The first 8 terms of the sequence are 12, 16, 22, 30, 40, 52, 66, 82 respectively.
(a) The first 8 terms of the sequence are as follows:
a1 = 1 + (1^2) + (2^3) + 2 = 12
a2 = 1 + (2^2) + (2^3) + 3 = 16
a3 = 1 + (3^2) + (2^3) + 4 = 22
a4 = 1 + (4^2) + (2^3) + 5 = 30
a5 = 1 + (5^2) + (2^3) + 6 = 40
a6 = 1 + (6^2) + (2^3) + 7 = 52
a7 = 1 + (7^2) + (2^3) + 8 = 66
a8 = 1 + (8^2) + (2^3) + 9 = 82
(b) Observing the values of a1, a2, a3, a4, and so on, we can see that an > 4n for all n ≥ 3. Hence, a = 3.
(c) We will prove the statement "an > 4n for all integers n ≥ a" using strong induction.
Base case (n=3): a3 = 28 and 28 > 4 * 3 = 12, which is true.
Inductive step: Suppose an > 4n for all integers n = k (k ≥ 3). We need to show that a(k+1) > 4(k+1).
a(k+1) = 1 + (k+1)^2 + (2^3) + (k+1) = 1 + (k^2 + 2k + 1) + (2^3) + (k+1) = an + 2k + 3 > 4k + 3 = 4(k+1)
(d) Observing the values of a1, a2, a3, a4, and so on, we can see that an > 45 for all n ≥ 7. Hence, b = 7.
(e) We will prove the statement "an > 5n for all integers n ≥ b" using strong induction.
Base case (n=7): a7 = 120 and 120 > 7 * 5 = 35, which is true.
Inductive step: Suppose an > 5n for all integers n = k (k ≥ 7). We need to show that a(k+1) > 5(k+1).
a(k+1) = 1 + (k+1)^2 + (2^3) + (k+1) = 1 + (k^2 + 2k + 1) + (2^3) + (k+1) = an + 2k + 3 > 5k + 3 = 5(k+1)
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One afternoon Mr. And Mrs. Baxter and their 3 children were busy working outside in their garden. Mrs. Baxter was feeling hungry, so she went inside to the kitchen where there was a plate full of cookies. She ate 1/6 of the cookies and then went back to work. Mr. Baxter was feeling hungry, so he went inside and ate 1/5 of the remaining cookies and then went back to work. Next, Anna was feeling hungry, so she went inside and ate 1/4 of the cookies that were left on the plate and then went back to work. Then Tyler was feeling hungry, so he went inside, ate 1/3 of the cookies left on the plate and then went back to work. Finally, little Adam was feeling hungry, and he went inside and ate 1/2 of the remaining cookies, leaving 4 cookies on the plate. How many cookies were on the plate before anyone started feeling hungry?
24 cookies were on the plate before anyone started feeling hungry.
Let X be the number of cookies on the plate before anyone started feeling hungry.
After Mrs. Baxter ate 1/6 of the cookies, there were,
X - (X * 1/6) = 5/6X cookies left.
After Mr. Baxter ate 1/5 of the remaining cookies, there were,
5/6X - (5/6X * 1/5) = 5/6X- 1/6X = 4/6X cookies left.
After Anna ate 1/4 of the remaining cookies, there were,
4/6X - (4/6X * 1/4) = 4/6X- 1/6X = 3/6X cookies left.
After Tyler ate 1/3 of the remaining cookies, there were
3/6X - (3/6X * 1/3) = 3/6X- 1/6X = 2/6X cookies left.
After Adam ate 1/2 of the remaining cookies, there were,
2/6X - (2/6X * 1/2) = 2/6X- 1/6X = 1/6 cookies left.
We know, after Adam eats, 4 cookies are left, therefore,
1/6X = 4
So X = 4 *6 = 24
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Jamila ha p pencil. Haan ha 3 fewer pencil than Jamila. Naomi ha 4 time a many pencil a Haan. Write an expreion, in term of p, for the total number of pencil that the three children have. Write your expreion in it implet form.
The total number of pencils that the three children have is represented by the expression p + 3 + 4p. This expression can be simplified to p + 3 + 4p.
The total number of pencils that the three children have is represented by the expression p + 3 + 4p. This expression can be simplified to p + 3 + 4p.
Jamila has p pencils. Haan has 3 fewer pencils than Jamila, so Haan has (p - 3) pencils. Naomi has 4 times as many pencils as Haan, so Naomi has (4(p - 3)) pencils.
Therefore, the total number of pencils the three children have is p + (p - 3) + (4(p - 3)). This can be simplified to p + 3 + 4p.
The total number of pencils that the three children have is represented by the expression p + 3 + 4p. This expression can be simplified to p + 3 + 4p.
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since 2005 the amount of money spent on restaurants in a certain country has increased at a rate of 4% each year in 2005 about 670 billion was spent on restaurants if the trend continues about how much money will be spent on restaurants in 2013 
Answer: Since 2005 the amount of money spent on restaurants in a certain country has increased at a rate of 4% each year. In 2005, about 670 billion was spent on restaurants.
To find the amount spent on restaurants in 2013, we can use the formula:
A = P(1 + r)^t
Where:
A = the final amount
P = the initial amount (670 billion)
r = the rate of increase (0.04 or 4%)
t = the number of years since the initial amount (8 years)
Plugging in the values, we get:
A = 670 billion (1 + 0.04)^8
A = 670 billion (1.04)^8
A = 670 billion * 1.4304
A = 951.8 billion
So, if the trend continues, about 951.8 billion will be spent on restaurants in 2013.
Step-by-step explanation:
Write a quadratic given a vertex (11,2) and a point (9,3)
The equation of a quadratic equation with a vertex and a point is
y = 1/4 (x - 11)² + 2
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 6 is an equation.
We have,
Vertex = (11, 2)
Point = (9, 3)
The equation of a quadratic equation with a vertex and a point is
y = a (x - h)² + k
Where (h, k) is the vertex and (x, y) is the point.
Now,
(11, 2) = (h, k)
(9, 3) = (x, y)
Substitute.
y = a (x - h)² + k
3 = a (9 - 11)² + 2
3 = a x 4 + 2
3 = 4a + 2
3 - 2 = 4a
a = 1/4
Now,
The equation of a quadratic equation is y = 1/4 (x - 11)² + 2.
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Write 0.000001365 in scientific notation.
Answer: 1.365 × 10-6
Step-by-step explanation:
All the numbers in scientific notation written in the form a × 10b, where b is an integer and the coefficient a is a non-zero real number between 1 and 10 in absolute value.
To convert 0.000001361 into scientific notation also known as standard form, follow these steps:
Move the decimal 6 times to right in the number so that the resulting number, m = 1.361, is greater than or equal to 1 but less than 10
Since we moved the decimal to the right the exponent n is negative
n = -6
Write in the scientific notation form, m × 10n
= 1.361 × 10-6
Therefore, 0.000001361 in scientific notation is 1.361 × 10-6 and It has 4 significant figures. In scientific e-notation it is written as 1.361e-6.
What fraction in simplest form is 28 hours out of a week
Consider the function f(x). Select all of the following that include a vertical stretch of f(x).
The vertical stretch of f(x) will be -2f(x), 5.5f(x) - 1, 7f(x)
As is well known, the vertical stretch of f(x) occurs when the value of the parameter is changed in the following general form:
When a constant c with a value larger than one is scaled across the board of a function, it results in a vertical stretch. It is written as the vertical stretch from the original function f(x) to the new, stretched function g(x)=cf(x) where c>1 g ( x ) = c f ( x ) where c > 1.
f(k(x-d)) = y = a+ c
when |a| exceeds 1 since The function is stretched vertically by a dilation factor of |a| when |a| > 1 (when a > 1).
Thus, we select F, D, and A.
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The full question:
Consider the function f(x). Select all of the following that includes a vertical stretch of f(x)
A. -2f(x)
B. 0.25f(x)
C. -f(x) + 6
D. 5.5f(x) - 1
E. f(x) + 9
F. 7f(x)
Can’t figure this out need help
Answer: Hi! I believe your answer would be y = −5/7x − 5
Step-by-step explanation: I hope this helps!
3 (2 x - 7) = 3
How many solutions does this equation have?
Answer:
It has one solution, x = 4
Step-by-step explanation:
[tex]3(2x-7) = 3\\2x-7 = \frac{3}{3} \\2x-7 = 1\\2x = 8\\x = 4[/tex]
No other answer would work
Find the percent of change when 51 is increased to 98. If necessary, round your answer to the nearest tenth.
a
48% decrease
b
48% increase
оооо
Ос
92. 296 decrease
Od
92. 29 increase
The percent of change when 51 is increased to 98, then there have increment of 92%. So the option d is correct.
The difference between a new and old amount, given as a percentage, is known as a percent change.
Calculating percent change involves determining the difference between the old and new quantities, dividing that difference by the old quantity, and multiplying that result by 100.
The number 51 is raised to 98; calculate the percentage change.
To start, subtract the new quantity from the previous one to determine the difference. 98 - 51 = 47, therefore the difference is 47.
The difference, 47, is then divided by the old amount, 51, yielding a result of 47/51=0.92.
Lastly, multiply the decimal by 100 × 0.92 to get the percentage change, which is increment of 92%.
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I need help with my math homework
Answer:
6 = D
7 = F
8 = A
9 = B
10 = C
Have a nice day ;)
Please explain and answer!
The Pythagorean identity is proved to be 1 + cot2 = cosec2, which results in sin2 + cos2 = 1.
How to verify Pythagorean identity?With the value of sine and the quadrant in which the angle is placed, we can use the Pythagorean identity to get the angle of cosine.
If we are given the cosine value and the quadrant in which the angle is placed, we may similarly compute the angle of sine.
Cosec2 = 1 + cot2
Therefore,
cosec2 = 1/sin2
Tan2 = cos2 / sin2 = cot2 = 1
Hence,
1/sin2 = 1 + cos2 / sin2.
multiplied both sides by sin2 as a result.
1(sin2 ) + (cos2 / sin2 ) = (sin2 / sin2 )
cos2 + sin2 = 1.
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what is (5x+6)-(2x+5).
*PLEASE HELP SOON* Evaluate 5^x for each given value of x.
Each of the function should be evaluated as follows;
When x = 2, 5^x = 25.When x = 1, 5^1 = 5.When x = 0, 5^0 = 1.When x = -1, 5^-1 = 1/5.How to evaluate the function for each given value of x?In Mathematics, the valid and true solutions to any function are referred to as the domain (input value) and they can be determined by substituting the x-values contained in the table into the function.
When x = 2, the range (output value) for this function is given by:
Function, f(2) = 5^x = 5^2 = 25.
When x = 1, the range (output value) for this function is given by:
Function, f(1) = 5^x = 5^1 = 5.
When x = 0, the range (output value) for this function is given by:
Function, f(0) = 5^x = 5^0 = 1.
When x = -1, the range (output value) for this function is given by:
Function, f(x) = 5^x = 5^-1 = 1/5.
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The graph of y = In(x) was transformed by shifting 8 units left and 12 units up. Write an equation
The equation of the translated function for this problem is given as follows:
y = ln(x + 8) + 12.
What are the translation rules?The four translation rules are defined as follows:
Left a units: g(x) = f(x + a).Right a units: g(x) = f(x - a).Up a units: g(x) = f(x) + a.Down a units: g(x) = f(x) - a.The parent function in this problem is given as follows:
y = ln(x).
The translations in this problem, and their effects, are given as follows:
8 units left: x -> x + 8.12 units up: y -> y + 12.Hence the translated function is defined as follows:
y = ln(x + 8) + 12.
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#7
DE is a midsegment of AABC. Find the value of x .
X =
B
A 6 E
D
X
C
Using the Midsegment Theorem, the value of x is: 6.
What is the Midsegment Theorem of a Triangle?The Midsegment Theorem states that in a triangle, the segment that connects the midpoints of two sides of the triangle is parallel to the third side and is half as long as the third side. In other words, the midsegment of a triangle is parallel to the third side and has half the length of the third side.
Therefore the ratio of DE to AB would be: 1/2
Ratio of EC to AC would be: x/(x + 6)
Thus:
x/(x + 6) = 1/2
Cross multiply
2x = x + 6
2x - x = 6
x = 6
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A department store receives a shipment of
3 boxes of glass ornaments. There are
25 ornaments in each box. Lucas opens the
boxes and finds that 6 of the ornaments are
broken. Based on this shipment, what
prediction can Lucas make about the next
shipment of ornaments?
F A delivery of 4 boxes will contain 4 more
broken ornaments than a delivery of
3 boxes.
G A delivery of 5 boxes will contain 6 more
broken ornaments than a delivery of
3 boxes.
H A delivery of 6 boxes will contain 6 more
broken ornaments than a delivery of
3 boxes.
JA delivery of 7 boxes will contain 10 more
broken ornaments than a delivery of
3 boxes.
The prediction that Lucas can make about the next shipment of ornaments is that H. A delivery of 6 boxes will contain 6 more broken ornaments than a delivery of 3 boxes.
How to calculate the word problem?It should be noted that a word problem in mathematics simply refers to a question that is written as a sentence or in some cases more than one sentence which requires an individual to use his or her mathematics knowledge to solve the real life scenario given.
In this case, since the department store receives a shipment of 3 boxes of glass ornaments and there are 25 ornaments in each box.
Lucas then opens the boxes and finds that 6 of the ornaments are broken. This implies that there are 2 broken for each box.
In this case, a delivery of 6 boxes will contain 6 more broken ornaments than a delivery of 3 boxes.
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Determine whether each pair of equations are equivalent. Explain your answer. 5x − 4 + 3x = 8 and 8x = 12
Gary used to make bottles of glitter glue.
Bottles Glue Glitter
1 3.8 1.9
4
How much glue and glitter would he need to make 4 bottles?
Gary would need 15.2 oz glue and 3.8 oz of glitter to make 4 bottles.
Gary would need 6.8 oz glue and 4.9 oz of glitter to make 4 bottles.
Gary would need 7.8 oz glue and 5.9 oz of glitter to make 4 bottles.
Gary would need 15.2 oz glue and 7.6 oz of glitter to make 4 bottles.
Gary would need 15.2 oz glue and 7.6 oz of glitter to make 4 bottles.
What is Multiplication?Multiplication of two numbers is defined as the addition of one of the number repeatedly until the times of the other number.
a × b means that a is added to itself b times or b is added to itself a times.
We have the information that,
Amount of glue needed for 1 bottle = 3.8 oz
Amount of glitter needed for 1 bottle = 1.9 oz
We have to find the amount of glue and glitter for 4 bottles.
Finding the product for 4 bottles.
Amount of glue needed for 4 bottles = 3.8 × 4 oz = 15.2 oz
Amount of glitter needed for 4 bottles = 1.9 × 4 oz = 7.6 oz
Hence Gary will need 15.2 oz glue and 7.6 oz of glitter to make 4 bottles.
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Determine if the conditions stated would result in a unique quadrilateral.
Select Yes or No for each statement.
Four sides and one angle, three sides and two included angles, or two neighboring sides and three angles, to construct a unique quadrilateral.
What is meant by unique quadrilateral?
If the two diagonals and three sides of a quadrilateral are known, it is possible to create it in a singular way. If the two neighboring sides and three angles of a quadrilateral are known, it can be built in an original fashion.
A polygon called a quadrilateral has four sides, four angles, and four vertices. If the size of a quadrilateral's sides and diagonal are known, it is possible to create it in a singular way.
They must have the same number of sides as angles because they are polygons and have four angles. Every quadrilateral has inner angles that sum to 360 degrees.
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Which functions show a positive rate of change? Select all that apply
The functions that show a positive rate of change include the following:
A. y = 2x/3 - 2.
B. y = 7 - (-3x).
What is the slope-intercept form?In Mathematics, the slope-intercept form of a line can be represented or modeled by using this linear equation:
y = mx + c
Where:
m represents the slope or rate of change.c represents the y-intercept.x and y are the data points.Generally speaking, a function with a positive rate of change represents an increasing function because its numerical value is increasing over time such as in the following increasing function:
y = 2x/3 - 2.
Slope, m = 2/3 (positive slope).
y = 7 - (-3x)
y = 7 + 3x
Slope, m = 3 (positive slope).
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The functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).
What is a positive function?A positive function is a function that has the values of x that are greater or higher than zero on the graph. It means that they are always increasing on the positive y-axis i.e. f(x) > 0.
The function's outputs known as the y-values, however, must be larger than zero even though the domain values i.e. (x-values) might be negative. A positive function, then, has values that are positive for all of its domain's parameters.
To determine the positivity of the function on the graph, the slope must also be positive. Using the linear function form y = mx + b.
where;
m = slopeb = y-interceptFrom the given information, the functions that demonstrated a positive slope are:
y = 2/3x - 2 andy = 7 - (-3x)Therefore, we can conclude that the functions that show a positive rate of change are y = 2/3x - 2 and y = 7 - (-3x).
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I need help asap. I attached a imagine of the question.
Answer:
[tex]\boxed{n = - 5}[/tex]
Step-by-step explanation:
[tex]\textrm{We have the following equation:}\\\\\dfrac{n}{5}+\dfrac{\left(n-3\right)}{2}=n\\\\\textrm{ and we are asked to solve for n}[/tex]
Solution Steps
Multiply throughout by 10 which is the LCM of the denominators, 5 and 2
[tex]\dfrac{n}{5}\cdot \:10+\dfrac{n-3}{2}\cdot \:10=n\cdot \:10\\\\2n + 5(n-3) = 10n\\[/tex]
Expand the brackets: [tex]5(n-3) = 5n - 15\\\\[/tex]
The equation becomes:
2n + 5n - 15 = 10n
Add similar terms:
7n - 15 = 10n
Move 15 to the right side:
7n = 10n + 15
Move 10 to the left side
7n - 10n = 15
- 3n = 15
n = -15/3
[tex]\boxed{n = - 5}[/tex]
Use the long division method to find the result when 8x^{3}+6x^{2}+26x-128x
3
+6x
2
+26x−12 is divided by 4x-14x−1.
Answer:
4x^2-3x+1+18?=2x+3
Step-by-step explanation:
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Hace 8 años, Maria tenía 10 años y Luis 11 años. Dentro de 7 años, Nico tendrá 27 años ¿cuanto es la suma de las edades actuales de Maria Luis y Nico?
Their ages are given below:
Maria 18, Luis 19 Nico 20The sum of their ages would be 57 years.
How to find their ages?The current age of Nico is 27 - 7 = 20 years.
Therefore, Maria and Luis's current ages would be 8 years older than when they were 10 and 11 respectively,
so Maria is currently 18 years old and Luis is currently 19 years old.
Therefore, the sum of their current ages is 18 + 19 + 20 = 57 years.
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8 years ago, Maria was 10 years old and Luis was 11 years old. In 7 years Nico will be 27 years old, what is the sum of the current ages of Maria, Luis and Nico?
The surface area of a cylinder is 24π m². The distance around the base of the cylinder is 3π meters and the diameter of a base of the cylinder is 4 meters. What is the height of the cylinder? Enter your answer, as a simplified fraction, in the box.
The height of the cylinder will be 6 meters.
What is a cylinder?One of the most fundamental curvilinear geometric shapes, a cylinder has traditionally been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. In some modern branches of geometry, a cylinder may also be defined as an infinite curvilinear surface.
Given that the surface area of a cylinder is 24π m². The distance around the base of the cylinder is 3π meters and the diameter of the base of the cylinder is 4 meters.
The height of the cylinder will be calculated as:-
Surface area = 2πrh
24π = 2πrh
r x h = 12
The diameter is 4 meters so the radius is 2 cm.
r x h =- 12
h = 12 / r
h = 12 / 2 = 6 meters
The height of the cylinder will be 6 meters.
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Please help ASAP!!!!!!!!!
The correct linear model for the table is f(x) = 5x + 1.
What is the model for the table?We have to note that when we have a table of values, it is possible that we can be able to obtain the equation that can be able to model the function. We know that we can be able to write the line equation in the form y = mx + c
Where y = g(x) we can write the equation to obtain the slope of the line as;
m = y2 - y1/x2 - x1
Then we have;
1 - (-4)/0 - (-1)
5/1
m = 5
The correct equation of the line is therefore;
f(x) = 5x + 1
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An initial population of 430 quail increases at an annual rate of 9%. Write an exponential function to model the
quail population.
Answer:
An exponential function has the form y = ab^x, where a is the initial value, b is the base of the exponential function (in this case, 1 + the growth rate as a decimal) and x is the number of years.
Given that the initial population is 430 and the annual rate of increase is 9%, we can write the exponential function as:
y = 430 (1 + 0.09)^x
Where y is the quail population after x years, and the base (1 + 0.09) represents the growth rate of 9%. This function gives the population of the quail for any number of years x.
It's worth noting that the function is in the form of y = a*(1+r)^x where a is the initial value, r is the annual growth rate, and x is the number of years
a researcher reported that for 100 college students, the mean was 80 and the standard deviation was 12.6. how is the standard deviation best interpreted?
It can be interpreted as the average distance between each data point and the mean value of 80 for the 100 college students.
Standard deviation is a measure of the spread of values in a set of data, calculated as the square root of its variance. It indicates how much variation or dispersion there is from the mean of the set.
The standard deviation is a measure of the variability or dispersion of the data set around the mean. In this case, it can be interpreted as the average distance between each data point and the mean value of 80 for the 100 college students.
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a player of a video game is confronted with a series of 4 opponents and a(n) 73% probability of defeating each opponent. assume that the results from opponents are independent (and that when the player is defeated by an opponent the game ends). round your answers to 4 decimal places. (a) what is the probability that a player defeats all 4 opponents in a game? enter your answer in accordance to the item a) of the question statement (b) what is the probability that a player defeats at least 2 opponents in a game? enter your answer in accordance to the item b) of the question statement (c) if the game is played 3 times, what is the probability that the player defeats all 4 opponents at least once? enter your answer in accordance to the item c) of the question statement
The Probability that the player will win against each of the four opponents at least once is 0.7942.
The player has a P(W) = 0.80 winning chance.
The likelihood that the player will lose is hence P(L)= 1 - P(W) = 0.20.
The player is up against 4 different opponents in the video game.
It is stipulated that the game finishes when a player is defeated by an opponent.
The player can win in any of the following ways: L, WL, WWL, WWWL, and WWWW.
The results from each of the four opponents are distinct, meaning that the outcome of a game played against one opponent has no bearing on the outcome of a game against another.
P (Player defeats all 4 opponents) = 0.4096 is the probability that a player will win a game against all four opponents.
Thus, the probability that the player defeats all four opponents in a game is 0.4096.
(b) The likelihood that a player will win a game against at least two opponents is,
P = 1 (Player defeats at least 2) P = 0.64 when the player loses the first and second games.
Therefore, there is a 0.64 percent chance that the player will win at least two games against opponents.
Let X equal the number of times the player triumphs over all four opponents.
The likelihood that a player will win a game against all four opponents is,
P(WWWW) = 0.4096
The probability that a player will triumph over each of the four opponents at least once is,
P (X ≥ 1) = 1 - P (X < 1)
= 1 - P (X = 0)\s = 0.7942
Therefore, there is a 0.7942 probability that the player will win against all four opponents at least once.
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I NEED HELP ON THIS ASAP!!!!
Answer:
see below
Step-by-step explanation:
v=πr² (h/3)
h=3v/πr²
original equation measures v using bottom circle area multiply the height.
the equation for the height reverses the multiplication using volume divided by the area, so the result is a linear measurement.