7, On the occasion of birthday, Pemba distributed (p² + 29p-96) bottles of sanitizer equally among (p + 32) people of his village. (a) Find the number of bottles of sanitizer received by each people. (b) If x = 8, find the actual number of bottles of sanitizer, number of people, and share of each people​

Step-by-step explanation:

Might be a little easier to visualize if yo re-arrange it into y = mx + b form:

4x+ 5y = 20

5y = -4x + 20

y = - 4/5 x + 4         y-axis intercept at y = 4

                                x axis intercept is found by:

                                     0 = -4/5 x + 4

                                             - 4 = -4/5 x

                                                 x = 5

So===> plot the two intercept points ( 0,4)  and  ( 5,0)  and connect the dots

Segment C has a length of √10, which is the closest to √10 compared to the other segments.

Segment is a customer data platform (CDP) that enables companies to collect, store, and analyze customer data from multiple sources. It helps companies build customer profiles and create personalized experiences for their customers. Segment allows businesses to track website visits, user actions, and other events in real-time, as well as to create custom events and store customer data in a secure and unified data warehouse. With Segment, companies can create powerful customer segmentation, which allows them to target customers with personalized messages and offers. Segment also integrates with various marketing, analytics, and CRM tools to provide a complete picture of customer behavior. It enables companies to build cohesive customer journeys, run campaigns, and optimize their customer experience.

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Complete Question.

Answer: There is a 5.32544378% Chance of a face card being pulled twice

Step-by-step explanation: If there is 52 cards in a deck, and 12 of them are face cards, there is roughly a 23% (23.0769%) chance of pulling one in the first draw. Multiply .230769 x .230769 and you get .0532544378 which equals 5.32544378%

Here the values -0.94 and 2.12 falls between the points -2.58 and 2.58. The area between is the acceptance region. So we cannot reject the null hypothesis.

The given is an example for two tailed test. A two tailed test is used to determine whether the value is greater than or less than the mean value of the population. It represents the area under both tails or sides on a normal distribution curve.

Here the value of the test statistic lies between -2.58 and 2.58. So the values less than -2.58 and greater than 2.58 fall in the rejection region, where the null hypothesis can be rejected.

a) -0.94 falls between -2.58 and 2.58. So it is in the acceptance region. So null hypothesis is accepted.

b) 2.12 lies between -2.58 and 2.58. It is also in acceptance region. So null hypothesis is accepted.

So in both cases null hypothesis cannot be rejected.

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The complete question is :

f the cutoffs for a z test are -2.58 and 2.58, determine whether you would reject or fail to reject the null hypothesis in each of the following cases and explain why:

a. z = −0.94

b. z = 2.12

Answer:16

Step-by-step explanation:

average is similar to mean

you get the Total value and divide it with the number of values added

a) The expression for the stable fixed point is P* = (1 + sqrt(1 - 4h)) / 2

b) There exists a fixed point for all values of h less than or equal to 1/4.

The fixed points of the model are the values of P at which dP/dt = 0. Therefore, we need to solve the equation

P(1-P) - h = 0

Expanding the left-hand side, we get

P - P^2 - h = 0

Rearranging, we get a quadratic equation

P^2 - P + h = 0

Using the quadratic formula, the two solutions for P are

P = (1 ± sqrt(1 - 4h)) / 2

a) The larger root is the stable fixed point, as it corresponds to a minimum of the fish population growth function. Therefore, the expression for the stable fixed point is

P* = (1 + sqrt(1 - 4h)) / 2

b) For the model to have a fixed point, the quadratic equation must have real roots. This occurs when the discriminant (the expression inside the square root) is non-negative

1 - 4h ≥ 0

Solving for h, we get

h ≤ 1/4

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The given question is incomplete, the complete question is:

This problem explores some questions regarding the fishery model

dP/dt =P(1−P)−h

If you have not yet run the Jupyter notebook please do so now. Find analytical expressions for the two fixed points of the model, in terms of h

a) Give an expression for the stable fixed point. You may assume that the larger fixed point is the stable one. b) For what values of h does there exist a fixed point?

Where F_pivot is the magnitude of the force exerted on the pivot point.

What are perpendicular lines?

Perpendicular lines are lines that intersect at a right angle (90 degrees). In other words, if you draw a line perpendicular to another line, the two lines will form four right angles at the point where they intersect.

(a) The torque applied by the adult rider (on the left) can be calculated as the product of the force applied by the rider and the perpendicular distance from the pivot to the force. Let F_a be the force applied by the adult rider and let r_a be the perpendicular distance from the pivot to the force. Then, the torque applied by the adult rider is:

τ_a = F_a * r_a

The perpendicular distance r_a can be calculated using the Pythagorean theorem:

r_a = sqrt((L/2 - d)^2 + Xoffset^2)

Therefore, the torque applied by the adult rider is:

τ_a = F_a * sqrt((L/2 - d)^2 + Xoffset^2)

(b) Similarly, the torque applied by the child rider (on the right) can be calculated as:

τ_c = F_c * sqrt((L/2 + d)^2 + Xoffset^2)

where F_c is the force applied by the child rider and r_c is the perpendicular distance from the pivot to the force, which can be calculated using the Pythagorean theorem.

(c) The torque applied by the board can be calculated as the sum of the torques applied by the two riders:

τ_b = τ_a + τ_c

Substituting the expressions for τ_a and τ_c, we get:

τ_b = F_a * sqrt((L/2 - d)^2 + Xoffset^2) + F_c * sqrt((L/2 + d)^2 + Xoffset^2)

(d) To determine the distance d in terms of given quantities and variables, we can use the condition for static equilibrium, which requires that the sum of the torques about the pivot point is zero:

τ_a + τ_c = 0

Substituting the expressions for τ_a and τ_c and simplifying, we get:

F_a * (L/2 - d) = F_c * (L/2 + d)

Solving for d, we get:

d = (F_a/F_c - 1) * L/4

Substituting F_a = Mg and F_c = mg, where M is the mass of the adult rider, m is the mass of the child rider, and g is the acceleration due to gravity, we get:

d = (M/m - 1) * L/4

(e) The magnitude of the force exerted on the pivot point by the see-saw while in use can be calculated using the condition for static equilibrium, which requires that the sum of the forces in the vertical direction is zero:

F_a + F_c = Mg + mg

Substituting F_a = Mg and F_c = mg, we get:

F_pivot = Mg + mg

Therefore, where F_pivot is the magnitude of the force exerted on the pivot point.

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(1) 4y-7z is a binomial.

(2) 8-xy² is a binomial.

(3) ab-a-b can be written as ab - (a + b) which is a binomial.

(4) z²-3z+8 is a trinomial.

In algebra, monomials, binomials, and trinomials are expressions that contain one, two, and three terms, respectively.

A monomial is an algebraic expression with only one term. A monomial can be a number, a variable, or a product of numbers and variables.

A binomial is an algebraic expression with two terms that are connected by a plus or minus sign. For example, 2x + 3y and 4a - 5b are both binomials.

A trinomial is an algebraic expression with three terms that are connected by plus or minus signs.

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Classify into monomials, binomials and trinomials.

(1) 4y-7z

(1) 8-xy²

(v) ab-a-b

(ix) z2-3z+8

Answer:

19

Step-by-step explanation:

We are given the following two functions of s

[tex]p(s) = s^3 + 10s\\f(s) = 6s - 3\\\\\text{To find p(2) substitute 2 for s in p(s)}\\p(2) = (2)^3 + 10(2) = 8 + 20 = 28\\\\[/tex]

[tex]\text{To find f(2) substitute 2 for s in f(s)}\\f(2) = 6(2) - 3= 12 - 3= 9\\[/tex]

[tex]p(2) - f(2) = 28 - 9 = 19[/tex]

The solution of the graphs are as follows

first graph

second graph

The graphs consist of two sets of equations plotted, each has shade peculiar to the equation.

The solution of the graph consist of the ordered pair that fall within the parts covered by the two shades

For the first graph by the left, the solutions are

For the second graph by the left, the solutions are

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The value of the expression 2.1 × 1.6 = 3.36.

Decimals are a collection of numbers falling between integers on a number line. They are only an additional mathematical representation of fractions. Decimals allow us to express quantifiable quantities like length, weight, distance, money, etc. with more accuracy. Integers, also known as whole numbers, are represented to the left of the decimal point, while decimal fractions are shown to the right of the decimal point.

Given that the expression is: 2.1 × 1.6.

2.1 × 1.6 can be written as:

2.1 × 1.6 = 21/10 × 16/10

Multiply the numerator and denominator:

21/10 × 16/10 = 336/100

Covert the fraction into decimal:

336/100 = 3.36

Hence, the value of the expression 2.1 × 1.6 = 3.36.

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The Amount from contributions = R * n

The future value refers to the value of an asset or investment at a specified time in the future, based on a specific interest rate or rate of return.

Without knowing the specific values of R, interest rate, and number of years, we cannot calculate the amounts from contributions and interest. However, we can provide the general formula for calculating the future value of an ordinary annuity:

FV = R * [(1 + i)ⁿ - 1] / i

where FV is the future value of the annuity, R is the periodic payment, i is the interest rate per period, and n is the number of periods.

To calculate the amount from contributions, we can multiply the periodic payment R by the number of periods n.

Amount from contributions = R * n

To calculate the amount from interest, we can subtract the amount from contributions from the future value of the annuity.

Amount from interest = FV - R * n

Once the specific values for R, interest rate, and number of years are provided, we can use these formulas to calculate the amounts from contributions and interest.

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Therefore, the coefficient of variation for the given data is approximately 24.71%.

A box plot, also known as a box-and-whisker plot, is a graphical representation of a data set that shows the distribution of the data using quartiles. It is a standardized way of displaying the distribution of data based on the five-number summary: minimum, first quartile, median, third quartile, and maximum. The box represents the middle 50% of the data, with the median marked by a line inside the box. The whiskers extend from the box to the minimum and maximum values, or to a certain range if there are outliers. Box plots are useful for comparing the distributions of different data sets and identifying potential outliers.

Here,

1. Stem-and-Leaf Plot:

A stem-and-leaf plot is a way to display data that separates the tens digit of each number from the ones digit. Here is the stem-and-leaf plot for the given data:

3 | 9 9

4 | 2 2 6 6

5 | 1 4 4 6

6 | 1 4 5 5 5 5 5 5

7 | 0 2 3 4

8 | 1 2 2 3 5

9 | 3 4

In this plot, the stem represents the tens digit and the leaves represent the ones digit.

2. Coefficient of Variation:

The coefficient of variation is a measure of the relative variability of a data set. It is calculated as the ratio of the standard deviation to the mean, expressed as a percentage. Here is how to calculate the coefficient of variation for the given data:

Calculate the mean of the data:

 Mean = (83+64+51+46+82+91+73+82+65+61+74+64+75+81+94+65+42+81+56+61+72+65+54+39+70+93+42+46+54+72)/20 = 68.25

Calculate the standard deviation of the data:

 Standard deviation = sqrt((1/20) * ((83-68.25)^2 + (64-68.25)^2 + ... + (72-68.25)^2))

Standard deviation ≈ 16.88

Calculate the coefficient of variation:

Coefficient of variation = (Standard deviation / Mean) * 100

Coefficient of variation ≈ 24.71%

3. Box Plot:

A box plot is a way to visualize the distribution of data. It displays the minimum value, first quartile (Q1), median, third quartile (Q3), and maximum value of the data. Here is the box plot for the given data:

  |              +----------+

94 |                        |

  |              +----------+

93 |              |

  |        +-----+-----+

82 |        |           |

  |        |           |

81 |        |           |

  |  +-----+           |

80 |  |                 |

  |  |                 |

79 |  |                 |

  |  |                 |

78 |  |                 |

  |  |                 |

77 |  |                 |

  |  |                 |

76 |  |                 |

  |  |                 |

75 |  |                 |

  |  |                 |

74 |  |                 |

  |  |                 |

73 |  |                 |

  |  |                 |

72 |  +-----------------+

  |                    

  +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

     1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20

In this plot, the horizontal line inside the box represents the median, the bottom and top edges of the box represent the first and third quartiles (Q1 and Q3), respectively, and the vertical lines extending from the box represent the minimum and maximum values, excluding outliers.

4. To find the 85th percentile, we need to arrange the data in order from smallest to largest:

39, 42, 42, 46, 46, 51, 54, 56, 61, 61, 64, 64, 65, 65, 65, 70, 72, 72, 73, 74, 75, 81, 81, 82, 82, 83, 91, 93, 94

There are a total of 20 scores, so the 85th percentile would be the score at the 0.85(20) = 17th position:

85th percentile = 72

To find the 2nd decile, we first need to calculate the number of scores in each decile. Since there are 20 scores, each decile would have 2 scores. The 2nd decile would be the score at the 0.2(20) = 4th position:

2nd decile = 46

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Complete question:

Please, I need help with this, it's about statistics. Thank you very much. The grades of 20 students who took an admission exam to a certain faculty, using a scale of 0 to 100, were: 83 64 51 46 82 91 73 82 65 61 74 64 75 81 94 65 42 81 56 61 72 65 54 39 70 93 42 46 54 72

• Make: stem-and-leaf plot

• Calculate: coefficient of variation

• Create a box plot

• 85th percentile, 2nd decile.

Answer:

50

Step-by-step explanation:

A Venn diagram is very helpful for this picture and I've included one in the attached.

If we look at the numbers we're given, we see that the numbers do not add up to 500 as 350 + 300 + 200 = 850.

However, we can work through the numbers to find the exact values and eventually the number of people that liked neither rock nor country.

Since 200 people like both rock and country, these people are part of the 350 people that like rock.

We can find the number of people who like rock only by subtracting 200 from 350:

350 - 200 = 150 (Rock only)

Using the same logic from above, we know that the 200 people who like both rock and country are a part of the 300 people who like country.

We can find the number of people who like country only by subtracting 200 from 300:

300 - 200 = 100 (Country only)

Currently, we have 450/500 as 150 + 200 + 100 = 450.

Now, we can find the number of people who like neither rock nor country by subtracting 450 from 500:

500 - 450 = 50 (Neither rock nor country)

We can check that the numbers we found equal 500:

Rock only + Both rock and country + Country only + Neither rock nor country = Total amount of music lovers surveyed

150 + 200 + 100 + 50 = 500

500 = 500

(**In the attached Venn diagram, M stands for the total set of music lovers, R stands for rock only, B stands for both, C stands for country only, and N stands for neither)

Step-by-step explanation:

4 i guess... sry i m not good at maths

The probability that X = 3 is 1/12.

To compute the probability that X = 3, we need to consider all possible ways of drawing three chips and count the number of ways in which we obtain three blue chips.

The total number of ways of drawing three chips from the urn without replacement is:

10C3 = (10!)/(3!7!) = 120

This is because we need to choose 3 chips out of the 10 in the urn, and the order in which we draw them does not matter.

Now, we need to count the number of ways in which we can obtain three blue chips. Since there are 5 blue chips in the urn, the number of ways of choosing 3 blue chips out of 5 is:

5C3 = (5!)/(3!2!) = 10

Therefore, the probability of obtaining three blue chips is:

P(X = 3) = 10/120 = 1/12

Hence, the probability that X = 3 is 1/12.

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The probability that X which denotes the total number of blue chips obtained when 3 consecutive chips are drawn from the urn without replacement is 1/12.

To calculate the probability that X = 3, the first step is to consider all the possible ways in which three chips can be drawn and count the number of ways in which we obtain three blue chips.

The total number of ways of drawing three chips from the urn without replacement is:

¹⁰C₃ = (10!)/(3!)(7!) = 120

This is because we need to choose 3 chips out of the 10 in the urn, and the order in which we draw them does not matter. Now, we need to count the number of ways in which we can obtain three blue chips. Since there are 5 blue chips in the urn, the number of ways of choosing 3 blue chips out of 5 is:

⁵C₃ = (5!)/(3!)(2!) = 10

Therefore, the probability of obtaining three blue chips is:

P(X = 3) = 10/120 = 1/12

Hence, the probability that X = 3 is 1/12.

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Answer:

9 sides

Step-by-step explanation:

180 - 140 = 40

360 ÷ 40 = 9

The values of the function from the graph are h(7) = 10, h(0) = 9, h(t) = 8, t = 5 and  h(t) = 0, t = 4

Given that the graph is the graph of height as a function of time

To calculate the values of h(t) from the graph of g(x), we need to follow these steps

Using the above, we have the following

h(7) = 10

h(0) = 9

h(t) = 8, t = 5

h(t) = 0, t = 4

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Answer: Zemāk

Step-by-step explanation:

1)

a. Le prix du manteau après la réduction de 15% est:

78€ - (15/100)*78€ = 66,30€

Le prix du manteau après la réduction est de 66,30€.

b. Soit x le prix initial du manteau.

Le prix du manteau après la réduction de 15% est:

x - (15/100)*x = 55,25€

Simplifions cette équation:

0,85x = 55,25€

x = 65€

Le prix initial du manteau était de 65€.

2)

Pour les lunettes, le prix initial est de 45€ et la réduction appliquée est de 20%:

45€ - (20/100)*45€ = 36€

Pour la montre, le prix initial est de 35,55€ et la réduction appliquée est également de 20%:

35,55€ - (20/100)*35,55€ = 28,44€

On constate que le pourcentage de réduction est le même pour les deux articles, donc Manu a tort.

Answer:

shelf area = 19018.33 square inches

Step-by-step explanation:

shelf area calculation:

Ignacio makes a display shelf from 4 wooden boards. All angles formed by the boards are right angles. Ignacio plans to stain all faces of the shelf, except the back face, which will be against the wall. What is the total area Ignacio will stain? Show your work. 512 7 in. 1 2 32 in. 30 in. 72 17 56 9 in. 4 in. 512 24 A 329-30.7 72

To find the total area that Ignacio will stain, we first need to determine the area of each face that will be stained.

Let's label the boards as follows:

Board 1: 512 in x 7 in

Board 2: 12 in x 32 in

Board 3: 30 in x 72 in

Board 4: 17 56/9 in x 4 in

For each board, we need to find the total area of all the faces that will be stained.

Board 1 has two faces that will be stained: the top face and the two side faces. The area of the top face is 512 in x 7 in = 3584 in^2. The area of each side face is 512 in x 12 in = 6144 in^2. So the total area of all three faces is 3584 in^2 + 2 x 6144 in^2 = 15872 in^2.

Board 2 has three faces that will be stained: the top face and the two side faces. The area of the top face is 12 in x 32 in = 384 in^2. The area of each side face is 12 in x 4 in = 48 in^2. So the total area of all three faces is 384 in^2 + 2 x 48 in^2 = 480 in^2.

Board 3 has three faces that will be stained: the top face and the two side faces. The area of the top face is 30 in x 72 in = 2160 in^2. The area of each side face is 72 in x 4 in = 288 in^2. So the total area of all three faces is 2160 in^2 + 2 x 288 in^2 = 2736 in^2.

Board 4 has two faces that will be stained: the top face and the side face. The area of the top face is 17 56/9 in x 4 in = 71 2/3 in^2. The area of the side face is 17 56/9 in x 9 in = 158 2/3 in^2. So the total area of both faces is 71 2/3 in^2 + 158 2/3 in^2 = 230 1/3 in^2.

To find the total area that Ignacio will stain, we just need to add up the areas of all the faces that will be stained:

15872 in^2 + 480 in^2 + 2736 in^2 + 230 1/3 in^2 = 19018 1/3 in^2

Therefore, the total area that Ignacio will stain is approximately 19018.33 square inches.

original question :

Ignacio makes a display shelf from 4 wooden boards. All angles formed by the boards are right angles. Ignacio plans to stain all faces of the shelf, except the back face, which will be against the wall. What is the total area Ignacio will stain? Show your work. 512 7 in. 1 2 32 in. 30 in. 72 17 56 9 in. 4 in. 512 24 A 329-30.7 72

chatgpt

The sample mean X is an unbiased estimator for θ.

To find a suitable statistic as an unbiased estimator for θ, we need to find a function of sample Y, YS, ..., Yn whose expected value is equal to θ.

X = (Y + YS + ... + Yn) / n

To show that X is unbiased, we need to calculate its expected value and show that is equal to θ:

E[X] = E[(Y + YS + ... + Yn) / n]

= (1/n) E[Y + YS + ... + Yn]

= (1/n) [E[Y] + E[YS] + ... + E[Yn]]

= (1/n) [nθ] (by the given density function)

= θ

Therefore, sample mean X is an unbiased estimator for θ.

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Answer:

m = 6

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points on the table (4,6) (5,12)

We see the y increase by 6 and the x increase by 1, so the slope is

m = 6

So, the slope is 6

To solve the equation -12x + 24 = 0, we want to get x by itself on one side of the equation.

First, we can subtract 24 from both sides:

- 12x + 24 - 24 = 0 - 24

This simplifies to:

- 12x = -24

Next, we can divide both sides by -12:

- 12x / -12 = -24 / -12

This simplifies to:

x = 2

Therefore, the solution to the equation -12x + 24 = 0 is x = 2.

Answer: 51.25 = 4.75c + 3.25p

Step-by-step explanation:

1. Since she spent $51.25, we can start our equation with this: 51.25=

2. Since she bought candies and pretzels, we can make 2 new variables, c for candies, and p for pretzels.

3. Since she spent $4.75 per candy, we can add this in to our equation:

51.25 = 4.75c +

4. We can do the same for the pretzels, which she spent $3.25 per piece. Adding this into our equation will leave us with: 51.25 = 4.75c + 3.25p.

5. Now we have to find out what c and p are, given the info that she bought 13 altogether.

6. If we c=6 and p=7, (because they add up to 13) we will get: 51.25!

7. Now we know what c and p are.

8. The answers would be 51.25 = 4.75c + 3.25p, or 51.25=28.5+22.75.

Answer:

The farmer uses 221 pipes in total: 124 plastic pipes and 97 metal pipes.

Answer:

To find out how many pipes the farmer used in total, we simply add the number of plastic pipes to the number of metal pipes:

Total pipes = Plastic pipes + Metal pipes

Total pipes = 124 + 97

Total pipes = 221

Therefore, the farmer used a total of 221 pipes for the irrigation system.

The temperature changed by 41°F during the overnight hours, decreasing from 28°F to -13°F.

The difference in temperature from 28°F to -13°F can be shown in one of two ways:

To calculate the change in temperature, the first exponent equation subtracts the initial temperature from the end temperature. The change in this instance is negative and points to a drop in temperature.

The absolute value function is employed in the second statement to guarantee a positive result regardless of whether the temperature rose or fell. The result is the same as the first expression in this instance, but with a positive sign.

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Answer:

$4.50

Step-by-step explanation:

There are 12 inches in a foot. 18 inches is 1.5 feet.

1 foot of wire is $3.00 and a half of foot should be $1.50.

1.5 feet of wire should cost $4.50

1.5 ft × $3/ft

= $4.50

The two cars are approaching each other at a rate of 36 km/hr at the given instant.

We can solve this problem by using the Pythagorean theorem and differentiating with respect to time. Let's call the distance of the first car from the intersection "x" and the distance of the second car from the intersection "y". We want to find the rate at which the two cars are approaching each other, which we'll call "r".

At any moment, the distance between the two cars is the hypotenuse of a right triangle with legs x and y, so we can use the Pythagorean theorem

r^2 = x^2 + y^2

To find the rates of change of x and y, we differentiate both sides of this equation with respect to time

2r(dr/dt) = 2x(dx/dt) + 2y(dy/dt)

Simplifying and plugging in the given values

dr/dt = (x(dx/dt) + y(dy/dt)) / r

dr/dt = (0.2 x 90 + 0.15 x (-60)) / sqrt((0.2)^2 + (0.15)^2)

dr/dt = (18 - 9) / sqrt(0.04 + 0.0225)

dr/dt = 9 / sqrt(0.0625)

dr/dt ≈ 36 km/hr

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The probability that at least 3 minutes will elapse before the next telephone order is 0.797.

The probability that less than 15 minutes will elapse between orders is 0.677.

The probability that between 15 and 30 minutes will elapse between orders is 0.2275

To solve the following problem, we need to use the Poisson distribution, which is a probability distribution that describes the number of events that occur in a fixed interval of time or space, given the average rate of occurrence of those events.

The Poisson distribution has the following formula:

               [tex]P(X = k) = (\lambda\times ex^{-\lambda}) / k![/tex]    

Where:

P(X = k) is the probability that there are exactly k events in the interval

λ is the average rate of occurrence of events in the interval

e is the mathematical constant e (approximately 2.71828)

k! is the factorial of k (i.e., k * (k-1) * (k-2) * ... * 2 * 1)

Here we have

Between 11 pm and midnight on Thursday night Mystery pizza gets an average of 4.2 telephone orders per hour

A. The probability that at least 3 minutes will elapse before the next telephone order, using the complement rule:

=> P(at least 3 minutes) = 1 - P(less than 3 minutes)

Assume that the time between telephone orders follows an exponential distribution with a mean of 1/4.2 = 0.2381 hours (or 14.28 minutes).

Therefore, the Poisson distribution is λ = 1/0.2381 = 4.2/1.0 = 4.2.

Using the exponential distribution, we can find the probability of less than 3 minutes elapsing between orders as follows:

P(less than 3 minutes) = [tex]1 - e ^{(-\lambda \times t) }[/tex]

Where t = 3/60 = 0.05 hours

P(less than 3 minutes) = [tex]1 - e^{(-4.2\times 0.05) } = 0.203[/tex]

Therefore,

P(at least 3 minutes) = 1 - 0.203 = 0.797

The probability that at least 3 minutes will elapse before the next telephone order is 0.797.

B. To find the probability that less than 15 minutes will elapse between orders, we can use the same exponential distribution as before and set t = 15/60 = 0.25 hours:

P(less than 15 minutes) = [tex]1 - e ^{(-\lambda \times t) }[/tex]

P(less than 15 minutes) = [tex]1 - e^{(-4.2 \times 0.25)} = 0.677[/tex]

Hence, The probability that less than 15 minutes will elapse between orders is 0.677.

C. To find the probability that between 15 and 30 minutes will elapse between orders, we can subtract the probabilities found in less than 15 minutes and less than 30 minutes.

P(15 to 30 minutes) = P(less than 15 minutes) - P(less than 30 minutes) -

P(15 to 30 minutes) = [tex]e^{ (-\lambda0.5)} - e^{ (-\lambda 0.25)}[/tex]  

= 0.3499 - 0.1224  = 0.2275

Therefore,

The probability that at least 3 minutes will elapse before the next telephone order is 0.797.

The probability that less than 15 minutes will elapse between orders is 0.677.

The probability that between 15 and 30 minutes will elapse between orders is 0.2275

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The equation for a cosecant function with vertical asymptotes found at x equals pi over 2 plus pi over 2 times n, where n is an integer, is [tex]f(x) = csc(x - \pi/2)[/tex] .

What is the cosecant function ?

The cosecant function is a trigonometric function that is defined as the reciprocal of the sine function. It is denoted as csc(x) and is defined for all values of x except where sin(x) is equal to zero. The graph of the cosecant function shows a series of vertical lines where the function is undefined, called vertical asymptotes. The value of the cosecant function oscillates between positive and negative infinity as it approaches these asymptotes. The cosecant function is used in trigonometry and calculus to model periodic phenomena such as sound and light waves.

Determining the equation for a cosecant function with vertical asymptotes :

The cosecant function has vertical asymptotes at the zeros of the sine function, which are given by

[tex]x = \pi/2 + n\times\pi[/tex], where n is an integer.

To shift the graph of the cosecant function horizontally by [tex]\pi/2[/tex] units to the right, we subtract [tex]\pi/2[/tex] from the input variable x, so the equation becomes [tex]f(x) = csc(x - \pi/2)[/tex].
[tex]f(x) = csc(x - \pi/2)[/tex] is the equation for a cosecant function with vertical asymptotes found at [tex]x = \pi/2 + n\pi[/tex], where n is an integer.

[tex]g(x) = 4csc(2x)[/tex] is the equation for a cosecant function with period pi, amplitude 4, and vertical asymptotes found at [tex]x = \pi/2 + n\pi[/tex], where n is an integer.

[tex]h(x) = 4csc(3x)[/tex] is the equation for a cosecant function with period [tex]2\pi/3[/tex], amplitude 4, and vertical asymptotes found at [tex]x = \pi/6 + n\pi,[/tex] where n is an integer.

[tex]j(x) = 2csc(x/2)[/tex] is the equation for a cosecant function with period 4pi, amplitude 2, and vertical asymptotes found at [tex]x = 2n\pi[/tex], where n is an integer.

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