Six different integers are picked from the numbers 1 through 10. How many possible combinations are there, if the the second smallest integer in the group is 3?


Please solve ASAP

Answer with explanation:

The meaning of bijection for two sets X and Y is each and every element of X is uniquely related with element of set Y and when you take the inverse mapping every element of set Y is uniquely related with each and every element of Set X.

The two sets given are

X=(0,1]---------Semi open or Semi closed Set

Y=(0,1)---------Closed set

Between any two real numbers there are infinite number of real numbers.So cardinal number of both the sets is infinite.

There can be infinite bijection between these two sets as both sets have infinite number of elements.

   X                      Y

0.1  -------------      0.01

0.2     -------          0.02

0.3     --------         0.03

0.4     ---------          0.04

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

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

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

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

One example of a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex] is the function [tex]\(f(x) = \frac{x}{1 + x}\)[/tex]. This function is injective (one-to-one) and surjective (onto), making it a bijection.

To establish a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex], consider the function [tex]\(f(x) = \frac{x}{1 + x}\).[/tex]

1. Injectivity (One-to-One): Assume [tex]\(f(x_1) = f(x_2)\)[/tex] for some [tex]\(x_1, x_2 \in (0,1]\)[/tex]. By solving the equation [tex]\(\frac{x_1}{1 + x_1} = \frac{x_2}{1 + x_2}\)[/tex], you find [tex]\(x_1 = x_2\)[/tex], showing that distinct elements in the domain map to distinct elements in the codomain.

2. Surjectivity (Onto): For any y in the codomain [tex]\((0,1)\)[/tex], solve [tex]\(f(x) = y\) for \(x\)[/tex]. This results in [tex]\(x = \frac{y}{1 - y}\)[/tex], which is well-defined for [tex]\(y \in (0,1)\)[/tex]. Therefore, every element in the codomain has a preimage in the domain.

Hence, [tex]\(f(x)\)[/tex] is a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex].

Answer:

2330

Step-by-step explanation:

In rounding nearest 10,

If the ones place digit is 5 or more then 5 then we round the number to upper 10, while if the ones place digit is less than 5 then we round the number to lower 10,

For example : 34 ≈ 30 while 36 ≈ 40,

Now, given expression,

961 + 27.8 + 693.0 + 573 +76.4

By the above statement 961≈ 960, 27.8 ≈ 30, 693.0 ≈ 690, 573 ≈ 570 and 76.4 ≈ 80,

Hence, 961 + 27.8 + 693.0 + 573 +76.4 = 960 + 30 + 690 + 570 + 80 = 2330.

Answer:

2330

Step-by-step explanation:

961 + 27.8 + 693.0 + 573 +76.4 =

Rounding to the nearest 10

960 + 30 + 690 + 570 + 80

2330

Answer:

Step-by-step explanation:

Remember that in the geometric serie if | r | < 1 the serie converges and if | r | ≥1 the serie diverges.

I suppose that the serie starts at 0, so using the geometric serie with r = | [tex]\frac{-3}{2}[/tex] | > 1 the serie diverges.

Answer: e. 0.3849

Step-by-step explanation:

We know that the mean lies exactly at the middle of the normal curve .

The z-score of the mean value is 0.

Also According to the standard normal probability table, the probability value of z=0 is P(z<0)=0.5.

And the probability value of z=-1.20 is P(z<-1.2) =0.1150697.

Now, the  area under the normal curve between the mean and a score for which z= - 1.20 is  given by :-

[tex]P(z<0)-P(z<-1.2)=0.5-0.1150697=0.3849303\approx0.3849[/tex]

Hence, the area under the normal curve between the mean and a score for which z= - 1.20 is 0.3849 .

Answer: Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.

Step-by-step explanation:

Since we have given that

Integers between 10000 and 99999 = 99999-10000+1=90000

n( divisible by 3) = [tex]\dfrac{90000}{3}=30000[/tex]

n( divisible by 5) = [tex]\dfrac{90000}{5}=18000[/tex]

n( divisible by 7) = [tex]\dfrac{90000}{7}=12857.14[/tex]

n( divisible by 3 and 5) = n(3∩5)=[tex]\dfrac{90000}{15}=6000[/tex]

n( divisible by 5 and 7) = n(5∩7) = [tex]\dfrac{90000}{35}=2571.42[/tex]

n( divisible by 3 and 7) = n(3∩7) = [tex]\dfrac{90000}{21}=4285.71[/tex]

n( divisible by 3,5 and 7) = n(3∩5∩7) = [tex]\dfrac{90000}{105}=857.14[/tex]

As we know the formula,

n(3∪5∪7)=n(3)+n(5)+n(7)-n(3∩5)-n(5∩7)-n(3∩7)+n(3∩5∩7)

[tex]=30000+18000+12857.14-6000-2571.42-4258.71+857.14\\\\=48884.15[/tex]

Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.

Answer:

2.3775

Step-by-step explanation:

Answer:

The lines L1 and L2 neither parallel nor perpendicular

Step-by-step explanation:

* Lets revise how to find a slope of a line

- If a line passes through points (x1 , y1) and (x2 , y2), then the slope

 of the line is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

- Parallel lines have same slopes

- Perpendicular lines have additive, multiplicative slopes

 ( the product of their slopes is -1)

* Lets solve the problem

∵ L1 passes through the point (1 , 10) and (-1 , 7)

- Let (1 , 10) is (x1 , y1) and (-1 , 7) is (x2 , y2)

∴ x1 = 1 , x2 = -1 and y1 = 10 , y2 = 7

∴ The slope of L1 is [tex]m1 = \frac{7-10}{-1-1}=\frac{-3}{-2}=\frac{3}{2}[/tex]

∵ L2 passes through the point (0 , 3) and (1 , 5)

- Let (0 , 3) is (x1 , y1) and (1 , 5) is (x2 , y2)

∴ x1 = 0 , x2 = 1 and y1 = 3 , y2 = 5

∴ The slope of L2 is [tex]m2=\frac{5-3}{1-0}=\frac{2}{1}=2[/tex]

∵ m1 = 3/2 and m2 = 2

- The two lines have different slopes and their product not equal -1

∴ The lines L1 and L2 neither parallel nor perpendicular

By calculating the slopes of L1 and L2, we find that they are 1.5 and 2 respectively. Since they are neither the same nor negative reciprocals, L1 and L2 are neither parallel nor perpendicular.

To determine if lines L1 and L2 are parallel, perpendicular, or neither, we need to calculate the slopes of both lines using the slope formula:

Slope formula: (y2 - y1) / (x2 - x1)

Calculating the slope of L1:

Points on L1: (1, 10) and (-1, 7)

Slope of L1 = (7 - 10) / (-1 - 1) = (-3) / (-2) = 1.5

Calculating the slope of L2:

Points on L2: (0, 3) and (1, 5)

Slope of L2 = (5 - 3) / (1 - 0) = 2 / 1 = 2

Since the slopes of L1 (1.5) and L2 (2) are neither the same nor negative reciprocals of each other, the lines L1 and L2 are neither parallel nor perpendicular.

I guess the series is

[tex]\displaystyle\sum_{n=3}^\infty\frac8{(n-1)(n+1)}[/tex]

Splitting into partial fractions gives

[tex]\dfrac8{(n-1)(n+1)}=\dfrac4{n-1}-\dfrac4{n+1}[/tex]

The series telescopes, which we can see by considering the [tex]k[/tex]-th partial sum:

[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=(2-1)+\left(\frac43-\frac45\right)+\cdots+\left(\frac4{k-2}-\frac4k\right)+\left(\frac4{k-1}-\frac4{k+1}\right)[/tex]

(feel free to write out more terms to convince yourself that the following is true)

[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=2+\frac43-\frac4k-\frac4{k+1}[/tex]

Then as [tex]k\to\infty[/tex], we end up with

[tex]\displaystyle\lim_{k\to\infty}\sum_{n=3}^k\frac8{(n-1)(n+1)}=\boxed{\frac{10}3}[/tex]

The series is convergent with a sum of -1.143

The given series is:
∞ 8 (n-1)(n+1) / n = 3

First, we can use partial fraction decomposition to simplify the series. Factoring (n-1)(n+1) gives us [tex]n^2[/tex] - 1. So, the series becomes:

∞ 8 ([tex]n^2[/tex] - 1) / n = 3

Next, we can rewrite the series as two separate series:

∞ 8 [tex]n^2[/tex] / n - 8 / n = 3

Simplifying each series:

∞ 8n - 8 / n = 8∞ n³ - 1∞

The first series is a geometric series with a common ratio of 8 and a first term of 8. Using the formula for the sum of a geometric series, we can find its sum:

S = a / (1-r)

Substituting the values:

S = 8 / (1-8) = 8 / -7 = -1.143

Therefore, the series is convergent, and its sum is -1.143.

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Answer: 462

Step-by-step explanation:

The general theorem of combination says that there are [tex]C(n+r-1, r)[/tex], with r-combinations from a set having n elements when repetition of elements is allowed.

Here the number of denomination: [tex]n = 7[/tex] , r =5

Also order doesn't matters.

Then the number of different possible ways can you choose the five bills is given by :-[tex]C(7+5-1, 5)= C(11,5)\\\\=\dfrac{11!}{5!(11-5)!}\\\\=462[/tex]

Hence, the number of different possible ways can you choose the five bills is  462.

Answer:

-18

Step-by-step explanation:

Simplify the following:

abs(3 - 10) - (12/4 + 2)^2

The gcd of 12 and 4 is 4, so 12/4 = (4×3)/(4×1) = 4/4×3 = 3:

abs(3 - 10) - (3 + 2)^2

3 + 2 = 5:

abs(3 - 10) - 5^2

3 - 10 = -7:

abs(-7) - 5^2

5^2 = 25:

abs(-7) - 25

Since -7<=0, then abs(-7) = 7:

7 - 25

7 - 25 = -18:

Answer:  -18

For this case we must resolve the expression:

[tex]| 3-10 | - (\frac {12} {4}+2) ^ 2[/tex]

We have the following definitions:

Different signs are subtracted and the sign of the major is placed.

Absolute value definition:[tex]| -a | = a[/tex]

Resolving we have:

[tex]| 3-10 | = | -7 | = 7[/tex]

[tex](\frac {12} {4}+2) ^ 2 = (3+2) ^ 2 = 5 ^ 2 = 25[/tex]

Then, rewriting the expression:

[tex]7-25 = -18[/tex]

Answer:

-18

Answer:

a is not an equivalence relation.

b is an equivalence relation.

Step-by-step explanation:

a.

A = {1,2,3,4, 5) R={(1,1), (1,2), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,4), (5,5)

To see if is an equivalence relation you need to see if you have these 3 things:

Part 1: xRx for all x in A. This is the reflexive property.

Do we? Yes we have all these points in R: (1,1), (2,2) ,(3,3) ,(4,4), and (5,5).

Part 2: If xRy then yRx. This is the symmetic property.

Do we? We have (1,2) but not (2,1). So it isn't symmetric.

Part 3: If xRy and yRz then xRz.

Do we? We are not going to check this because there is no point. We have to have all 3 parts fot it be an equivalence relation.

b.

A = {a, b, c} R={(a, a), (a, c), (b, b), (c, a), (c, c)}

To see if is an equivalence relation you need to see if you have these 3 things:

Part 1: xRx for all x in A. This is the reflexive property.

Do we? Yes we have all these points in R: (a,a),(b,b), and (c,c).

Part 2: If xRy then yRx. This is the symmetric property.

Do we? We have (a,c) and (c,a). We don't need to worry about any other (x,y) since there are no more with x and y being different. This is symmetric.

Part 3: If xRy and yRz then xRz.

Do we? We do have (a,c), (c,a), and (a,a).

We do have (c,a), (a,c), and (c,c).

So it is transitive.

Question b has all 3 parts so it is an equivalence relation.

Answer:

A. negatively skewed

Step-by-step explanation:

Plot the values of salary against years

You will notice that;

This mean that this graph is a negative skewed graph/left-skewed distribution

In the attached sketch of the plot, join the points with a smooth curve and observe the above mentioned properties.

The x-axis scale ranges from 2000 to 2015 where 0=2000, 1=2001,2=2002......,15=2015

Hope this will give you a visual picture of the negatively- skewed distribution

Answer:

The width is 16 inches while the length is 24 inches.

Step-by-step explanation:

We are given that the length (L) is 8 more than it's width (W).

The equation for this is: L=8+W.

We are given the perimeter is 80.

The equation for this is 2L+2W=80.

Each term here is even so I'm going to divide both sides by 2: L+W=40.

My system to solve is:

L=8+W

L+W=40

I'm going to plug the first equation into the second giving me:

(8+W)+W=40

Drop the (), don't really need:

8+W+W=40

Combine like terms:

8+2W=40

Subtract 8 on both sides:

    2W=32

Divide both sides by 2:

      W=32/2

Simplify:

     W=16

So if W=16 and L=8+W then L=8+16=24.

The width is 16 inches while the length is 24 inches.

Perimeter is the sum of the length of the sides used to make the given figure. The length and the width of the rectangle are 24 inches and 16 inches, respectively.

Perimeter is the sum of the length of the sides used to make the given figure.

Let the width of the rectangle be represented by x.

Given that the length of a rectangle exceeds its width by 8 inches. Therefore, the length and the width of a rectangle can be written as,

Width = x inches

Length = x inches + 8 inches

Now, as it is given that the perimeter of the rectangle is 80 inches. Therefore, the perimeter of the given rectangle can be written as,

Perimeter of rectangle = 2(Length + Width)

80 inches = 2[(x+8) + x]

80 = 2(x + 8 + x)

80 = 2(2x + 8)

80/2 = 2x + 8

40 = 2x + 8

40 - 8 = 2x

32 = 2x

x = 32/2

x = 16 inches

Further, the length and the width of the rectangle can be written as,

Width = x = 16 inches

Length = x + 8 = 16 + 8 = 24 inches

Hence, the length and the width of the rectangle are 24 inches and 16 inches, respectively.

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Answer:

242.

Step-by-step explanation:

C = 0.03 x 2 − 8 x + 700

When the marginal cost = $522 we have:

522 = 0.03 x 2 − 8 x + 700

0.03x^2 - 8x + 700 - 522 = 0

0.03x^2 - 8x + 178 = 0

Using the quadratic formula:

x =  [ -(-8) +/- sqrt( 64 - 4 * 0.03 * 178)] / 0.06

= 8 +/- sqrt42.64 / 0.06

= 242  players.

242 DVD/Blu-ray players were produced

Marginal cost is the change in cost as a result of producing additional number of units.

Given that the marginal cost is:

C = 0.03x² - 8x + 700

Therefore for a marginal cost of $522, the number of DVD players is:

522 = 0.03x² - 8x + 700

0.03x² - 8x + 178 = 0

x = 24 and x = 242

Therefore 242 DVD/Blu-ray players were produced

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To find the outward flow through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] integrate the dot product of velocity and surface normal. The outward flux is [tex]\(810 \times 32 \pi\).[/tex]

To calculate the rate of flow outward through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] let's compute the flux of the given vector field through the cylindrical surface. The outward flux through a closed surface is given by:

[tex]\[\Phi = \int_S \mathb{v} \cdot \mathb{dS}.\][/tex]

Surface Parameterization and Surface Normal.

The cylinder has radius r = 2,  so the general parameterization is:

- [tex]\(x = 2 \cos(\theta)\),[/tex]

- [tex]\(y = 2 \sin(\theta)\),[/tex]

- z = z,

where [tex]\(0 \leq \theta \leq 2\pi\) and \(0 \leq z \leq 4\).[/tex]

The outward normal for the cylindrical surface is [tex]\(\mathb{n} = \cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}\).[/tex]  

The differential surface element is:

[tex]\[\mathb{dS} = 2 \, d\theta \, dz \, (\cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}).\][/tex]

Dot Product of Velocity and Normal :

The given velocity field is:

[tex]\[\mathb{v} = z \mathb{i} + y^2 \mathb{j} + x^2 \mathb{k}.\][/tex]

The dot product of the velocity with the surface normal is:

[tex]\[\mathb{v} \cdot \mathb{dS} = 2 \, dz \, d\theta \, (z \cos(\theta) + (4 \sin^2(\theta)) \sin(\theta)).\][/tex]

Integrate to Find the Flux :

The flux through the cylindrical surface is given by:

[tex]\[\int_0^4 \int_0^{2\pi} 2 \, (z \cos(\theta) + 4 \sin^2(\theta)) \, dz \, d\theta,[/tex]  

Separate and compute the integral:

- The integral of [tex]\(z \cos(\theta)\)[/tex]  over[tex]\((0, 2\pi)\)[/tex]  is zero (because [tex]\(\cos(\theta)\)[/tex] has symmetric oscillations).

- The integral of [tex]\(4 \sin^2(\theta)\) over \((0, 2\pi)\) is \(2\pi \cdot 4\),[/tex] since [tex]\(\sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta))\).[/tex]

This results in [tex]\(8\pi\).[/tex]

To compute the flux, multiply by 4 : [tex]\[8\pi \times 4 = 32\pi.\][/tex]

Since the density of the fluid is 810 kg/m³, the outward flux of the fluid through the cylinder, considering the density, is : [tex]\[810 \times 32 \pi.[/tex]

This would be the answer, with the expression [tex]\(810 \times 32 \pi\)[/tex] giving the rate of flow outward through the cylinder.

Complete question : A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 4, 0 ≤ z ≤ 4.

Answer:

2.0

Step-by-step explanation:

x = age

y = average systolic blood pressure

x₁ = 40 years old, y₁ = 110 systolic blood pressure

x₂ = 50 years old, y₂ = 130 systolic blood pressure

[tex]\text{Slope of a line}=m=\frac{y_2-y_1}{x_2-x_1}\\\Rightarrow m= \frac{130-110}{50-40}\\\Rightarrow\frac{20}{10}\\\Rightarrow m= 2.0[/tex]

∴ Slope of the regression line is 2.0

Final answer:

The slope of the regression line relating average systolic blood pressure to age is 2.0 when rounded to one decimal place.

Explanation:

To find the slope of the regression line relating average systolic blood pressure (y) to age (x), we can use the two points provided (x1, y1) = (40, 110) and (x2, y2) = (50, 130). The slope (m) of a line passing through two points is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the given values:

m = (130 - 110) / (50 - 40) = 20 / 10 = 2

Therefore, the slope of the regression line is 2.0 when rounded to one decimal place.

Answer:

The sample standard deviation is 15.3.

Step-by-step explanation:

Given data items,

84, 85, 83, 63, 61, 100, 98,

Number of data items, N = 7,

Let x represents the data item,

Mean of the data points,

[tex]\bar{x}=\frac{84+85+83+63+61+100+98}{7}[/tex]

[tex]=82[/tex]

Hence, sample standard deviation would be,

[tex]\sigma= \sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (x_i-\bar{x})^2}[/tex]

[tex]=\sqrt{\frac{1}{6}\sum_{i=1}^{7} (x_i-82)^2}[/tex]

[tex]=\sqrt{\frac{1}{6}\times 1396}[/tex]

[tex]=\sqrt{232.666666667}[/tex]

[tex]=15.2534149182[/tex]

[tex]\approx 15.3[/tex]

The sample standard deviation of the dataset: 84, 85, 83, 63, 61, 100, 98 is approximately 15.3 when rounded to the nearest tenth.

To find the sample standard deviation of the given set, we first need to calculate the mean of the data set. Then, each number in the data set should be subtracted from the mean, and the results squared. These squared differences should be summed and divided by the number of data values minus one, which gives the variance. Taking the square root of the variance gives the sample standard deviation.

Let's do this step by step for the given dataset: 84, 85, 83, 63, 61, 100, 98.

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Answer:

Given:

(a.) ㏑( x + 5 ) − ㏑( x − 5 )

(b.) ㏑( x + 5 ) + ㏑( x − 5 )

To compute the above expression, we'll use the properties of natural logarithm. i.e.

㏑(a) − ㏑(b) =  ㏑[tex]\frac{a}{b}[/tex]

∴ ㏑( x + 5 ) − ㏑( x − 5 ) = ㏑[tex]\frac{x+5}{x-5}[/tex]

Similarly

㏑(a) + ㏑(b) =  ㏑[tex](a\times b)[/tex]

∴ ㏑( x + 5 ) + ㏑( x − 5 ) = ㏑([tex]x^{2}[/tex]-25)

Answer:

11 dance tickets were sold. ( approx )

Step-by-step explanation:

Let x represents the number of car wash tickets, y represents the number of silly string fight tickets and z represents number of dance tickets,

Total tickets = 380,

⇒ x + y + z = 380 -----(1),

The car wash tickets were $5 each, the silly string fight tickets were $3 each and the dance tickets were $10 each, also total cost is $1460,

⇒ 5x + 3y + 10z = 1460 ----(2)

Also, there are twice as many silly string tickets as car wash tickets,

⇒ y = 2x -----(3)

From equation (1) and (2),

x + 2x + z = 380 ⇒ 3x + z = 380 -----(4)

5x + 3(2x) + 10z = 1460 ⇒ 11x + 10z = 1460 ------(5)

For finding the value of z,

3 × equation (5) - 11 × equation (4),

We get,

30z - 11z = 4380 - 4180

19z = 200

z = 10.5263157895 ≈ 11

Hence, 11 dance tickets were sold.

Answer: Psychology test score is relatively better than economics test score.

Step-by-step explanation:

For this question, we are using Z-score to compare the psychology test score and economics test score.

Z-scores are an approach to compare results from a test with a "normal" population.

Z = [tex]\frac{X - u}{S.D}[/tex]

where,

X - Test score

u - Mean

S.D - Standard deviation

Psychology Test:

Z = [tex]\frac{87 - 92}{5}[/tex]

= [tex]\frac{-5}{5}[/tex]

= -1

Economics Test:

Z = [tex]\frac{52 - 62}{6}[/tex]

= [tex]\frac{-10}{6}[/tex]

= -1.6

Hence, above calculation shows that z- score for pshychology test is greater than the z- score for economics test. so, psychology test score is better than economics test score.

To determine which score is relatively better, we need to use the concept of z-scores, which measure how many standard deviations a particular score is from the mean.

To determine which score is relatively better, we need to use the concept of z-scores, which measure how many standard deviations a particular score is from the mean. The formula for calculating the z-score is:

z = (X - μ) / σ

where X is the score, μ is the mean, and σ is the standard deviation.

For the 87 on the psychology test:

z = (87 - 92) / 5 = -1

For the 52 on the economics test:

z = (52 - 62) / 6 = -1.67

Since a higher z-score indicates a score that is relatively better, we can conclude that the score of 87 on the psychology test is relatively better than the score of 52 on the economics test.

Answer:

Step-by-step explanation:

1. Multiply both sides of the equation by the reciprocal of the coefficient of r.

[tex]V\cdot\dfrac{3}{2\pi h^2}=r\\\\r=\dfrac{3V}{2\pi h^2}[/tex]

__

2. Multiply both sides of the equation by the reciprocal of the coefficient of h.

[tex]V\cdot\dfrac{3}{b^2}=h\\\\h=\dfrac{3V}{b^2}[/tex]

__

3. Solve the circumference formula for r, then substitute the given information.

[tex]C=2\pi r\\\\r=\dfrac{C}{2\pi}\qquad\text{divide by the coefficient of r}\\\\r=\dfrac{50\,\text{cm}}{2\pi}=\dfrac{25}{\pi}\,\text{cm}\approx 7.9577\,\text{cm}[/tex]

__

4. Solve the perimeter formula for width, the substitute the given information and do the arithmetic.

[tex]P=2(L+W)\\\\\dfrac{P}{2}=L+W\qquad\text{divide by 2}\\\\\dfrac{P}{2}-L=W\qquad\text{subtract L}\\\\\dfrac{40\,\text{cm}}{2}-5\,\text{cm}=W=15\,\text{cm}[/tex]

_____

In general, solving for a particular variable involves "undoing" what has been done to the variable, usually in the reverse order. In part 4, the variable W has L added and the sum is multiplied by 2. We "undo" those operations, last operation first, by dividing by 2 and subtracting L.

The properties of equality say you can do what you like to an equation as long as you do the same thing to both sides of the equation. So, when we say "divide by 2", we mean "divide both sides of the equation by 2." Likewise, "subtract L" means "subtract L from both sides of the equation."

Answer:

Proofs are in the explantion.

Step-by-step explanation:

We are given the following:

1) [tex]a \equi b (mod n) \rightarrow a-b=kn[/tex] for integer [tex]k[/tex].

1) [tex] c \equi  d (mod n) \rightarrow c-d=mn[/tex] for integer [tex]m[/tex].

a)

Proof:

We want to show [tex]a+c \equiv b+d (mod n)[/tex].

So we have the two equations:

a-b=kn and c-d=mn and we want to show for some integer r that we have

(a+c)-(b+d)=rn. If we do that we would have shown that [tex]a+c \equiv b+d (mod n)[/tex].

kn+mn   =  (a-b)+(c-d)

(k+m)n   =   a-b+ c-d

(k+m)n   =   (a+c)+(-b-d)

(k+m)n  =    (a+c)-(b+d)

k+m is is just an integer

So we found integer r such that (a+c)-(b+d)=rn.

Therefore, [tex]a+c \equiv b+d (mod n)[/tex].

//

b) Proof:

We want to show [tex]ac \equiv bd (mod n)[/tex].

So we have the two equations:

a-b=kn and c-d=mn and we want to show for some integer r that we have

(ac)-(bd)=tn. If we do that we would have shown that [tex]ac \equiv bd (mod n)[/tex].

If a-b=kn, then a=b+kn.

If c-d=mn, then c=d+mn.

ac-bd  =  (b+kn)(d+mn)-bd

          =    bd+bmn+dkn+kmn^2-bd

          =           bmn+dkn+kmn^2

          =            n(bm+dk+kmn)

So the integer t such that (ac)-(bd)=tn is bm+dk+kmn.  

Therefore, [tex]ac \equiv bd (mod n)[/tex].

//

Answer: 325

Step-by-step explanation:

Combination is a way to calculate the total outcomes of an event where order of the outcomes does not matter where as a Permutation is a way of arranging the elements of a set into a order or a sequence . Here order matters.

If we want to create password with 6 different letters then order matters.

Hence, we use permutations.

The number of passwords created is given by :-

[tex]^{26}P_6=\dfrac{26!}{2!(26-2)!}\\\\=\dfrac{26\times25\times24!}{2\times24!}=325[/tex]

Hence, the number of passwords created = 325

Answer:

[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]

Step-by-step explanation:

The logistic equation is

[tex]N(x)=\frac{c}{1+ae^{-rx}}[/tex]

where:

c/(1+a) is the initial value.

c is the limiting value

r is constant determined by growth rate

So we are given that:

N(0)=500 or that c/(1+a)=500

If your not sure about his initial value of c/(1+a) then replace x with 0 in the function N:

[tex]N(0)=\frac{c}{1+ae^{-r \cdot 0}}[/tex]

Simplify:

[tex]N(0)=\frac{c}{1+ae^{0}}[/tex]

[tex]N(0)=\frac{c}{1+a(1)}[/tex]

[tex]N(0)=\frac{c}{1+a}[/tex]

Anyways we are given:

[tex]\frac{c}{1+a}=500[/tex].

Cross multiplying gives you [tex]c=500(1+a)[/tex].

We are also giving that N(1)=1000 so plug this in:

[tex]N(1)=\frac{c}{1+ae^{-r \cdot 1}}[/tex]

Simplify:

[tex]N(1)=\frac{c}{1+ae^{-r}}[/tex]

So this means

[tex]1000=\frac{c}{1+ae^{-r}}[/tex]

Cross multiplying gives you [tex]c=1000(1+ae^{-r})[/tex]

We are giving that c=50000 so we have these two equations to solve:

[tex]50000=500(1+a)[/tex]

and

[tex]50000=1000(1+ae^{-r})[/tex]

I'm going to solve [tex]50000=500(1+a)[/tex] first because there is only one constant variable here,[tex]a[/tex].

[tex]50000=500(1+a)[/tex]

Divide both sides by 500:

[tex]100=1+a[/tex]

Subtract 1 on both sides:

[tex]99=a[/tex]

Now since we have [tex]a[/tex] we can find [tex]r[/tex] in the second equation:

[tex]50000=1000(1+ae^{-r})[/tex] with [tex]a=99[/tex]

[tex]50000=1000(1+99e^{-r})[/tex]

Divide both sides by 1000

[tex]50=1+99e^{-r}[/tex]

Subtract 1 on both sides:

[tex]49=99e^{-r}[/tex]

Divide both sides by 99:

[tex]\frac{49}{99}=e^{-r}[/tex]

Take natural log of both sides:

[tex]\ln(\frac{49}{99})=-r[/tex]

Multiply both sides by -1:

[tex]-\ln(\frac{49}{99})=r[/tex]

So the function N with all the write values plugged into the constant variables is:

[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]

Final answer:

The question involves applying the logistic growth equation to determine the number of people who will see an advertisement over time, given initial conditions and the carrying capacity. The process includes finding the growth rate from the provided data and using it to solve the logistic growth formula for any time t.

Explanation:

The number of people in a community who are exposed to a particular advertisement is described by the logistic growth equation. Given that initially N(0) = 500, and after one unit of time N(1) = 1000, and the carrying capacity is 50,000, we want to solve for N(t), the number of people who will see the advertisement at any time t.

The logistic growth model can be written as:

N(t) = K / (1 + (K - N_0) / N_0 ×[tex]e^{(-rt)}[/tex]

Where:

N(t) is the number of individuals at time t

K is the carrying capacity of the environment

N_0 is the initial number of individuals

r is the growth rate

e is the base of the natural logarithms

We are given that K = 50,000, N_0 = 500, and N(1) = 1000. From N(1), we can find the growth rate r. Re-arranging the logistic equation and substituting the values for N(1), t = 1, K, and N_0, we get an equation that we can solve for r. Once we have found r, we can substitute all known values back into the logistic equation to solve for N(t) for any given value of t.

To find the solution for this kind of problem it might require numerical methods or algebraic manipulation which is beyond this explanation, but once the value of r is found, the N(t) formula can be applied to predict the number of people who will see the advertisement at any given time.

Answer: There are 52 men we can expect to have been involved in a minor traffic accident.

Step-by-step explanation:

Since we have given that

Percentage of men has been involved in a minor traffic accident = 13%

Number of men surveyed = 400

So, Number of sample men we can expect to have been involved in a minor traffic accident is given by

[tex]\dfrac{13}{100}\times 400\\\\=0.13\times 400\\\\=52[/tex]

Hence, there are 52 men we can expect to have been involved in a minor traffic accident.

Answer:

9/17

Step-by-step explanation:

Chance of being over 40:

[tex] \frac{20 + 30 + 35}{255} = \frac{1}{3} [/tex]

Chance of drinking

root beer:

[tex] \frac{25 + 20 + 30}{255} = \frac{75}{255} [/tex]

Chance of drinking root beer and being over 40

[tex] \frac{1}{3} \times \frac{75}{255} = \frac{25}{255} [/tex]

Chance of drinking root beer OR being over 40

[tex] \frac{1}{3} + \frac{75}{255} - \frac{25}{255} = \frac{135}{255} = \frac{9}{17} [/tex]

20+30+35/255

1/3 chance of being 40+ years old

20+25+30/255

75/255 chance of drinking root beer

75/255 * 1/3

25/255 chance of drinking root beer being 40+ years old

75/255 - 25/255 * 1/3

135/255

9/17 chance of drinking root beer under the age of 40 years old.

Best of Luck!

[tex](x-a)^2+(y-b)^2=r^2[/tex]

center - [tex](a,b)[/tex]

[tex]x^2 + y^2 + 6x -8y +21 = 0 \\x^2+6x+9+y^2-8y+16-4=0\\(x+3)^2+(y-4)^2=4[/tex]

center - [tex](-3,4)[/tex]

[tex]r=2[/tex]

Answer:

The center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]

Step-by-step explanation:

The general equation of a circle has the following formula:

[tex](x-h)^2 + (y-k)^2 = r^2\\[/tex]

Where r is the radius of the circle and (h, k) is the center of the circle

In this case we have the following equation

[tex]x^2 + y^2 + 6x -8y +21 = 0[/tex]

To find the radius and the center of this cicunference we must rewrite the equation in the general form of a circumference completing the Square

[tex](x^2 + 6x)+ (y^2 -8y) +21 = 0\\\\(x^2 + 6x+9)+ (y^2 -8y+16) +21 = 9+16\\\\(x^2 + 6x+9)+ (y^2 -8y+16) = 9+16-21\\\\(x+3)^2+ (y-4)^2 = 4\\\\(x+3)^2+ (y-4)^2 = 2^2[/tex]

Then the center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]

Answer:

The answer is 0 and no.

Step-by-step explanation:

Consider the provided information.

If the event is impossible then the probability of that event will be 0.

Now, consider the statement 'Suppose that a probability is approximated to be zero-based on empirical results.'  

Empirical probability is based on observations and if you do more observation then the probability of an event might be increased.

Thus, this means that the event is not​ impossible.

Answer:

THE ANSWER IS WHAT THE GUY ABOVE ME SAID....0

Step-by-step explanation:

The correct option is B.

Any prism volume is V = BH where B is an area of the base and H is the height of the prism, so find the area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.

How to Find the volume of the prism?

The triangles on the two ends are congruent.

So, to find the prism area, it is enough to take the area of one of the triangles and multiply it by the prism depth, which is 20 units.

Use the Pythagorean theorem to find the height of the triangle.

5^2 - 3^2 = 4^2, so the height is 4.

Multiply half times height times base: 1/2 x 4 x 3 = 6.

Multiply that by the depth, 20.

6x20=120 units^3.

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Answer:

12.1%

Step-by-step explanation:

First calculate the z-score:

z = (x − μ) / σ

z = (55000 − 48000) / 6000

z = 1.17

Look up in a z-score table.

P(z>1.17) = 1 − 0.8790

P(z>1.17) = 0.1210

There's a 12.1% probability that a randomly selected individual with a master's in statistics will have a starting salary of at least $55,000.

The probability that a randomly selected individual with a Master's in statistics will get a starting salary of at least $55,000 is 12.17%.

The problem you're dealing with involves the normal distribution which is a type of continuous probability distribution for a real-valued random variable. Given that we have a mean (μ) of $48,000, and a standard deviation (σ) of $6,000, we are asked to find the probability that a randomly selected individual with a master's in statistics will get a starting salary of at least $55,000.

This involves calculating a 'Z-score', which tells us how many standard deviations an element is from the mean. To find it, you would subtract the mean from the value of interest and divide by the standard deviation.

So, in this case, the Z-score would be: Z = (55000 - 48000) / 6000 = 1.167. Using standard Z-score tables, this corresponds to a probability of 0.8783.

However, as we are interested in the probability of a salary being at least $55,000, we need to subtract this value from 1 to get the final probability. So, the answer is: 1 - 0.8783 = 0.1217 or 12.17%.

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