Find all values of x for which the series converges. (Enter your answer using interval notation.)
∑(6x)^n

The probability that the tennis player will make exactly 3 first serves out of 8 attempts is 0.278%.

To solve this problem, we can use the binomial distribution. The binomial distribution is used to calculate the probability of a certain number of successes (in this case, first serves) in a fixed number of independent trials (in this case, serves). The formula for the binomial distribution is:

P(X = x) = (n choose x) x pˣ x (1 - p)ⁿ⁻ˣ

where P(X = x) is the probability of getting x successes, n is the number of trials, p is the probability of success in each trial, and (n choose x) is the binomial coefficient, which represents the number of ways to choose x successes out of n trials.

Using this formula, we can plug in the values from our problem:

P(X = 3) = (8 choose 3) x 0.6³ x (1 - 0.6)⁸⁻³

P(X = 3) = (8! / (3! x 5!)) x 0.216 x 0.32768

P(X = 3) = 0.278%

This means that out of 1000 attempts, we can expect the player to make exactly 3 first serves around 2-3 times. It's important to note that this is just an estimation, and the actual number of successful serves may vary.

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Answer:

It is a trapezoid

Step-by-step explanation:

Yes, the given points represent the vertices of a trapezoid.

A trapezoid is a quadrilateral with one set of parallel sides. In this case, the parallel sides are AB and CD. The other two sides, AD and BC, are not parallel.

The trapezoid is not isosceles because the two non-parallel sides are not congruent. AD has a length of 6 units, while BC has a length of 4 units.

Here is a diagram of the trapezoid:

A(-4,-1)

B((-4,6)

C(2,6)

D(2,-4)

A trapezoid is a quadrilateral with at least one pair of parallel sides. In this case, sides AB and CD are parallel because they have the same slope. So, the given points do represent the vertices of a trapezoid.

An isosceles trapezoid has two congruent legs (non-parallel sides). In this case, the length of side AD is `sqrt((-4-2)^2+(-1+4)^2)=sqrt(36+9)=sqrt(45)` and the length of side BC is `sqrt((-4-2)^2+(6-6)^2)=sqrt(36+0)=sqrt(36)`. Since `sqrt(45)` is not equal to `sqrt(36)`, the trapezoid is not isosceles.

The measure of an angle that goes through 2/8 of a circle is 90°

A circle is a 2-dimensional shape that is round in shape it is equidistant from the center.

A circle has a total angle of 360°

That is the whole complete angle of the circle = 360°

The 2/8 th of the complete angle of the circle = 360 * 2/8

= 360 * 1/4

= 360/4

=90°

Thus, the 90° of the circle is given as 2/8th of an angle of the circle or we can say that the quarter angle of a circle comes out to 90°.

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In a bag containing 5 red, 3 blue, and 12 yellow marbles, we will predict the number of times Hazel will select a blue marble out of 500 trials.

Step 1: Calculate the total number of marbles in the bag:
Total marbles = 5 red + 3 blue + 12 yellow = 20 marbles

Step 2: Determine the probability of selecting a blue marble:
Probability of selecting a blue marble = (number of blue marbles) / (total marbles) = 3 blue / 20 marbles = 3/20

Step 3: Predict the number of times Hazel will select a blue marble in 500 trials:
Predicted blue marbles selected = (probability of selecting a blue marble) x (total trials) = (3/20) x 500

Step 4: Perform the calculation:
(3/20) x 500 = 75

In conclusion, we predict that Hazel will select a blue marble 75 times out of 500 trials, given that the bag contains 5 red, 3 blue, and 12 yellow marbles.

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Dilating the figure by a scale factor of 3, the new coordinates are:

A' (-3, 12)

B' (9, 6)

C' (5886, -6)

D (unknown)

To dilate a figure by a scale factor, we need to multiply the coordinates of each point by the scale factor. Given the scale factor K = 3, we can dilate the figure using the formula:

New x-coordinate = K * original x-coordinate

New y-coordinate = K * original y-coordinate

Let's apply this to the given coordinates:

(-1, 4)

New x-coordinate = 3 * (-1) = -3

New y-coordinate = 3 * 4 = 12

A' (-3, 12)

(3, 2)

New x-coordinate = 3 * 3 = 9

New y-coordinate = 3 * 2 = 6

B' (9, 6)

(1962, -2)

New x-coordinate = 3 * 1962 = 5886

New y-coordinate = 3 * (-2) = -6

C' (5886, -6)

Dilating the figure by a scale factor of 3, the new coordinates are:

A' (-3, 12)

B' (9, 6)

C' (5886, -6)

D (unknown)

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The cost of apples and bananas are $ 0.70 and $ 1.10 respectively if Patricia bought 4 apples and 9 bananas for $12.70 and Jose got 8 apples and 11 bananas for $17.70

Let the cost of one apple be a

the cost of one banana be b

In the case of Patricia,

12.70 = cost of 4 apples + cost of 9 bananas

Cost of 4 apples = 4a

Cost of 9 bananas = 9b

The equation we get is

4a + 9b = 12.70 ----(i)

In the case of Jose,

17.70 = cost of 8 apples + cost of 11 bananas

Cost of 8 apples = 8a

Cost of 11 bananas = 11b

The equation we get is

8a + 11b = 17.70 ----(ii)

Multiply (i) by 2

8a + 18b = 25.40 --- (iii)

Subtract (ii) and (iii)

7b = 7.70

b = $ 1.10

4a + 9 (1.10) = 12.70

4a + 9.90 = 12.70

4a = 2.80

a = $ 0.70

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Answer:

22.86 centimeters

Step-by-step explanation:

How many centimeters are in 9 inches?

1 inch = 2.54 centimeters

9 inches = 2.54 x 9 = 22.86 centimeters

So, there are 22.86 centimeters in 9 inches.

Answer:

22.86 centimeters

Step-by-step explanation:

To convert inches to centimeters, multiply the inches by 2.54:

9·2.54=22.86

So, there are 22.86 centimeters in 9 inches.

Hope this helps :)

The probability in the two fair dice problem is given as [tex]P(A|B) = 1/6[/tex].

To find [tex]P(A|B)[/tex], we first need to find [tex]P(B)[/tex], which is the probability that two dice are showing different faces.

The total number of possible outcomes when rolling two dice is [tex]6x6 = 36[/tex]. Out of these [tex]36[/tex] possible outcomes, there are [tex]6[/tex] outcomes where both dice show the same face (e.g., both dice show a 1). Therefore, there are [tex]36-6=30[/tex] outcomes where two dice show different faces.

Hence, P(B) = [tex]30/36 = 5/6[/tex].

Next, we need to find the probability of A and B occurring together, i.e., P(A and B).

The possible pairs of faces that add up to 10 are [tex](4,6), (6,4),[/tex] and [tex](5,5)[/tex]. Each of these pairs can occur in 2 ways (e.g., the pair [tex](4,6)[/tex] can occur as [tex](4,6) or (6,4))[/tex]. Therefore, there are 6 ways in total for the sum of two dice to be 10.

Out of these 6 outcomes, only one outcome (the pair (5,5)) violates condition B (i.e., both dice showing the same face). Therefore, there are [tex]6-1=5[/tex] outcomes where the sum of the two dice is 10 and the two dice show different faces.

Hence, P(A and B) [tex]= 5/36[/tex].

Using the formula for conditional probability, we can find P(A|B) as:

[tex]P(A|B) = P(A and B) / P(B) = (5/36) / (5/6) = 1/6.[/tex]

Therefore, [tex]P(A|B) = 1/6[/tex], which is the required probability.

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The volume of the rectangular prism is 200 in³.

The volume of a rectangular prism is found by multiplying its length, width, and height. In this case, the prism has a length of 8 inches, width of 2 inches, and height of 12 and one-half inches. To calculate the volume, you would use the following formula:

Volume = Length × Width × Height

Now, plug in the given dimensions:

Volume = 8 in × 2 in × 12.5 in

Perform the calculations:

Volume = 16 in² × 12.5 in

Volume = 200 in³

So, the rectangular prism has a volume of 200 cubic inches. This means that the space occupied by the prism is equal to 200 cubic inches. The other options provided, such as 16 in³, 22 and one-half in³, and 212 and one-half in³, are not correct because they do not represent the product of the length, width, and height of the given prism. In conclusion, the correct answer for the volume of this rectangular prism is 200 in³.

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In the given circle, measure of angle m is 44° and the measure of angle n is 39°. Thus, the value of m is 44 and the value of n is 39

From the question, we are to determine the values of m and n in the given circle

From one of the circle theorems, we have that

The angles at the circumference subtended by the same arc are equal. That is, angles in the same segment are equal.

In the given diagram,

Angle m is in the same segment as the angle that measures 44°

Since angles in the same segment are equal,

Measure of angle m = 44°

Also,

Angle n is in the same segment as the angle that measures 39°

Since angles in the same segment are equal,

Measure of angle n = 39°

Hence,

m ∠m = 44°

m ∠n = 39°

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Answer:

2

Step-by-step explanation:

The median of a data set is the middle value when the data is arranged in order of size.

The given table of data is:

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\frac{3}{4}&3&1&8\\\cline{1-4}\vphantom{\dfrac12}5&\frac{1}{4}&\frac{3}{5}&\frac{7}{2}\\\cline{1-4}\end{array}[/tex]

Arrange the data in order of size:

[tex]\dfrac{1}{4},\;\dfrac{3}{5},\;\dfrac{3}{4},\;1,\;3,\;\dfrac{7}{2},\;5,\;8[/tex]

As there are 8 data values, the median is the average of the two middle values.

The two middle values are 1 and 3.

The average of 1 and 3 is 2:  

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

Therefore, the median of the given data is 2.

The given linear equation in slope intercept form is y = -19x/18 - 17/18.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + c

Where:

By making "y" the subject of formula, we have the following:

19x + 18y = –17

18y = -19x - 17

y = -19x/18 - 17/18

By comparison, we have the following:

Slope, m = -19/18.

y-intercept, c = -17/18.

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The measure of angle x is 77 degrees.

We have two angles given: 65 degrees and 38 degrees. Let's call the measure of angle x as "x".

From the information, we have the measure of the missing angle of the angles in a triangle are 38°, 65° and x. Then we can set up an equation:

x + 65 + 38 = 180 (the sum of the measures of the angles in a triangle is 180)

Simplifying this equation, we get:

x + 103 = 180

x = 77

Therefore, the measure of angle x is 77 degrees.

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Complete question

Find the measure of the missing angle of the angles in a triangle are 38°, 65° and x.

The measure of the largest interior angle is  105°.

What is the measure of angle?

When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.

Here, we have

Given: The measures of 5 of the interior angles of a hexagon are: 130, 120°, 80, 160, and 155. we have to find the measure of the largest interior angle.

The sum of all the six interior angles of a hexagon is 720°.

As sum of five angles is 130° + 120° + 80° + 160° +155° = 165°

The sixth angle is 720° - 165° = 75°

So the smallest interior angle of the hexagon is 75°.

and the largest exterior angle is 180° -  75° =  105°.

Hence, the measure of the largest interior angle is  105°.

Question: The measures of 5 of the interior angles of a hexagon are: 130, 120°, 80, 160, and 155 What is the measure of the largest exterior angle?

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Mr. Smith should make 3 rows on the table so that each brownie has its own square.

We shall use mathematical operations to determine the number of rows Mr. Smith would use on the table.

Some mathematical operations include addition, subtractions, multiplications, division, etc., to find out the number of rows Mr. Smith would make.

First, let's find the total number of brownies brought by the students:

12 + (4 x 3) = 24

Next, we shall divide the table into squares so that each brownie has its own square.

Since there are 24 brownies, we need 24 squares.

Then, since the table has 8 columns, we can divide the brownies equally among these columns to get the number of rows needed.

24 ÷ 8 = 3

Therefore, Mr. Smith should make 3 rows on the table so that each brownie has its own square.

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The police dog spends 1/6 of its workday at the police station.

To find the fraction of the police dog's workday spent at the police station, we need to add up the fractions of time spent in each location and subtract them from 1, since the dog spends the rest of the day at the police station.

Fraction of time spent in police car =  [tex]1/3[/tex]

Fraction of time spent in public = [tex]1/2[/tex]

To add these fractions, we need to find a common denominator:

[tex]1/3 = 2/6\\1/2 = 3/6[/tex]

So, the fraction of the dog's day spent at the police station is:

[tex]1 - (2/6 + 3/6) = 1 - 5/6[/tex]

                        = [tex]1/6[/tex]

Therefore, the police dog spends 1/6 of its workday at the police station.

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Step-by-step explanation:

Use Pythagorean theorem for right triangles

c^2 = a^2 + b^2      where c = hypotenuse and   a   and b are the legs

8.6^2 = 5^2 + r^2

8.6^2 - 5^2 = r^2

r = ~ 7  units

The amount of water needed for 1/3 cup of rice, is 3/4 cups of water.

The problem asks us to find out how much water is needed for 1/3 cup of rice, given that 2 1/4 cups of water are needed for 1 cup of rice. To solve this problem, we can use a proportion.

A proportion is an equation that says two ratios are equal. In this case, we want to set up a proportion that relates the amount of water needed to the amount of rice.

Let's start by writing down what we know. We know that for 1 cup of rice, we need 2 1/4 cups of water. We can write this as a ratio:

2 1/4 cups water : 1 cup rice

Now we want to figure out how much water we need for 1/3 cup of rice. Let's call the amount of water we need "x" (we don't know what it is yet), and set up another ratio:

x cups water : 1/3 cup rice

We can now set up our proportion by equating these two ratios:

2 1/4 cups water : 1 cup rice = x cups water : 1/3 cup rice

To solve for x, we can cross-multiply and simplify. Cross-multiplying means we multiply the numerator of one ratio by the denominator of the other ratio, like this:

(2 1/4 cups water) * (1/3 cup rice) = (x cups water) * (1 cup rice)

To simplify this, we can convert the mixed number 2 1/4 to an improper fraction:

2 1/4 = 9/4

Now we can substitute these values and multiply:

(9/4 cups water) * (1/3 cup rice) = (x cups water) * (1/1 cup rice)

Multiplying the fractions on the left-hand side gives:

9/12 cups water = (x cups water) * (1/1 cup rice)

Simplifying the fraction on the left-hand side gives:

3/4 cups water = x cups water

So we have found that x, the amount of water needed for 1/3 cup of rice, is 3/4 cups of water. Therefore, if you have 1/3 cup of rice, you would need to use 3/4 cups of water to cook it.

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Therefore, the trader should sell the article for GHC 4440.00 to make a profit of 48%.

Percent is a way of expressing a number as a fraction of 100. The term "percent" means "per hundred". Percentages are usually denoted by the symbol %, which is placed after the numerical value. Percentages are used in many fields, including finance, science, and everyday life, to represent proportions, rates, and changes in quantities.

Here,

Let's call the original cost of the article "C".

We know that the trader made a profit of 24%, which means that he sold the article for 100% + 24% = 124% of its cost:

124% of C = GHC 3720.00

To find C, we can divide both sides by 1.24:

C = GHC 3720.00 / 1.24

C = GHC 3000.00

So the trader originally purchased the article for GHC 3000.00.

Now we want to know how much the trader should sell the article for to make a profit of 48%. This means that he wants to sell the article for 100% + 48% = 148% of its cost:

148% of C = ?

Substituting C = GHC 3000.00, we get:

148% of GHC 3000.00 = (148/100) x GHC 3000.00

= GHC 4440.00

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The value of S2 is 50, under the condition that {sn} is  a geometric sequence that starts with an initial index of 0.

Here we have to apply the principles of geometric progression.
The derived formula for regarding the nth term concerning the geometric sequence is
[tex]= ar^{n-1 }[/tex]
Here
a = first term and r is the common ratio.
For the given case from the question
a = 2
r = 5.
Then,
s2 = a× r²
= 2×5²
= 50.
A geometric sequence refers to a particular sequence of numbers that compromises each term after the first is evaluated by multiplying the previous one by a fixed one , non-zero number known as  the common ratio.

For instance, if the first term of a geometric sequence is 2 and the common ratio is 5, then the sequence would be 2, 10, 50, 250.
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The area of the region bounded by the curves is 4/3 square units.

To find the area of the region bounded by the curves y = 10 - [tex]x^2[/tex] and y = [tex]x^2[/tex] + 8, we need to find the points of intersection between the two curves.
Setting the two equations equal to each other, we have:
10 - [tex]x^2[/tex] = [tex]x^2[/tex] + 8
Simplifying, we get:
[tex]2x^2[/tex] = 2
[tex]x^2[/tex] = 1
x = ±1
Substituting x = 1 into either equation gives us:
y = 10 - [tex]1^2[/tex] = 9
And substituting x = -1 gives us:
y = 10 - [tex](-1)^2[/tex] = 10
So the two curves intersect at the points (1, 9) and (-1, 10).
To find the area of the region bounded by the curves, we need to integrate the difference between the two equations with respect to x, from x = -1 to x = 1:
∫[10 - [tex]x^2[/tex]] - [[tex]x^2[/tex] + 8] dx, from x = -1 to x = 1
= ∫(2 - 2[tex]x^2[/tex]) dx, from x = -1 to x = 1
= [2x - (2/3)[tex]x^3[/tex]] from x = -1 to x = 1
= 4/3
So

The area of the region bounded by the curves y = 10 - [tex]x^2[/tex] and y = [tex]x^2[/tex] + 8 is 4/3 square units.

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The value of s is, 27 yards

:: Volume of cube with side s, is equal to s³

So, as the given volume is 19,683 cubic yards.

Therefore, it can related as,

s³ = 19,683 (yards)³

So,

s = ∛(19,683) yards

s = 27 yards

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To find the value(s) of the variable(s), you need to have an equation or problem statement that relates the variable(s) to other known quantities. Once you have this equation or statement, you can solve for the variable(s) by manipulating the equation algebraically.

For example, if the problem states that 2x + 5 = 17, you can solve for x by first subtracting 5 from both sides to get 2x = 12. Then, you can divide both sides by 2 to get x = 6. So, the value of the variable x is 6.

In some cases, you may need to use more advanced methods such as factoring or the quadratic formula to solve for the variable(s). Regardless of the method used, it's important to check your answer(s) by plugging them back into the original equation to make sure they satisfy the given conditions.

In terms of rounding decimal answers to the nearest tenth, this means that if the answer is a decimal with more than one digit after the decimal point, you would round to the nearest tenth place (i.e. the digit immediately to the right of the decimal point). For example, if the answer is 3.456, you would round to 3.5.

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The number of left-handed students in the first trial of the simulation can be found by following the above steps and counting the occurrences of two-digit numbers within the 00-14 range on line 1 of the random-number table.

To find out how many left-handed students occur in the first trial of the simulation, you'll need to follow these steps,
1. Identify the probability range for left-handed students, which is 00-14 as you've mentioned.
2. Start on line 1 of the random-number table.
3. Read each two-digit number on the line and check if it falls within the range 00-14.
4. Count the number of times a number within the 00-14 range appears in the first 10 two-digit numbers (since you're selecting a random sample of 10 students).
5. The count of numbers within the 00-14 range represents the number of left-handed students in the first trial of the simulation.

By doing the aforementioned processes and counting the occurrences of two-digit numbers between the ranges of 00 and 14 on line 1 of the random-number table, it is possible to determine the number of left-handed pupils in the simulation's first trial.

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Answer:

x = 150

Step-by-step explanation:

We know that the total amount of degrees in a circle is 360°.

We also know that a right angle is 90°.

Using this information, and the given 120° angle, we can form the following equation to solve for x:

90° + 120° + x° = 360°

210° + x° = 360°

x° = 360° - 210°

x° = 150°

x = 150

Step-by-step explanation:

120° + 90° + x = 360°

210° + x = 360°

x = 360° - 210°

= 150°

The area of the triangle ∆WXY is derived to be 5.7 to the nearest tenth.

When two side length of a triangle and the angle between them is given, the area is half the multiplication of the two sides and the sine of the angle.

Area of the triangle = 1/2 × 4.7 × sin162

Area of the triangle = 11.4738/2

Area of the triangle = 5.7369.

Therefore, the area of the triangle ∆WXY is derived to be 5.7 to the nearest tenth.

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The amount of more sauce needed is 5/12.

Thus we are given that the total amount of sauce required for making the pizza is 2/3 cup.

Thus we already have 1/4 cup of tomato sauce present.

Hence, for making the recipe the leftover amount of sauce will be the difference in the sauce we have got to the sauce required.

Tomato sauce required= Total tomato sauce needed - Sauce already present

Therefore,

Tomato sauce required= 2/3-1/4

Thus we have to make the denominators qual by taking their LCM as the denominator.

The LCM of the denominators comes out to be 12.

Therefore,

[tex]=\frac{8-3}{12}[/tex]

[tex]=\frac{5}{12}[/tex]

Therefore, the amount of sauce required is 5/12.

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The recursive formula for an, the nth term of the sequence is a(n) = a(n - 1) * 2 where a(1) = -1

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

-1, -3, -9, -3³ ...

The above definitions imply that we simply multiply 3 to the previous term to get the current term

Using the above as a guide,

So, we have the following representation

a(n) = a(n - 1) * 3

Where

a(1) = -1

Hence, the sequence is a(n) = a(n - 1) * 2 where a(1) = -1

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∠AOD≅∠COB,∠AOB≅∠COD by vertical angle theorem. ΔAOD≅ΔCOB,ΔAOB≅ΔCOD by SAS. ∠DAC≅∠BCA,∠BAC≅∠DCA by CPCTC. AB║CD,AD║BC by converse alternate interior angles theorem

Vertical angles theorem states that perpendicular angles, angles that are contrary each other and formed by two cutting straight lines, are harmonious.

Alternate angle theorem states that when two resemblant lines are cut by a transversal, also the performing alternate interior angles or alternate surface angles are harmonious.

SAS Side angle side

CPCTC Corresponding corridor of harmonious triangles are harmonious.

discourse of alternate interior angle theorem If two alternate interior angles are harmonious also the two lines are resemblant.

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