100 points for this one

[tex] \Large{\boxed{\sf \dfrac{1 + 2i}{3 - 4i} + \dfrac{2 - i}{5i } = - \dfrac{2}{5}}} [/tex]

[tex] \\ [/tex]

Given sum:

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i}[/tex]

[tex] \\ [/tex]

We can simplify the sum only if the denominators of the two fractions are the same. Since they are different, we have to multiply the numerator and the denominator of each fraction by the denominator of the other one.

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i} = \dfrac{5i(1 + 2i)}{5i(3 - 4i)} + \dfrac{(3 - 4i)(2 - i)}{(3 - 4i)5i} \\ \\ \\ \sf \: = \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } [/tex]

[tex] \\ [/tex]

Replace i² with -1:

[tex] \sf \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } \: \\ \\ \\ \\ \sf \: = \dfrac{5i + 10( - 1)}{15i - 20( - 1)} + \dfrac{ 6 - 11i + 4( - 1)}{15i - 20( - 1)} \\ \\ \\ \\ \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} [/tex]

[tex] \\ [/tex]

Simplify the expression:

[tex] \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} \\ \\ \\ \\ \sf \: = \dfrac{5i - 10 + 2 - 11i}{ 20 + 15i} = \sf \dfrac{ - 8 - 6i}{20 + 15i}[/tex]

[tex] \\ [/tex]

To write our solution in the x + iy form, also known as the algebraic form, we have to understand what the conjugate of a complex number is.

[tex] \textsf{Let z be our complex number, and} \: \overline{\sf z} \: \textsf{its conjugate.} [/tex]

[tex] \\ [/tex]

The conjugate of z, [tex] \overline{ \sf z}, [/tex] is the complex number formed of the same real part as z but of the opposite imaginary part.

Since x is the real part of z, and y is its imaginary part, this can be expressed as:

[tex] \sf If \: z = x + iy \:, then \: \overline{ \sf z} = x - iy [/tex]

[tex] \\ [/tex]

Now, we have to multiple both the denominator and the numerator of our fraction by the conjugate of its denominator:

[tex]\sf \dfrac{ - 8 - 6i}{20 + 15i} = \dfrac{( - 8 - 6i)( \overbrace{20 - 15i}^{ \overline{z}}) }{ (20 + 15i)( \underbrace{20 - 15i}_{ \overline{z}}) } \\ \\ \\ \sf = \dfrac{ - 160 + 120i - 120i + 90 {i}^{2} }{400 - 300i + 300i - 225 {i}^{2} } \\ \\ \\ \sf = \dfrac{ - 160 + 90 {i}^{2} }{400 - 225 {i}^{2} }[/tex]

[tex] \\ [/tex]

One more time, substitute -1 for i²:

[tex] \sf \: \dfrac{ - 160+ 90 {i}^{2} }{400 - 225 {i}^{2} } \: = \dfrac{ - 160 + 90( - 1)}{400 - 225( - 1)} \\ \\ \\ \sf = \boxed{\sf - \dfrac{ 250}{625}} [/tex]

[tex] \\ [/tex]

Finally, let's simplify our result:

[tex] \sf - \dfrac{250}{625} = - \dfrac{2 \times 125}{5 \times 125} = \boxed{ \boxed{ \sf - \dfrac{2}{5}}}[/tex]

[tex] \\ \\ [/tex]

▪️Learn more about the algebraic form of a complex number and the conjugate of a complex number here:

↣https://brainly.com/question/25633236

The distribution of the sample mean for all possible simple random samples of size 15 from the population of net worth in the US will be approximately normal, with a mean of μ and a standard deviation of σ/√15.

Courtney's estimate of the net worth of all people in the US is strongly skewed to the right. This means that the majority of the population has a net worth that is lower than the average net worth. When Courtney selects an SRS of size n = 15 from this population and calculates the sample mean, the shape of the distribution of the sample mean for all possible simple random samples of size 15 from this population will be approximately normal.

The Central Limit Theorem states that if a population has a mean of μ and a standard deviation of σ, then the sampling distribution of the sample mean will be approximately normal, no matter what the shape of the population distribution is, as long as the sample size is large enough. Since Courtney's sample size of 15 is large enough, the sampling distribution of the sample mean will be approximately normal.

The sample mean will have a mean of μ, which is the population mean, and a standard deviation of σ/√n, where n is the sample size. Since Courtney's sample size is 15, the standard deviation of the sample mean will be σ/√15. This means that the sample mean will be normally distributed with a mean of μ and a standard deviation of σ/√15.

Learn more about Central Limit Theorem at : https://brainly.com/question/18403552

#SPJ4

The first, second, third, and fourth consecutive even integers must be greater than 10, 12, 14, and 16 respectively.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

Let the four consecutive even integers be n, (n + 2), (n + 4) and (n + 6).

Such that twice the sum of the second and third exceeds 3 times the first by 32.

2{(n + 2) + (n + 4)} > 3(n) + 32.

2n + 4 + 2n + 8 > 3n + 32.

4n - 3n > 32 - 12.

n > 10.

So, the consecutive even numbers must all be greater than 10.

learn more about numerical expressions here :

https://brainly.com/question/29199574

#SPJ1

Answer:

2

Step-by-step explanation:

Answer:

a)1/216

b)1/36

Step-by-step explanation:

p(2r)and p(3b) and p(4g)

1/6×1/6×1/6=1/216

p(4r) and p(even b) and p(no less than 3)

1/6×3/6×2/6=1/36

Answer:

Step-by-step explanation:

To find the linearization of the function f(x) = sqrt(x) at x = a, we can use the linearization formula:

f(x) ≈ f(a) + f'(a) (x - a)

To use this formula, we first need to find the value of f(a) and f'(a). At x = a = 4, the value of f(x) is f(4) = sqrt(4) = 2. The derivative of f(x) is f'(x) = 1/(2 * sqrt(x)), so the value of f'(a) is f'(4) = 1/(2 * sqrt(4)) = 1/4.

Substituting these values into the linearization formula, we get:

f(x) ≈ 2 + 1/4 (x - 4)

This is the linearization of f(x) = sqrt(x) at x = 4. It is an approximation of the function f(x) that is valid for values of x that are close to 4.

Step-by-step explanation:

since it is geometric sequence we will use the formula

[tex]tn = {ar}^{n - 1} [/tex]

a = 6

r = 4

The first term

T1(a) = 6

The second term

[tex]t2 = {a \times r}^{2 - 1} [/tex]

[tex]t2 = {6 \times 4}^{1 } = 24[/tex]

The Third Term

[tex]t3 = {a \times r}^{3 - 1} = {a \times r}^{2} [/tex]

[tex]t3 = {6 \times 4}^{2} = 6 \times 16 = 96[/tex]

The fourth term

[tex]t4 = {a \times r}^{4 - 1} = {a \times r}^{3} [/tex]

[tex]t4 = {6 \times 4}^{3} = 6 \times 64 = 384[/tex]

The fifth term

[tex]t5 = {a \times r}^{5 - 1} = {a \times r}^{4} [/tex]

[tex]t5 = {6 \times 4}^{4} = 6 \times 256 = 1,536[/tex]

i hope these helped

Answer:

6, 24, 96, 384, 1536, ...

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the given values of a and r into the formula to create an equation for the nth term:

[tex]a_n=6(4)^{n-1}[/tex]

To find the first 5 terms of the geometric sequence, substitute n = 1 through 5 into the equation.

[tex]\begin{aligned}\implies a_1&=6(4)^{1-1}\\&=6(4)^{0}\\&=6(1)\\&=6\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_2&=6(4)^{2-1}\\&=6(4)^{1}\\&=6(4)\\&=24\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_3&=6(4)^{3-1}\\&=6(4)^{2}\\&=6(16)\\&=96\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_4&=6(4)^{4-1}\\&=6(4)^{3}\\&=6(64)\\&=384\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_5&=6(4)^{5-1}\\&=6(4)^{4}\\&=6(256)\\&=1536\end{aligned}[/tex]

Therefore, the first 5 terms of the given geometric sequence are:

Answer:

  £46

Step-by-step explanation:

You want to know the cost of luggage in the hold when 2 in the hand and 1 in the hold is £86, and 1 in the hand and 1 in the hold is £66.

Let 'a' and 'b' represent the costs of a piece of hand luggage and one in the hold, respectively. Then the given relations are ...

  2a +b = 86

  a +b = 66

Subtracting the first equation from twice the second gives ...

  2(a +b) -(2a +b) = 2(66) -(86)

  b = 46 . . . . . . . simplify

It costs £46 to put an item of luggage in the hold.

Answer:

Step-by-step explanation: So First you  would need to multiply 2x3 because if you need to triple two cups you have to multiply so the answer is 6 cups

Answer:

Below

Step-by-step explanation:

w = 7 - sqrt (g)    ?? ( syntax is unclear in your post)

sqrt(g) = 7-w

g = (7-w)^2

Answer:

128

Step-by-step explanation:

DO the "E"

  125 + 28/7   -1    do the  division next

  125  +   4     - 1     now do the + or -  or both

     128

Answer:

5x + 100 = 7x + 30

100 - 30 = 7x - 5x

70 = 2x

x = 35

3y + 16 = 5y

16 = 5y - 3y

16 = 2y

y = 8

Step-by-step explanation:

________________________________

=> 5x + 100 = 7x + 30

=> 100 – 30 = 7x – 5x

=> 70 = 2x

=> x = 70/2

=> x = 35

The value of x is 35.

________________________________

=> 16 = 5y – 3y

=> 16 = 2y

=> y = 16/2

=> y = 8

The value of y is 8.

_______________________________

To expand the expression (x + 10)(3x² + 5x - 2), we can use the distributive property, which states that for any numbers a, b, and c, the product (a + b)(c) can be written as a * c + b * c. Applying this property to the given expression, we get (x + 10)(3x² + 5x - 2) = (x * 3x² + 10 * 3x²) + (x * 5x + 10 * 5x) + (x * -2 + 10 * -2) = 3x³ + 30x² + 5x² + 50x - 2x - 20. Combining like terms, we get 3x³ + 35x² + 51x - 20. Thus, the expanded form of the expression (x + 10)(3x² + 5x - 2) is 3x³ + 35x² + 51x - 20.

The length of one side of the pentagon after translation is 9 units

Step-by-step explanation:

To be clear, "A translation is an isometry that translates all points of a figure the same distance in the same direction." The dimensions of any geometric figure will never change as a result of translation. The figure's measurements won't change; it will simply be moved from one location to another.

This means that even after being translated 2 units to the left, a regular pentagon with sides of 9 units will still have a side length of 9 units.

The length of one side of the pentagon after translation is 9 units(option B)

Know more about translation of objects: https://brainly.com/question/29011957

#SPJ4

Answer:

Step-by-step explanation:

Let, the Speed of the water be 'x'

and the Radius be 'r'

Now, the speed of Water Varies inversely with the Radius

So, x ∝ 1/r

      x = k/r   -----------(i)

Here, k is the proportionality constant

Now,

Given that, x = 2.5 ft per sec

then, 30 ft,

So, from eq. (i)

               2.5 = k/30

          So, k = 75

Again From eq. (i)

                    x = 75/r

          Now, if r = 3ft

           then, x = 75/3

                     x = 25

Hence, Speed of the water (x) = 25 ft per sec.

183  tickets must be sold to break even. The price at which a good or service must be provided in order to recoup its expenses of production.

The band costs $650,

decorations cost $45

refreshments cost $2.20 per person

let x be the  no of ticket

Total cost = 650+ 45+ 2.20x

break even

tickets cost = $6

6x =   650+ 45+ 2.20x

6 x - 2.20x = 695

3.8 x  = 695

x = 182.89

183  tickets must be sold to break even.

tickets must be sold to clear $700

700/6 =116.6

700/ 7.5 = 93.33

117 tickets must be sold to clear $700

94 tickets must be sold to clear $700,  If admission tickets are $7.50 each.

To learn more about break-even price refer,

https://brainly.com/question/21137380

#SPJ4

The number 'n' is equal to 14

In the question, we have been given that the number n is equal to 7 more than half of the number 'n'.

So in these linear equation-solving types of problems, first of all, we convert the written sentence into an equation.

So the equation for the following problem statement is -

n = 7 + n/2  

{since we have been given the number equals 7 more than half of the number }

Solving the equation we have,

n = 14.

We can easily see that 14 is the number which is 7 more than half of itself.

Know more about linear equation solving at - https://brainly.com/question/25858757

Answer:

Step-by-step explanation:

There is a scale factor of 2 so since the trapezoid is 9 we can divide by 2 so AD is 4.5 and CB is 4.5

YZ is the same as ZW so 8

Using the scale factor we can divide 8 by 2 to get 4

Perimeters can be found from the sums so

ABCD would be 19 and WXYZ would be 38

Answer: the answer is A

Step-by-step explanation:

Answer:

No solution

Step-by-step explanation:

Answer:

y = (-2/3)x + 6

Step-by-step explanation:

To find the equation of the line that passes through the points (-3, 8) and (6, 2), we can use the slope-intercept form of the equation of a line, which is:

y = mx + b

where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.
To find the slope of the line, we can use the following formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. Plugging in the coordinates of the two points given in the problem, we get:

m = (2 - 8) / (6 - (-3)) = -6/9 = -2/3

Next, we need to find the y-intercept of the line. This is the point where the line crosses the y-axis, which means that the x-coordinate of this point is 0. Since we know the slope of the line and one of the points that it passes through, we can use the point-slope form of the equation of a line to find the y-intercept:

y - y1 = m(x - x1)

Substituting the values we have calculated above and setting x = 0, we get:

y - 8 = (-2/3)(0 - (-3))

Solving for y, we find that the y-intercept is 8 - (2/3) * 3 = 8 - 2 = 6.

Therefore, the equation of the line that passes through the points (-3, 8) and (6, 2) is:

y = (-2/3)x + 6

Answer:

[tex]w \approx 96.6 \, \textrm{m}[/tex]

Step-by-step explanation:

See the attached image for my labeled diagram.

To solve for [tex]w[/tex], we can use the trigonometric ratio tangent.

[tex]\tan(\theta)=\dfrac{\textrm{opposite}}{\textrm{adjacent}}[/tex]

[tex]\tan(44\textdegree)=\dfrac{w}{100 \, \textrm{m}}[/tex]

↓ multiply both sides by 100m

[tex]100 \, \textrm{m} \cdot \tan(44\textdegree)=w[/tex]

↓ plug [tex]100\, \textrm{m} \cdot \tan(44\textdegree)[/tex] into a calculator

[tex]96.56887 \, \textrm{m} \approx w[/tex]

[tex]w \approx 96.56887 \, \textrm{m}[/tex]

↓ round to 3 significant figures

[tex]w \approx 96.6 \, \textrm{m}[/tex]

Answer: 24

Step-by-step explanation:

Answer:

h = 24

Step-by-step explanation:

Answer:

33/100

Step-by-step explanation:

The 100 natural numbers are from 1-100.

In order to find the total numbers divisible by 3, divide 100 by 3:

100/3 = 33.33

Put 33.33 into integer form:

33.33 ==> 33 numbers divisible by 3.

Divide 33 by 100 to get the probability: 33/100

To learn more about probabilities, click here:

https://brainly.com/question/25870256

Answer:

[tex]\frac{10}{30}[/tex]

Step-by-step explanation:

Answer:

10/30

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let total mass be x

(3/4)*x=15

on solving

x=20

now,

mass of 2/5 of piece of metal is-

=> (2/5)*20

=> 8

Ans- 8kg

Gonzalez Family Sprinklers used 15 liters and Walkers Family Sprinklers used 40 liters.

From the information, the Gonzalez family's sprinkler was used for 35 hours. The Walker family's sprinkler was used for 40 hours. There was a combined total output of 2125L of water.

Let Gonzalez = g

Let Walker = w

The equation will be:

g + w = 55 .... i

35g + 40w = 2125 ....

From equation i g = 55 - w

Put the above into equation ii

35g + 40w = 2125

35(55 - w) + 40w = 2125

1925 - 35w + 40w = 2125

Collect like terms

5w = 2125 - 1925

5w = 200

Divide

w = 200 / 5

w = 40

Walkers Family Sprinklers used 40 liters.

Since g + w = 55

g + 40 = 55

g = 55 - 40

g = 15

Gonzalez Family Sprinklers used 15 liters.

Learn more about equations on:

brainly.com/question/2972832

#SPJ1

Answer:

Step-by-step explanation: for the first part, 525/7 = 75.

for the second part, 55y = x, which means 55 words per day for a total of (55*7) or 385

for the third part, 1200-525-385=290, not totally sure, but I think this is the answer, the text was a bit blurry and I couldn't understand everything, try not to following this blindly and just see if it makes sense. Hope I could help

Answer:30x^2 - 54x - 18

Step-by-step explanation: To find the product of f and g, we can simply multiply the expressions for f and g together. The product of f and g is:

(f * g)(x) = (6x + 2)(-5x - 9)

          = -30x^2 - 54x - 18

So, the product of f and g is -30x^2 - 54x - 18

Answer:

72kg

Step-by-step explanation:

Stephen = 72

Saravanan = x

Shamsul = (x • 1.5) since Shamsul's mass is 1.5 times Saravanan's mass

1. make an equation:

( 72 + x + (x • 1.5) ) ÷ 3 = 64

2. solve it:

( 72 + x + (x • 1.5) ) ÷ 3 = 64

( 72 + x + (x • 1.5) ) = 64 • 3

( 72 + x + (x • 1.5) ) = 192

x + 1.5x = 192 - 72

2.5x = 120

x = 120 ÷ 2.5

x = 48

Thus shamsul's mass is:

48 • 1.5 = 72kg