The relationship modeled that has the function of f(x) = 4x³ - 72x² + 320x is known to be the volume of a right prism that has its dimensions to be : 4 times of its desired length, 10 units lower that its desired length, and 8 units lower than its desired length.
What is the volume of the prism about?For us to know the relationship modeled by the above function, the best way to do it is for one to factor it.
The function is: f(x) = 4x³ - 72x² + 320x
First. we factor it so as to extract the common factor 4x:
Secondly, we factor the quadratic trinomial.
This is often done by writing it as a product of two binomials where the two constant terms are said to be sum up - 18 and its product is said to be 80.
The above terms are known to be -10 and - 8; so the two factors are said to be (x - 10) and (x - 8).
Therefore The relationship modeled that has the function of f(x) = 4x³ - 72x² + 320x is known to be the volume of a right prism that has its dimensions to be : 4 times of its desired length, 10 units lower that its desired length, and 8 units lower than its desired length.
Where it has :
x = the desired length,
4x = 4 times its desired length,
x - 10 = 10 lower than its desired length,
x - 8 = 8 lower than its desired length,
We can therefore say that the volume of the prism is said to be the product of the three factors which are:
V = (4x)(x-10)(x-8)
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A cylindrical can, open at the top, is to hold 230 cm3 of liquid. find the height and radius that minimize the amount of material needed to manufacture the can. enter answer with rational exponents.
The height and the radius that minimize the material needed to manufacture the can is 4.18336 cm and 4.18337 cm respectively.
Computed using differentiation.
We assume the radius of the can be r cm, and its height to be h cm.
We are given that the can is to hold 230 cm³ of liquid, thus the volume of the can is:
V = 230,
or, πr²h = 230 {Since, the volume of a cylinder is πr²h, where r is the radius, and h is the height},
or, h = 230/(πr²) ... (i) .
We are asked to find the height and radius that minimize the amount of the material.
To calculate the amount of the material, we calculate the surface area of the can (A).
The surface area of the can = area of the base + area of the side,
or, A = πr² + 2πrh = πr(r + 2h).
Substituting h = 230/(πr²), we get:
A = πr(r + 2{230/(πr²)}),
or, A = {πr(πr³ + 460)}/πr² = πr² + 460/r.
Differentiating both sides with respect to the radius r, we get:
dA/dr = 2πr - 460/r² ...(ii)
To find the point of inflection, we equate this to zero, to get:
2πr - 460/r² = 0,
or, 2πr = 460/r²,
or, r³ = 460/2π = 73.2113,
or, r = 4.18337 cm.
To check whether the point of inflection shows the point of maximum or minimum, we differentiate (ii) with respect to the radius r, to get:
d²A/dr² = 2π + 920/r³, which is always positive when the radius r is positive.
Thus, the area is minimum when the radius r = 4.18337 cm.
Height, h = 230/(πr²) = 4.18336 cm.
Thus, the height and the radius that minimize the material needed to manufacture the can is 4.18336 cm and 4.18337 cm respectively.
Computed using differentiation.
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Determine the domain and range of (g ○ f)(x) if f of x is equal to 2 over the quantity x squared minus 1 end quantity and g(x) = x + 1.
The domain of the function is (-∞, 1) U(1, ∞) while the domains are values greater than or equal to 1 that is y ≥1
Domain and range of a functionThe domain are independent variable for which the equation exist while the range are dependent variable for which the equation exist
Given the following functions
f(x) =2/x-1
g(x) = x+1
Determine (g ○ f)(x)
(g ○ f)(x) = g(f(x))
(g ○ f)(x) = g(2/x-1)
(g ○ f)(x) = (2/x-1) + 1
(g ○ f)(x) = 2+(x-1)/x-1
(g ○ f)(x) = x+1/x-1
Hence the domain of the function is (-∞, 1) U(1, ∞) while the domains are values greater than or equal to 1 that is y ≥1
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[Will Mark Brainliest]
Make tables to find the solution to 8x = 2x + 2. Take the integer values of x between -3 and 3.
Answer:
i'm not realy sure what you mean by a table
8x=2x+2
subtract 2x from each side
6x=2
divide by 6
x=1/3
Hope This Helps!!!
Step-by-step explanation:
So if you want to make a table to find the solution, you'll need 3 columns, one will be the value of x, the other two will be the value of the equation (left and right) and then you'll need to look where the two equations are equal, and the solution is whatever the x-value is. It'll look something like this:
[tex]x\ \ \ \ |\ 2x+2\ |8x\\-3\ \| -4\ \ \ \ \ | -24\\-2\ \| -2\ \ \ \ \ |-16\\-1\ \| 0\ \ \ \ \ \ \ \ \ | -8\\0\ \ \ \ \|2\ \ \ \ \ \ \ \ \ | 0 \\1\ \ \ \ \| 4\ \ \ \ \ \ \ \ \ | 8\\2\ \ \ \ \|6\ \ \ \ \ \ \ \ \ | 16\\3\ \ \ \ \|8\ \ \ \ \ \ \ \ \ | 24[/tex]
As you can see, none of rows (the two right most columns with the equations) are ever equal, on the same row. This is because the solution is not an integer. If you solve it algebraically. You'll get this:
Original equation
8x = 2x+2
Subtract 2x from both sieds
6x = 2
Divide both sides by 6
x = 2/6
Simplify the fraction
x=1/6
Which equation can be used to represent "six added to twice the sum of a number and four is equal to one-half of
the difference of three and the number"?
O6 + 2(x + 4) = (3-x)
O 6+ 2(x+4)=(x-3)
O (6+2)(x+4)=(3-x)
O (6+2)(x+4)=(x-3)
Answer:
Step-by-step explanation:
Comment
Start with the right hand side.
1/2(3 - x)
This reads as The difference between 3 and a number. You put it the way you read it difference between 3 and a number = 3 - x The 1/2 affects both terms. So that makes b and d incorrect.
The left side is a bit harder. You want the sum of 6 + something. Read the rest of the given
6 + 2(x +4)
Now that is all equal to
6 + 2(x + 4) = 1/2 (x - x)
The answer is A
15 Which equation represents a line with slope of
1/2
and y-intercept of 3?
1
A y = 3x + ²/²
B. y=-1⁄2x+3
C. y = x+3
D. y=1/2x-3
[tex]\Large\maltese\underline{\textsf{Our problem:}}[/tex]
Which equation represents a line with slope: [tex]\bf{\dfrac{1}{2}[/tex] and y-intercept: [tex]\bf 3[/tex]?
[tex]\Large\maltese\underline{\textsf{This problem has been solved!}}[/tex]
[tex]\bf{Equation: y=mx+b[/tex]
[tex]\bf{Equation\:with\:the\:given\:information: y=\dfrac{1}{2}x+3}[/tex]
[tex]\rule{300}{1.7}[/tex]
[tex]\bf{Result:}[/tex]
[tex]\bf{y=\dfrac{1}{2}x+3}[/tex]
[tex]\boxed{\bf{aesthetic \not101}}[/tex]
Find the gradient of the line segment between the points (4,3) and (5,6).
[tex]\Large\maltese\underline{\textsf{A. \space What is Asked}}[/tex]
Find the gradient of the line segment between the points [tex]\bf{(4,3)\:and\:(5,6)}[/tex]
[tex]\Large\maltese\underline{\textsf{B. This problem has been solved!}}[/tex]
The formula used, here is
[tex]\bf{m=\dfrac{y2-y1}{x2-x1}[/tex]
[tex]\cline{1-2}[/tex]
Now let's put in the co-ordinates from these two points.
[tex]\bf{m=\dfrac{6-3}{5-4}}[/tex] | subtract on top & bottom
[tex]\bf{m=\dfrac{3}{1}}[/tex] | divide
[tex]\bf{m=3}[/tex]
[tex]\cline{1-2}[/tex]
[tex]\bf{Result:}[/tex]
[tex]\bf{=m=3}[/tex]
[tex]\boxed{\bf{ aesthetic\not101}}[/tex]
Answer:
g=3
Step-by-step explanation:
gradient=change ofY/change of X
3-6/4-5
-3/-1 simply
gradient =3
Kenyan formula easy
Imagine the center of the Ferris wheel is located at (0,0) on a coordinate grid, and the radius lies in the x-axis. Write an equation of a circle for your Ferris wheel and sketch an image of what your Ferris wheel would look like on the grid.
Answer:
(0,-0)
Step-by-step explanation:
P(0,0)=P(0,-0)
SOMEONE PLEASE HELP!
a = 4, b = 7, c = 10. What is the measure of angle A to the
nearest tenth?
Answer:
17.2°
Explanations
To find angle A, use the cosine rule.
a^2 = b^2 + c^2 - 2 × a × c cos A
4^2 = 7^2 + 10^2 - 2 × 7 × 10 cos A
16 = 49+ 100 - 140 × cosA
16 = 149 - 140cosA
16- 149 = - 140cosA
-133 = - 140cosA
cosA = 133/140
cosA = 0.95
A = 17.2°
On a coordinate plane, a curved line with a minimum value of (negative 1.25, negative 3.25) and a maximum value of (0.25, negative 1.75), crosses the x-axis at (negative 2.25, 0), and crosses the y-axis at (0, negative 2). The line exits the plane at (negative 2.75, 6) and (1.5, 6).
Which statement is true about the end behavior of the graphed function?
As the x-values go to positive infinity, the function's values go to negative infinity.
As the x-values go to zero, the function's values go to positive infinity.
As the x-values go to negative infinity, the function's values are equal to zero.
As the x-values go to positive infinity, the function's values go to positive infinity.
The correct option is:
As the x-values go to positive infinity, the function's values go to positive infinity.
What is the end behavior of the function?
By the description, We know that the function has the points:
(-1.25, -3.25), (0.25, -1.75) (-2.25, 0), (0, -2), (-2.75, 6), (1.5, 6)
Notice that from x = 0.25 to x = 1.5 the function goes upwards, then in the right side, the function goes up, so as x goes to positive infinity, also does f(x).
On the left side, we can see that from x = -2.75 to x = -2.25 the function goes downwards.
So, as x goes to the left (negative infinity) f(x) goes upwards.
Then as x tends to negative infinity, f(x) tends to infinity.
The correct option is:
As the x-values go to positive infinity, the function's values go to positive infinity.
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Answer:
D
Step-by-step explanation:
edge 23
Triangle RST is circumscribed about the circle below. What is the
perimeter of the triangle?
Answer:
I think the answer is B.36
Step-by-step explanation:
hope this helps if not let me know have a blessed day
(1/2 + 3/2)² + (2/3 x 1/2)³
The result of the mathematical expression given as follows: (1/2 + 3/2)² + (2/3 x 1/2)³ is 5.59 or 5⅔.
How to calculate mathematical expressions?According to this question, the following expression is given to solve: (1/2 + 3/2)² + (2/3 x 1/2)³
First, we solve the both parts as follows:
½ + 3/2 = 2⅔ + ½ = 1..16Next, we solve the exponential function as follows:
= (2)² + (1.16)³
= 4 + 1.59
= 5.59 or 5⅔
Therefore, the result of the mathematical expression given as follows: (1/2 + 3/2)² + (2/3 x 1/2)³ is 5.59 or 5⅔.
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A composition of transformations maps ΔKLM to ΔK"L"M".
Triangle K L M is rotated 270 degrees about point P to form Triangle K prime L prime M prime. Triangle K prime L prime M prime is shifted down and to the right to form triangle K double-prime L double-prime M double-prime.
The first transformation for this composition is [________], and the second transformation is a translation down and to the right.
a 90° rotation about point L
a 270° rotation about point L
a 90° rotation about point P
a 270° rotation about point P
The blank space in the task content should be filled with; a 270° rotation about point P.
What transformation is first to effect the image formation?It follows from the transformation described in the task content that; Triangle KLM from which triangle K'L'M' is rotated 270 degrees about point P and then translated down and to the right.
It therefore follows from observation. that the first transformation is; a 270° rotation about point P.
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TG and TH are tangent to circle Z. If TG measures 4.2 cm, what is the measure of TH?
Based on the two-tangent theorem, the measure of TH is also: 4.2 cm.
What is the Two-Tangent Theorem?According to the two-tangent theorem, the length of two tangents (TG and TH) that are drawn from a circle to intersect at an external point are equal. That is, they are congruent to each other.
Thus, we are given that, TG = 4.2 cm. Therefore, based on the two-tangent theorem, the measure of TH is also: 4.2 cm.
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14 cm perimeter of circle
Step-by-step explanation:
Radius = 14 cm
π = 22/7 (Because the value of the circle's radius is the multiples of 7).
• Perimeter :
= 2 . π . r
= 2 . 22/7 . 14
= 28 . 22/7 (Divide 28 and 7 with 7)
= 4 . 22/1
= 4 . 22
= 88 cm is the answer
What epoch do we currently live in?
A. Archaean
B. Holocene
C. Pleistocene
D. Mesozoic
Answer:
Holocene
Step-by-step explanation:
The Holocene began 11,700 years ago after the last major ice age.
Which classification best represents a triangle with side lengths 6 cm, 10 cm, and 12 cm?
acute, because 62 + 102 < 122
acute, because 6 + 10 > 12
obtuse, because 62 + 102 < 122
obtuse, because 6 + 10 > 12
Answer:
obtuse because 62 + 102 < 122
S
its c, obtuse, becuase 6^2+10^2<12^2
Juan, Carlos and Manu take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Manu will win
Answer:
⅓ ×½ which will give you ⅙
Step-by-step explanation:
because
The probability that Manu will win the coin flip amongst others is 4/7
What is the probability that Manu will win?To determine the probability that Manu will win, we need to consider the possible outcomes of the coin flips.
Let's break down the scenarios:
Scenario 1: Manu wins on the first flip
- The probability of Manu winning on the first flip is 1/2, as he needs to flip heads right away.
Scenario 2: Manu wins on the second flip
- Juan flips tails (1/2 probability)
- Carlos flips tails (1/2 probability)
- Manu flips heads (1/2 probability)
- The probability of Manu winning on the second flip is (1/2) * (1/2) * (1/2) = 1/8.
Scenario 3: Manu wins on the third flip
- Juan flips tails (1/2 probability)
- Carlos flips tails (1/2 probability)
- Manu flips tails (1/2 probability)
- Juan flips heads (1/2 probability)
- Carlos flips heads (1/2 probability)
- Manu flips heads (1/2 probability)
- The probability of Manu winning on the third flip is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/64.
The pattern continues, and we can see that Manu's probability of winning decreases exponentially with each subsequent flip.
Therefore, the total probability of Manu winning is the sum of the probabilities from each scenario:
1/2 + 1/8 + 1/64 + ...
This is an infinite geometric series with a first term (a) of 1/2 and a common ratio (r) of 1/8.
Using the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
Sum = (1/2) / (1 - 1/8)
Sum = (1/2) / (7/8)
Sum = (1/2) * (8/7)
Sum = 4/7
Therefore, the probability that Manu will win is 4/7.
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Twelve of the 20 students in Mr. Skinner’s class brought lunch from home. Fourteen of the 21 students in Ms. Cho’s class brought lunch from home. Siloni is using two 15-section spinners to simulate randomly selecting students from each class and predicting whether they brought lunch from home or will buy lunch in the cafeteria.
If each spinner is divided into 15 congruent sectors, how does the spinner representing Mr. Skinner’s class differ from the spinner representing Ms. Cho’s class?
For Ms. Cho's class, the formula is =randBetween(1,21). The result of interest is 14 or less.
12/20 reduces to 4/5 --> 4/5 of 15 is 12.
So using the spinner to model Mr. Skinner's class, an outcome of the select 12 sections of interest represent, symbolize, and model a student bringing lunch from home.
LIkewise, 14/21 reduces to 2/3 --> 2/3 of 15 is 10.
So using the spinner to model Ms. CHo's class, an outcome of only 10 of the 15 sections of interest represent, model, and symbolize a student who brings lunch from home.
The same can be done in Excel: The function is =randBEtween( minVal, maxVal)
For example =randBetween(1,20) will model Mr. Skinner's class. A result of 12 or less represents, model, simulates, and symbolizes a student who packed their lunch.
For Ms. Cho's class, the formula is =randBetween(1,21). The result of interest is 14 or less.
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Write an expression equivalent to the following problems using the fewest number of terms. 2x + 10) + (−3x + 15)
Combining the like terms, the equivalent expression is given as follows:
-x + 25.
How to add polynomials?
To add polynomials, we combine the like terms, that is, those with the same variable and exponent.
In this problem, the expression is:
(2x + 10) + (-3x + 15).
2x + 10 - 3x + 15
Combining the like terms:
2x - 3x + 10 + 15 = -x + 25.
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Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Match each graph to the inequality it represents.
The graphs and their inequalities are
Graph 1 ⇒ 3x + 4y > 6Graph 2 ⇒ 2x - 7y ≥ 14Graph 3 ⇒ 3x + 4y < 6Graph 4 ⇒ 7x + 2y ≥ 14How to match the inequalities?The inequalities are given as:
2x - 7y ≤ 14
7x + 2y ≥ 14
2x - 7y ≥ 14
3x + 4y < 6
4x - 3y > 6
3x + 4y > 6
To represent the inequalities on a graph, we use the following guides:
> uses a dotted line, and the upper parts are shaded< uses a dotted line, and the lower parts are shaded≥ uses a thick line, and the upper parts are shaded≤ uses a thick line, and the lower parts are shadedUsing the above as a guide, we have:
Graph 1 ⇒ 3x + 4y > 6
Graph 2 ⇒ 2x - 7y ≥ 14
Graph 3 ⇒ 3x + 4y < 6
Graph 4 ⇒ 7x + 2y ≥ 14
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Using f(x) = log x, what is the x-intercept of g(x) = log (x + 4)?
X5
Answer:
a
Step-by-step explanation:
Pierre found a new store where he hopes to save
money on his supplies. He bought 4 brushes and 7
tubes of paint for $32. He has another order for 6
brushes and 8 tubes of paint that will cost $43.
What is the cost of one brush? What is the cost of
one tube of paint?
Answer:
A brush: $4.50
A tube of paint: $2
Step-by-step explanation:
Let b = brushes
Let p = paint
1) Set up equations.
4b + 7p = 32
6b + 8p = 43
2) Solve the system of equations using elimination.
(4b + 7p = 32)3
(6b + 8p = 43)2
-----------------------
12b + 21p = 96
12b + 16p = 86
----------------------
5p = 10
p = 10/5
p = 2
Substitute 2 into p of any of the two equations.
4b + 7(2) = 32
4b + 14 = 32
4b = 32 - 14
4b = 18
b = 18/4
b = 4.50
Therefore, a brush costs $4.50, while a tube of paint costs $2.
Which linear inequality is represented by the graph?
y < 3x + 2
y > 3x + 2
y < One-thirdx + 2
y > One-thirdx + 2
The linear equality y < 3x + 2 is represented by the graph attached.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
An inequality is the non equal comparison of two or more numbers and variables.
The linear equality y < 3x + 2 is represented by the graph attached.
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Next question
a man borrowed $3400 from a bank for 3 months. a friend was cosigner of the man's personal note. the bank collected 3% simple interest on the date of maturity.
a) how much did the man pay for the use of the money?
b) determine the amount he repaid to the bank on the due date of the note.
a) the man paid $ for the use of the money.
b) on the due date of the note, the man repaid to the bank.
The man paid $ 25.5 for the use of money.
The amount repaid to the bank on the due date is $3425.5
Calculate the total amount as well as the simple interest.Calculating the amount of simple interest that will be charged on a loan can be done quickly and easily using the simple interest formula. To compute simple interest, multiply the principle, the number of days between payments, and the daily interest rate.
Principal(P)=$3400
Time(T)=3 months=1/4 yrs
Rate(R)=3%
The simple interest and final amount are calculated as follows:
Simple Interest(SI)=P*R*T/100
[tex]=\frac{3400*3*\frac{1}{4} }{100}[/tex]
=25.5
Net amount repaid=P+SI
=3400+25.5
=3425.5
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An observatory is 150 feet high. when a person is standing in the observatory looking at an island, the angle of depression is . what is the approximate horizontal distance from the observatory to the island?
Using Trigonometry, the approximate distance from the observatory to the island is evaluated to be 322 feet.
Given Information
It is given that,
The height of the observatory = 150 feet
The angle of depression from the top of the observatory = 25°
We can calculate the horizontal distance of the observatory from the island using Trigonometry
What is Trigonometry?
The area of mathematics known as Trigonometry is concerned with certain functions of angles and how to use them in computations. There are six popular Trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their respective names and acronyms.
Calculating the Horizontal Distance Using Trigonometry
As per the rules of the Trigonometry,
[tex]tan \alpha = \frac{Perpendicular/ Opposite Side}{Base}[/tex]
Here,
[tex]\alpha[/tex] = 25°
Perpendicular = 150 ft.
Base = Horizontal Distance
Thus, [tex]tan 25 =\frac{150}{Horizontal Distance}[/tex]
∴ x ≈ 322 ft.
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Please help ill give whoever answers the best the brainliest answer :)
Answer: Quadratic
Step-by-step explanation:
I'm too lazy to explain. Trust me
On a coordinate plane, 2 triangles are shown. Triangle A B C has points (negative 3, negative 1), (negative 1, 2), and (negative 5, 3). Triangle R S T has points (1, 1), (3, 4), and (5, 0).
Triangle RST is rotated 180° about the origin, and then translated up 3 units. Which congruency statement describes the figures?
ΔRST ≅ ΔACB
ΔRST ≅ ΔABC
ΔRST ≅ ΔBCA
ΔRST ≅ ΔBAC
The congruency statement that describes the figure is: D. ΔRST ≅ ΔBAC.
What is a Congruency Statement?A congruency statement is a statement that states that two triangles are congruent to each other.
180 degrees rotation around the origin will give us the rule: (x, y) → (-x, -y)
Therefore, applying this rule to triangle RST that was rotated, we would have:
R'(-1, -1)
S'(-3, -4)
T'(-5, 0)
Translating R'S'T' 3 units up, using the rule, (x, y + 3) will give us:
B(-1, 2)
A(-3, -1)
C(-5, 3)
Therefore, we can conclude that, D. ΔRST ≅ ΔBAC.
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Answer:its D just took test:)
Step-by-step explanation:
Solve this system of equations by using the elimination method -3x+y=6 -3x+y=12
Answer:
None
Step-by-step explanation:
We can eliminate the x's by add the 2 equations :
2y = 18
Divide both sides by 2 :
y = 9
Substitute this value into equation 1 to solve for x :
-3x + (9) = 6
Subtract 9 from both sides :
-3x = -3
Divide both sides by -3 :
x = 1
Substitute these values into equation 2 to check :
-3(1) + 6 = 12 ❌❌
So it has no solution
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins
The probability that, at the tip of the fourth round, each of the players has four coins is 5/192.
Given that game consists of 4 rounds and every round, four balls are placed in an urn one green, one red, and two white.
It amounts to filling in an exceedingly 4×4 matrix. Columns C₁-C₄ are random draws each round; row of every player.
Also, let [tex]\%R_{A}[/tex] be the quantity of nonzero elements in [tex]R_{A}.[/tex]
Let [tex]C_{1}=\left(\begin{array}{l}1\\ -1\\ 0\\ 0\end{array}\right)[/tex].
Parity demands that [tex]\%R_{A}[/tex] and[tex]\%R_{B}[/tex] must equal 2 or 4.
Case 1: [tex]\%R_{A}[/tex]=4 and [tex]\%R_B[/tex]=4. There are [tex]\left(\begin{array}{l}3\\ 2\end{array}\right)[/tex]=3 ways to put 2-1's in [tex]R_A[/tex], so there are 3 ways.
Case 2: [tex]\%R_{A}[/tex]=2 and [tex]\%R_B[/tex]=4. There are 3 ways to position the -1 in [tex]R_A[/tex], 2 ways to put the remaining -1 in [tex]R_B[/tex] (just don't put it under the -1 on top of it!), and a pair of ways for one among the opposite two players to draw the green ball. (We know it's green because Bernardo drew the red one.) we are able to just double to hide the case of [tex]\%R_{A}=4,\%R_{B}=2[/tex] for a complete of 24 ways.
Case 3: [tex]\%R_A=\%R_B=2[/tex]. There are 3 ways to put the -1 in [tex]R_{A}[/tex]. Now, there are two cases on what happens next.
The 1 in [tex]R_B[/tex] goes directly under the -1 in[tex]R_A[/tex]. There's obviously 1 way for that to happen. Then, there are 2 ways to permute the 2 pairs of 1,-1 in [tex]R_C[/tex] and[tex]R_D[/tex]. (Either the 1 comes first in[tex]R_C[/tex] or the 1 comes first in [tex]R_D[/tex].)The 1 in [tex]R_B[/tex] doesn't go directly under the -1 in [tex]R_A[/tex]. There are 2 ways to put the 1, and a couple of ways to try and do the identical permutation as within the above case.Hence, there are 3(2+2×2)=18 ways for this case. There's a grand total of 45 ways for this to happen, together with 12³ total cases. The probability we're soliciting for is thus 45/(12³)=5/192
Hence, at the top of the fourth round, each of the players has four coins probability is 5/192.
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4. The two figures shown are congruent. Which statement is true?
One figure is a translation image of the other.
One figure is a reflection image of the other.
One figure is a rotation image of the other