Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation (in lb/min) for the amount of salt A(t) (in lb) in the tank at time t > 0. (Use A for A(t).)

Answer:

To compute the probabilities, just divide the number of times a number was landed by the total amount of outcomes.

Spinning 2: 30/120 = 0.25  

Spinning 3: 19/120  = 0.16

Spinning 4:  25/120  = 0.21

Spinning 5:  29/120  = 0.24

Spinning 6:  17/120  = 0.14

Stacy would spin a 6 approximately 0.14*50 = 7 times in 50 tries.

Answer:

32/25 walls per gallon.

Step-by-step explanation:

Martin uses 5/8 Of a gallon of paint to cover 4/5 Of a wall

Hence 1 gallon would be 4/5÷5/8=

4/5 × 8/5= 32/25 walls per gallon.

Answer:

The coordinates of point A after both transformations becomes (0, 3)

Step-by-step explanation:

Translation 4 units down is equivalent to -4 units in the y direction, therefore, we have;

A(-3, 4), B(6, 4), and C(2, -3) becomes

A(-3, 4 - 4), B(6, 4 - 4), and C(2, -3 - 4) which is A(-3, 0), B(6, 0), and C(2, -7)

Rule on operation on coordinates due to rotation of a triangle 90° clockwise is as follows;

For vertex coordinate, (x, y), we change it to (y, -x)

Therefore, we have the coordinates of the point A(-3, 0) after a rotation of 90° clockwise becomes A(0, 3).

Answer:

Option D,four is correct

Step-by-step explanation:

The tax withholding from the gross income of $951 is the gross income itself minus the income after tax withholding i.e $189  ($951-$762)

The percentage of the withholding =189/951=20% approximately

Going by the multiple  choices provided,option with 4,189 dollars seems to the correct option as that is the exact of the tax withholding on Robert's gross income and his earnings fall in between $950 and $960

The number of allowances that Robert has claimed is four. Option d is correct.

A withholding tax is a tax that an employer deducts from an employee's paycheck and delivers it straight to the government (federal income tax).

From the given information:

The tax allowance = Gross income - savings amount

The tax allowance = $951 - $762

The tax allowance = $189

From the weekly income between 950 and 960 data, we can see that the number of allowances claimed related to the withholding allowances are:

0  → $259

1   →  $242

2   → $224

3   → $207

4  → $189

5   → $173

6  →  $162

7   → $151

8  →  $140

Therefore, we can conclude that the number of allowances that Robert has claimed is four.

Learn more about tax withholding here:

https://brainly.com/question/25927192

Answer:

  7 and 8

Step-by-step explanation:

  54 is between 7^2 = 49 and 8^2 = 64.

The square roots have the same relationship.

  49 < 54 < 64

  √49 < √54 < √64

  7 < √54 < 8

Answer:

YlThe higher the grader the more hours they can work.

Step-by-step explanation:

We can see that the higher the grader the more hours they can work ; which could mean less academic work if the work defined is that with which to earn money but if the work defined is academic it means more hours of academic work

Answer:

  the sum shown is (20/21)a -(1/7)

Step-by-step explanation:

As written, the sum is ...

  [tex]a-\dfrac{\frac{3}{7}}{3}-\dfrac{a}{21}=\boxed{\dfrac{20}{21}a-\dfrac{1}{7}}[/tex]

__

We wonder if you mean the quotient ...

  ((a-3)/7)/((3-a)/21)

  [tex]\dfrac{\left(\dfrac{a-3}{7}\right)}{\left(\dfrac{3-a}{21}\right)}=\dfrac{a-3}{7}\cdot\dfrac{21}{3-a}=\dfrac{-21(3-a)}{7(3-a)}=\boxed{-3}[/tex]

_____

Comment on the problem presentation

Parentheses are required when plain text is used to represent fractions. The symbols ÷, /, and "over" all mean the same thing: "divided by." The denominator is the next item in the expression. If arithmetic of any kind is involved in the denominator, parentheses are needed. This is the interpretation required by the Order of Operations.

When the expression is typeset, fraction bars and text formatting (superscript) serve to group items that require parentheses in plain text.

__

Please note that some authors make a distinction between the various forms of division symbol. Some use ÷ to mean ...

  (everything to the left)/(everything to the right)

and they reserve / solely for use in fractions. Using this interpretation, your expression would be ...

  (a -(3/7))/(3 -(a/21)) = (21a -9)/(63 -a)

That distinction is not supported by the Order of Operations.

Answer:

V =2744 cm^3

Step-by-step explanation:

The volume of a cube is given by

V = s^3 where s is the side length

V = 14^3

V =2744 cm^3

Answer: 2,744 cm³

Step-by-step explanation: Since the length, width, and height of a cube are all equal, we can find the volume of a cube by multiplying side × side × side.

So we can find the volume of a cube using the formula s³.

Notice that we have a side length of 14cm.

So plugging into the formula, we have (14 cm)³ or

(14 cm)(14 cm)(14 cm) which is 2,744 cm³.

So the volume of the cube is 2,744 cm³.

Answer:

Hey!

Your answer is 150 square meters!

Step-by-step explanation:

13*3=39 (the tilted face)

12*3=36 (the flat face)

5*3=15 (the base)

TO FIND THE TRINGLULAR AREA:

1/2 base x height...

1/2 x 5 x 12 = 30

(WE DOUBLE THIS BECAUSE THERE ARE TWO TRIANGLES)

SO 60...

ADD THESE TOGETHER...

39 + 36 + 15 + 30 + 30...

GIVES US 150 square metres

Answer:

150 square meters

Step-by-step explanation:

Answer:

a) The height decreases at a rate of [tex]\frac{7}{12}[/tex] ft/sec.

b) The area increases at a rate of [tex]\frac{527}{24}[/tex] ft^2/sec

c) The angle is increasing at a rate of [tex]\frac{1}{12}[/tex] rad/sec

Step-by-step explanation:

Attached you will find a sketch of the situation. The ladder forms a triangle of base b and height h with the house. The key to any type of problem is to identify the formula we want to differentiate, by having in mind the rules of differentiation.

a) Using pythagorean theorem, we have that [tex] 25^2 = h^2+b^2[/tex]. From here, we have that

[tex]h^2 = 25^2-b^2[/tex]

if we differentiate with respecto to t (t is time), by implicit differentiation we get

[tex]2h \frac{dh}{dt} = -2b\frac{db}{dt}[/tex]

Then,

[tex]\frac{dh}{dt} = -\frac{b}{h}\frac{db}{dt}[/tex].

We are told that the base is increasing at a rate of 2 ft/s (that is the value of db/dt). Using the pythagorean theorem, when b = 7, then h = 24. So,

[tex]\frac{dh}{dt} = -\frac{2\cdot 7}{24}= \frac{-7}{12}[/tex]

b) The area of the triangle is given by

[tex]A = \frac{1}{2}bh[/tex]

By differentiating with respect to t, using the product formula we get

[tex] \frac{dA}{dt} = \frac{1}{2} (\frac{db}{dt}h+b\frac{dh}{dt})[/tex]

when b=7,  we know that h=24 and dh/dt = -1/12. Then

[tex]\frac{dA}{dt} = \frac{1}{2}(2\cdot 24- 7\frac{7}{12}) = \frac{527}{24}[/tex]

c) Based on the drawing, we have that

[tex]\sin(\theta)= \frac{b}{25}[/tex]

If we differentiate with respect of t, and recalling that the derivative of sine is cosine, we get

[tex] \cos(\theta)\frac{d\theta}{dt}=\frac{1}{25}\frac{db}{dt}[/tex] or, by replacing the value of db/dt

[tex]\frac{d\theta}{dt}=\frac{2}{25\cos(\theta)}[/tex]

when b = 7, we have that h = 24, then [tex]\cos(\theta) = \frac{24}{25}[/tex], then

[tex]\frac{d\theta}{dt} = \frac{2}{25\frac{24}{25}} = \frac{2}{24} = \frac{1}{12}[/tex]

Answer:

T = £85.87

the cost of the petrol the car used in November is £85.87

Step-by-step explanation:

Given;

In November Hillary drove 580 miles in her car;

Distance travelled d = 580 miles

the car travelled 33.5 miles for each gallon of petrol used;

Fuel consumption rate r = 33.5 miles per gallon

Number of gallons N consumed by the car is;

N = distance travelled/fuel consumption rate

N = d/r = 580/33.5 = 17.3134 gallons

Given that;

Petrol cost £1.09 per litre

Cost per litre c = £1.09

1 gallon = 4.55 litres

Converting the amount of fuel used to litres;

N = 17.3134 gallons × 4.55 litres per gallon

N = 78.77612 litres

The total cost T = amount of fuel consumption N × fuel cost per litre c

T = N × c

T = 78.77612 litres × £1.09 per litre

T = £85.87

the cost of the petrol the car used in November is £85.87

Answer:

1.8

Step-by-step explanation:

Answer:

   {-7, -4, -1, 3, 7}

Step-by-step explanation:

The range is the list of y-coordinates of the points:

  range = {-7, -4, -1, 3, 7}

Answer:

x = pi/6

x = 11pi/6  

x = 5pi/6  

x =7pi/6

Step-by-step explanation:

2 sec^2 (x) + tan ^2 (x) -3 =0

We know tan^2(x) = sec^2 (x) -1

2 sec^2 (x) +sec^2(x) -1 -3 =0

Combine like terms

3 sec^2(x) -4 = 0

Add 4 to each side

3 sec^2 (x) = 4

Divide by 3

sec^2 (x) = 4/3

Take the square root of each side

sqrt(sec^2 (x)) = ±sqrt(4/3)

sec(x) = ±sqrt(4)/sqrt(3)

sec(x) = ±2 /sqrt(3)

Take the inverse sec on each side

sec^-1 sec(x) = sec^-1(±2 /sqrt(3))

x = pi/6 + 2 pi n    where n is an integer

x = 11pi/6 + 2 pi n  

x = 5pi/6 + 2 pi n  

x =7pi/6 + 2 pi n  

We only want the solutions between 0 and 2pi

Answer:

1/52

Step-by-step explanation:

Answer:

The Probability that the sample average was 289 yards or less

P(x⁻≤ 289) = P( Z≤ -1.909) =  0.0287

Step-by-step explanation:

step(i):-

Mean of the Population = 292.5 yards

Standard deviation of the Population = 14.2 yards

sample size 'n' =60 drives

               N = 4244 drives

Step(ii):-

Let X⁻ be random sample average

[tex]Z = \frac{x^{-} -mean}{\frac{S.D}{\sqrt{n} } }[/tex]

Let  X⁻ = 289

[tex]Z = \frac{289 -292.5}{\frac{14,2}{\sqrt{60} } }[/tex]

Z  = - 1.909

The Probability that the sample average was 289 yards or less

P(x⁻≤ 289) = P( Z≤ -1.909)

                = 0.5 -A(1.909)

                = 0.5 -0.4713

               = 0.0287

Conclusion:-

The Probability that the sample average was 289 yards or less = 0.0287

Answer:

Step-by-step explanation:

Hello!

The variable of interest is:

X: Impact force needed for Sap Stone tires to blow out. (foot-pounds)

n= 29, S= 1358 foot-pound

a)

Soap Stone claims that "The tires will blow out at an average pressure of μ= 26000 foot-pounds with a standard deviation of σ= 1020 foot-pounds.

According to the consumer's complaint, the variability of the blown out forces is greater than the value determined by the company.

I)

Then the parameter of interest is the population variance (or population standard deviation) and to test the consumer's complaint you have to conduct a Chi-Square test for σ².

σ²= (1020)²= 1040400 foot-pounds²

H₀: σ² ≤ 1040400

H₁: σ² > 1040400

α: 0.01

II)

[tex]X^2= \frac{(n-1)S^2}{Sigma^2} ~~X^2_{n-1}[/tex]

[tex]X^2_{H_0}= \frac{(n-1)S^2}{Sigma^2}= \frac{(29-1)*(1358)^2}{1040400} = 49.63[/tex]

III)

This test is one-tailed to the right and so is the p-value. This distribution has n-1= 29-1= 28 degrees of freedom, so you can calculate the p-value as:

P(X²₂₈≥49.63)= 1 - P(X²₂₈<49.63)= 1 - 0.99289= 0.00711

⇒ The p-value is less than the significance level so the test is significant at 1%. You can conclude that the population variance of the blowout forces is less than 1040400 foot-pounds², at the same level the population standard deviation of the blow out forces is less than 1020 foot-pounds.

b)

99% CI for the variance. Using the X² statistic you can calculate it as:

[tex][\frac{(n-1)S^2}{X^2_{n-1;1-\alpha /2}} ;\frac{(n-1)S^2}{X^2_{n-1;\alpha /2}} ][/tex]

[tex]X^2_{n-1;\alpha /2}= X^2_{28; 0.005}= 13.121[/tex]

[tex]X^2_{n-1;1-\alpha /2}= X^2_{28; 0.995}= 49.588[/tex]

[tex][\frac{28*(1358)^2}{49.588} ;\frac{28*(1358)^2}{13.121} ][/tex]

[1041312.253; 3935415.898] foot-pounds²

I hope this helps!

Answer:

200.96 units

Step-by-step explanation:

Use the formula for the volume of a cone [tex]V=\pi r^{2} \frac{h}{3}[/tex]

Plug in the values ([tex]\pi[/tex]=3.14) and multiply them all out

Answer:

≈ 201

Step-by-step explanation:

V= πr²h/3

V= 3.14*4²*12/3= 200.96 ≈ 201

Answer: The area of circle V is 196π ft² (196Pi feet squared)

Step-by-step explanation:

From the equation for area of a circle,

A = πr²

Where A is the area of the circle

r is the radius of the circle

In Circle V, the radius, r of the circle is 14 feet

That is,

r = 14ft

Hence, Area is

A = π × (14ft)²

A = π × 14ft × 14ft

A = 196π ft²

Hence, the area of circle V is 196π ft² (196Pi feet squared)

Answer:

The answer is D on Edge 2020

Step-by-step explanation:

I did the Quiz

Answer:

Step-by-step explanation:

Surface Area is the combined area of all two-dimensional surfaces of a shape. Just like ordinary area, the units are the squares of the units of length (that is, if a shape's sides are measured in meters, then the shape's area is measured in square meters

Answer:

X=-2or,-3

Step-by-step explanation:

X^2+5x+6=0

or,x^2+(2+3)x+6=0

or,x^2+2x+3x+6=0

or,x(x+2)+3(x+2)=0

or,(x+3) (x+2)=0

Either,

x+3=0 x+2=0

or,x=-3 or,x=-2

Therefore,the value if x is -2 or -3 .

I HOPE IT WILL HELP YOU

Answer:

Check Explanation

Step-by-step explanation:

According to the Central limit theorem, the population mean (μ) is approximately equal to the mean of sampling distribution (μₓ).

And the standard deviation of the sampling distribution (σₓ) is related to the population standard deviation (σ) through

Standard deviation of the sampling distribution = (Population standard deviation)/(√N)

where N = Sample size

σₓ = (σ/√N)

So, population mean (μ) = Mean of sampling distribution (μₓ)

Population Standard deviation = (Standard deviation of the sampling distribution) × √N

= σ × √N

A) The expected value of a given distribution is simply equal to the mean of that distribution.

Hence, the expected value of random variable Y thay varies with different samples is given as

E(Y) = Mean of sampling distribution = μₓ

But μₓ = μ

Hence, E(Y) = μ (Proved)

B) Var (Y) is given as the square of the random distribution's standard deviation.

Var (Y) = (standard deviation of the sampling distribution)²

= (σ/√N)²

= (σ²/N) (Proved)

Hope this Helps!!!

Answer:

Step-by-step explanation:

Using the formula for finding range in projectile, Since range is the distance covered in the horizontal direction;

Range [tex]R = U\sqrt{\frac{H}{g} }[/tex]

U is the velocity of the arrow

H is the maximum height reached = distance below the bullseye reached by the arrow.

R is the horizontal distance covered i.e the distance of the target from the archer.

g is the acceleration due to gravity.

Given R = 60ft, U = 250ft/s, g = 32ft/s H = ?

On substitution,

[tex]60 = 250\sqrt{\frac{H}{32}} \\\frac{60}{250} = \sqrt{\frac{H}{32}}\\\frac{6}{25} = \sqrt{\frac{H}{32}[/tex]

Squaring both sides we have;

[tex](\frac{6}{25} )^{2} = (\sqrt{\frac{H}{32} } )^{2} \\\frac{36}{625} = \frac{H}{32} \\625H = 36*32\\H = \frac{36*32}{625} \\H = 1.84feet[/tex]

The arrow will hit the target 1.84feet below the bullseye.

Answer:

8.7

Step-by-step explanation:

on edge . You're welcome

Answer:

i think it's c.)3.5 i may be wrong

Step-by-step explanation:

Answer:

The differential equation which describes the mixing process is [tex]\frac{dc_{salt,out}}{dt} + \frac{2}{125}\cdot c_{salt,out} = \frac{71}{100000}[/tex].

Step-by-step explanation:

The mixing process within the tank is modelled after the Principle of Mass Conservation, which states that:

[tex]\dot m_{salt,in} - \dot m_{salt,out} = \frac{dm_{tank}}{dt}[/tex]

Physically speaking, mass flow of salt is equal to the product of volume flow of water and salt concentration. Then:

[tex]\dot V_{water, in, 1}\cdot c_{salt, in,1} + \dot V_{water, in, 2} \cdot c_{salt,in, 2} - \dot V_{water, out}\cdot c_{salt, out} = V_{tank}\cdot \frac{dc_{salt,out}}{dt}[/tex]

Given that [tex]\dot V_{water, in, 1} = 9\,\frac{L}{min}[/tex], [tex]\dot V_{water, in, 2} = 7\,\frac{L}{min}[/tex], [tex]c_{salt,in,1} = 0.04\,\frac{kg}{L}[/tex], [tex]c_{salt, in, 2} = 0.05\,\frac{kg}{L}[/tex], [tex]\dot V_{water, out} = 16\,\frac{L}{min}[/tex] and [tex]V_{tank} = 1000\,L[/tex], the differential equation that describes the system is:

[tex]0.71 - 16\cdot c_{salt,out} = 1000\cdot \frac{dc_{salt,out}}{dt}[/tex]

[tex]1000\cdot \frac{dc_{salt, out}}{dt} + 16\cdot c_{salt, out} = 0.71[/tex]

[tex]\frac{dc_{salt,out}}{dt} + \frac{2}{125}\cdot c_{salt,out} = \frac{71}{100000}[/tex]

Answer:

0.30

Step-by-step explanation:

Data provided in the question

Uniform density function for a friend = x minutes late

The Friend is at least 21 minutes late

Based on the above information, the probability that the friend is at least 21 minutes late is

[tex]= \frac{Total\ minutes - minimum\ minutes}{Total\ minutes}[/tex]

[tex]= \frac{30 - 21}{30}[/tex]

= 0.30

Based on the above formula we can easily find out the probability for the friend who is at least 21 minutes late

The probability of the friend to be at least 21 minutes late for the uniform density function shown in the graph is 0.30.

Probability of an event is the ratio of number of favorable outcome to the total number of outcome of that event.

The graph is attached below shows the uniform density function for a friend who is x minutes late.

The friend is at least 21 minutes late. This means that the friend is 21 minutes late or more than it. 21 or more minutes goes from 21 to 30. Thus, the difference is,

[tex]d=30-21\\d=9[/tex]

The density of the graph is 1/30. The probability will be equal to the area under the curve.

In this, the length of the rectangle will be 9 and width will be 1/30 for the probability of at least 21 minutes late. The probability is,

[tex]P=9\times\dfrac{1}{30}\\P=0.30[/tex]

Thus, the probability of the friend to be at least 21 minutes late for the uniform density function shown in the graph is 0.30.

Learn more about the probability here;

https://brainly.com/question/24756209

Answer:

y = - 3x + 9    or     (y - 6) = - 3(x - 1)

Step-by-step explanation:

I'm assuming you are looking for the equation of the line.

First, let's find the slope of the line, which we use equation (y1-y2) / (x1-x2)

x1: 1      x2: 2

y1: 6     y2: 3

(6-3) / (1-2) = 3 / -1 = -3

The slope of the line is -3.

The equation for slope-intercept form is y = mx + b, where m is slope and b is the y - intercept.

y = -3x+b

Now substitute a set of points in (any coordinates work). I'll use (1, 6)

6 = -3(1)+b

6 = - 3 + b

b = 9

So our equation in slope intercept form is y = - 3x + 9

You can also write this in point slope form,   (y-y1) = m (x-x1)     m is still slope

(y - 6) = - 3(x - 1)

Answer:

C

Step-by-step explanation:

[tex]\frac{6}{13}[/tex]÷[tex]\frac{6}{12}[/tex]  can also be [tex]\frac{6}{13}[/tex]×[tex]\frac{12}{3}[/tex] and [tex]\frac{6}{13}[/tex]×[tex]\frac{12}{3}[/tex]=13/12

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

\frac{6}{13}136 ÷\frac{6}{12}126  can also be \frac{6}{13}136 ×\frac{12}{3}312 and \frac{6}{13}136 ×\frac{12}{3}312 =13/12

Answer:

(a) See below

(b) r = 0.9879  

(c) y = -12.629 + 0.0654x

(d) See below

(e) No.

Step-by-step explanation:

(a) Plot the data

I used Excel to plot your data and got the graph in Fig 1 below.

(b) Correlation coefficient

One formula for the correlation coefficient is  

[tex]r = \dfrac{\sum{xy} - \sum{x} \sum{y}}{\sqrt{\left [n\sum{x}^{2}-\left (\sum{x}\right )^{2}\right]\left [n\sum{y}^{2} -\left (\sum{y}\right )^{2}\right]}}[/tex]

The calculation is not difficult, but it is tedious.

(i) Calculate the intermediate numbers

We can display them in a table.

   x            y             xy                      x²             y²    

   36       0.22              7.92               1296           0.05

   67        0.62            42.21              4489           0.40

   93         1.00            93.00           20164           3.46

 433        11.8          5699.4          233289        139.24

 887      29.3         25989.1          786769       858.49

1785      82.0        146370          3186225      6724

2797     163.0         455911         7823209    26569

3675   248.0         911400        13505625   61504        

9965   537.81     1545776.75  25569715   95799.63

(ii) Calculate the correlation coefficient

[tex]r = \dfrac{\sum{xy} - \sum{x} \sum{y}}{\sqrt{\left [n\sum{x}^{2}-\left (\sum{x}\right )^{2}\right]\left [n\sum{y}^{2} -\left (\sum{y}\right )^{2}\right]}}\\\\= \dfrac{9\times 1545776.75 - 9965\times 537.81}{\sqrt{[9\times 25569715 -9965^{2}][9\times 95799.63 - 537.81^{2}]}} \approx \mathbf{0.9879}[/tex]

(c) Regression line

The equation for the regression line is

y = a + bx where

[tex]a = \dfrac{\sum y \sum x^{2} - \sum x \sum xy}{n\sum x^{2}- \left (\sum x\right )^{2}}\\\\= \dfrac{537.81\times 25569715 - 9965 \times 1545776.75}{9\times 25569715 - 9965^{2}} \approx \mathbf{-12.629}\\\\b = \dfrac{n \sum xy - \sum x \sum y}{n\sum x^{2}- \left (\sum x\right )^{2}} - \dfrac{9\times 1545776.75 - 9965 \times 537.81}{9\times 25569715 - 9965^{2}} \approx\mathbf{0.0654}\\\\\\\text{The equation for the regression line is $\large \boxed{\mathbf{y = -12.629 + 0.0654x}}$}[/tex]

(d) Residuals

Insert the values of x into the regression equation to get the estimated values of y.

Then take the difference between the actual and estimated values to get the residuals.

   x            y       Estimated   Residual

    36        0.22        -10                 10

    67        0.62          -8                  9

    93        1.00           -7                  8

   142        1.86           -3                  5

  433       11.8             19               -  7

  887     29.3             45               -16  

 1785     82.0            104              -22

2797    163.0            170               -  7

3675   248.0            228               20

(e) Suitability of regression line

A linear model would have the residuals scattered randomly above and below a horizontal line.

Instead, they appear to lie along a parabola (Fig. 2).

This suggests that linear regression is not a good model for the data.

Answer:

a. The Poisson approximation is good because n is large, p is small, and np < 10.

The parameter of thr Poisson distribution is:

[tex]\lambda =np\approx1.6[/tex]

b. P(r=0)=0.2019

c. P(r>1)=0.4751

d. P(r>2)=0.2167

e. P(r>3)=0.0789

Step-by-step explanation:

a. The Poisson distribution is appropiate to represent binomial events with low probability and many repetitions (small p and large n).

The approximation that the Poisson distribution does to the real model is adequate if the product np is equal or lower than 10.

In this case, n=963 and p=1/596, so we have:

[tex]np=963*(1/596)\approx1.6[/tex]

The Poisson approximation is good because n is large, p is small, and np < 10.

The parameter of thr Poisson distribution is:

[tex]\lambda =np\approx1.6[/tex]

We can calculate the probability for k events as:

[tex]P(r=k)=\dfrac{\lambda^ke^{-\lambda}}{k!}[/tex]

b. P(r=0). We use the formula above with λ=1.6 and r=0.

[tex]P(0)=1.6^{0} \cdot e^{-1.6}/0!=1*0.2019/1=0.2019\\\\[/tex]

c. P(r>1). In this case, is simpler to calculate the complementary probability to P(r<=1), that is the sum of P(r=0) and P(r=1).

[tex]P(r>1)=1-P(r\leq1)=1-[P(r=0)+P(r=1)]\\\\\\P(0)=1.6^{0} \cdot e^{-1.6}/0!=1*0.2019/1=0.2019\\\\P(1)=1.6^{1} \cdot e^{-1.6}/1!=1.6*0.2019/1=0.3230\\\\\\P(r>1)=1-(0.2019+0.3230)=1-0.5249=0.4751[/tex]

d. P(r>2)

[tex]P(r>2)=1-P(r\leq2)=1-[P(r=0)+P(r=1)+P(r=2)]\\\\\\P(0)=1.6^{0} \cdot e^{-1.6}/0!=1*0.2019/1=0.2019\\\\P(1)=1.6^{1} \cdot e^{-1.6}/1!=1.6*0.2019/1=0.3230\\\\P(2)=1.6^{2} \cdot e^{-1.6}/2!=2.56*0.2019/2=0.2584\\\\\\P(r>2)=1-(0.2019+0.3230+0.2584)=1-0.7833=0.2167[/tex]

e. P(r>3)

[tex]P(r>3)=1-P(r\leq2)=1-[P(r=0)+P(r=1)+P(r=2)+P(r=3)]\\\\\\P(0)=1.6^{0} \cdot e^{-1.6}/0!=1*0.2019/1=0.2019\\\\P(1)=1.6^{1} \cdot e^{-1.6}/1!=1.6*0.2019/1=0.3230\\\\P(2)=1.6^{2} \cdot e^{-1.6}/2!=2.56*0.2019/2=0.2584\\\\P(3)=1.6^{3} \cdot e^{-1.6}/3!=4.096*0.2019/6=0.1378\\\\\\P(r>3)=1-(0.2019+0.3230+0.2584+0.1378)=1-0.9211=0.0789[/tex]