what percentage of the area under the normal curve lies (a) to the left of m? (b) between m s and m 1 s? (c) between m 3s and m 1 3s

Step-by-step explanation:

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

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.

The closed formula for this particular sequence is an = n² + 2.

Take note that the odd numbers 3, 5, 7, 9, and 11 are separate consecutive terms. This shows that the first n odd numbers can be added to the initial term, az, to get the nth term. Hence, the following is how we may represent the nth term a = az + 1 + 3 + 5 + ... + (2n-3) (2n-3). We may utilize the formula for the sum of an arithmetic series to make the sum of odd integers simpler that is 1 + 3 + 5 + ... + (2n-3) = n².

As a result, we get a = az + n^2 - 1. In conclusion, the equation for the series (an)n21, where a1 = az and an is the result of adding the first n odd numbers to az, is as a = az + n^2 - 1. We have the following for the given series where a1 = az = 3.

So, the closed formula for this particular sequence is an = n² + 2.

To learn more about arithmetic sequences, refer to:

Your question is incomplete. The complete question is:

Find the closed formula for the sequence (an)n21. Assume the first term given is az. an = 3, 6, 11, 18, 27... Hint: Think about the perfect squares.

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

(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

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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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:

9 cm

Step-by-step explanation:

By the Triangle Inequality, any two sides of a triangle must be greater than the remaining side.

In order to minimize the perimeter, we will assume that 4 cm is the longest side.

Thus, the two remaining sides must be greater than 4.

Since we are given that the sum of the two remaining sides is a whole number, the smallest whole number value greater than 4 is 5.

Hence, the smallest perimeter possible 9 cm.

Answer:

Total cost for groceries = ($3.75, $1.09,

$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. add them all). = $ 31.3

the amount she paid= $ 35

balance =$ 3.7

therefore she have enough money

Answer:

9 sides

Step-by-step explanation:

180 - 140 = 40

360 ÷ 40 = 9

Answer:

[tex]\sqrt {45}[/tex]

Step-by-step explanation:

Distance between two points, (x1, y1) and (x2, y2)  in a 2D cartesian coordinate is given by
[tex]d = \sqrt {(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

here the points are (6, - 2) and (3, 4)

We get

[tex]d = \sqrt {(3 - 6)^2 + (4 - (-2))^2}\\\\d = \sqrt {(-3)^2 + (6)^2}\\\\d = \sqrt {{9} + {36}}\\\\d = \sqrt {45}[/tex]

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 calculation of the interest received by Investor A and Investor B is as follows:

Compound interest is based on the system of charging interest on accumulated interest and principal for each period.

Simple interest is charged on only the principal for each period.

N (# of periods) = 3 years

I/Y (Interest per year) = 8%

PV (Present Value) = $4,500

PMT (Periodic Payment) = $0

Results:

Future Value (FV) = $5,668.70

Total Interest = $1,168.70

N (# of periods) = 3 years

r = 1 paisa per rupee per month

= 1÷100*100 = 1%

Annually rate =1%*12 = 12%

PV (Present Value) = $4,500

Results:

Total Interest = $1,620 (R4,500 x 12% x 3)

Future Value (FV) = $6,120.

Thus, while Investor A receives $1,168.70 at the end of 3 years, Investor B receives $1,620.

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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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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.

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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Answer: It's a 80% increase

Step-by-step explanation:

Answer:

 [tex]30\leq x[/tex]  AND [tex]x \leq 106[/tex]

Notice the valid answer is the one with the AND since need to be both at the same time.

Step-by-step explanation:

Is the one is market, you add 6 to each side and you obtain that answer

[tex]24 \leq x-6\leq 100[/tex]

[tex]24 +6\leq x\leq 100+6[/tex]

[tex]30\leq x\leq 106[/tex]

The answer of the given question based on the limits the answers are as follows, (a)  lim f(x) = 1 , (b)  lim f(x) = 3 , (c)  lim f(x) = 3.

A graph is  visual representation of data that shows the relationship between two or more variables. Graphs can be used to display  wide variety of information, including numerical data, functions, and networks. The most common types of graphs like line graphs, bar graphs, scatter plots, and pie charts.

Graphs are widely used in many fields, like science, economics, engineering, and social sciences, to help people understand and analyze complex data. They are  powerful tool for visualizing trends, patterns, and relationships, and are often used to communicate findings to wider audience.

a) The limit of f(x) as x approaches 2 from the left:

We can see from the graph that as x approaches 2 from the left, f(x) approaches 1. Therefore, we can write:

lim f(x) = 1

x→2-

b) The limit of f(x) as x approaches 2 from the right:

Similarly, as x approaches 2 from the right, f(x) approaches 3. Therefore:

lim f(x) = 3

x→2+

c) The limit of f(x) as x approaches 2:

Since the limit from the left and the limit from the right exist and are equal, we can say that the limit of f(x) as x approaches 2 exists and equals the common value of the left and right limits. Therefore:

lim f(x) = 3

x→2

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Answer:

1/3

Step-by-step explanation:

using rise over run fron the two dots, we can find 2/6, which simplifies down to 1/3

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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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

10 is the value of k of  the slope of the line .

What are slopes called?

Slope, usually referred to as rise over run, is a line's perceived steepness. By dividing the difference between the y-values at two places by the difference between the x-values, we can determine slope.

                                   You may determine a line's slope by looking at how steep it is or how much y grows as x grows. slope categories. When lines are inclined from left to right, they are said to have a positive slope, a negative slope, or a zero slope (when lines are horizontal).

the points  (2,4) and (5,k)

formula from slope of two points

        slope = y₂ - y₁/x₂ - x₁

substitute the values in formula

 slope = 2

 slope =  k - 4/5- 2

   2 =k - 4/3

   6 = k - 4

    k = 6 + 4  

      k = 10

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a) Number of one-to-one functions are equal to the zero, because n< m.

b) Number of one-to-one functions are equal to the ⁵P₅ = 120.

c) Number of one-to-one functions are equal to the ⁶P₅ = 720.

c) Number of one-to-one functions are equal to the ⁷P₅ = 2250.

One to one function is a special form of function that defined from one set to another and maps every element of the range to exactly one element of its domain unique output. As we know a set A has m elements and set B has n elements, then

Now, we have a domain set with five elements, m = 5

a) Here, another set (co-domain) has 4 elements, n = 4. So, Number of one-to-one functions = 0 , n<m.

b) number of elements in another set,n= 5

So, Number of one-to-one functions = ⁵P₅ = 5!/(5 - 5 )! ( permutation formula)

= 5!/0! = 120

c) Number of elements in another set, n= 6

So, Number of one-to-one functions= ⁶P₅

= 6!/(6 - 5)!

= 6!/1! = 720

d) Number of elements in another set, n

= 7

So, Number of one-to-one functions

= ⁷P₅ = 7!(7 - 5)!

= 7!/2! = 2250

Hence, required value is 2250.

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The answer of the given question based on statistics to find the  most appropriate size of the bin from the data the answer is , the most appropriate bin size for this data set would be A. 69.

Statistics is  the practice of collecting, analyzing, and interpreting the data. It involves  use of mathematical tools and techniques to gather insights and knowledge from numerical and categorical information. Statistics is essential in many fields, like  business, medicine, social sciences, and engineering, as it enables researchers to draw conclusions from data and make informed decisions based on evidence.

It includes topics like probability, hypothesis testing, regression analysis, and data visualization. The application of statistical methods can help identify patterns, relationships, and trends in data, allowing researchers to make predictions and solve problems.

To determine the appropriate bin size, we need to consider the range of values in the data. The range is the difference between the largest and smallest values, which in this case is 192 - 123 = 69.

To get the bin size, we divide the range by the number of bins. So the bin size would be 69/4 = 17.25. However, since we can't have a fraction of a unit for bin size, we should round up to the nearest whole number. Therefore, the most appropriate bin size for this data set would be A. 69.

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The answer of the given question based on statistics to find the  most appropriate size of the bin from the data the answer is , the most appropriate bin size for this data set would be A. 69.

Statistics is  the practice of collecting, analyzing, and interpreting the data. It involves  use of mathematical tools and techniques to gather insights and knowledge from numerical and categorical information. Statistics is essential in many fields, like  business, medicine, social sciences, and engineering, as it enables researchers to draw conclusions from data and make informed decisions based on evidence.

It includes topics like probability, hypothesis testing, regression analysis, and data visualization. The application of statistical methods can help identify patterns, relationships, and trends in data, allowing researchers to make predictions and solve problems.

To determine the appropriate bin size, we need to consider the range of values in the data. The range is the difference between the largest and smallest values, which in this case is 192 - 123 = 69.

To get the bin size, we divide the range by the number of bins. So the bin size would be 69/4 = 17.25. However, since we can't have a fraction of a unit for bin size, we should round up to the nearest whole number. Therefore, the most appropriate bin size for this data set would be A. 69.

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The complete question is as fpllows:

123, 185, 143, 137, 192, 185, 129, 143, 154, 165, 143, 138, 187, 176

A bin size of is most appropriate for the data shown above.

A. 69

B. 2

C. 10

D. 1​

it is false. 0 is not a natural number. it is a whole number

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

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Complete Question.

The required answers are 1) [tex]$$A = x^2 + 28x + 192$$[/tex] 2) 300000 3) [tex]$$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$[/tex].

area of the land purchased is given as 480m², and the dimensions of the land are (x+12)mx(x+16)m. Therefore, the area of the land can be expressed as:

[tex]$$A = (x+12)(x+16)$$[/tex]

Expanding this expression, we get:

[tex]$$A = x^2 + 28x + 192$$[/tex]

Hence, the area of the land purchased is given by the polynomial expression [tex]$x^2 + 28x + 192$[/tex].

The total budget for the expenses is Rs. 5,00,000. If Mr. Chand's daughter is ready to share 3/5 of the expenses, then the fraction of the expenses she will pay is:

[tex]$\frac{3}{5}=\frac{x}{500000}$$[/tex]

Simplifying this expression, we get:

[tex]$x = \frac{3}{5}\times 500000 = 300000$$[/tex]

Therefore, Mr. Chand's daughter will pay Rs. 3,00,000 towards the expenses.

We can solve the polynomial [tex]$x^2 + 28x + 192$[/tex] into different factors by using the quadratic formula:

[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$[/tex]

Here, the coefficients of the polynomial are:

[tex]$$a = 1, \quad b = 28, \quad c = 192$$[/tex]

Substituting these values in the quadratic formula, we get:

[tex]$x = \frac{-28 \pm \sqrt{28^2 - 4\times 1 \times 192}}{2\times 1}$$[/tex]

Simplifying this expression, we get:

[tex]$$x = -14 \pm 2\sqrt{19}$$[/tex]

Therefore, the polynomial [tex]$x^2 + 28x + 192$[/tex] can be factored as:

[tex]$$x^2 + 28x + 192 = (x - (-14 + 2\sqrt{19}))(x - (-14 - 2\sqrt{19}))$$[/tex]

or

[tex]$$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$[/tex]

So, we have factored the polynomial into two factors.

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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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