Simplify:

[tex]\frac{4-i\sqrt{2}}{4+i\sqrt{2}}[/tex]

Answer:

the answer is 45/100=9/50 according to my solution

The cost of one cup of coffee in terms of x and y is x/12y.

Given in question,

Cost of 12 grams coffee = x dollars

   Cost of 1 gram coffee = x/12 dollars

1 gram coffee make cups = y

                   Cost of 1 cup = (x/12)/y

                                          = x/12y

Hence, the cost of one cup of coffee in terms of x and y is x/12y.

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Answer: 204 miles

Step-by-step explanation:

She received a discount of 75 percent from the sale of the scarf.

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

Given that the original price of the scarf is $16, the money saved during the sale is $12.

Therefore:

The Amount the scarf was bought (Discounted price) = Original Price - Money saved

Discounted price =$16 - $12 = $4

Now the discount is given as;

Discount = (Amount saved / Original price) × 100%

Discount = ($12/ $16) × 100% = 75%

Therefore the discount of 75% was received.

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The equation that describes the proportional relationship between the distance she runs in miles, and her running speed, in miles per hour, when she runs, is:

d = 0.5v.

A proportional relationship is a special case of a linear function, with intercept zero, in which the output variable is given by the multiplication of the input variable and the constant of proportionality, given as follows:

y = kx

The distance as a function of the velocity and of the time in hours, is given as follows:

d = vt.

30 minutes is half an hour, hence the constant of the proportional relationship is given as follows:

v = 0.5.

Then the equation is:

d = 0.5v.

In which:

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The value of  tan(45°+Θ) after using trigonometric ratio is  

1+tanΘ/ 1-tanΘ.

What is trigonometric ratios?

   There are six trigonometric ratios used in trigonometry: sine, cosine, tangent, secant, and cotangent. The abbreviations for these ratios are sin, cos, tan, sec, cosec(or csc), and cot. Look at the below-displayed right-angled triangle. Any two of the three sides of a right-angled triangle can be compared in terms of their relative angles using trigonometric ratios.

Here the given,

=> tan(45°+Θ)

Now using formula ,

=> tan(a+b) = [tex]\frac{tan a+tan b}{1-tan a*tanb}[/tex]

=> tan(45°+Θ)= (tan 45°+tanΘ)/(1-tan45°*tanΘ)

We know that tan 45°=1 then ,

=> tan (45°+Θ)= 1+tanΘ/ 1-tanΘ

Hence the formula for  tan (45°+Θ) is  1+tanΘ/ 1-tanΘ.

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The correct coefficient is 147

In scientific notation, a number is represented by an exponent, or the power of 10, multiplied by the coefficient, which appears before the times sign. For instance, 4.3 is the coefficient and 6 is the exponent in the equation 4.3 x 10^6, which is equivalent to 4,300,000 (four million three hundred thousand).

From the question, we have

147,000 = 147 *10^3

The correct coefficient is 147

Multiplication:

Mathematicians multiply two or more numbers to determine the sum of the parts. It is a fundamental mathematical procedure that is commonly used in real-world situations. Multiplication is used when groups of like sizes need to be combined. Multiplication is a mathematical operation that illustrates the fundamental idea of adding the same number again. The product of two or more numbers is what occurs from multiplying them, while the components are the quantities that are multiplied. Multiplying the numbers makes it simpler to repeatedly add the same quantity.

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Kelly's knitting speed is 2.92 feet per hour

From the question, the given parameters are:

Length of knot = 38 foot

Time to cover this length = 13 hours

The speed of the knitting the scarf is calculated as

Speed = Length of knot /Time to cover this length

Substitute the known values in the above equation

So, we have the following equation

Speed = 38/13

Evaluating the quotient in the above equation, we have

Speed = 2.92

Hence, the speed of knotting the scarf is 2.92 feet per hour

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Correlation coefficient = 0.945 ( approx) and weight of a student increases as the height of the student increases.

What is correlation coefficient?

A statistical indicator of the strength of a linear link between two variables is the correlation coefficient. Its values may be between -1 and 1.

Main Body:

Here, value of x are  =58, 59, 60, 62, 63, 64, 66, 68, 70

Value of y are,  =   122, 128, 126, 133, 145, 136, 144, 150, 151.

∑x = 570, ∑y= 1236, ∑ = 36234, ∑ = 170694

∑xy=78617

Thus the correlation coefficient r =

formula is in the image,

Where n is the number of the term.

here n = 9

thus, correlation coefficient

r= [tex]\frac{9*78617- 570*1236}\sqrt{{9*36234- (570^{2} )}\sqrt{9*170794- (1235)^{2} }[/tex]

Thus, r= 0.94452987512

Which is closed to 0.95 and less than 1.

Therefore, weight of a student increases as the height of the student increases.

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35x + 65y = 815 and x + 7 = y are the system of equations that can be used to find the number of boxes.

What is an algebra?

One of the many different mathematical disciplines is algebra. Algebra is, broadly stated, the study of mathematical symbols and the guidelines for applying those symbols to formulas.

Given,

The weight of each small box = 35 pounds
The weight of each large box = 65pounds
The total weight of the boxes = 815pounds
There were7 times as many small boxes shipped as large boxes
Let 'x' be the number of small boxes and 'y' be the number of large boxes
The weight of 'x' small boxes = 35x pounds
The weight of 'y' large boxes = 65y pounds
The total weight of the two types of  boxes together = 35x+65y

Since the total weight of the boxes is 815 pounds, we have
35x+65y = 815---------------------(1)

Since 7 more small boxes =   large boxes, hence we have,
x + 7 = y ------------------------(2)
Solving both the equations we get,

x = 3.95 or 4
and, y = 10.95 or 11

The number of large boxes shipped = 11

The number of small boxes shipped  = 4

∴ The number of small boxes shipped = 4 and the number of large boxes shipped = 11.

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Answer:

Step-by-step explanation:

138

x > 5 represents a line with an open circle at 5 and going indefinitely toward infinity.

Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.

The symbol a < b indicates that a is smaller than b.

When a > b is used, it indicates that a is bigger than b.

a is less than or equal to b when a notation like a ≤ b.

a is bigger or equal value of an is indicated by the notation a ≥ b.

Given inequality is  - 3(2x - 5) < 5(2 - x).

- 6x + 15 < 10 - 5x.

- x < - 5.

x > 5 (think of moving x to make it positive).

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The area of the floor in the scale drawing is 20 square inches.

Firstly calculating the width of scale drawing. For this we will equate the two fractions.

Length/width (rectangle) = length/width (scale drawing)

Let the width of scale drawing be x

24/30 = 4/x

x = 4×30/24

Performing multiplication on Right Hand Side of the equation

x = 120/24

Performing division on Right Hand Side of the equation

x = 5

So, the area of scale drawing = length × breadth

Area of scale drawing = 4 × 5

Performing multiplication on Right Hand Side of the equation

Area of scale drawing = 20 square inches

Hence, the area is 20 square inches.

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The Bar graph, Line graph and Report displays the counter in decimal format.

Selecting a counter displays information about that counter's collected data as a chart type. The bar graph, line graph and report displays the current value of each performance counter in decimal format.

Line graph: - Line graph, also known as a line chart or line plot, is commonly drawn to show how something changes over a period of time. scientists and engineers use these graphs to understand and derive meaning from large chunks of data.

Bar graph: - This are the pictorial representation of data, in the form of vertical or horizontal rectangular bars, where the  length of bars are proportional to the measure of data. They are mostly used in business and finance, but they can also be found in other contexts. This can be used to show the distribution of different heights of students in a class.

Report: -  A report is a document that presents information in an organized format for a specific audience and purpose. All the skills and the knowledge of the professionals are communicated through reports. Reports help the top line in decision making. A rule and balanced report also helps in problem solving.

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The equation 3n = 6 has a solution n = 2.

An equation is an expression that shows the relationship between two or more numbers and variables.

We have to find that which equation below is n = 2 a solution.

Then substitute n = 2;

3n = 6

3(2) = 6

This equation has a solution n = 2.

2. n-1=2

Then substitute n = 2;

2 - 1 = 1

The solution of the equation is not n = 2

3. 9+n=12

Then substitute n = 2;

9 + 2 = 11

The solution of the equation is not n = 2

4. 3+n=8

Then substitute n = 2;

3 + 2 = 5

The solution of the equation is not n = 2

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The line of equation that passes through (5,2) and (6,4) is 4x+y=22.

An equation is a combination of different variables, in which two mathematical expressions are equal to each other.

The given points are (5,2) and (6,4)

The equation that passes from two points (x₁,y₁) and (x₂,y₂) is,

y-y₁=m.(x-x₁)

Where m=(y₂-y₁)/(x₂-x₁)

Then linear equation,

y-y₁=(y₂-y₁)/(x₂-x₁)×(x-x₁)

Substitute the values here,

y-2=(6-2)/(4-5) ×(x-5)

y-2=4/(-1)(x-5)

y-2=20-4x

4x+y=22

So the required linear Equation is 4x+y=22.

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For the given sequence of transformation mapping of pentagon ABCDE onto pentagon A"B"C"D"E" is given by Line reflection followed by a translation.

As given in the question,

Given sequence of transformation mapping of pentagon ABCDE onto pentagon A"B"C"D"E" is given by :

a. dilation followed by a rotation :

Here dilation means enlarge or reduced the original shape which is not true for the given pentagon ABCDE.

b. translation followed by a rotation :

Translation means moves around a fixed point by certain degrees that is horizontal or vertical shift.

Not true.

c. Line reflection followed by a translation:

In this case preimage is flipped across the line or get reflected.

True.

d. Line reflection followed by a line reflection:

Here  fixed line is or a line is used as mirror . You get a image of the original.

Not true.

Therefore, for the given sequence of transformation mapping of pentagon ABCDE onto pentagon A"B"C"D"E" is given by Line reflection followed by a translation.

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Answer:

30

Step-by-step explanation:

in index form the nth if a root number is X and that is X to the power so the seventh root of 30 is 30 to the power of 1/7

The probability of more than 3 of the flips turn up tails is 0.0528 i.e 5.28%.

The binomial distribution in contains n independent Bernoulli trials. A Bernoulli trial is a trial that happens to succeed (probability p) or fail (probability 1- p = q )

Suppose we have a variable X for a Bernoulli distribution with parameters n and p, which is written as The probability of x successes in n trials is P(X = x)

= ⁿCₓ pˣ(q)ⁿ ⁻ ˣ

Toss a fair coin 12 times.

Next is the probability that 3 or more flips will be a number.

The probability of success is (P) = 1/2 = 0.5

total number of tosses (n) = 12

we have x = 3

P(X = 3) = ¹²C₃(0.5)³x (1 - 0.5)¹²⁻³

= 220 (0.5)³ (1 - 0.5)⁹

P(X = 3) = 0.0528

P(X = 3) = 5.28%

Hence, the probability that more than 3 of flips turn up tails is 0.0528.

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From the given condition of number of tote bags ordered and there costs represents linear equation then cost of ordering 110 tote bags is equal to $915.

As given in the question,

Let us consider y be the cost of ordering a bag.

And x represents the number of bags to be ordered as per the cost.

It is given that Cost of ordering bags and number of tote bags represents the linear equation,

Standard form of Equation is :

y = mx + c

For order of x = 75 tote bags cost is y = $635.

635 = 75m + c   __(1)

For order of x = 125 tote bags cost is y = $1035.

1035 = 125m + c ____(2)

Subtract (1) from (2) we get,

1035 = 125m + c

635 = 75m  + c

400 = 50m

⇒m = 8

Substitute value of m in (1) we get,

635 = 75(8) + c

⇒c = 35

Equation is:

y = 8x + 35

When number of ordering bags x = 110 then,

y = 8(110) + 35

⇒ y = $915

Therefore, from the given condition of number of tote bags ordered and there costs represents linear equation then cost of ordering 110 tote bags is equal to $915.

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Answer: 8 months

Step-by-step explanation:

Inequality can be used to represent the situation is s + l <= 2 and 20s + 15l >= 185.

Given:

Edith tutors after school for which she gets paid at a rate of $20 an hour. she has also accepted a job as a library assistant that pays $15 an hour. she will work both jobs, but she is able to work no more than a total of ll hours a week, due to school commitments.

Edith wants to earn at least $185 a week working a combination of both jobs.

Let s and l be the hours of school and library.

s + l <= 2

20s + 15l >= 185.

Therefore Inequality can be used to represent the situation is s + l <= 2 and 20s + 15l >= 185.

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Answer:

- [tex]\sqrt{5}[/tex]

Step-by-step explanation:

using the rule of radicals

[tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] ⇔ [tex]\sqrt{ab}[/tex]

simplifying radicals

[tex]\sqrt{20}[/tex]

= [tex]\sqrt{4(5)}[/tex]

= [tex]\sqrt{4}[/tex] × [tex]\sqrt{5}[/tex]

= 2[tex]\sqrt{5}[/tex]

and

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

= [tex]\sqrt{9(5)}[/tex]

= [tex]\sqrt{9}[/tex] × [tex]\sqrt{5}[/tex]

= 3[tex]\sqrt{5}[/tex]

Then

4[tex]\sqrt{20}[/tex] - 3[tex]\sqrt{45}[/tex]

= 4(2[tex]\sqrt{5}[/tex] ) - 3(3[tex]\sqrt{5}[/tex] )

= 8[tex]\sqrt{5}[/tex] - 9[tex]\sqrt{5}[/tex]

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

Answer:

below

Step-by-step explanation:

The angle dimensions given for each triangle sum to 180 °

so

(x+10)  +  y+15   + 70 = 180     Simplifies to

x +y = 85

and

x + 20    + 50 + 60 = 180 °    simplifies to

x = 50°     use this value in the underlined equation to find y = 35°

Differentiation of   [tex]\frac{2x-3}{2x+1}[/tex]  is [tex]\frac{8}{4x^{2}+4x+1}[/tex] .

Differentiation formula for  [tex]\frac{d}{dx}(\frac{u}{v}) = \frac{v\frac{d}{dx} - v\frac{d}{dx}}{v^{2} }[/tex]

Here, u = 2x - 3

          v = 2x + 1

[tex]\frac{d}{dx}u[/tex] = 2

[tex]\frac{d}{dx}v[/tex] = 2

[tex]v^{2} = (2x+1)^{2}[/tex]

    [tex]= 4x^{2}+4x+1[/tex]

Putting the values,

[tex]\frac{d}{dx}(\frac{2x-3}{2x+1}) = \frac{(2x+1)(2)-(2x-3)(2)}{4x^{2}+4x+1}[/tex]

              [tex]=\frac{4x+2-(4x-6)}{4x^{2}+4x+1}[/tex]

              [tex]=\frac{4x+2-4x+6}{4x^{2}+4x+1}[/tex]

              [tex]=\frac{8}{4x^{2}+4x+1}[/tex]

Hence, Differentiation of   [tex]\frac{2x-3}{2x+1}[/tex]  is [tex]\frac{8}{4x^{2}+4x+1}[/tex] .

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The possible days Wendy goes to the library are Monday and Thursday.

The number of books that Wendy reads is 12 books.

From the information, when she goes to the library she always reads 3 books and she also goes to the bookstore to read books twice a week.

In this case, the total books read in two weeks will be:

= (2 × 3) × Number of weeks

= (2 × 3) × 2

= 6 × 2

= 12

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The solutions are;

The critical value with a level of significance ‘of 0.10’ and degrees of freedom ‘of 3’ is 1.533.

The critical value with a level of significance ‘of 0.01’ and degrees of freedom ‘of 3’ is 3.747.

The critical value with a level of significance ‘of 0.05’ and degrees of freedom ‘of 3’ is 2.132.

Given;

Let's say you randomly choose n = 4 measurements from a normal distribution. If you were constructing the following confidence intervals, compare the standard normal z values with the appropriate t values.

(a) 80% confidence interval:

The z-value is obtained as shown below:

The level of significance is;

Locate the probability in the z table. Then, the corresponding row value is 1.2 and the column value is 08.

The critical value with a level of significance is 1.28.

α = 1 - 0.80 = 0.20

α/2 = 0.10

The t-value is obtained as shown below;

From the information, the sample size is 4.

df = n - 1 = 4 - 1

              = 3

The critical value with a level of significance ‘of 0.10’ and degrees of freedom ‘of 3’ is 1.533.

(b) 98% confidence interval:

The z-value is obtained as shown below:

The level of significance is;

Locate the probability in the z table. Then, the corresponding row value is 2.3 and the column value is 03.

The critical value with a level of significance is 2.33.

α = 1 - 0.98 = 0.02

α/2 = 0.01

The t-value is obtained as shown below;

From the information, the sample size is 4.

df = n - 1 = 4 - 1

              = 3

The critical value with level of significance ‘0.01’ and degrees of freedom ‘3’ is 3.747

(C) 90% confidence interval:

The z-value is obtained as shown below:

The level of significance is;

Locate the probability in the z table. Then, the corresponding row value is 1.6 and the column value is 04.

The critical value with a level of significance is 1.64.

α = 1 - 0.90 = 0.10

α/2 = 0.05

The t-value is obtained as shown below;

From the information, the sample size is 4.

df = n - 1 = 4 - 1

              = 3

The critical value with a level of significance ‘of 0.05’ and degrees of freedom ‘of 3’ is 2.132.

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Answer:

The numbers are:

4 and 11

Step-by-step explanation:

a + b = 15      Eq. 1

3a - 1 = b        Eq. 2

Replacing Eq. 2 in Eq, 1

a + (3a - 1) = 15

a + 3a = 15 + 1

4a = 16

a = 16/4

a = 4

From Eq. 1:

a + b = 15

4 + b = 15

b = 15 - 4

b = 11

Check:

From Eq. 2:

3a - 1 = b

3*4 - 1 = 11

12 - 1 = 11