A cube has edges of length 'n', where 'n' is a positive integer. Three faces, meeting at a corner, are painted red. The cube is then cut into n³ smaller cubes of unit length. That is, the side lengths of the new cubes are 1 unit. f exactly 125 of these cubes have no faces painted red, determine the value of n.

Please do show workings, I would really appreciate it. Thank you so much!

The solution to the equation is a = 6 and a = 4.

There is no extraneous solution.

To find the solution for the given equation, follow these steps:
1. Rewrite the equation: [tex]15 + a^2 - 1 = 5(2a - 2)[/tex].
2. Simplify both sides: [tex]a^2 + 14 = 10a - 10.[/tex]
3. Move all terms to one side:[tex]a^2 - 10a + 24 = 0.[/tex]
4. Factor the quadratic equation: (a - 6)(a - 4) = 0.
5. Set each factor equal to zero: a - 6 = 0 or a - 4 = 0.
6. Solve for 'a': a = 6 or a = 4.
Now, we need to check if both solutions are valid or if there is an extraneous solution.
Since there are no restrictions on the domain of the original equation, both solutions are valid.

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Answer:

There were 24 chocolates in the box to begin with.

Step-by-step explanation:

Let's use working backwards to solve this problem.

We know that there are 12 chocolates left in the box. Let's call the number of chocolates that were in the box to begin with "x".

After eating 1/4 of what was left, Forest had 3/4 of the chocolates left. So, we can set up an equation:

3/4 (x - 1/3x) = 12

We can simplify this equation by multiplying both sides by 4/3:

x - 1/3x = 16

Multiplying both sides by 3:

2x = 48

x = 24

Therefore, there were 24 chocolates in the box to begin with.

Using the formula, we now know that the surface area of the rectangular prism is 22 in².

A rectangular prism is a polyhedron in geometry that has two parallel and congruent bases.

It also goes by the name cuboid.

Six faces, each with a rectangle shape and twelve edges, make up a rectangular prism.

Its length, width, and height are its three dimensions.

Rectangular tissue boxes, school notebooks, laptops, fish tanks, and big buildings like freight containers, rooms, storage sheds, etc. are a few instances of rectangular prisms in daily life.

So, the formula for the surface area of the rectangular prism:

A = 2(wl+hl+hw)

Now, insert values as follows:

A = 2(wl+hl+hw)

A = 2·(1·1+5·1+5·1)

A = 22 in²

Therefore, using the formula, we now know that the surface area of the rectangular prism is 22 in².

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Answer:

Jayla selected 14 bean bags randomly with replacement from a box of 100 bean bags, and 6 of them were red. To estimate how many red bean bags are in the box, we can use proportion:

Let's call "x" the number of red bean bags in the box. Then we can set up a proportion:

(red bean bags in the box) / (total bean bags in the box) = (red bean bags selected) / (total bean bags selected)

x / 100 = 6 / 14

Simplifying the right side, we get:

x / 100 = 3 / 7

Multiplying both sides by 100, we get:

x = 300 / 7

Rounding to the nearest whole number, we get:

x ≈ 43

Therefore, based on the data, we can estimate that there are approximately 43 red bean bags in the box.

Answer + Explanation:

To estimate how many red bean bags are in the box, we can use a sampling method called "balloting." Balloting is a way of estimating the number of items of a specific type in a larger population by taking a random sample and counting how many of that type are in the sample.

In this case, we have a population of 100 bean bags and a sample size of 14. We can calculate the sample proportion of red bean bags by dividing the number of red bean bags in the sample by the total number of bean bags in the sample, which gives:

(4/14) × 100% ≈ 28.6%

This means that, on average, we would expect 28.6% of the 100 bean bags in the box to be red. To estimate the actual number of red bean bags, we can multiply the sample proportion by the total number of bean bags in the box:

28.6% × 100 = 28.6

So the closest estimate for the number of red bean bags in the box is 29.

Note that this is just an estimate, and the actual number of red bean bags in the box may be higher or lower than the estimated value.

Answer: D

Step-by-step explanation:
To narrow it out, Histograms dont have spaces meaning this chart wouldnt work as a histogram, and A and C are just plain incorrect

The radical expression √(-4x) = 4 when evaluated has a value of x = -4

From the question, the equation is given as

√(-4x) = 4

We can start by squaring both sides of the equation to eliminate the square root:

(√(-4x))^2 = 4^2

So, we have

-4x = 16

Dividing both sides by -4, we get:

x = -4

However, we need to check if this solution satisfies the original equation.

√(-4x) = √(-4(-4)) = √(16) = 4

So, the only solution to the equation is x = -4.

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The area of a square is given by the formula A = s^2, where s is the length of a side of the square. If the area of the square is 100 sq meters, then:

s^2 = 100

Taking the square root of both sides, we get:

s = 10

Therefore, the length of a side of the square is 10 meters. Since all four sides of a square are equal, the perimeter of the square is:

4s = 4(10) = 40 meters

Answer:

Step-by-step explanation:

If you don't know, the Pythagorean Theorem states that a^2+b^2=c^2 in a right triangle where a and b are the legs and c is the hypotenuse.

Imagine unfolding a cone. It would become part of a circle. Therefore, to find the lateral surface area(the surface are not including bases), you have to find the surface area of this part of the circle. The circumference of the entire circle is 6*pi*sqrt10. The circumference of this partial circle is the circumference of the base of the cone which is 2*3*pi=6pi. therefore, the percent of the circle is 6*pi/6*pi*sqrt10=1/sqrt10. 1/sqrt10*90pi=90*sqrt10*pi/10=9*sqrt10*pi.

the area of the base is 3^2*pi=9pi. the answer is then 9pi(1+sqrt10)

The sum of an angle measures of any trapezoid is greater than the sum of the angle measures of any parallelogram contradicts the rule which states that the sum of all angles in any quadrilateral is 360°.

What is a quadrilateral?

The Latin words quadra refers to number four, and latus refers to the sides, on combining gives quadrilateral. Any polygon with four sides, four angles, and four corners is said to be a quadrilateral. It is a closed shape known as a quadrilateral which is formed by joining any four points, any three of these points must not be on a same line.The quadrilateral has interior angles at each of its four corners. The sum of all for interior angles is always 360°.

Given that The sum of an angle measures of any trapezoid is greater than the sum of the angle measures of any parallelogram:

As we know that trapezoid and parallelogram both are quadrilaterals. so both will have interior angles sum equal to 360°

The given statement is false.

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The equations model the possible dimensions of the playground is option (A) x(x + 8) = 256.

A statement that affirms the equality of two expressions connected by the equals symbol "=" is known as an equation. For illustration, 2x - 5 = 13. 2x - 5 and 13 are expressions in this case. These two expressions are joined together by the sign "=".

The area of the rectangular playground is given as 256 square yards. Let x be the length of the shorter side of the playground in yards. Then the length of the longer side can be expressed as x + 8 yards, since it is 8 yards longer than the shorter side.

The area of the rectangle is given by the product of its length and width. Therefore, we can write:

x(x + 8) = 256

This equation represents the possible dimensions of the playground, where x is the length of the shorter side and x + 8 is the length of the longer side.

So, the correct equation that models the possible dimensions of the playground is:

x(x + 8) = 256.

Therefore, the equations model the possible dimensions of the playground is option (A) x(x + 8) = 256.

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Based on the information, the height of point A is 6 feet above ground and the length of OB is approximately 88 feet.

b) At t = 0.25 seconds, the approximate coordinates of the ball are (11, 13.5).

At t = 0.5 seconds, the approximate coordinates of the ball are (22, 12).

At t = 0.75 seconds, the approximate coordinates of the ball are (33, 9).

At t = 1 second, the approximate coordinates of the ball are (44, 6).

At t = 1.25 seconds, the approximate coordinates of the ball are (55, 3).

At t = 1.5 seconds, the approximate coordinates of the ball are (66, 0).

At t = 1.75 seconds, the approximate coordinates of the ball are (77, -3).

At t = 2 seconds, the approximate coordinates of the ball are (88, -6).

The height of the ball will be negative when it reaches point B again after traveling from A to B, assuming it follows the same path.

The time it takes for the ball to reach its maximum height and return to the ground will be approximately 2 seconds.

When the ball has coordinates (88, 0), it has reached the ground at point B, and its horizontal distance traveled is 88 feet.

At t = 0, the ball is at point A, which is 6 feet above the ground. This is because it was thrown from that height, and the time elapsed is not enough for gravity to affect its height yet.

Yes, the graph allows us to make predictions about the height of the ball at all points. However, the accuracy of the predictions may depend on the assumptions and approximations made about the ball's initial velocity, air resistance, and other factors.

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A) linear inequality, 0.5 s + 1 m = 30 < 75 where, 's' represents a small cup, and 'm' represents a medium cup.

We know that Sarah was selling lemonade and she makes 0.50 dollars on a small cup, and 1 dollar on a medium cup but she wants to make at least 30 dollars and only has 75 cups.

We need to show a linear inequality where, 's' represents a small cup, and 'm' represents a medium cup:

Linear inequalities are the expressions where any two values are compared by the inequality symbols ‘<’, ‘>’, ‘≤’ or ‘≥’

Linear equations are a combination of constants and variables.

let number of small cups be s,

therefore, total amount of small cups = 0.5 s

let number of medium cups be m,

therefore, total amount of medium cups = 1 m

with this given information, we can form our equation as:

0.5 s + 1 m = 30 < 75.

Therefore, a linear inequality, 0.5 s + 1 m = 30 < 75 where, 's' represents a small cup, and 'm' represents a medium cup.

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Complete question: Sarah was selling lemonade. she makes .50 dollars on a small cup, and 1 dollar on a medium cup. She wants to make at least 30 dollars and only has 75 cups. If s represents a small cup, and m represents a medium cup, which system of linear inequalities correctly represents her situation?

A. 0.5 s + 1 m = 30 < 75

B. 1s + 3m + 5l

C. $75

D. $93

The probability that the marble drawn from the bag will be either white or black is 2/3 or approximately 0.667.

To calculate this probability, we first need to find the total number of marbles in the bag, which is 10 + 20 + 5 = 35. Then, we need to find the total number of marbles that are either white or black, which is 10 + 20 = 30.

Finally, we divide the number of white or black marbles by the total number of marbles to get the probability:

30/35 = 2/3 ≈ 0.667

Therefore, the probability is approximately 0.667 or 2/3. This means that there is a higher chance of drawing a white or black marble than a grey marble, as there are more white and black marbles in the bag than grey marbles.

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The fraction of the check and tip did John pay is 5/24.

What are rational numbers?

It is possible to express rational numbers in the form pq, where p and q are integers and q0. The distinction between fractions and rational numbers is that the numerator or denominator of a fraction cannot be negative. As a result, the numerator and denominator of a fraction are whole numbers (denominator 0), as opposed to integers in the case of rational numbers.

Here, we have

Given: A group of three friends ate dinner at a restaurant. When they settled the check and tip, Peter paid 4/5 as much as John paid, and John paid 1/3 as much as Ralph paid.

we have to find a fraction of the check and tip did John pay.

p, j, r, the fractions that each paid.

p + j + r = 1....(1)

p = (4/5)j and j = (1/3)r

From equation(1), we get

(4/5)j + j + 3j = 1

j(4/5 + 1 + 3) = 1

(24/5)j = 1

j = 5/24

We have

r = 3j

r = 3(5/24)

r = 5/8

Hence, the fraction of the check and tip did John pay is 5/24.

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Answer:

66 hours.

Step-by-step explanation:

6 cars x 34 hours per car = 204

6 cars x 23 hours per car= 138 hours

now subtract:

204 - 138 = 66

therefore, it took Theo 66 hours longer to wash all the cars last week than last week.

The equation y=x²+2 (letter C) describes the given function.

In math, there are different functions: linear, quadratic, cubic, exponential and others. Each type of function presents its characteristics. See below:

A linear function can be represented by the standard form for the linear equation is: y= mx+b , for example, y=5x+6. Where: m= the slope and b= the constant term that represents the y-intercept.

Since, the quadratic function can represent a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function, the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.

For answering that the question asks you should test the values of x that represent the value y in the given table.

For x=-2 -> y=-3*(-2)=6 . Then, (-2,6) is a point of the function.

For x=5 -> y=-3*(5)=-15.   Then, (5,27) is not a point of the function.

Thus, the equation y=-3x does not describe the function.

For x=-2 -> y=5*(-2)+16=-10+16=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=5*(5)+16=-25+16=41 .   Then, (5,27) is not a point of the function.

Thus, the equation y=5x+16 does not describe the function.

For x=-2 -> y=(-2)²+2=4+2=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=(5)²+2=25+2=27.   Then, (5,27) is a point of the function.

For x=4 ->  y=(4)²+2=16+2=18.   Then, (4,18) is a point of the function.

For x=7 ->  y=(7)²+2=49+2=51.   Then, (7,51) is a point of the function.

Thus, the equation y=x²+2 describes the function.

For x=-2 -> y=2*(-2)²-2=2*4-2=8-2=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=2*(5)²-2=2*25-2=50-2=48.   Then, (5,27) is not a point of the function.

Thus, the equation y=2x²-2 does not describe the function.

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Answer: y= 4x+2

Step-by-step explanation: meaning of slope - the amount of bacteria multiplies by 4 as each minute passes

(sorry English isn't my first language)

The value of the expression using PEDMAS is 7.

The acronym PEMDAS, which can alternatively stand for "Please Forgive My Dear Aunt Sally," is a mnemonic tool for remembering the order of operations in mathematical and algebraic formulas. Its acronym is:

P - Parenthesis (operations inside parentheses are done first)

E: Exponents (operations involving exponents are done next)

Divide and multiply is abbreviated MD (these operations are done left to right, whichever comes first)

AS stands for addition and subtraction (these operations are done left to right, whichever comes first). No matter how they approach it or how they understand it, PEMDAS helps to ensure that the same statement is assessed in the same way by various people. In mathematics, it is crucial to understand this idea, especially when working with complicated expressions or formulae.

The given expression is 1/2 x (48 ÷ 6) – 2 + 5.

Simplifying the expressions we have:

1/2 x (48 ÷ 6) – 2 + 5

= 1/2 x 8 - 2 + 5

= 4 - 2 + 5

= 7

Hence, the value of the expression using PEDMAS is 7.

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The temperature at 7:00 a.m. was 67 degrees.

To find the temperature at 7:00 a.m., we need to consider that it was 20 degrees cooler than at 1:00 p.m.
1. Identify the given information: At 1:00 p.m., the temperature was 87 degrees, and it was 20 degrees cooler six hours earlier.
2. We need to find the temperature at 7:00 a.m., which is six hours before 1:00 p.m.
3. Since the temperature was 20 degrees cooler at 7:00 a.m.

we will subtract 20 from the temperature at 1:00 p.m.
4. Perform the subtraction:

87 degrees (1:00 p.m. temperature) - 20 degrees (cooler difference) = 67 degrees.
5. The result, 67 degrees, is the temperature at 7:00 a.m.
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If the probability of failure is 0.001, then the probability that

(a) lot is accepted is 0.9277,

(b) lot is rejected on 20th test is 0.000981,

(c) lot is rejected in 10 or fewer trials is 0.01.

Let the random-variable "X" represent number of batteries to test to get the first-battery which fails; in other words, to get first success.

Since, the trials are independent, X has a geometric distribution

Part (a) :

Let X =  the number of trials,

So, the probability that the lot is accepted is :

⇒ P(X=75) = (0.999)⁷⁵×(0.001)⁰ = 0.9277,

Part (b) :

let Y = the numbers of trials before the first failure (geometric distribution and

So, the probability that lot is rejected on 20th-test is :

⇒ P(Y=20) = (0.001)⁰×(0.999)¹⁹ = 0.000981,

Part (c) :

The probability that it is rejected in 10 or fewer trials is :

⇒ 1- P(no Failures)

⇒ 1 – (0.001)⁰(0.999)¹⁰ = 0.01.

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The volume of the cone is approximately 2941.6 cubic inches.

To begin with, let's recall the formula for the volume of a cone. The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius of the base, and h is the height of the cone.

We can see that the radius of the base of the cone is 15 inches. However, we do not know the height of the cone yet. To find the height, we can use the fact that the sphere is inscribed in the cone.

We have a right triangle with the height of the cone, the radius of the base of the cone (which is one leg of the triangle), and the diameter of the sphere (which is the hypotenuse of the triangle). Using the Pythagorean theorem, we get:

h² + 15² = 20²

Simplifying this equation, we get:

h² + 225 = 400

h² = 175

h ≈ 13.23

Now that we know the height of the cone, we can plug in the values of the radius and the height in the formula for the volume of the cone. We get:

V = (1/3)π(15²)(13.23) ≈ 2941.6 cubic inches

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Answer:

To calculate the final cost of the meal, we need to first calculate the discount and then add the sales tax. Discount = 10% of $5.75 = $0.575 Price after discount = $5.75 - $0.575 = $5.175 Sales tax = 4% of $5.175 = $0.207 Final cost = Price after discount + Sales tax = $5.175 + $0.207 = $5.382 Therefore, you should charge the customer $5.382.

well, let's hmmm apply the 10% discount first

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{10\% of 5.75}}{\left( \cfrac{10}{100} \right)5.75} ~~ \approx ~~ 0.58~\hfill \stackrel{ 5.75~~ - ~~0.58 }{\approx\text{\LARGE 5.17}}[/tex]

now let's apply the 4% tax to it

[tex]\stackrel{\textit{4\% of 5.17}}{\left( \cfrac{4}{100} \right)5.17} ~~ \approx ~~ 0.21\hspace{5em}\stackrel{ 5.17~~ + ~~0.21 }{\text{\LARGE 5.38}}\qquad \textit{final charge}[/tex]

The value οf y fοr pοint P οn the unit circle in the third quadrant with x-cοοrdinate -6/7 is -sqrt(13)/7.

PWe knοw that the pοint  is in the third quadrant and is distance 1 away frοm the οrigin, which is at (0,0). We alsο knοw that there is a pοint at (1,0) οn the x-axis, which is tο the right οf the οrigin.

Tο sοlve fοr y, we can use the Pythagοrean theοrem tο set up an equatiοn based οn the distance fοrmula:

[tex]\sqrt{(x^2 + y^2)} = 1[/tex]

Squaring bοth sides, we get:

[tex]x^2 + y^2 = 1[/tex]

Since we knοw that x = -6/7, we can substitute that in and sοlve fοr y:

[tex](-6/7)^ + y^2 = 1[/tex]

Simplifying, we get:

[tex]36/49 + y^2 = 1[/tex]

Subtracting 36/49 frοm bοth sides, we get:

y^2 = 13/49

Taking the square rοοt -

οf bοth sides yields:

[tex]y = +/- \sqrt{(13)} /7[/tex]

Because P is lοcated in the third quadrant, y must be negative. Therefοre, the value οf y in simplest fοrm is:

[tex]y = - \sqrt{(13)}/7[/tex]

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The length of the line can be made is 5x/8 inch, where 'x' is the number of strips.

Length is a measure of the size of an object or distance between two points, usually measured in units such as meters, centimeters, inches, feet, or miles.

In the given problem to find the length of a line we need to multiply the length of each strip by the number of strips

Here we have

The length of the strip = 5/8

The length of the line will depend on the number of strips

Let there be 'x' number of strips

The length of the line can be made = x × 5/8

=> 5x/8

Therefore,

The length of the line can be made is 5x/8 inch, where 'x' is the number of strips.

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The total number of campers surveyed is 150.

The symbol for percentage is "%". Percentages are commonly used in a wide range of contexts, such as in finance, statistics, and everyday life.

We can use the information provided about the percentage of campers in each time interval to set up an equation to solve for the total number of campers surveyed.

Let n be the total number of campers surveyed. Then:

15% of n spend more than 3 hours reading daily.

10% of n spend between 2 and 3 hours reading daily.

35% of n spend less than 1 hour reading daily.

40% of n spend between 1 and 2 hours reading daily.

We know that 60 campers spend between 1 and 2 hours reading daily, which represents 40% of the total number of campers surveyed. So we can set up the following equation:

0.4n = 60

Solving for n, we get:

n = 60 / 0.4 = 150

Therefore, the total number of campers surveyed is 150.

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The properties of the conic section are Ellipse, Domain = [-2, 2] and the range is [-3, 3]

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

The conic section

From the graph, we have

The conic section is an ellipse

The domain of this equation is all possible values of x and the range is for y

From the graph, we have

x values = -2 to 2

y values = -3 to 3

These are the domain and the range

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we conclude that △ACO is congruent to △BCO by the side-angle-side (SAS) congruence theorem.

We are given that angle AOB is a central angle of circle O and that angle ACB is a circumscribed angle of circle O. We see that AO ≅ BO because they are radii of the same circle.

We also know that AC ≅ BC since they are both tangent to the same circle and intersect at point C.

Thus, angle ACB and angle AOB have the same measure, which means that triangle ACO and triangle BCO are isosceles with AC ≅ BC and ∠ACO ≅ ∠BCO.

Using the reflexive property, we see that angle COA is congruent to angle COB since they are both vertical angles.

Therefore, we conclude that △ACO is congruent to △BCO by the side-angle-side (SAS) congruence theorem.

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The probability that a standard normal variable exceeds 2 is approximately 0.0228

To solve this problem, we need to use the central limit theorem, which states that the sum of a large number of independent and identically distributed random variables tends to follow a normal distribution, regardless of the distribution of the individual variables.

In this case, we have 25 independent individuals, each with a normal distribution of weights with mean 160 pounds and standard deviation 30 pounds. The total weight of the 25 individuals is the sum of these 25 random variables.

The mean weight of a single person is 160 pounds, so the mean weight of 25 people is 25 times 160, which is 4000 pounds. The standard deviation of the sum of 25 random variables is the square root of the sum of the variances of the individual variables, which is the square root of 25 times 30 squared, or 150 pounds.

The probability that the total weight of the 25 people exceeds the design limit of 4300 pounds can be calculated using the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. We can convert the total weight of the 25 people to a z-score using the formula

z = (x - mu) / sigma

where x is the total weight, mu is the mean weight, and sigma is the standard deviation of the sum of the individual weights.

z = (4300 - 4000) / 150 = 2

The probability that a standard normal variable exceeds 2 is approximately 0.0228.

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