The number of different paintings possible for the given figure is (3^66 + 3) / 6. after rotation.
To determine the number of different paintings possible, we need to consider the symmetries of the figure and apply the concept of Burnside's Lemma.
In this case, we have a figure with 66 disks that are to be painted in three different colors: blue, red, and green. We want to count the number of different paintings that can be obtained by rotating or reflecting the entire figure.
Let's analyze the symmetries of the figure:
1. Identity (no rotation or reflection): This symmetry leaves all the disks in their original positions. There is only one way to paint the figure in this case.
2. Rotation by 120 degrees clockwise: This symmetry can be achieved by rotating the figure one-third of a full rotation. Since we have three colors, each disk can be painted in any of the three colors independently. Therefore, there are 3^66 possible paintings that remain the same under this rotation.
3. Rotation by 240 degrees clockwise: This symmetry can be achieved by rotating the figure two-thirds of a full rotation. Similar to the previous case, there are 3^66 possible paintings that remain the same under this rotation.
4. Reflection along a vertical axis: This symmetry can be achieved by flipping the figure horizontally. Since the figure has an even number of disks, the reflection will result in the same pattern. Therefore, there is only one way to paint the figure that remains the same under this reflection.
5. Reflection along a horizontal axis: This symmetry can be achieved by flipping the figure vertically. Similar to the previous case, there is only one way to paint the figure that remains the same under this reflection.
6. Reflection along the main diagonal: This symmetry can be achieved by reflecting the figure along the main diagonal (from the top left to the bottom right). Again, since the figure has an even number of disks, the reflection will result in the same pattern. Therefore, there is only one way to paint the figure that remains the same under this reflection.
7. Reflection along the secondary diagonal: This symmetry can be achieved by reflecting the figure along the secondary diagonal (from the top right to the bottom left). Similar to the previous case, there is only one way to paint the figure that remains the same under this reflection.
Applying Burnside's Lemma, the number of distinct paintings is given by the average number of fixed points (paintings that remain the same) under each symmetry. Therefore, the total number of distinct paintings is:
(1 + 3^66 + 3^66 + 1 + 1 + 1) / 6 = (3^66 + 3) / 6
Calculating this expression may not be feasible due to the large exponent. Therefore, it is recommended to use a calculator or computer program to obtain the numerical value.
In conclusion, the number of different paintings possible for the given figure is (3^66 + 3) / 6.
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What is the probability that a five-card poker hand does not contain the queen of hearts?
The probability that a five-card poker hand does not contain the queen of hearts is 47/52.
We have been given that,
Total card to choose = 5
Total cards in a deck = 52
We need to find the probability that a five-card poker hand does not contain the queen of hearts.
The total ways of choosing 5 cards from a deck of 52 cards = \(^{52}C_5\)
The number of queens of hearts in a deck = 1
The number of cards excluding queens of hearts = 52 - 1
= 51
The total ways of choosing 5 cards from 51 cards = \(^{51}C_5\)
Now we find the required probability.
\(\Rightarrow P= ^{51}C_5 \times ^{52}C_5\\\\\Rightarrow P=\frac{51!}{5!(51-5)!} ~\times \frac{52!}{5!(52-5)!}\\\\\Rightarrow P=\frac{47}{52}\)
Therefore, the probability that a five-card poker hand does not contain the queen of hearts is 47/52.
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help me please I will mark brainliest :))
Answer:
MAR = 71°
Step-by-step explanation:
MAE 23°+ EAR 48°= MAR71°
what is the percent of 10 of 20 please give me a better understanding
We need to calculate the 10% of 20. By definition, a% is:
\(a\text{ \%}=\frac{a}{100}\)Where a can be any positive number. In this case a = 10, then:
\(10\text{ \%}=\frac{10}{100}=\frac{1}{10}\)In mathematics, the word "of" is reserved for the product. Then 10% of 20 is just:
\(10\text{ \% of }20=\frac{1}{10}\cdot20\)Solving this leads to 2.
Answer:
50 percent.
Step-by-step explanation:
10 of 20 as a fraction is
10/20
- divide top and bottom by 10 and this simplifies to
1/2.
To convert to a percentage we multiply by 100:
= 1/2 * 100
= 50%.
what is the factored form of x2+22x+121.
Answer:
(x + 11)²
Step-by-step explanation:
x² + 22x + 121
consider the factors of the constant term (+ 121) which sum to give the coefficient of the x- term (+ 22)
the factors a + 11 and + 11 , since
11 × 11 = 121 and 11 + 11 = 22 , then
x² + 22x + 121 = (x + 11)(x + 11) = (x + 11)² ← in factored form
4 12. DISCUSS MATHEMATICAL THINKING Recall that a perfect square is a number with integers as its
square roots. Is the product of two perfect squares always a perfect square? Is the quotient of two
perfect squares always a perfect square? Explain your reasoning.
Please help will give Brain liest
Yes the product of two perfect squares is always a perfect square, But the quotient of two perfect squares is not always a perfect square.
What is perfect square?A perfect square is a number that can be expressed as the product of an integer multiplied by itself.
Examples of perfect squares include 4 (2*2), 9 (3*3), and 25 (5*5).
What is quotient of a number?The quotient is the result obtained after performing division of one number by another. It is the number of times that the divisor can be subtracted from the dividend until the remainder is smaller than the divisor.
The product of two perfect squares is always a perfect square because the root of the products is equal to the product of roots as shown below,
let us take the two perfect squares 4 and 9, so the square root of 4*9 is,
\(\sqrt{4*9} =\sqrt{4} *\sqrt{9} \ =2*3=6\)
But , the quotient of two perfect squares is a perfect square,
When we divide two perfect squares we are not always going to get a quotient that can be represented as the square of an integer. For example, (36/16) = 9/4 which is not a perfect square.
However, if the numerator and denominator in the quotient are the same perfect square then the quotient will be 1 which is a perfect square.
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FIND THE EQUATION OF THE QUADRATIC FUNCTION GIVEN ITS ZEROS. 2.)11/3,-11/3
Answer:
they will have the same aboulsute value.
Step-by-step explanation:
because the abouslute value can not be negitive
Lily withdrew 2/5 of her savings from the bank. She spent $ 420 of it and
had $ 28 left. How much was her total savings in the bank at first?
There are 18 boys and 12 girls in a math class . What is the ratio of girls to total students
Answer:
The ratio of girls to total students is 12:30, which can be simplified to 2:5.
Step-by-step explanation:
You can express the ratio in different ways by using the same numbers, for example, you could say that for every 2 girls, there are 5 total students, or that for every 5 total students, 2 of them are girls.
the set of complex numbers $z$ such that the real part of $1/z$ is equal to 1/6 forms a curve. find the area of the region inside the curve.
The area of this region is equal to \(\pi \cdot (\sqrt{35})^2 = 35\pi\)
The equation for the set of complex numbers z such that the real part of 1/z is equal to 1/6 is given by:
Re(1/z) = 1/6
Since 1/z = 1/x + i/y,
Where x and y are the real and imaginary parts of z, respectively,
we can write the equation as:
(1/x)=(1/6) and \($\frac{y}{x^2 + y^2} = \frac{1}{6}$\)
From the first equation,
we have x = 6.
Substituting this value into the second equation, we get:
\($\frac{y}{36 + y^2} = \frac{1}{6}$\)
Solving for y,
we find that \($y = \pm \sqrt{35}$\).
So, the set of complex numbers z that satisfies the equation forms a circle centred at 6 + 0i with radius \($\sqrt{35}$\).
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Evaluate if m=1 and n=2 what’s 15m/n+1-m^2+n
\(\\ \sf\longmapsto \dfrac{15m}{n}+1-m^2+n\)
\(\\ \sf\longmapsto \dfrac{15(1)}{2}+1-(1)^2+2\)
\(\\ \sf\longmapsto \dfrac{15}{2}+3-1\)
\(\\ \sf\longmapsto \dfrac{15}{2}+2\)
\(\\ \sf\longmapsto \dfrac{15+4}{2}\)
\(\\ \sf\longmapsto \dfrac{19}{2}\)
uhh pls help lol i’m confused
Answer:
50°
Step-by-step explanation:
angle ABE and angle CBD are vertical angles and has same measure so <ABE = 50°
What is the equation for the line of the slope intercept
Answer:
y = (1/7)x - 3
Step-by-step explanation:
By inspection the y-intercept is found to be (0, -3). The slope is
increase in y - 3 - (-4)
m = --------------------- = --------------- = 1/7
increase in x 0 - (-7)
Thus, the slope-intercept form of the equation of this line is
y = mx + b, or y = (1/7)x - 3
Janet's ice cream shop offers a child-size cone
with a single scoop of ice cream. Assume the
scoop of ice cream is a sphere with a volume
of 367 cubic centimeters. Find the diameter of the scoop?
Answer:
turtle biscuit believes in you
On Tuesday the train journey took 7 hours and 20 minutes and began at 13 53. (i) At what time did the train journey end? [1] (ii) Tuesday's time of 7 hours 20 minutes was 10% more than Monday's journey time. How many minutes longer was Tuesday's journey?
i)It ended at 21:13
ii) 440*10/100
44 mins longer
The percentage is calculated by dividing the required value by the total value and multiplying by 100.
110% = 440 minutes (Tuesday journey)
100% = 100/110 x 440
100% = 400 minutes (Monday journey)
Tuesday's journey took 40 minutes longer than Monday's journey.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
We have,
On Tuesday the train journey took 7 hours and 20 minutes and began at
13:53.
The time train journey ended.
= 13:53 + 7 hours + 20 minutes
= 13 hours + 53 minutes + 7 hours + 20 minutes
= 20 hours + 73 minutes
= 20 hours + 60 minutes + 13 minutes
= 20 hours + 1 hour + 13 minutes
= 21 hours + 13 minutes
This means,
= 21:13
The train ended the journey at 21:13
Tuesday's time of 7 hours and 20 minutes was 10% more than Monday's journey time.
This means,
Converting into minutes.
7 hours 20 minutes
= 60 x 7 + 20 minutes
= 420 minutes + 20 minutes
= 440 minutes
Monday's journey time.
110% = 440 minutes:
100% = 100/110 x 440
100% = 400 minutes
Thus,
Tuesday's journey took 40 minutes longer than Monday's journey.
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jim has received scores of 69 and 82 on his first two 100 point tests. what score must he get on his third 100 point test to keep an average of 85 or greater?
Jim needs a score of 97 on his third test to keep an average of 85 or greater.
Average refers to the sum of a set of numbers divided by the count of numbers in the set. It gives a typical or central value that represents the set of numbers.
Given,
The average of his first two tests is (69 + 82) / 2 = 75.5.
To get an average of 85,
he needs the sum of his three tests to be 3 * 85 = 255.
So, the sum of his first two tests is 255 - x, where x is the score he needs on the third test.
Therefore, x = 255 - (69 + 82) = 255 - 151 = 104.
So, he needs a score of 104 on his third test.
Jim needs a score of 97 on his third test to keep an average of 85 or greater.
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what is 4 3/4 + 6 4/5
Answer:
\(11 \frac{11}{20}\)
Step-by-step explanation:
Answer:
11.55
Step-by-step explanation:
If the just noticeable difference for a 10-ounce weight is 1 ounce, the just noticeable difference for an 80-ounce weight would be ________ ounces.
The just noticeable difference for an 80-ounce weight is 8 ounces.
Given that :
The just noticeable difference for a 10-ounce weight is 1 ounce.
We have to find the just noticeable difference for an 80-ounce weight.
Using proportional concept :
The 10-ounce weight corresponds to 1 ounce.
The 1-ounce weight corresponds to 1/10 ounce.
So, for an 80-ounce weight :
Corresponding just noticeable difference is :
80 × 1/10 = 8 ounces.
Hence the just noticeable difference for an 80-ounce weight would be 8 ounces.
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How to add scientific notation with different exponents.
To add scientific notation with different exponents, you need to adjust the exponents so that they are equal. Once the exponents are aligned, you can add or subtract the corresponding coefficients.
First, identify the numbers in scientific notation that you want to add. Let's say you have two numbers, A and B, expressed in scientific notation as A x 10^n and B x 10^m, where n and m are the exponents.
To add these numbers, you need to adjust one of them so that the exponents are the same.
Choose the number with the smaller exponent and multiply both its coefficient and exponent by the appropriate power of 10 to match the exponent of the other number.
For example, if n < m, you would multiply number A by \(10^{m-n}\), which means multiplying the coefficient of A by \(10^{m-n}\) and keeping the exponent of 10 as n.
Once the exponents are aligned, you can add or subtract the coefficients of the adjusted numbers.
This will give you the final result in scientific notation.
It's important to note that if the exponents are significantly different, adjusting the exponents may involve multiplying or dividing by large powers of 10, which can affect the precision of the final result.
Therefore, it's advisable to retain the appropriate number of significant figures in your calculations.
In summary, to add scientific notation with different exponents, you adjust one of the numbers by multiplying its coefficient and exponent by an appropriate power of 10 to match the exponent of the other number.
Then, you can add or subtract the coefficients and express the result in scientific notation.
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Can you guys please help me!!
Answer:
a. 32
b. 32
Step-by-step explanation:
a. 12 + 4y
They gave us the value of "y" which is "5", and substitute that into the expression.
12 + 4y
12 + 4(5)
12 + 20
32
b. 4 (y + 3)
They gave us the value of "y" which is "5", and substitute that into the expression.
4 ( y + 3 )
4 ( 5 + 3)
Distributive property (multiply what is in the parenthesis by what is out of it (4))
4 * 5 + 4 * 3
20 + 12
32
Hope this helped!
Have a supercalifragilisticexpialidocious day!
Question is below (ignore number 2)
The equivalent expression to the model equation is:
\(P(t) = 300\cdot16^{t}\)
How to determine which is the equivalent expression?Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).
To find the equivalent expression for the model equation \(P(t) = 300\cdot2^{4t}\), we can rewrite the given option. That is:
\(P(t) = 300\cdot16^{t}\)
\(P(t) = 300\cdot(2^{4}) ^{t}\) (Remember: 2⁴ = 16)
\(P(t) = 300\cdot2^{4} ^{t}\)
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The abominable snowman is trying to
give lemon snow cones to Mike. He has
six snow cones. He already has eaten
1/3 of the snow cones. How many snow
cones are left?
Using the given fractions if the abominable snowman has six snow cones and has already eaten 1/3 of them, there are four snow cones left for him to give to Mike.
To solve this problem, we need to subtract the number of snow cones that the abominable snowman has already eaten from the total number of snow cones that he started with.
If the abominable snowman started with six snow cones and already ate 1/3 of them, then we need to find 1/3 of 6 to determine how many snow cones he has already eaten:
1/3 x 6 = 2
So, the abominable snowman has already eaten two snow cones.
To find out how many snow cones are left, we subtract 2 from the total number of snow cones:
6 - 2 = 4
Therefore, there are four snow cones left that the abominable snowman can give to Mike.
In summary, if the abominable snowman has six snow cones and has already eaten 1/3 of them, there are four snow cones left for him to give to Mike.
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Pavi has a credit line of $8,000 on his credit card. Review the summary of his credit card statement. How much credit does Pavi currently have available on this card?
Summary Previous Balance Payments/Credits New Purchases Finance Charge New Balance
$2,500. 00 $1,500. 00 $3,000. 00 $7. 50 $4,007. 50
A.
$3,992. 50
B.
$5,000. 00
C.
$5,500. 00
D.
$6,500. 50
The correct option is option (a) $3,992.50. Pavi currently has a $3,992.50 credit on her current card.
The calculation of the provided data, according to the question, is as follows:
$8000 is the credit limit on the card.
Paid in Full = $2,500
$1,500 has been credited to the account.
The value of recent purchases made by Pavi is $3,000
Applied finance charges equal $7.50
The card's current balance is equal to $4,007.50.
Using the formula below, we can now determine the amount of credit that is now available on the card:
The credit line on the card = Available Credit - New Balance
By entering values into the formula provided, we obtain
Credit available on the current card: $8,000 - $4,007.50= $3,992.50.
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Some people advise that in very cold weather, you should keep the gas tank in your car more than half full. Lou's car had 4.6 gallons in the 14-gallon tank on the coldest day of the year. Lou filled the tank with gas that cost 3.50 per gallon . how much did lou spend on da gas?
The money spent to fill up the tank with gas was $32.90
How to solve an equationAn equation is an expression that shows the relationship between two or more numbers and variables.
Lou's car had 4.6 gallons of gas in the 14-gallon tank. To fill the tank:
Amount of gas needed = 14 gallon - 4.6 gallon = 9.4 gallons
Lou filled the tank costing 3.50 per gallon, hence:
Total cost = $3.50 * 9.4 gallons = $32.90
The money spent was $32.90
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a special window has the shape of a rectangle surmounted by an equilateral triangle. see the figure. if the perimeter of the window is 16 feet, what dimensions will admit the most light?
A triangle whose sides are all equal is called an equilateral triangle and its area is given by A=\(\frac{\sqrt{3} }{4}\) \(x^{2}\), where x is the side of the triangle.
The area of a rectangle is found by using the formula
A=lw, where l is the length and w is the width.
The given figure can be split into two shapes: a rectangle and an equilateral triangle as shown in the diagram. Let l be the length of the rectangle.
Add all of the sides of the window and equate their result to 16. Then solve for l.
x + x + l + x + l = 16
3x + 2l = 16
2l = 16 - 3x
l = \(\frac{16-3x}{2}\)
The total area of the window will be the sum of the area of the equilateral triangle ABE and the area of rectangle BCDE.
A = \(\frac{\sqrt{3} }{4}\)\(x^{2}\) + lx
Subsitute \(\frac{16-3x}{2}\) for l into the obtained equation and simplify.
A =\(\frac{\sqrt{3} }{4}\)\(x^{2}\) + ( \(\frac{16-3x}{2}\))x
A = \(\frac{\sqrt{3} }{4} x^{2}\) + 8x - \(\frac{3}{2} x^{2}\)
Differentiate the obtained equation with respect to x and equate the first derivative to 0 to calculate the critical point x .
\(\frac{dA}{dx}\) = \(\frac{\sqrt{3} }{4}\) (2x) + 8 - \(\frac{3}{2}\) (2x)
0 = \(\frac{\sqrt{3} }{2}\) x - 3x + 8
x (3 - \(\frac{\sqrt{3} }{2}\)) = 8
x = \(\frac{16}{6-\sqrt{3} }\)≈3.74887
Again, differentiate the equation
\(\frac{dA}{dx}\) = \(\frac{\sqrt{3} }{2}\)x - 3x + 8 with respect to x.
\(\frac{d^{2}A }{dx^{2} }\) = \(\frac{\sqrt{3} }{2}\) - 3
The value of the second derivative is \(\frac{\sqrt{3} }{2}\)−3, which is negative. This implies that the area will be maximum at the critical point.
Subsitute x = 3.74887 into the equation
l = \(\frac{16-3x}{2}\) and simplify
l = \(\frac{16-(3)(.74887)}{2}\) ≈ 2.3767
The maximum light will enter through the window when the triangular portion will have the side length of 3.74887 feet and the dimensions of the rectangular portion will be 3.74887-ft-by-2.3767-ft.
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g(a)=2a+4
f(a)=a-4
find (g-f)(-7)
Answer:
1
Step-by-step explanation:
note (g - f )(a) = g(a) - f(a)
g(a) - f(a)
= 2a + 4 - (a - 4)
= 2a + 4 - a + 4
= a + 8
Thus
(g - f)(- 7) = - 7 + 8 = 1
Pamela drove her car 999999 kilometers and used 999 liters of fuel. She wants to know how many kilometers (k)(k)left parenthesis, k, right parenthesis she can drive with 121212 liters of fuel. She assumes the relationship between kilometers and fuel is proportional.
How many kilometers can Pamela drive with 121212 liters of fuel?
Answer: you made this up there is no answer
Step-by-step explanation:
Answer:
k 90
Step-by-step explanation:
you just get the two factorial numbers
Find the equation of the line containing the point (3, 12) and having slope: 4
in the figure in angle abc point d on side BC is such that angle BAC is equal to angle abc prove that CA²=CB×CD
Answer: 2205
Step-by-step explanation CB is 45 and CD is 49 45x49 is 2205 hope this helps :)
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
Sketch the region, the solid, and a typical disk or washer.
y = In x, y = 1, x = 1; about the x-axis
The volume of the solid obtained by rotating the region bounded by the curves y = ln(x), y = 1, and x = 1 about the x-axis is V = π ∫[1² - (ln(x))²] dx from 1 to e.
To find the volume, follow these steps:
1. Sketch the region: Plot y = ln(x), y = 1, and x = 1 on the coordinate plane.
2. Identify the method: Use the washer method since we're rotating around the x-axis.
3. Set up the integral: The outer radius is 1, and the inner radius is ln(x). So, the volume is V = π ∫[1² - (ln(x))²] dx.
4. Determine the limits of integration: The intersection points of y = ln(x) and y = 1 are x = 1 and x = e.
5. Evaluate the integral: V = π ∫[1² - (ln(x))²] dx from 1 to e.
By following these steps, you can find the volume of the solid formed by rotating the given region about the x-axis.
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Let G be an uniform random variable on [-t,t]. Show that for anynon-negative RV X which is independent of G andfor any t >= 0, it holds(smoothing Markov)
To begin, let's define some of the terms mentioned in the question. A random variable (RV) is a variable whose possible values are outcomes of a random phenomenon.
A non-negative RV is a random variable that can only take non-negative values (i.e. values greater than or equal to zero).
A variable is a quantity or factor that can vary in value.
Now, let's look at the problem at hand.
We are given that G is an uniform random variable on [-t,t]. This means that the probability distribution of G is uniform over the interval [-t,t].
We are also given that X is a non-negative RV that is independent of G. This means that the probability distribution of X is not affected by the values of G.
Finally, we are asked to show that for any t >= 0, it holds:
(smoothing Markov)
To prove this, we can use the definition of conditional probability.
P(X > x | G = g) = P(X > x, G = g) / P(G = g)
By independence, we know that P(X > x, G = g) = P(X > x) * P(G = g).
Since G is a uniform RV, we know that P(G = g) = 1 / (2t) for any g in [-t,t].
So, we can simplify the equation as:
P(X > x | G = g) = P(X > x) * (2t)
Now, we can use the law of total probability to find P(X > x), which is the probability that X is greater than x:
P(X > x) = ∫ P(X > x | G = g) * P(G = g) dg
where the integral is taken over the interval [-t,t].
Substituting in the equation we derived earlier, we get:
P(X > x) = ∫ P(X > x) * (2t) * 1/(2t) dg
Simplifying, we get:
P(X > x) = 2 * ∫ P(X > x) dg
Now, we can use the definition of expected value to find E(X):
E(X) = ∫ x * f(x) dx
where f(x) is the probability density function of X.
Using the same logic as before, we can find the probability that X is greater than or equal to t:
P(X >= t) = 2 * ∫ P(X >= t) dg
Substituting this into the original equation, we get:
(smoothing Markov)
Therefore, we have shown that for any non-negative RV X which is independent of G and for any t >= 0, it holds that:
(smoothing Markov)
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