Let z= -5 sqrt 3/2 + 5/2i and w=1 + sqrt 3i

a. Convert z and w to polar form.

b. Calculate zw using De Moivre’s Theorem.

c. Calculate (z / w) using De Moivre’s Theorem.

Answer:

The area of a sector GHJ is [tex]36.3\ cm^{2}[/tex]

Step-by-step explanation:

step 1

we know that

The area of a circle is equal to

[tex]A=\pi r^{2}[/tex]

we have

[tex]r=8\ cm[/tex]

substitute

[tex]A=\pi (8)^{2}[/tex]

[tex]A=64\pi\ cm^{2}[/tex]

step 2

Remember that the area of a complete circle subtends a central angle of 360 degrees

so

by proportion find the area of a sector by a central angle of 65 degrees

[tex]\frac{64\pi}{360}=\frac{x}{65}\\ \\x=64\pi (65)/360[/tex]

Use [tex]\pi =3.14[/tex]

[tex]x=64(3.14)(65)/360=36.3\ cm^{2}[/tex]

Answer:

[tex]\boxed{ \text{B. }\frac{2}{ 3}}[/tex]

Step-by-step explanation:

The scale factor (C) is the ratio of corresponding sides of the two pyramids.

The ratio of the areas is the square of the scale factor.

[tex]\frac{A_{1}}{ A_{2}} =C^{2}\\ \\\frac{48 }{108} = C^{2}\\\\ \frac{4}{9} = C^{2}\\\\ C = \frac{2}{3}\\\\[/tex]

The scale factor is [tex]\boxed {\frac{2}{3}}[/tex].

Answer:

60.2 m

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you ...

Tan = Opposite/Adjacent

The width of the river is the side of a right triangle opposite the angle of 76°. The 15 m distance is the length of the side adjacent to the measured angle. So, we have ...

tan(76°) = width/(15 m)

Multiplying by 15 m, we get ...

width = (15 m)tan(76°) ≈ 60.2 m

The width of the river is about 60.2 m.

Answer:

the answer is 60.16

Step-by-step explanation:

got it right

1) Convert 4% to decimal = .04

2) Multiply the purchase by that decimal

.04(56.70) = $2.27

C

The amount of tax on a $56.70 purchase is $2.27

How to find the amount of tax?

A tax exists as a mandatory fee or financial charge levied by any government on a person or an organization to manage revenue for public works providing the most suitable facilities and infrastructure. The collected fund exists then utilized to fund various public expenditure programs.

Sales tax in state = 4%

The amount tax on $1 purchase = $ [tex]\frac{4}{100}[/tex]  =  $ 0.04

The amount tax on a $56.70 purchase

= 56.70 * 0.04

= $2.27

Therefore option  c. $2.27 is the correct answer

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Answer:

The correct answer is the first one listed.

Step-by-step explanation:

First you have to determine what the equation for that circle is.  The standard form of a circle is

[tex](x-h)^2+(y-k)^2=r^2[/tex] where h and k are the coordinates of the center and the radius is squared.  Using the given info, our equation will look like this:

[tex](x-5)^2+(y+1)^2=25[/tex]

Now we use the coordinates given and plug in 2 for x and 3 for y and do the math and see if the 2 sides are equal.

[tex](2-5)^2+(3+1)^2=25[/tex]

and 9 + 16 = 25, right?  So that's how you can check the other coordinate pairs to verify that they DON'T work out!

The point Y(5, -4) lies on the circle with center O(5, -1) and radius 5, as per the distance formula and the equation of a circle.

To determine which point lies on the circle with center O(5, -1) and radius 5, we can use the distance formula to check if any of the given points are exactly radius 5 units away from center O. The distance formula, which is derived from the Pythagorean theorem, states that the distance between two points (x₁, y₁) and (x₂, y₂) in a 2-dimensional plane is given by √[(x₂ - x₁)² + (y₂ - y₁)²].

Thus, if a point (x, y) is on the circle, the following must be true: (x - 5)² + (y + 1)² = 52.

Checking for each point:

For point V(2, 3), we find (√[(2 - 5)² + (3 + 1)²]) which does not equal 5.

For point X(-3, -2), we calculate (√[(-3 - 5)² + (-2 + 1)²]) which does not equal 5.

For point Y(5, -4), we get (√[(5 - 5)² + (-4 + 1)²]) which equals to 5, hence Y lies on the circle.

For point Z(6, 9), we determine (√[(6 - 5)² + (9 + 1)²]) which does not equal 5.

Therefore, the point Y(5, -4) is on the circle.

Answer:

answer is C

Step-by-step explanation:

Answer:

Definitely C.) Pythagorean Theorem

Step-by-step explanation:

Answer:

[tex]\$7,204.85[/tex]  

Step-by-step explanation:

we know that    

The compound interest formula is equal to  

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest  in decimal

t is Number of Time Periods  

n is the number of times interest is compounded per year

in this problem we have  

[tex]t=10\ years\\ P=\$6,000\\ r=0.0183\\n=365[/tex]  

substitute in the formula above  

[tex]A=\$6,000(1+\frac{0.0183}{365})^{365*10}=\$7,204.85[/tex]  

The investment will be worth approximately $6,960.47 after 10 years.

To solve this problem, we can use the formula for compound interest, which is:

[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]

where:

-  A  is the amount of money accumulated after n years, including interest.

-  P  is the principal amount (the initial amount of money).

-  r  is the annual interest rate (decimal).

-  n  is the number of times that interest is compounded per year.

-  t  is the time the money is invested for, in years.

 Given:

[tex]- \( P = $6,000 \) \\-( r = 1.83\% = 0.0183 \) (as a decimal) \\- \( n = 365 \) (since the interest is compounded daily) \\- \( t = 10 \) years[/tex]

Plugging these values into the compound interest formula, we get:

[tex]\[ A = 6000 \left(1 + \frac{0.0183}{365}\right)^{365 \times 10} \][/tex]

Now, we calculate the value inside the parentheses first:

[tex]\[ \frac{r}{n} = \frac{0.0183}{365} \approx 0.000050137 \][/tex]

[tex]\[ 1 + \frac{r}{n} = 1 + 0.000050137 \approx 1.000050137 \][/tex]

[tex]\[ (1.000050137)^{365 \times 10} \] \[ = (1.000050137)^{3650} \][/tex]

Using a calculator or a software tool to compute this value, we find:

[tex]\[ (1.000050137)^{3650} \approx 1.1600785 \] \[ A = 6000 \times 1.1600785 \approx 6960.47 \][/tex]

Therefore, the investment will be worth approximately $6,960.47 after 10 years.

Answer:

  C.  The field will be covered 25% of the way in 2014.

Step-by-step explanation:

Since the field was completely covered in 2016, and the blueberries doubled in size in one year to do that, the field was half-covered in 2015. Similar reasoning tells you the field was 1/4 covered in 2014.

Answer: OPTION B

Step-by-step explanation:

Since the base is a square, you need to use the following formula:

[tex]V=\frac{s^2h}{3}[/tex]

Where "s" is the lenght of any side of the base and "h" is the height of the pyramid.

 You can identify in the figure that:

[tex]s=12units\\h=8units[/tex]

Then you can substitute these values into the formula to calculte the volume of this pyramid. Therefore, the result is:

[tex]V=\frac{(12units)^2(8units)}{3}=384units^3[/tex]

Answer:

The correct answer is option B.   384 units³

Step-by-step explanation:

Formula:-

Volume of pyramid = (a²h)/3

Where a -  side of base

h - height of pyramid

To find the volume of given pyramid

Here a = 12 units  and h = 8 units

Volume =  (a²h)/3

  =  (12² * 8)/3

  = (12 * 12 * 8)/3

  = (4 * 12 * 8) = 384 units³

Therefore the correct answer is option B.   384 units³

Answer:

64

Step-by-step explanation:

Answer:

Step-by-step explanation:

6/4=1.5 (price per can)

10/8=1.25 (price per can)

1.25 is cheaper than 1.5

the second one is the better buy

The better buying option is 8 cans for $10 and this can be determined by using the unitary method and the given data.

Given :

4 cans for $6 or 8 cans for $10.

The following steps can be used in order to determine the correct buying option:

Step 1 - The unitary method can be used in order to determine the better buying option.

Step 2 - If the value of 4 cans is $6 then the value of 1 can is:

4 cans = $6

1 can = [tex]\rm \dfrac{6}{4}[/tex]

= $1.5

Step 3 - If the value of 8 cans is $10 then the value of 1 can is:

= $1.25

So, the best buying option is 8 cans for $10.

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Answer:

see explanation

Step-by-step explanation:

Given x < 5

The graph will have an open circle at 5 indicating that x is not equal to 5, which would be a closed circle.

The ray will move to the left since values less than 5 are to the left of 5

The characteristics of the graph of the inequality x < 5 tells that value of x is less then the 5. It will use an open circle and the ray will move to the left.

Inequality of a graph is represented with the greater then(<), less then(>) or with the other inequity signs. The inequality line on the graph is represented with the dotted lines.

Characteristics of the graph of the inequality-

The inequality equation given in the problem is,

[tex]x<5[/tex]

This equation represent that the value of variable x is less then the number 5.

Hence, the characteristics of the graph of the inequality x < 5 tells that value of x is less then the 5. It will use an open circle and the ray will move to the left.

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Answer:

[tex]a=\sqrt{c^2-b^2}[/tex]

Step-by-step explanation:

Subtract b² from both sides of the equation.

a² = c² -b²

Now, take the square root of both sides of the equation.

a = √(c² -b²)

_____

Please note that when writing the square root (or any function) in plain text, it only applies to the first operand after the function name. That is ...

square root [ ] - [ ]

is evaluated as ...

(square root [ ]) - [ ]

If you want an entire expression to be considered to be "under the radical", then it must be enclosed in parentheses:

square root ([ ] -[ ])

This is the interpretation demanded by the Order of Operations. It can only be modified by using parentheses.

Answer:

[tex]A'(10,-5)\\\\B'(15,20)\\\\C'(-35,10)[/tex]

Step-by-step explanation:

You know that the triangle ABC has these points:

A(2, -1), B(3, 4), C(-7, 2)

If the triangle ABC is dilated by a factor of 5 to create the triangle A'B'C', you can find the coordinates of this new triangle by multiplying by 5 the coordinates of the triangle ABC given.

Therefore, the coordinates of the triangle A'B'C' are:

[tex]A'=(2(5),\ -1(5))=(10,-5)\\\\B'=(3(5),\ 4(5))=(15,20)\\\\C'=(-7(5),\ 2(5))=(-35,10)[/tex]

Using the numbers in the given equation:

a =-2, b = 6 and c = -5

The vertex form is written as : a(x+d)^2 + e

we need to find d and e:

d = b/2a = 6/2(-2) = -3/2

e = c-b^2/4a = -5 - 6^2/4(-2) = -1/2

Now substitute the letters for their values in the vertex form formula above:

-2(x-3/2)^2 -1/2

The vertex is (3/2, -1/2)

2. The formula begins with a negative number ( -2) so the Parabola opens downwards.

3. To find the intercept replace x with 0 and solve for y:

the intercept is (0,-5)

Answer:

Step-by-step explanation:

The first thing you should do is get a graph of this quadratic. The graph will answer the location of the vertex and it will also tell you if it opens up or down. It (finally) shows the intercept although that is easily found.

Question 3

You find the intercept by making x = 0

When you do that y becomes

y = - 2(0)^2 + 6(0) - 5

y = 0 - 5

y = - 5

Question 2

This too, is just a visual answer. Just look at the number in front of x^2

y = ax^2

if a < 0 then the graph always opens down

if a > 0 then the graph always opens up

In this case, y = - 2x^2. The graph opens down. The 6x + 5 does not affect the answer at all.

Question 1

This part is the tricky part and you have to complete the square.

Put brackets around the first 2 terms.

y = ( - 2* x^2 + 6x ) - 5    Pull out the common factor of - 2

y = -2 (x^2 - 3x) - 5         divide - 3 by 2 and square. Add inside the brackets.

y = -2(x^2 - 3x +  (3/2)^2  ) - 5   Add 2 times the squared amount outside the brackets.

y = -2(x^2 - 3x + (3/2)^2 ) - 5 + 2*(3/2)^2

y = -2(x^2 - 3x  + 9/4)     - 5 + 9/2     Show what is inside the brackets as a perfect square. Combine - 5 and 9/2

y = -2(x - 3/2)^2 - 5 + 4.5

y = -2(x - 3/2)^2 - 0.5

The vertex should be at (3/2, - 0.5)

Answer: The graph confirms (3/2, - 0.5) as the vertex.

Answer:

To quickly solve this problem, we can use a graphing tool or a calculator to plot the equation.

Please see the attached image below, to find more information about the graph

The equation is:

f(x) = 1/x - 1

Domain

All real numbers except for {0}

Answer:

It would be the fourth graph, D

Explanation:

When faced with these problems, you can either plot it on your own or you can use a graph generator online.

Let's call the left side of this tirangle y and downside z.

3² + 6² = y²

y² = 45

y² + z² = (3 + x)²

45 + z² = 9 + 6x + x²

x² + 6² = z²

45 + x² + 6² = 9 + 6x + x²

45 + x² + 36 = 9 + 6x + x²

45 + 36 - 9 = 6x

72 = 6x

x = 12

Answer:

A = $12831.8

Step-by-step explanation:

We know that the formula for compound interest is given by:

[tex]A=P(\frac{1+r}{n} )^{nt}[/tex]

where [tex]A[/tex] is unknown which is the amount of investment with interest,

[tex]P=9000[/tex] which is the initial amount,

[tex]r=12/100=0.12[/tex] is the interest rate,

[tex]n=4[/tex] which is the number of compoundings a year; and

[tex]t=3[/tex] which is the number of times that interest is compounded per unit t.

So substituting these values in the above formula to find A:

[tex]A=P(\frac{1+r}{n} )^{nt}[/tex]

[tex]A=9000(\frac{1+0.12}{4} )^{(4.3)}[/tex]

[tex]A = 9000(1 + 0.03)^{12}[/tex]

A = $12831.8

Answer:

The amount after 3 years = $12381.85

Step-by-step explanation:

Points to remember

Compound interest

A = P[1 +R/n]^nt

Where A - amount

P - principle amount

R = rate of interest

t - number of times compounded yearly

n  number of years

To find the amount

Here,

P = $9000.00, n = 3 years, t = 4, n = 3 and R = 12% = 0.12

A = P[1 +R/n]^nt

 = 9000[1 + 0.12/4]^(3 * 4)

 = 9000[1 + 0.03]^12

 = 12831.85

Therefore the amount after 3 years = $12381.85

Answer: 7 units

Explanation:

We have two points, (-4,-3) and (-4,4). The X values are both the same (-4). This means we will be working with the Y values. We need to find the distance between -3 and 4. We can use inverse operations to assist us. -3 + 7 = 4. Therefore, we know the distance between these two points is 7 units.

The distance between point A and point B is simply the difference in their y-coordinates since they share the same x-coordinate, resulting in a distance of 7 units.

To find the distance between two points on a coordinate plane, you can use the distance formula. However, in this case, since points A and B have the same x-coordinate, we can calculate the distance by looking at the difference in their y-coordinates. Point A is at (-4, -3) and Point B is at (-4, 4).

The difference in the y-coordinates is 4 - (-3) = 4 + 3 = 7. Therefore, the distance between point A and point B is 7 units, which means the correct answer is C. 7 units.

A = {3, 4, 7, 9}

B = {8, 9, 10, 11}

C = {4, 9, 11, 13, 15}

A∩(B∪C)

First let's solve parentheses, we want the union between B and C.

B∪C = {4, 8, 9, 10, 11, 13, 15}

Now we want the interception between A and this, which means we want just the value which appears in both.

A∩{4, 8, 9, 10, 11, 13, 15} = {4, 9}

Final answer:

To determine A∩(B∪C), first calculate the union of B and C which is {4, 8, 9, 10, 11, 13, 15}. Then find the intersection of this union with set A, which results in {4, 9}.

Explanation:

To find the intersection of set A with the union of sets B and C, denoted as A∩(B∪C), we first identify the union of sets B and C. The union of two sets contains all elements that are members of either set, without duplicates. Therefore, the union of set B = {8, 9, 10, 11} and set C = {4, 9, 11, 13, 15} is the set that contains all distinct elements from both, which is {4, 8, 9, 10, 11, 13, 15}.

Next, we identify the intersection of set A with this union. The intersection of two sets contains only the elements that are members of both sets. Set A = {3, 4, 7, 9}. We check which elements in set A are also in the union of B and C. The elements 4 and 9 appear in both set A and the union of B and C. Consequently, the intersection A∩(B∪C) is {4, 9}.

Answer:

The second graph.

Step-by-step explanation:

The second dotted graph represents viable values because the weights of cans are discrete data. We deal with a whole number of cans ( not parts of a can) so a continuous graph like the first one is not appropriate here.

Answer:

Step-by-step explanation:

The second graph represents viable values to these variables, because the independent variable, which it's the horizontal axis, represents cans of tomato past, and that it's only represented by a discrete variables, this means that cans can be counted only in natural numbers 1, 2, 3, 4, 5, ... and the second graph represent these discrete values, because it shows points for each can.

On the other hand, the first graph represents a continuous variable, which admits decimal numbers that cannot represent cans, because we cannot say "I have 2.345 cans", it's not possible, because each can is a whole, 1 can, 2 cans, and so on.

Therefore, the second graph is the viable.

BA and BD are radii of the circle, so triangle ABD is isosceles. Then angles BDA and BAD are congruent, and the remaining (central) angle ABD has measure

[tex]m\angle ABD=(180-2\cdot20)^\circ=140^\circ[/tex]

Angles ABD and CBD are supplementary, so

[tex]m\angle CBD=(180-140)^\circ=40^\circ[/tex]

and the answer is E.

The measure of the angle CBD is 40°.

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.

According to the given problem,

BA and BD are radii of the circle.

Triangle ABD is isosceles.

Angles BDA and BAD are congruent.

m∠ABD = (180 - 2*20)

              = 140°

Angles ABD and CBD are supplementary, so,

∠CBD = 180 - 140

           = 40°

Hence, the measure of the angle CBD is 40°.

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Answer:

The most reasonable estimate for the number of Auto Wheel magazine subscribers who own fewer than 2 vehicles is 102.

Step-by-step explanation:

Consider the provided information.

A random sample of 50 subscribers to Auto Wheel magazine about the number of cars that they own. Of the subscribers surveyed, 15 own fewer than 2 vehicles.

We need to find the most reasonable estimate for the number of Auto Wheel magazine subscribers who own fewer than 2 vehicles if there are 340 subscribers.

Here, the sample used is 50 and the population size is 340.

Now use the formula to find the reasonable estimate.

[tex]\frac{Part}{Sample}=\frac{x}{Population}[/tex]

Substitute the respective values in the above formula as shown;

[tex]\frac{15}{50}=\frac{x}{340}[/tex]

[tex]\frac{15}{50}\times 340=x[/tex]

[tex]x=3\times 34[/tex]

[tex]x=102[/tex]

Hence, the most reasonable estimate for the number of Auto Wheel magazine subscribers who own fewer than 2 vehicles is 102.

Answer:

w = 13.3, x= 10.3

First option is correct.

Step-by-step explanation:

In triangle ABC, we have

[tex]\tan42^{\circ}=\frac{AB}{BC}\\\\\tan42^{\circ}=\frac{12}{w}\\\\w=\frac{12}{\tan42^{\circ}}\\\\w=13.3[/tex]

Now, in triangle ABD

[tex]\tan27^{\circ}=\frac{AB}{BD}\\\\\tan27^{\circ}=\frac{12}{w+x}\\\\13.3+x=\frac{12}{\tan27^{\circ}}\\\\13.3+x=23.6\\\\x=10.3[/tex]

Thus, we have

w = 13.3, x= 10.3

First option is correct.

Answer:

[tex]\large\boxed{a=10.92\ and\ b=14.52}[/tex]

Step-by-step explanation:

Step 1:

Calculate a measure of angle C.

We know: The sum of measures of angles in a triangle is equal 180°.

Therefore we have the equation:

[tex]m\angle C+43^o+115^o=180^o[/tex]

[tex]m\angle C+158^o=180^o[/tex]          subtract 158° from both sides

[tex]m\angle C=22^o[/tex]

Step 2:

Use the law of sines to calculate the length a:

[tex]\dfrac{6}{\sin22^o}=\dfrac{a}{\sin43^o}[/tex]

[tex]\sin22^o\approx0.3746\\\\\sin43^o\approx0.6820[/tex]

[tex]\dfrac{6}{0.3746}=\dfrac{a}{0.6820}[/tex]          multiply both sides by 0.6820

[tex]a\approx10.92[/tex]

Step 3:

Use the law of sines to calculate the length b:

[tex]\dfrac{6}{\sin22^o}=\dfrac{b}{\sin115^o}[/tex]

[tex]\sin22^o\approx0.3746[/tex]

To calculate sin115°, use the formula:

[tex]\sin(180^o-\theta)=\sin\theta[/tex]

[tex]\sin115^o=\sin(180^o-65^o)=\sin65^o[/tex]

[tex]\sin65^o\approx0.9063[/tex]

[tex]\dfrac{6}{0.3746}=\dfrac{b}{0.9063}[/tex]         multiply both sides by 0.9063

[tex]b\approx14.52[/tex]

The answer is D: (-1,-1).

The solution to the system graphed below is (-1,-1) which is option D.

It is a collection of various points and it is used to draw lines of different functions or equations by finding out two points and by joining them.

We have to check solution of the system graphed in the question. There are two lines graphed. Both are intersecting each other at one point. A solution of a line exists at a point where they are intersecting. If we try to find the point then we need to identify origin first and then according to x and y axis points will be found. The required points will be (-1,-1).

Hence the solution of graphed system is (-1,-1).

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blue marbles =2/3

red marbles = 6/7

The required probability to pick the blue marbles is 40%.

Given that,
A jar contains 12 red marbles numbered 1 to 12 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar.

Probability can be defined as the ratio of favorable outcomes to the total number of events.

here,
Number of red marbles = 12
Number of blue marbles = 8
total number of ball = 12 + 8 = 20
probability =  Number of favorable outcomes / total outcome
Probability of the picking blue marble = 8 / 20 = 0.40

Thus, the required probability to pick the blue marbles is 40%.

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Answer:

First question C

C) after drawing the shape

Second question C

C). Solved the equation and then put 6 in place of X

x+17=3x+5. 3x-x= 17-5. 2x=12. 2x/2=12/2. x=6

NM=6+17= 23. OL=3(6)+5=23

Step-by-step explanation:

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Answer:

D

Step-by-step explanation:

Answer:

The volume of oblique cylinder is 252π cm³

D is correct.

Step-by-step explanation:

We are given a oblique cylinder whose radius 6 and slant height 8, vertical height 7

Volume of cylinder = Base area x vertical height.

Because volume of cylinder can't be change if we change oblique to vertical cylinder.

Base area [tex]=\pi r^2[/tex]

                 [tex]=\pi \times 6^2[/tex]

                [tex]=36\pi\text{ cm}^2[/tex]

Vertical height = 7 cm

Volume of cone [tex]=36\pi \times 7[/tex]

                           [tex]=252\pi\text{ cm}^3[/tex]

Hence, The volume of oblique cylinder is 252π cm³