nearest whole= 141 inch^2 I think

as the quadrilateral is a square, all the sides will be equal to each other so DE is equal to DG.

3t + 46 = 8t - 19

-5t = -65

5t = 65

so t = 13

Answer: 600

Step-by-step explanation:150=.25*X Divide both sides by .25 to get rid of it on the right and move it to the left so now 150/.25=X  600=X

The Bisection Method is a numerical method used to find the roots of a function within a given interval. It works by repeatedly dividing the interval in half and checking which half contains the root until the desired level of accuracy is achieved. In this case, we are using 17 iterations to extract the root of the function f(x) = x/3 + In(x) - 1 on the closed interval [1,2].

To locate the root of f(x) = x/3 + In(x) - 1 on a closed interval [1,2] using the Bisection Method with 17 iterations, we will follow the steps below:

1. Define the function f(x) = x/3 + In(x) - 1
2. Set the initial values for the interval: a = 1 and b = 2
3. Start the iteration process by finding the midpoint of the interval: c = (a+b)/2
4. Evaluate the function at the midpoint: f(c) = f((a+b)/2)
5. Determine if the root lies in the left or right half of the interval by checking the sign of f(c)
6. If f(c) is positive, then the root lies in the left half of the interval, so we set b = c and repeat the process from step 3.
7. If f(c) is negative, then the root lies in the right half of the interval, so we set a = c and repeat the process from step 3.
8. Continue the iteration process until we have completed 17 iterations.

After 17 iterations, we will have narrowed down the interval to a very small range and the midpoint of this interval will be an approximation of the root of the function.

The Bisection Method is a numerical method used to find the roots of a function within a given interval. It works by repeatedly dividing the interval in half and checking which half contains the root until the desired level of accuracy is achieved. In this case, we are using 17 iterations to extract the root of the function f(x) = x/3 + In(x) - 1 on the closed interval [1,2].

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Answer:

First, find the total number of cups of liquid in the punch:

24 cups pineapple juice + 46 cups orange juice + 72 cups seltzer = 142 cups

Then, divide the total cups of liquid by the amount of liquid in each serving:

108 cups / 142 cups per serving = 0.76 servings

Therefore, approximately 0.76 servings can be made from 108 cups of punch.

We can conclude that the students did not have any time left to work on homework.

The duration of Mr. K's math class is 1 and 1/4 hours, or 5/4 hours.

We need to determine how much time is left for the kids to do their assignment after working on problems on the board for 55 minutes and 11/12 hours. To achieve this, we must take the time spent solving issues out of the overall class time:

Total class time = 5/4 hours

Time spent on working problems = 55 minutes + 11/12 hours

= (55/60) hours + (11/12) hours

= (11/12 + 55/60) hours

= (11/12 + 11/12) hours

= (22/12) hours

= (11/6) hours

Overall class time minus the amount of time spent solving problems equals the amount of time remaining for homework.

= 5/4 hours - 11/6 hours

= (15/12) hours - (22/12) hours

= (-7/12) hours

We can infer that the kids ran out of time to finish their homework because the response is illogical because we cannot have negative time.

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a. The probability that x=10 in a Poisson distribution with µ = 7 is 0.0942.
b. The probability that x > 5 in a Poisson distribution with µ = 7 is 0.4790.

A discontinuous probability distribution is a Poisson distribution. It provides the likelihood that an occurrence will occur a specific number of times (k) over a predetermined period of time or place. The mean number of occurrences, denoted by the letter "lambda," is the only component of the Poisson distribution.

When a discontinuous count variable is the subject of concern, poisson distributions are used. Many financial and economic data are count variables, such as how frequently someone is laid off in a given year, which makes them amenable to study using a Poisson distribution.

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Answer:

this is awnser

Step-by-step explanation:

Answer:

X=5 y=27

Step-by-step explanation:

2x+17=6x-3

Take away 2x both sides
17=4x-3
Add 3 to get x on one side
20=4x
20/4 =5
x=5
Subsitute 5 in, 6 x 5 = 30 30-3=27

The missing value in this equation is 4, or (4)/(1) in fraction form.

The missing value in this equation is the exponent of n in the final expression. To find this value, we can use the rule of exponents that states that when raising a power to another power, we multiply the exponents. In this case, we have (n^(2))^(2), so we multiply the exponents 2 and 2 to get a final exponent of 4. Therefore, the missing value is 4 and the equation can be written as (n^(2))^(2)=n^(4).

In fraction form, the missing value would be (4)/(1), since any whole number can be written as a fraction with a denominator of 1. So, the equation can also be written as (n^(2))^(2)=n^((4)/(1)).

Overall, the missing value in this equation is 4, or (4)/(1) in fraction form.

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The car travels 4 feet in 1 second, 16 feet in 4 seconds, 24 feet in 6 seconds, 32 feet in 8 seconds.

Distance is a measure of the amount of space between two objects or points. It is the length of the shortest path between the two points, and it is typically measured in units such as meters, kilometers, miles, or feet.

In mathematics, distance is often calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The car travels 4 feet in 1 second, which is equivalent to a rate of 4 feet per second.

The car also travels 16 feet in 4 seconds, which is also equivalent to a rate of 4 feet per second.

Similarly, the car travels 24 feet in 6 seconds, 32 feet in 8 seconds, and so on, all at a constant rate of 4 feet per second.

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The solutions to the equation (w-2)^(2)-54=0 are w = 2, w = 9, and w = -6.

The equation (w-2)2 - 54 = 0 can be solved by factoring.

Let's start by writing the equation as: (w-2)(w-2) - 54 = 0

We can factor out (w-2) to get: (w-2)(w2 - 4w - 54) = 0

Now we can factor the second part of the equation, by splitting the middle term (4w) to get: (w-2)(w-9)(w+6) = 0

Now we have three equations, which we can solve separately:

w - 2 = 0 → w = 2

w - 9 = 0 → w = 9

w + 6 = 0 → w = -6

Therefore, the solutions to the equation (w-2)2 - 54 = 0 are w = 2, w = 9, and w = -6.

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Using cοmpοund interest , the tοtal wοrth after 20 years is $2,330.48.

The interest that is calculated using bοth the principal and the interest that has accrued during the previοus periοd is called cοmpοund interest. It differs frοm simple interest in that the principal is nοt taken intο accοunt when determining the interest fοr the subsequent periοd with simple interest.

Here the principal P= $500

Rate οf interest r = 8%

Number οf years = 20 years,

Nοw using cοmpοund interest fοrmula then,

=> Total amount = [tex]P(1+\frac{r}{100})^t[/tex]

=> A = [tex]500(1+\frac{8}{100})^{20}[/tex]

=> A = [tex]500(1+0.08)^{20}[/tex]

=> A = [tex]500(1.08)^{20}[/tex]

=>A = $2,330.48

Hence after 20 years , nft's total worth is $2,330.48.

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Answer: The answer would be b<-9 for the nonsimplified version though.

Step-by-step explanation:

b+1>-8

first, -1 from each side.

then you will have -9>b

But you are going to divide by a negative number, so you are going to flip the inequality.

( b>-9 )

I remember learning this is like 7th grade haha. Let me know if you have any questions

The area of the shaded region which is a triangle will be 32 square inches.

Surface area refers to the area of an open surface or the border of a multi-object, whereas the area of a plane region or plane field refers to the area of a form or planar material.

From the graph, the height of the triangle is given as,

2h = L

2h = 16 inches

h = 8 inches

And the base of the triangle is half of the base of the rectangle. Then we have

b = L / 2

b = 16 / 2

b = 8 inches

Then the area of the shaded region is given as,

A = 1/2 x 8 x 8

A = 32 square inches

The area of the shaded region which is a triangle will be 32 square inches.

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Answer:

Let's assume that the price before adding the VAT is x.

We know that the VAT rate is 8%, which means that the VAT amount is 8% of x, or 0.08x.

The total price including VAT is the sum of the price before VAT and the VAT amount, so we can write:

Total price = price before VAT + VAT amount

or

2,700 = x + 0.08x

Simplifying this equation, we can combine like terms on the right-hand side to get:

2,700 = 1.08x

To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 1.08:

x = 2,700 / 1.08

x = 2,500

Therefore, the price before VAT was added is 2,500.

The sοlutiοn οf the given prοblem οf area cοmes οut tο be the side length rises by 1, the area grοws and then shrinks by an equal factοr.

Calculating hοw much space wοuld be needed tο fully cοver the οutside will reveal its οverall size. When determining the surface οf such a trapezοidal fοrm, the surrοundings are additiοnally taken intο accοunt. The surface area οf sοmething determines its οverall dimensiοns. The number οf edges here between cubοid's fοur trapezοidal extremities determines hοw much water it can hοld inside.

Here,

Optiοn B

The area grοws initially and then decreases by an amοunt equivalent tο the side length multiplied by 1. The area grοws befοre decreasing by an equivalent amοunt.

This is the case because a square's surface and side length are inversely prοpοrtiοnal. Where r is the οriginal side length, the area grοws by a factοr οf (1 + 2r + r2) as the side length rises by 1.

When r is big, hοwever, the term r2 dοminates and the area increase is less nοticeable.

Eventually, as r gets clοser tο infinite, the area increases οnly marginally and eventually reaches a limit.

As a result, as the side length rises by 1, the area grοws and then shrinks by an equal factοr.

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Complete question:

Graph is the collection of all points whose coordinates satisfy a given relation (such as a function).

Coordinates of a point in a 2D plane, also known as cartesian coordinates, are two numbers or sometimes a combination of a letter and a number that tell us the exact location of a specific point on a grid. Here, the grid is known as a coordinate plane.

Let tosses be x and and dollars be y

Then we have, (x, y) as (63, 6)

So if in 6 dollars you get 63 tosses then 1 dollar you get = 63/6 = 10.5 tosses.

Then in 2 dollars we have = 10.5 × 2 = 21 tosses

As 21 tosses = 2 dollars

Then 42 tosses = 2 × 2 = 4 dollars

Thus, we have the table:

[tex]\boxed{\begin{array}{cc}\underline{\text{Tosses}}&\underline{\text{dollars}}&\\ \boxed{10.5}&2&\\42&\boxed4&\\63&6&\end{array}}[/tex]

To graph the given points, just correspond x as tosses and y as dollars.

Graph given in attachment

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Yes, a probability of 4/5 indicates that an event is likely to happen.

Probability refers to the measure of the likelihood of an event occurring, typically expressed as a number between 0 and 1. It is used to quantify uncertainty and is widely used in various fields such as mathematics, statistics, economics, science, and engineering.

Yes, a probability of 4/5 indicates that an event is likely to happen. It means that out of 5 possible outcomes, 4 outcomes are favorable to the event. In other words, there is an 80% chance that the event will occur.

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