WILL THANK, GIVE FIVE STARS, AND BRAINLIEST!!!!!

(a) Let the function [tex]$f$[/tex] be defined on the complex numbers as [tex]$\[f(z) = (1+i)z.\]$[/tex] Prove that the distance between [tex]$f(z)$[/tex] and [tex]$0$[/tex] is a constant multiple of the distance between [tex]$f(z)$[/tex] and [tex]$z$[/tex], and find the value of this constant.


(b) Let the function [tex]$g$[/tex] be defined on the complex numbers as [tex]$\[g(z) = (a + 2 i)z\]$[/tex] for some real value of [tex]$a$[/tex]. Then if [tex]$g(z)$[/tex] is equidistant from [tex]$0$[/tex] and [tex]$z$[/tex] for all [tex]$z$[/tex], what is [tex]$a$[/tex] equal to?

Answer:

b. [tex]\csc(x)[/tex]

Step-by-step explanation:

The given expression is

[tex]\frac{\sec(x)}{\tan(x)}[/tex]

We express in terms of basic trigonometric ratios to obtain;

[tex]\frac{\frac{1}{\cos(x)} }{\frac{\sin(x)}{\cos(x)} }[/tex]

This is the same as

[tex]\frac{1}{\cos(x)}\div \frac{\sin(x)}{\cos(x)}[/tex]

[tex]\frac{1}{\cos(x)}\times \frac{\cos(x)}{\sin(x)}[/tex]

Cancel out the common factors;

[tex]\frac{1}{\sin(x)}=\csc(x)[/tex]

Answer:

[tex]\frac{secx}{tanx}[/tex]  = cscx

Step-by-step explanation:

We have given a trigonometric expression.

[tex]\frac{secx}{tanx}[/tex]

We have to simplify the above expression.

Since, we know that

secx is reciprocal of cosx.

secx  =  1/cosx

Tanx is the ratio of sinx and cosx.

Tanx  =  sinx / cosx

Given expression becomes

[tex]\frac{1/cosx}{sinx/cosx}[/tex]

[tex]\frac{1}{cosx}\frac{cosx}{sinx}[/tex]

[tex]\frac{1}{sinx}[/tex]

[tex]\frac{secx}{tanx}[/tex]  = cscx which is the answer.

The vertex, would be the tip of the pyramid. If it was sliced at the vertex, the shape would be a triangle. Since the slice doesn't intersect the vertex, the point of the pyramid would net be included, it would be as if the tip was cut off.

You would then have a trapezoid, because the top would be a straight horizontal line parallel with the bottom line.

Answer:

The volume of a rectangular prism is [tex]28\frac{7}{8}\ cm^{3}[/tex]

Step-by-step explanation:

we know that

The volume of the rectangular prism is equal to

[tex]V=BHL[/tex]

Convert the given dimensions to an improper fractions

[tex]B=3\frac{1}{2}\ cm=\frac{3*2+1}{2}=\frac{7}{2}\ cm[/tex]

[tex]H=1\frac{1}{2}\ cm=\frac{1*2+1}{2}=\frac{3}{2}\ cm[/tex]

[tex]L=5\frac{1}{2}\ cm=\frac{5*2+1}{2}=\frac{11}{2}\ cm[/tex]

substitute in the formula

[tex]V=(\frac{7}{2})(\frac{3}{2})(\frac{11}{2})=\frac{231}{8}\ cm^{3}[/tex]

Convert to mixed number

[tex]\frac{231}{8}=\frac{224}{8}+\frac{7}{8}=28\frac{7}{8}\ cm^{3}[/tex]

Answer:

2.5 %  

Step-by-step explanation:

The simple interest formula is

I = Prt

Data:

I = $75

P = $600

t = 5 yr

Calculation:

75 = 600 × r × 5

75 = 3000 r

Divide each side by 3000

r = 75/3000 = 0.025 = 2.5 % APR

The annual percentage rate is 2.5 %.

Final answer:

The annual rate of simple interest that Remi received from his savings account is 2.5%. This was calculated using the simple interest formula I = PRT, rearranged to solve for R, the annual interest rate.

Explanation:

To work out the annual rate of simple interest that Remi received from his savings account, we can use the formula for simple interest:

I = PRT

where I is the total interest earned, P is the principal amount invested, R is the annual interest rate, and T is the time the money is invested in years.

From the question, we know that Remi invested £600 (P=600), received £75 in interest (I=75), and the investment was for 5 years (T=5). We need to find the annual rate (R).

Rearranging the formula to get value of R, we get:

R = I / (PT)

Substituting the given values:

R = 75 / (600 × 5) (× is the multiplication symbol)

R = 75 / 3000

R = 0.025

To Convert decimal to a percentage, we multiply by 100:

R = 0.025 × 100

R = 2.5%

So, the annual rate of simple interest that Remi received is 2.5%.

Answer:

2

Step-by-step explanation:

Let's disect the problem. Three times a particular number plus fourteen equals 20. Let's make an equation, and replace the number with x.

We'll first minus 14 from each side to balance the equation, but with the intention of isolating the variable, x.

We have almost successfully isolated the variable. To isolate the variable, we can divide by three on each side.

Answer:

[tex]3,358\ people[/tex]  

Step-by-step explanation:

The formula to calculate the exponential growth function is equal to

[tex]f(x)=P(e)^{rx}[/tex]  

where  

f(x) is the population  

P is the population in the year 2000  

r is the rate in decimal  

x is number of years since 2000  

e is the mathematical constant number

we have  

[tex]x=2020-2000=20\ years\\ P=2,400\\ r=0.0168[/tex]  

substitute in the formula above  

[tex]f(x)=2,400(e)^{0.0168*20}[/tex]  

[tex]f(x)=3,358\ people[/tex]  

Answer:

a. 61.41%

b. 27 minutes

Step-by-step explanation:

a:  Find the z-score for the situation.  

µ = 19

x-bar = 20

σ = 3.5

  z = (20 - 19)/3.5 = 0.29

The p-value for z = 0.29 is 0.6141, so 61.41% of people will get this discount

b:  They want no more than 2% to get the discount, so they want less than 98% getting the discount.  The z-score for 98% (0.98 as a decimal) is 2.055

*You need to look at the chart and find where 0.98 would be.  It's between a z-score of 2.05 and 2.06.  

The z-score is    2.055 = (x - 19)/3.5      We are solving for the time for this one.  So solve for x...

              7.1925 = x - 19      (multiply both sides by 3.5)

             26.1925 = x    (add 19 to both sides)

So 26.1925 minutes, or about 26 minutes, 12 seconds, so round up to 27 minutes because they want less than 2%.   I chose 27 minutes because no places give odd wait times like 26 minutes 12 seconds.

A) The percent of customers receive the service for​ half-price is; 61.41%

B) The time to make the guaranteed limit is; 26 minutes 12 seconds

A) We are given;

Population mean; µ = 19

Sample mean; x' = 20

Standard deviation; σ = 3.5

Thus, z-score is;

z = (20 - 19)/3.5

z = 0.29

From online p-value from s-score calculator,  the p-value for z = 0.29 is;

p = 0.6141 = 61.41%

B) We are told that they now want more than 2% to get the discount. This means that they want less than 98% or 0.98 getting the discount.  

The z-score for 0.98 is; z = 2.055

Thus, using z-score formula, we have;

2.055 = (x' - 19)/3.5    

x' - 19 = 3.5 * 2.055

x' - 19 = 7.1925

x' = 26.1925 = 26 minutes 12 seconds

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Answer:

1/3

Step-by-step explanation:

1/3 since there are two numbers less than three, and there are 6 possible outcomes, so 2/6 = 1/3.

Our three numbers are...

3 3/10 = 3.3

3.1

3 1/4 = 3.25

So, if we order those from least to greatest, we have...

3.1, 3.25, 3.3

which, in the forms given, is...

3.1, 3-1/4, 3-3/10

Answer:

0.78

Step-by-step explanation:

There are 500 students in total. Thus,

The probability that one of the students on the trip is an athlete or a band memeber is

[tex]Pr=\dfrac{325+40+25}{500}=\dfrac{390}{500}=0.78.[/tex]

Answer: -0.6

Step-by-step explanation:

First thing to do is to solve for θ and ϕ from the given information

tan(θ) = 4/3,

θ = tan-¹4/3,

θ = 53.1°

Since tan is positive in quadrant III, θ = 53.1°

Also,

sin(ϕ) = − 10/10 ,

ϕ = sin-¹-1

ϕ = 270°

If ϕ is in the fourth quadrant, that gives 360 - ϕ i.e 360 - 270 = 90°

Substituting the values of θ and ϕ into sin(θ − ϕ), we have;

Sin(53.1 - 90)

= sin (-36.9°)

= -0.6

The given expression sin(θ - ϕ), with tan(θ) = 4/3, θ in Quadrant III, sin(ϕ) = -10/10, and ϕ in Quadrant IV, is evaluated by using trigonometric principles and identities. Upon calculations, sin(θ - ϕ) comes out to be -3/5.

The question is asking us to evaluate the expression sin(θ − ϕ), given that tan(θ) = 4/3, θ is in Quadrant III, sin(ϕ) = -10/10, and ϕ is in Quadrant IV. In trigonometry, tan θ = sin θ/cos θ. We have tan θ = 4/3 and we know that in Quadrant III, tangent is positive but sine and cosine are negative. So, we can make a right triangle where the opposite side is 4 (basing this on the absolute value of the tan θ) and the adjacent side is 3. The hypotenuse then, by using Pythagoras theorem, comes out to be 5. Then sin θ = -4/5 and cos θ = -3/5.

For sin(ϕ), we are given that it equals -1. In Quadrant IV, sine is negative and cosine is positive, so cos ϕ = √(1 - (-1)^2) = 0.

Finally, utilizing the formula sin (a ± ß) = sin a cos ß ± cos a sin ß, we plug in our values to come to the solution sin(θ - ϕ) = (sin θ cos ϕ) - (cos θ sin ϕ) = ((-4/5)*0) - ((-3/5)*-1) = -3/5

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Answer:

x     y

-2   7

-1    10.5

0    15.75

1     23.625

2    35.4375

Step-by-step explanation:

The general equation of the exponential function is [tex]y=ab^x[/tex].

We know from our table that when [tex]x=0[/tex], [tex]y=15.75[/tex]. Let's replace those values in our equation:

[tex]y=ab^x[/tex]

[tex]15.75=ab^0[/tex]

Remember that [tex]b^0=1[/tex], so:

[tex]15.75=a(1)[/tex]

[tex]15.75=a[/tex]

[tex]a=15.75[/tex]

We also know from our table that when [tex]x=-1[/tex], [tex]y=10.5[/tex]. Let's replace the values again:

[tex]y=ab^x[/tex]

[tex]10.5=ab^{-1}[/tex]

But we now know that [tex]a=15.75[/tex], so let's replace that value as well:

[tex]10.5=15.75b^{-1}[/tex]

Remember that [tex]b^{-1}=\frac{1}{b}[/tex], so:

[tex]10.5=\frac{15.75}{b}[/tex]

[tex]10.5b=15.75[/tex]

[tex]b=\frac{15.75}{10.5}[/tex]

[tex]b=1.5[/tex]

Now, we can put it all together to complete our exponential function:

[tex]y=ab^x[/tex]

[tex]y=15.75(1.5)^x[/tex]

To find the missing values, we just need to evaluate our function at [tex]x=1[/tex] and [tex]x=2[/tex]:

- For [tex]x=1[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^1[/tex]

[tex]y=23.625[/tex]

- For [tex]x=2[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^2[/tex]

[tex]y=35.4375[/tex]

D

generally,

[tex] \sin( \alpha ) = \cos(90 - \alpha )[/tex]

you can visualize this by drawing a right triangle with acute angles a and 90-a

Answer:

D.  Both a and c are true.

Step-by-step explanation:

a . sin 18 = cos (90 - 18) = cos 72 so a is True.

b. this is not true.

c. sin 72 = cos(90 - 72) = cos 18, so c is true.

Answer:

20

85%

Step-by-step explanation:

You are given the function [tex]S(n)=20\cdot b^n.[/tex]

If n  is the number of hours, then initially n=0 and

[tex]S(0)=20\cdot b^0=20\cdot 1=20.[/tex]

If S(n) is the function of exponential growth, then it can be represented as

[tex]S(n)=I\cdot (1+r)^n,[/tex]

where I is the initial amount, r -is the percent growth rate and n is the number of hours.

If b = 1.85, we can represent it as b = 1 + 0.85. Thus, the hourly percent growth rate of the bacteria would be 0.85=85%.

s(n) = 20b^n

n is the time in hours. At the beginning, the time is zero hours, so n = 0.

s(0) = 20 * b^0

s(0) = 20 * 1

s(0) = 20

The initial amount was 20.

For b = 1.85,

s(n) = 20(1.85)^n

s(n) = 20(1 + 0.85)^n

The hourly growth is 0.85.

0.85 * 100% = 85%

The hourly percent change is 85%.

Answer:

Step-by-step explanation:

Your car breaks down unexpectedly and you must have it repaired immediately.  That's an example of a situation where you might use money from a financial reserve.

Answer:

The answer is circle; (x')² + (y')² - 4 = 0

Step-by-step explanation:

* At first lets talk about the general form of the conic equation

- Ax² + Bxy + Cy²  + Dx + Ey + F = 0

∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.

∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.

∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.

* Now we will study our equation:

* 2x² + 2y² = 8

∵ A = 2 , B = 0 , C = 2

∴ B² - 4AC = (0) - 4(2)(2) = -16 < 0

∵ B² - 4AC < 0

∴  it will be either a circle or an ellipse

* Lets use this note to chose the correct figure

- If A and C are equal and nonzero and have the same sign,

 then the graph is a circle.

- If A and C are nonzero, have the same sign, and are not equal

 to each other, then the graph is an ellipse.

∵ A = 2 and C = 2

∴ The graph is a circle.

∵ D and E = 0

∴ The center of the circle is the origin (0 , 0)

∵ Ф = 30°

∴ The point (x , y) will be (x' , y')

- Where x = x'cosФ - y' sinФ and y = x'sinФ + y'cosФ

∴ x = x'cos(30°) - y'sin(30°)

∴ y = x'sin(30°) + y'cos(30°)

∴ x = (√3/2)x' - (1/2)y' and y = (1/2)x' + (√3/2)y'

∴ [tex]x=\frac{\sqrt{3}x'-y'}{2}[/tex]

∴ [tex]y=\frac{x'+\sqrt{3}y'}{2}[/tex]

* Lets substitute x and y in the first equation

∴ [tex]2(\frac{\sqrt{3}x'-y'}{2})^{2}+2(\frac{x'+\sqrt{3}y'}{2})^{2}=8[/tex]

* Use the foil method

∴ [tex]2(\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}}{4})+2(\frac{x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{4})=8[/tex]

* Open the brackets

∴ [tex]\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}+x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{2}=8[/tex]

* Collect the like terms

∴ [tex]\frac{4x'^{2}+4y'^{2}}{2}=8[/tex]

* Simplify the fraction

∴ 2(x')² + 2(y')²= 8

* Divide each side by 2

∴ (x')² + (y')² = 4

∴ The equation of the circle is (x')² + (y')² = 4

* The general equation of the circle is (x')² + (y')² - 4 = 0  

 after rotation 30° about the origin

* Look to the graph

- The blue circle for the equation 2x² + 2y² = 8

- The blue circle for equation (x')² + (y')² - 4 = 0

* That is because the two circles have same centers and radii

- The green line is x' and the purple line is y'

Answer:

The answer is D

Good luck on the Ed-genuity test

Answer:

x = 25 mm

Step-by-step explanation:

The perimeter of a rectangle is given by

P = 2(l+w) where l is the length and w is the width

We know the perimeter is 60 mm and the width is 5 mm and the length is x

60 = 2(x +5)

Divide each side by 2

60/2 = 2/2(x+5)

30 = x+5

Subtract 5 from each side

30-5 = x+5-5

25 =x

x = 25 mm

[tex]\huge\bold\red{Answer}[/tex]

☑The diagram which is shown above is a rectangle.

✍ Perimeter= 60mm

✍ breadth = 5mm

➡ Perimeter of rectangle = 2(l+b)

✍ 60 = 2(l +5)

✍ 60/2 = l+5

✍ 30 - 5 = l

✍ 25 = l

☑ L = 25mm

❣..hope it helps you..❣

Answer:

17.85 months

Step-by-step explanation:

original number of rabbits=50

% increase per month=12%

number of months=x

form equation for 1000 rabbits

(112/100) ×50× x = 1000

56x=1000

x=1000/56

x=17.857 months

Answer:

9.9 cm

Step-by-step explanation:

We can use the Pythagorean theorem to find the length of CY

a^2 + b^2 = c^2

CY^2 + YZ^2 = CZ^2

YZ = XY  since CZ is the perpendicular bisector

YZ = 5

CZ = 7

CY ^2 + 5^2 = 7^2

CY^2 +25 = 49

Subtract 25 from each side

CY^2 = 49-25

CY^2 = 24

Take the square root of each side

sqrt(CY^2) = sqrt(24)

CY = 4.898979

CY = 4.9 cm

We want the length of WY

WY = WC + CY

WC is a radius which is 5 cm

WY = 5cm + 4.9cm

WY = 9.9 cm

Answer:

We should work backwards we need to find YC+CW to get YW

angle bisector theorem means that ZC and XC are equal

then we can use the Pythagorean theorem to get YC

5^2 + x = 7^2

YC= √13

CW = 7 because they are both the radius of a circle

YW= 7+√13

YW=10.60 (rounded)

Answer:

Step-by-step explanation:

The outcome of the game is

Toss 1               Toss 2      Result

H                         H               18

H                         T                 -1

T                         H                -1

T                         T                -20

His only winning position is H ---- H

That means he has only a 1/4 chance in winning. He shouldn't play at all. It might be a fair coin, but it is not a fair game.

Answer:

I would say about 60 to 70 pounds. But I have no more information from you to better answer your question, so that's all I have right now.

Step-by-step explanation:

Answer:

Angles UWV and UWZ are complementary

Step-by-step explanation:

we know that

If the sum of the measures of two angles is equal to 90 degrees, then, the angles are complementary

so

In this problem

[tex]m<UWZ+m<UWV=90\°[/tex]

therefore

Angles UWV and UWZ are complementary

Answer:

x = log base 4 of 2

Step-by-step explanation:

I'm assuming this is truly an exponential function as you state and not a mistype.

6^x = 2

Take the log of each side:

x · log(6) = log(2)

Divide each side by log(6):

x = log(2) / log(6)

x = 0.30103 / 0.77815

x = 0.3869... (rounded)

Answer:

Option b

Step-by-step explanation:

We know that, by definition:

[tex]secx = \frac{1}{cosx}[/tex]

We also know that:

[tex]tanx = \frac{sinx}{cosx}[/tex]

Applying these identities we can simplify the given expression

[tex]\frac{secx}{tanx} = \frac{\frac{1}{cosx}}{\frac{sinx}{cosx}}\\\\\frac{secx}{tanx} = \frac{cosx}{sinxcosx}\\\\\frac{secx}{tanx} = \frac{1}{sinx}[/tex]

We know that, by definition:

[tex]\frac{1}{sinx} = cscx[/tex]

Final answer:

To simplify sec x/tan x, we use trigonometric identities to show that sec x/tan x equals csc x, which is the cosecant of x.

Explanation:

To simplify the expression sec x/tan x, we need to recall the trigonometric identities for secant (sec) and tangent (tan). We know that sec x is equal to 1/cos x and tan x is equal to sin x/cos x. When we divide sec x by tan x, we are essentially dividing 1/cos x by sin x/cos x. This simplifies to:

To summarize, the simplified form of the expression sec x/tan x is csc x.

Final answer:

The division algorithm for polynomials establishes that a polynomial p(x) can be divided by a non-zero polynomial d(x) (with degree less or equal to p(x)), to yield unique quotient q(x) and remainder r(x), with r(x) having a lower degree than d(x).

Explanation:

The division algorithm in the context of polynomials is a fundamental concept in algebra that stipulates for any two polynomials p(x) and d(x), with d(x) ≠ 0 and the degree of d(x) less than or equal to the degree of p(x), there exist unique polynomials q(x) and r(x) such that p(x) = d(x) × q(x) + r(x).

In this scenario, q(x) is referred to as the quotient and r(x) is the remainder. The degree of the remainder r(x) will always be less than the degree of d(x), following the division algorithm.

When applying this algorithm to special sets of polynomials like Legendre polynomials, additional properties can be observed, such as the roots of the polynomials or specific transformations like the Poisson bracket that could arise in mathematical physics. Moreover, the concept extends to rational functions, which are the quotients of polynomials.

Answer:

x = 5

Step-by-step explanation:

For the parallelogram to be a square we use the property

The diagonals of a square are congruent, hence

12x - 23 = 4x + 17 ( subtract 4x from both sides )

8x - 23 = 17 ( add 23 to both sides )

8x = 40 ( divide both sides by 8 )

x = 5

Answer:

C. sin(90° - θ)

Step-by-step explanation:

A trig function of an angle equals the cofunction of the angle's complement.

The cofunction of  cosine is sine, and the complement of θ is 90° - θ.

∴ cosθ = sin(90° - θ)

Answer:  Option 'C' is correct.

Step-by-step explanation:

Since we have given that

[tex]\cos \theta[/tex]

We need to find the correct cofunction identity for it.

Cofunction identities represent the relationship among the trigonometric functions.

The value of trigonometric function of an angle is equal to cofunction of its complement.

As we know that sine is a complement of cosine.

so, it becomes,

[tex]\cos \theta=\sin(90^\circ-\theta)[/tex]

Hence, Option 'C' is correct.

Answer:

The value of x = 0.41

Step-by-step explanation:

∵ [tex]6e^{2x}-5e^{x}=6[/tex]

Let [tex]e^{x}=y[/tex]

∴ [tex]e^{2x}=y^{2}[/tex]

∴ 6y² - 5y = 6

∴ 6y² - 5y - 6 = 0 ⇒ factorize

∴ (3y + 2)(2y - 3) = 0

∴ 3y + 2 = 0 ⇒ 3y = -2 ⇒ y = -2/3

∴ 2y - 3 = 0 ⇒ 2y = 3 ⇒ y = 3/2

∵ [tex]y=e^{x}[/tex]

∴ [tex]e^{x}=\frac{-2}{3}[/tex] ⇒ refused

  ([tex]e^{ax}[/tex] never gives -ve values)

∵ [tex]e^{x}=3/2[/tex] ⇒ insert ln in both sides

∵ [tex]lne^{ax}=axlne=ax[/tex] ⇒ ln(e) = 1

∴ [tex]xlne=ln(3/2)[/tex]

∴ x = ln(3/2) = 0.41

Answer:

15

Step-by-step explanation:

10 + 25= 35 and 35-20 = 15

please give branlest

Answer:

Step-by-step explanation: She Needs To Deposit $35 Dollars To Return To The Orginal Ammount She Started With On Monday.