Calculate the mean, median, and mode for each of the following populations of numbers: (a) 17, 23, 19, 20, 25, 18, 22, 15, 21, 20 N (Population) Mean Median Mode (b) 505, 497, 501, 500, 507, 510, 501 N (Population) Mean Median Mode

Answer:

(x,y,z)=(5,0,3)

[tex]((x,y,z)-(2,3,0))*(-1,1,0)=0[/tex]

Step-by-step explanation:

a)

The problem requires to find the intersection point of the lines, at that point the position 'r' of the lines is the same:

[tex]r_{1} =r_{2} \\(2,3,0)+(3,-3,3)t=(5,0,3)+(-3,3,0)s\\[/tex]

First, built the parametric equation system; this is just a simplification coordinate to coordinate of the vector equation:

[tex]2+3t=5-3s\\3-3t=3s\\3t=3[/tex]

From the last equation,

[tex]t=1[/tex]

And for whatever of the other two,

[tex]s=0[/tex]

You can check that replacing t=1 and s=0 the point gotten is (5,0,3), which is the intersection point (the point that belongs to both lines).

b) The plane is defined by an orthogonal direction. The equation of the plane uses the fact that the dot product between two orthogonal vectors is always zero.

The general equation of a plane is:

[tex]((x,y,z)-(x_{0},y_{0},z_{0}))*(n)=0[/tex]

Where (x,y,z) are the variables that may be part of the plane or not, [tex](x_{0},y_{0},z_{0})[/tex] is a point that belongs to the plane and n is a vector which is orthogonal to the plane.

Due that both lines belong to the plane, the cross product between their direction vectors will give us the orthogonal vector.

[tex]n=(3,-3,3)X(-3,3,0)=(-9,-9,0)[/tex]

We can divide (-9,-9,0) by nine, because we only need the direction and the division does not affect it.

[tex]n=(-1,-1,0)[/tex]

Finally, we know that both lines are inside the plane, so any point that belong to a line, belong to the plane. For this reason, let's select any point, for example: (2,3,0) (It could be another). So, the equation of the plane is:

[tex]((x,y,z)-(2,3,0))*(-1,-1,0)=0[/tex]

The intersection point of the lines can be obtained by equating the parametric forms of the lines and finding the values of parameters. The equation of the plane containing these lines can be obtained using the directional vectors of these lines, which essentially define the plane.

First, we need to find the common point at which the given lines intersect. We can do this by setting r = 2, 3, 0 + t 3, −3, 3 and r = 5, 0, 3 + s −3, 3, 0 to be equal, and finding the values of t and s that make this true. This gives us the (x, y, z) coordinates of the intersection point.

To find the equation of the plane that contains these lines, we know that any point on this plane can be expressed as a linear combination of the directional vectors of these lines, which are (3, -3, 3) and (-3, 3, 0). Therefore, the equation of the plane can be written in the form of Ax + By + Cz = D, where (A, B, C) is a normal vector to the plane, and D is a constant that can be determined by substituting the coordinates of any point on the plane.

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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: 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:

[tex]p=9900\\[/tex] bacterias in the initial two hours

Step-by-step explanation:

the growing rate is given by the ecuation

[tex]p(t)=0.8(t)^{3} +3.5 [/tex] thousand per hour

for t=2 we have

[tex]p(2)=0.8(2)^{3} +3.5 = 9.9[/tex] thousand

[tex]p=9900\\[/tex] bacterias

In two hours we have 9900 bacterias

[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:

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: 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:

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: 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

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:

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:386

Step-by-step explanation:

We have given

Smoothing parameter [tex]\left ( \alpha \right )=0.56[/tex]

Forecasted demand[tex]\left ( F_t\right )=385[/tex]

Actual demand[tex]\left ( D_t\right )=386[/tex]

And Forecast is given by

[tex]F_{t+1}=\alpha D_t+\left ( 1-\alpha \right )F_t[/tex]

[tex]F_{t+1}=0.56\cdot 386+\left ( 1-0.56\right )385=385.56\approx 386[/tex]

[tex]F_{t+2}=0.56\cdot 386+\left ( 1-0.56\right )385.56=385.806\approx 386[/tex]

[tex]F_{t+3}=0.56\cdot 386+\left ( 1-0.56\right )385.806=385.914\approx 386[/tex]

[tex]F_{t+4}=0.56\cdot 386+\left ( 1-0.56\right )385.914=385.962\approx 386[/tex]

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!

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:

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:

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.

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.

The Wronskian determinant is

[tex]\begin{vmatrix}\sin x&x\sin x\\\cos x&x\cos x+\sin x\end{vmatrix}=\sin x(x\cos x+\sin x) - x\sin x\cos x=\sin^2x[/tex]

which is non-zero for all [tex]x\in(0,1)[/tex], so the solutions are linearly independent.

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: 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:

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:

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:

68 chickens and 12 cows.

Step-by-step explanation:

Let x represents the number of chicken and y represents the number of cows in the barnyard,

Given,

Total heads = 80

⇒ x + y = 80 ------(1),

Also, total legs = 184,

Since, a chicken has two legs and cow has 4 legs,

⇒ 2x + 4y = 184 -----(2),

Equation (2) - 2 × equation (1),

We get,

4y - 2y = 184 - 160

2y = 24

y = 12

From equation (1),

x + 12 = 80 ⇒ x = 80 - 12 = 68

Hence, the number of chicken = 68,

And, the number of cows = 12

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:

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:

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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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:

2.3775

Step-by-step explanation:

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:

They have 45 quarters and 16 nickels.

Step-by-step explanation:

Let x be the number of quarters and y be the number of nickels in the jar,

Since, the jar contains 61 coins,

⇒ x + y = 61 ------(1)

Also, 1 quart = $ 0.25 and 1 nickel = $ 0.05,

So, the total cost = ( 0.25x + 0.05y ) dollars,

According to the question,

0.25x + 0.05y = 12.05,

⇒ 25x + 5y = 1205 -----(2),

Equation (2) - 5 × Equation (1),

20x = 1205 - 305

20x = 900

⇒ x = 45,

From equation (1)m,

y = 61 - 45 = 16,

Hence, they have 45 quarters and 16 nickels.

Let's solve this problem step by step.
We have two types of coins: quarters and nickels. Let's use two variables to represent the number of each type of coin in the jar.
Let \( Q \) represent the number of quarters and \( N \) represent the number of nickels. Since we have two unknowns, we'll need two equations to solve for them.
1. The total number of coins is 61:
\[ Q + N = 61 \]   (Equation 1)
2. The total value of the coins is $12.05. Since quarters are worth 25 cents each and nickels are worth 5 cents each, we can convert this total value into cents to avoid dealing with dollars and make the calculation easier.
\[ 12.05 dollars = 1205 cents \]
Now we set up an equation based on the value of the coins:
\[ 25Q + 5N = 1205 \]   (Equation 2)
These are our two equations:
\[ Q + N = 61 \]  
\[ 25Q + 5N = 1205 \]
Let's solve this system of linear equations.
First, we can simplify the second equation by dividing by 5 to make the numbers smaller and easier to work with:
\[ 5Q + N = 241 \]   (Simplified Equation 2)
Now, let's subtract Equation 1 from the Simplified Equation 2 to eliminate \( N \):
\[ (5Q + N) - (Q + N) = 241 - 61 \]
\[ 5Q + N - Q - N = 241 - 61 \]
\[ 4Q = 180 \]
Divide both sides by 4 to solve for \( Q \):
\[ Q = \frac{180}{4} \]
\[ Q = 45 \]
Now we know there are 45 quarters. To find the number of nickels, we plug the value of \( Q \) back into Equation 1:
\[ Q + N = 61 \]
\[ 45 + N = 61 \]
\[ N = 61 - 45 \]
\[ N = 16 \]
Therefore, there are 45 quarters and 16 nickels in the jar.