What is a probability ?


A number that measures the likelihood that the event will occur.

An activity involving chance in which results are observed

An observation of an experiment

The result of an experiment

Answer:

We want to find:

[tex]\lim_{n \to \infty} \frac{\sqrt[n]{n!} }{n}[/tex]

Here we can use Stirling's approximation, which says that for large values of n, we get:

[tex]n! = \sqrt{2*\pi*n} *(\frac{n}{e} )^n[/tex]

Because here we are taking the limit when n tends to infinity, we can use this approximation.

Then we get.

[tex]\lim_{n \to \infty} \frac{\sqrt[n]{n!} }{n} = \lim_{n \to \infty} \frac{\sqrt[n]{\sqrt{2*\pi*n} *(\frac{n}{e} )^n} }{n} = \lim_{n \to \infty} \frac{n}{e*n} *\sqrt[2*n]{2*\pi*n}[/tex]

Now we can just simplify this, so we get:

[tex]\lim_{n \to \infty} \frac{1}{e} *\sqrt[2*n]{2*\pi*n} \\[/tex]

And we can rewrite it as:

[tex]\lim_{n \to \infty} \frac{1}{e} *(2*\pi*n)^{1/2n}[/tex]

The important part here is the exponent, as n tends to infinite, the exponent tends to zero.

Thus:

[tex]\lim_{n \to \infty} \frac{1}{e} *(2*\pi*n)^{1/2n} = \frac{1}{e}*1 = \frac{1}{e}[/tex]

Answer:

d = [tex]\frac{2x + 3}{p}[/tex]  - x

Step-by-step explanation:

First divide both sides by p.

then subtract x from both sides

Answer:

[tex]\displaystyle d = \frac{2x + 3}{d} - x[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Algebra I

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle p(d + x) = 2x + 3[/tex]

Step 2: Solve for d

Answer: Rigid just means that the whole shape goes through the same transformation, so with rotations, reflections, and translations, the shape should not change at all, just in a different place or orientation.

In a translation, ALL of the points move the same distance in the same direction. A translation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image. ... In addition, the corresponding segment sides of the pre-image and image are parallel.

A rigid transformation preserves both the side lengths and the angle measures of a polygon. Step-by-step explanation: Rigid transformations are the transformations which does not affect the size and the shape of figure . The figure doesn't shrink or get enlarger.

There are three different types of transformations: translation, reflection, and rotation.

The distance from each point in the figure to the center of rotation is preserved. A rigid transformation is a transformation that preserves congruence.

More information for Rigid

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Answer:

7.56

Step-by-step explanation:

Kevin should expect to pay approximately $7.56 for 18 juice boxes based on the given information.

To find out how much Kevin should expect to pay for 18 juice boxes based on the given information, we can set up a proportion using the number of juice boxes and the cost:

In general, an expression refers to a combination of symbols, numbers, variables, and operators that represent a specific computation or value. Expressions are a fundamental concept in mathematics, programming, and logic.

Let "x" be the cost of 18 juice boxes.

We have the proportion:

6 juice boxes / $2.52 = 18 juice boxes / x

To solve for "x," we can cross-multiply:

6x = 18 x $2.52

6x = $45.36

Now, divide both sides by 6 to isolate "x":

x = $45.36 / 6

x ≈ $7.56

Therefore, According to the data provided, Kevin should budget about $7.56 for 18 juice cartons.

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Answer: $235.4

Step-by-step explanation:

Given

Price list on slacks is $22

Price list on jumpers is $37

Store ordered 30 pairs of slacks and 40 Jumpers

Total price becomes

[tex]\Rightarrow 22\times 30+37\times 40\\\Rightarrow \$2140[/tex]

for a discount of 11%

Trade discount is [tex]2140\times 11\%[/tex]

[tex]\Rightarrow 2140\times 0.11\\\Rightarrow \$235.4[/tex]

Answer:

4%

Step-by-step explanation:

264 interest/3 years=88 interest/year

principal x interest rate =interest/year

2200 x interest rate =88

interest rate =88/2200

interest rate =.04 or 4%

Step-by-step explanation:

Starting out with the Taylor series,

[tex]\displaystyle f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x-a)^n[/tex]

where [tex]f^{(n)}[/tex] is the nth derivative of f(x) and if we set a = 0, we get the special case of the Taylor series called the Maclaurin series:

[tex]\displaystyle f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(0)}{n!}x^n[/tex]

Expanding this series up to the 1st 3 terms at a = 0,

[tex]f(x) = f(0) + \dfrac{f'(0)}{1!}x + \dfrac{f''(0)}{2!}x^2[/tex]

Let's find the derivatives of [tex]e^{\frac{x}{2}}[/tex]:

[tex]f'(x) = \frac{d}{dx} (e^{\frac{x}{2}}) = \frac{1}{2}e^{\frac{x}{2}} \Rightarrow f'(0) = \frac{1}{2}[/tex]

[tex]f''(x) = \frac{1}{4}e^{\frac{x}{2}} \Rightarrow f''(0) = \frac{1}{4}[/tex]

We can now write the Maclaurin series for [tex]e^{\frac{x}{2}}[/tex]as

[tex]e^{\frac{x}{2}} = 1 + \frac{1}{2} x + \frac{1}{8} x^2[/tex]

Answer:

y = 2 / (r/3 - s/5)

Step-by-step explanation:

r/3 - 2/y = s/5

add 2/y to both sides

r/3 = s/5 + 2/y

Subtract s/5 from both sides

r/3 - s/5 = 2/y

multiply both sides by y

y(r/3 - s/5) = 2

Divide both sides by r/3 - s/5

y = 2 / (r/3 - s/5)

===========================================================

Use this formula

a & b/c = (a*c+b)/c

to convert from a mixed number to an improper fraction

So,

a & b/c = (a*c+b)/c

3 & 3/4 = (3*4+3)/4

3 & 3/4 = 15/4

---------------------------------------------------

Next, we add the fractions 15/4 and 5/6. The denominators aren't the same, so we can't add right away. We need to get the denominators to the LCD.

In this case, the LCD is 4*6 = 24

Now we can add the fractions:

90/24+20/24 = (90+20)/24 = 110/24

---------------------------------------------------

So far, we've added the 3&3/4 portion to the 5/6 portion to get the result 110/24

The last thing to do is to add on the two 1/3 ft pieces, which means we're adding on 2/3 of a foot

2/3 = 16/24 after multiplying top and bottom by 8

Add this to the result of the last section

110/24+16/24 = (110+16)/24 = 126/24

---------------------------------------------------

From here, we convert the improper fraction 126/24 to decimal form

Using a calculator, you should find that 126/24 = 5.25

The result is larger than 5, which means she used too much ribbon. There won't be enough ribbon to do everything she wants to do.

---------------------------------------------------

The shortcut we can take is to type the following into the calculator

3+3/4+5/6+2/3

doing so should lead to 5.25

Answer:

Yes

Step-by-step explanation:

So at the begining, Cherly has 60 in (12inx5ft) of ribbon. She uses 45 in(3.75x12) to make the hair bow, leaving 15 in of ribbon. She, then, uses 10 in (5/6x12) of ribbon, leaving 5 in. After that, Cheryl uses 8 in [2(1/3x12)] to put on the picture frame, leaving two inches of ribbon.

Answer:

50%

Step-by-step explanation:

50% of the student population chooses instramural volleyball, meaning 50% doesn't. So if a random student is chosen, then it is a 50% chance that they will choose volleyball and 50% chance they won't.

Given:

Marginal cost for making 1 mask = $0.50

Total cost to make 100 masks = $62

Revenue per mask = $3

To find:

(a) Linear cost function C(x)

(b) Revenue function R(x)

(c) Break even point

(d) Interval of profit & loss

Solution:

(a) We know that linear cost function is given by,

Total cost = Fixed cost per unit + (Marginal cost per unit)*(Number of units)

Let 'x' denote the number of units

[tex]\Rightarrow[/tex] C(x) = b + mx

It is given that, m = $0.50

[tex]\Rightarrow[/tex] C(x) = b + 0.5x

It is also given that the total cost of making 100 masks is $62

[tex]\Rightarrow[/tex] C(100) = $62

[tex]\Rightarrow[/tex] b +0.5(100) = 62

[tex]\Rightarrow[/tex] b + 50 =62

[tex]\Rightarrow[/tex] b = 62 - 50

[tex]\Rightarrow[/tex] b = 12

[tex]\Rightarrow[/tex] C(x) = 12 + 0.5x

This is the linear cost function, C(x)

(b) We know that,

Total Revenue = Revenue per unit * Number of units

It is given that a mask is sold for $3, i.e., revenue per mask is $3

Let 'x' denote the number of units

Then the revenue function is given by,

R(x) = 3x

This is the revenue function, R(x)

(c) We know that, break even point refers to the point where total cost is equal to the total revenue. Thus, at break even point, we have,

C(x) = R(x)

[tex]\Rightarrow[/tex] 12 + 0.5x = 3x

[tex]\Rightarrow[/tex] 3x - 0.5x = 12

[tex]\Rightarrow[/tex] 2.5x = 12

[tex]\Rightarrow x=\frac{12}{2.5}[/tex]

[tex]\Rightarrow[/tex] x = 4.8

Since, 'x' denotes the number of masks, it must be a whole number and not a fraction. Thus, we will round off our value to get the break even point as 5 masks.

That is, 5 masks must be made to break even.

(d) We know that the profit function is given as the difference of revenue function and cost function. That is, we have,

P(x) = R(x) - C(x)

[tex]\Rightarrow[/tex] P(x) = 3x - (12 + 0.5x)

[tex]\Rightarrow[/tex] P(x) = 3x - 12 - 0.5x

[tex]\Rightarrow[/tex] P(x) = 2.5x -12

Now, we know that there is profit when the value of the profit function is positive & there is loss when the value of profit function is negative. Thus, we can calculate the intervals of profit and loss by finding the intervals where the profit function is positive & negative respectively. Alternatively, since the break even point denotes the point where the value of the profit function is 0, we can find the intervals of profit and loss as the intervals greater than and lesser than the break even point respectively.

That is, since the break even point is x = 4.8, the interval of profit is given as x > 4.8 & the interval of loss is given as x < 4.8

Taking into account that 'x' denotes the number of masks and thus must be a whole number, we have the intervals of profit and loss as,

Loss:= [tex]x \in [0,4][/tex]

Profit:= [tex]x \in [5, \infty)[/tex]

Final answer:

(a) Linear Cost function: C(x) = 12 + 0.5x

(b) Revenue function: R(x) = 3x

(c) 5 masks must be made to break even

(d) Interval of profit: [tex]x \in [5, \infty)[/tex], Interval of loss: [tex]x \in [0,4][/tex]

Answer:

36 meters

Step-by-step explanation:

A=1/2b×h

108sq. m=1/2.6m×h

both side canceling meter

h=2×108m/6

h=36m. answer

The height of the triangle is 12 meters, and the base of the triangle is 18 meters.

Let's denote the height of the triangle as "h" meters. According to the given information, the base of the triangle is 6 meters longer than the height, so the base can be represented as "(h + 6)" meters.

The area of a triangle is given by the formula: Area = (1/2) * base * height

We are given that the area of the triangle is 108 square meters, so we can write the equation as:

108 = (1/2) * (h + 6) * h

Now, let's solve for "h":

108 = (1/2) * (h² + 6h)

Multiply both sides by 2 to eliminate the fraction:

216 = h² + 6h

Rearrange the equation in standard quadratic form:

h² + 6h - 216 = 0

Now, let's solve this quadratic equation for "h." We can factor it or use the quadratic formula. Factoring, we get:

(h + 18)(h - 12) = 0

Setting each factor to zero:

h + 18 = 0     or     h - 12 = 0

Solving for "h" in each case:

h = -18 (discard this negative value as height cannot be negative)     or     h = 12

Since height cannot be negative, we take the positive value of "h," which is 12 meters.

Now, we can find the base by using the given relationship: base = height + 6

base = 12 + 6 = 18 meters

So, the height of the triangle is 12 meters, and the base of the triangle is 18 meters.

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Answer:

x=-4  and y=2  Those 2 equations when graphed give a vertical line and a horizontal line that intersect at the point (-4,2)

x+4=0 and y-2=0  are the same two lines.

or if you want more complicated systems of equations, with the same solution:

x+y=-2 and x-y=-6

in standard form, that's y=-x-2 and y=x+6  Graph those two lines and they intersect at (-4,2)

Those 2 lines are with slopes of -1 and +1.  Those 2 equations are a system of equations

whose solution is x=-4 and y=2.

Rotate two perpendicular lines around the point (-4,2) and you get more systems of  

equations with the same solution.

They don't have to be perpendicular though.  x+y=-2 and x=-4 are two lines forming

a 45 degree angle, with the same solution (-4,2)

They don't have to be linear either.  Take a parabola with the vertex (-4,2) and the line x=-4.

They intersect at the point (-4,2).  That parabola could be the simple y=x2 shifted up and

to the left, to y-2 = (x+4)2  or y=x2+8x+18   or it could be any one of a family of parabolas with

the same vertex.

You could have 2 circles tangent at the point (-4,2) Endless circles could have that point

as a tangent and their equations having that same solution (-4,2)

Step-by-step explanation:

Hope this helps <3

Step-by-step explanation:

Here is the answer for your question

i think hhjs is how it goes

If we try to use L'Hopital's Rule to find the limit we cannot apply L’Hopital’s Rule because the denominator equals zero for some value x = a, the correct option is D.

We are given that;

Function [tex]x/√x^2 +6[/tex]

Now,

The limit of the function as x approaches infinity is an indeterminate form of type ∞/∞.

However, L’Hopital’s Rule can only be applied when the limit is in the form 0/0 or ∞/∞.

So, L’Hopital’s Rule cannot be applied in this case

Therefore, by L’Hopital’s rule answer will be  You cannot apply L’Hopital’s Rule because the denominator equals zero for some value x = a.

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Answer:

1) the planning value for the population standard deviation is 10,000

2)

a) Margin of error E = 500, n = 1536.64 ≈ 1537

b) Margin of error E = 200, n = 9604

c) Margin of error E = 100, n = 38416

3)

As we can see, sample size corresponding to margin of error of $100 is too large and may not be feasible.

Hence, I will not recommend trying to obtain the $100 margin of error in the present case.

Step-by-step explanation:

Given the data in the question;

1) Planning Value for the population standard deviation will be;

⇒ ( 50,000 - 10,000 ) / 4

= 40,000 / 4

σ = 10,000

Hence, the planning value for the population standard deviation is 10,000

2) how large a sample should be taken if the desired margin of error is;

we know that, n = [ ([tex]z_{\alpha /2[/tex] × σ ) / E ]²

given that confidence level = 95%, so [tex]z_{\alpha /2[/tex]  = 1.96

Now,

a) Margin of error E = 500

n = [ ([tex]z_{\alpha /2[/tex] × σ ) / E ]²

n = [ ( 1.96 × 10000 ) / 500 ]²

n = [ 19600 / 500 ]²

n = 1536.64 ≈ 1537

b) Margin of error E = 200

n = [ ([tex]z_{\alpha /2[/tex] × σ ) / E ]²

n = [ ( 1.96 × 10000 ) / 200 ]²

n = [ 19600 / 200 ]²

n = 9604

c)  Margin of error E = 100

n = [ ([tex]z_{\alpha /2[/tex] × σ ) / E ]²

n = [ ( 1.96 × 10000 ) / 100 ]²

n = [ 19600 / 100 ]²

n = 38416

3) Would you recommend trying to obtain the $100 margin of error?

As we can see, sample size corresponding to margin of error of $100 is too large and may not be feasible.

Hence, I will not recommend trying to obtain the $100 margin of error in the present case.

========================================================

Explanation:

The observations are in ascending order, which means the set is sorted from smallest to largest.

The X is at the right-most endpoint, so X is the largest value (aka the max).

The min is 33, so,

max - min = range

X - 33 = 58

X = 58+33

X = 91 is the max

Answer:

Step-by-step explanation:

The range of the data is the highest value - the lowest value.

The lowest in the list is 33 so the highest must be 33 + 58

= 91.

Answer:

30. A right triangle has one angle that is 90°

All angles add to 180°

180 - 90 - 60 = 30

Step-by-step explanation:

Answer:

f ( -4 ) = 1024 + 8 + 9

Step-by-step explanation:

f ( x ) = 4x⁴ - x² + 9

If f ( - 4 ) then we get

f ( -4 ) = 4 ( -4)⁴ - ( - 4)² + 9

Expand the exponents

f ( - 4 ) = 4 ( 256 ) + 8 + 9

multiply the numbers

f ( -4 ) = 1024 + 8 + 9

Answer:

DM=38\ cmDM=38 cm

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

Triangles DEF and OEN are similar by AA Similarity Postulate

Remember that if two triangles are similar, then the ratio of its corresponding sides is proportional

In this problem

\frac{DE}{OE}=\frac{DF}{ON}

OE

DE

=

ON

DF

substitute the given values

\frac{44}{22}=\frac{DF}{38}

22

44

=

38

DF

2=\frac{DF}{38}2=

38

DF

DF=2(38)=76\ cmDF=2(38)=76 cm

\begin{gathered}DF=2DM\\76=2DM\\DM=38\ cm\end{gathered}

DF=2DM

76=2DM

DM=38 cm

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Answer:

C

Step-by-step explanation:

Answer:

sssssssss

Step-by-step explanation:

ssssssss

Answer:

Patrice does not have sufficient evidence to reject the fast-food chain's claim.

Step-by-step explanation:

unless she already ate some fries, not enough evidence

Answer: $15219.20

Step-by-step explanation:

Based on the information given, firstly we need to calculate the discount which will be:

= 18% × $18560

= 0.18 × $18560

= $3340.80

Then, the amount that Dao will pay will be:

= Sticker price - Discount

= $18560 - $3340.80

= $15219.20

Therefore, Dao will pay $15219.20

Answer:

The modulus of z is [tex]\sqrt{10}[/tex]

Step-by-step explanation:

Complex number:

A complex number, given by:

[tex]z = x + iy[/tex]

Can be represented also as a point (x,y).

The modulus is given by:

[tex]|z| = \sqrt{x^2 + y^2}[/tex]

Point z is at (negative 3, 1). What is the modulus of z?

[tex]x = -3, y = 1[/tex]. So

[tex]|z| = \sqrt{x^2 + y^2} = \sqrt{(-3)^2+1^2} = \sqrt{10}[/tex]

The modulus of z is [tex]\sqrt{10}[/tex]

Answer:

-2

Step-by-step explanation:

if this is the right problem it is -2

Answer:

x = 28

Step-by-step explanation:

HFG = EFI

6x - 4 = 164

6x = 164 + 4

6x = 168

x = 168/6

x = 28

Answer:

x = 35

Step-by-step explanation:

x and 3x+40 are same side interior angles and same side interior angles are supplementary when the lines are parallel

x + 3x+40 = 180

Combine like terms

4x+40 =180

4x = 180-40

4x = 140

Divide by 4

4x/4 = 140/4

x = 35

Answer and Step-by-step explanation:

To solve for x, we add together the angles, and equal it to 180.

This is because the angles shown are same side interior angles, and add up to 180 degrees.

x + 3x + 40 = 180

Subtract 40 and combine like terms.

4x = 140

Divide 4 from both sides of the equation.

x = [tex]35[/tex]

The value of x is 35.

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