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1 + 2i in polar form is 5√2∠63.4°.

What is a polar form ?

In mathematics, the polar form is a way of representing complex numbers using their magnitude and angle. A complex number can be represented in the form r(cos θ + i sin θ), where r is the magnitude (or modulus) of the complex number, and θ is its argument (or phase angle). Alternatively, the polar form can be represented in terms of magnitude and angle as r ∠θ, where r is the magnitude and θ is the angle in radians. This form is also known as the exponential form of a complex number. The polar form is useful in calculations involving complex numbers, especially in multiplication and division, where it is often easier to work with the polar form rather than the rectangular form (a + bi).

D. 5√2∠63.4°

To convert a complex number from rectangular form to polar form, we use the following formulas:

r = √(a^2 + b^2)

θ = tan^-1 (b/a)

Where a and b are the real and imaginary parts of the complex number, respectively.

In this case, a = 1 and b = 2, so:

r = √(1^2 + 2^2) = √5

θ = tan^-1 (2/1) = 63.4°

Therefore, 1 + 2i in polar form is 5√2∠63.4°.

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Answer: If Jamie is selling bags of carrots for $5 each and already has $30 what would he use to calculate how much money he made.

The Converse of the Pythagorean Theorem is proved below.

What is Pythagoras Theorem ?

Pythagoras' theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The converse of the Pythagorean theorem states that if a triangle has sides of length a, b, and c, where c is the longest side, and a² + b² = c², then the triangle is a right triangle.

To prove the converse of the Pythagorean theorem, we can use the fact that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. That is, if a triangle has sides of length a, b, and c, where c is the longest side and c² = a² + b², then the triangle is a right triangle.

Proof:

Suppose we have a triangle ABC, with sides of length a, b, and c, where c is the longest side. Assume that a² + b² = c². We want to show that triangle ABC is a right triangle.

We will use the Law of Cosines to show that angle C is a right angle. The Law of Cosines states that for any triangle ABC with sides of length a, b, and c opposite angles A, B, and C, respectively:

c² = a² + b² - 2ab cos(C)

Substituting a² + b² = c² into this equation, we get:

c² = c² - 2ab cos(C)

2ab cos(C) = 0

cos(C) = 0

Since 0 is the cosine of 90 degrees, this implies that angle C is a right angle. Therefore, triangle ABC is a right triangle.

Thus, this completes the proof of the converse of the Pythagorean theorem.

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Answer:

4.

a-9=15

or. a=15+9

or. a=24

5.

n-9=13

or. n=13+9

or. n=22

6.

b-8=9

or. b=9+8

or. b=17

7

x-5=16

or. X=16+5

or X=21

8

n-6=17

or. n= 17+6

or. n=23

Step-by-step explanation:

minus sign changes into plus sign when changing sides

The coordinates of the point P on the directed line segment from A to B such that P is 2/3 the length of the line segment from A to B are (-1, 2).

The coordinates of point P are obtained applying the proportions in the context of the problem.

P is 2/3 the length of the line segment from A to B, hence the equation is given as follows:

P - A = 2/3(B - A)

Then the x-coordinate of P is obtained considering the x-coordinates of A and B as follows:

x - 9 = 2/3(-6 - 9)

x - 9 = -10

x = -1.

The y-coordinate of P is obtained considering the y-coordinates of A and B as follows:

y + 8 = 2/3(7 + 8)

y + 8 = 10

y = 2.

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The experimental probability that the next train to arrive at Danville Station will be on time is 3/4 or 0.75.

What is probability?

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can range from 0 to 1, with 0 denoting an impossibility and 1 denoting a certainty.

The experimental probability of the next train to arrive at Danville Station being on time is given by the number of on-time trains divided by the total number of trains that arrived -

experimental probability = number of on-time trains / total number of trains

From the information given, we know that out of the last 20 trains to arrive, 15 were on time. Therefore -

number of on-time trains = 15

total number of trains = 20

Substituting these values into the equation, we get -

experimental probability = 15 / 20

Simplifying the fraction, we get -

experimental probability = 3/4

Therefore, the experimental probability value is obtained to be 3/4 or 0.75.

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Answer:

Alright

Step-by-step explanation:

50/1.428

Multiply the numerator by the denominator to get 50000/1428

Set equation up in long division format

Divide 50000 by 1428 to get 3

Divide 7160 by 1428 to get 5

Divide 200 by 1428 to get 0

Divide 2000 by 1428 to get 1

Divide 5720 by 1428 to get 4

Divide 80 by 1428 to get 0

Divide 800 by 1428 to get 0

35.014

Answer:

C)  2, 7, 6 can form a triangle.

Step-by-step explanation:

The triangle inequality theorem states that the sum of any two sides of a triangle is greater than the length of the third side.

To determine which group of given side measures will form a triangle, sum each pair of sides and compare to the third side. If all three inequalities are true, the group will form a triangle.

Given side lengths: 9, 4, 3

  ⇒ 9 + 4 > 3

  ⇒ 9 + 3 > 4

  ⇒ 4 + 3 [tex]\ngtr[/tex] 9

Therefore, this group cannot form a triangle.

Given side lengths: 8, 1, 7

  ⇒ 8 + 1 > 7

  ⇒ 8 + 7 > 1

  ⇒ 1 + 7 [tex]\ngtr[/tex] 8

Therefore, this group cannot form a triangle.

Given side lengths: 2, 7, 6

  ⇒ 2 + 7 > 6

  ⇒ 2 + 6 > 7

  ⇒ 7 + 6 > 2

Therefore, this group can form a triangle.

Given side lengths: 12, 10, 22

  ⇒ 12 + 10  [tex]\ngtr[/tex] 22

  ⇒ 12 + 22 > 10

  ⇒ 10 + 22 > 12

Therefore, this group cannot form a triangle.

Answer:

Step-by-step explanation:

Ranking:

Compound the interest every month instead of every two months.

Increase the interest rate by 1%.

Let the account increase for one more year.

Increase the starting amount by $100.

Calculations:

If the interest is compounded monthly instead of every two months, the final value would be $1,357.36, which is an increase of $9.18 compared to the original value.

If the interest rate is increased to 6%, the final value would be $1,411.58, which is an increase of $63.40 compared to the original value.

If the account increases for one more year at the same interest rate, the final value would be $1,435.09, which is an increase of $86.91 compared to the original value.

If the starting amount is increased by $100, the final value would be $1,421.61, which is an increase of $73.43 compared to the original value.

The maximum revenue is achieved when she sells 33 photos at a price of 25 per photo.

Using quadratic regression, [tex]y = ax^2 + bx + c[/tex]

where a, b and c are constants and x is the independent variable we can determine the equation of the curve that best fits the data. The equation will be in the form of[tex]y = ax^2 + bx + c[/tex], where y is the revenue, x is the number of photos sold, and a, b, and c are constants. The constants can be determined by solving the system of equations formed by the three data points.

After solving for the constants, we can determine the maximum revenue by finding the vertex of the curve. The vertex is located at x = -b/2a, and the corresponding revenue is[tex]y = ax^2 + bx + c[/tex]. Plugging in the values for a, b, and c, we can calculate that the maximum revenue is achieved when she sells 33 photos at a price of 25 per photo.

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Complete question

Darlena has started taking photos at amateur dog racing events, later offering the photos for sale to the dog owners by email. The prices she has charged per photo at each of her first three events, and the corresponding number of photos sold and total revenue raised, appear in the table below. Treating revenue as a function of the number of photos sold, a graph of the three data points is also shown. If she uses quadratic regression to fit a curve to the data, what number of photos sold and what price per photo will maximize her revenue?

The volume of the cone in cubic centimeters in terms of π = 120π cm³. That is option B.

To calculate the volume of a cone, the formula that is to be used;

Volume = 1/3 πr²h

Where;

Height = 10 centimetres

Radius = 6 centimetres

Volume = 1/3 × π × 6²× 10

= 360π/3

= 120π cm³

Therefore the volume of the cone whose dimensions are given = 120π cm³.

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The quadratic equation on the graph can be written as:

y = x² + 6x + 8

Remember that a quadratic equation whose vertex is (h,k) and the leading coefficient is a, can be written as:

y = a*(x - h)² + k

Here we can see that the vertex is at (-3, -1), replacing that we will get:

y = a*(x + 3)² - 1

Now we can also see that it passes through the point (-2, 0), replacing these values we will get:

0 = a*(-2 + 3)² - 1

0 = a - 1

1 = a

Then the quadratic is:

y  = (x + 3)² - 1

Expanding that we get:

y = x² + 6x + 9 - 1

y = x² + 6x + 8

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Answer:

p=84

Step-by-step explanation:

p=6×14

p=84

answer: p=84

Therefore, the quadratic function in standard form that represents the data in the table is: y = 1/2 x² - 5/2 x + 5.

A function is a mathematical concept that describes the relationship between two sets of numbers. It is a rule that assigns to each input (or element in the domain) exactly one output (or element in the range). In simpler terms, a function is like a machine that takes in a number and produces another number as output, according to some specific rules. The input is usually denoted by x, and the output by f(x), which is read as "f of x". Functions can take many forms, such as linear, quadratic, trigonometric, exponential, logarithmic, and more. They are widely used in mathematics, science, engineering, and many other fields to model and analyze various phenomena.

Here,

To write a quadratic function in standard form, we need to use the general form of the quadratic equation:

y = ax²+ bx + c

where a, b, and c are constants. We can use the values in the table to find these constants.

When x = 2, y = 3

When x = 4, y = 1

When x = 6, y = 3

When x = 8, y = 9

When x = 10, y = 19

Substituting these values into the quadratic equation, we get:

3 = 4a + 2b + c

1 = 16a + 4b + c

3 = 36a + 6b + c

9 = 64a + 8b + c

19 = 100a + 10b + c

We can now use these equations to solve for a, b, and c. One way to do this is to use a matrix equation:

|16 4 1| |a| |1|

|36 6 1| |b| = |3|

|64 8 1| |c| |9|

Using a calculator or matrix software, we can solve for a, b, and c:

a = 1/2

b = -5/2

c = 5

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The order of the matrix is 2 x 2.

What is a matrix and what is meant by its order?

In the field of mathematics, matrices are rectangular arrangements of numerical values, symbols, or algebraic expressions, organized in rows and columns. Matrices are used in many areas of mathematics and science, as well as in engineering, physics, and computer graphics.

The term 'order' in the context of matrices refers to the dimensions of the matrix, which are determined by the number of rows and columns it contains. It is denoted by m x n, where m is the number of rows and n is the number of columns. To find the order of a matrix, we simply count the number of rows and columns. The order is usually written as a pair of numbers in the form m x n.

Determine the order of the given matrix:

In the given problem, we are asked to find the order of the matrix:

[tex]\begin{pmatrix}9 &6 \\-4 & 8\end{pmatrix}[/tex]

we count the number of rows and columns.

This matrix has 2 rows and 2 columns, so its order is 2 x 2.

Therefore, the order of the matrix is 2 x 2.

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Answer:

The midpoint of a line segment AB with endpoints A(x1, y1) and B(x2, y2) is given by the coordinates:

[(x1 + x2) / 2, (y1 + y2) / 2]

In this case, A is located at (1, 4) and B is located at (3, -6). So the midpoint M of AB is:

[(1 + 3) / 2, (4 + (-6)) / 2]

= [2, -1]

Therefore, the midpoint of AB is at the point (2, -1).

The coordinates of the midpoint of the line AB is [2, -1]

Section formula is used to find the ratio in which a line segment is divided by a point internally or externally.

It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc.

Given that, we need to find the midpoint of AB if A located at (1, 4) and B is located at (3, -6).​

The midpoint of a line segment AB with endpoints A(x1, y1) and B(x2, y2) is given by the coordinates:

[(x1 + x2) / 2, (y1 + y2) / 2]

In this case, A is located at (1, 4) and B is located at (3, -6).

So the midpoint M of AB is:

= [(1 + 3) / 2, (4 + (-6)) / 2]

= [2, -1]

Hence, the midpoint of AB is at the point (2, -1).

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Answer: A 48 x 24/16 = r

Step-by-step explanation:

48 x 24 = the total number of drinks

Since there are 16 drinks in each row you want divide to get the number of rows in each drink.

Ex: 48 x 24= 1,152 (total drinks)

16 drinks in a row

1,152/16 = 72

72 rows

Answer:

a-no for the first triangle, because the total of the inner angles must be 180, the last angle must be 90.

b-yes- a triangle with angles 90, 50, and 40 is possible

c-yes. 65+45+70=180.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Simplify the expressionApply the logarithm power identityApply the logarithm product identitySimplify the expression2Exponentiate both sides

log10(4x8)=2log10(48)=2

48=102

x=258=258√

The function that expresses f(x) is f(x) = 1.99x + 3, and its graph is continuous.

A continuous function is one that doesn't have any sudden or unexpected changes in value. It means that if we make small enough changes to its input, the output will also change by a small amount.

We can express the total cost of an order as a function of the number of books sold, x, using the formula:

f(x) = 1.99x + 3

This function takes the number of books sold, multiplies it by the cost per book ($1.99), and adds a fixed fee of $3.

For example, if we sold 3 books, we can use the function to find the total cost:

f(3) = 1.99(3) + 3 = 9.97

This means that selling 3 books would cost $9.97 in total.

The graph of this function is continuous, which means that as we move along the x-axis, the y-values change smoothly and without any sudden jumps.

The term "continuous" refers to a function that can take on any value between two given points and can be broken down into infinitely many fractions and decimals.

In contrast, a discrete function can only take on specific values, usually whole numbers or integers.

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1. Calculating the output when using -6 as the input, we have:

Rule 1: subtract 7  = -13

Rule 2: square the input = 36.

Rule 3: divide by 3 = 12

Rule 4: divide the input into 3 = -2

Rule 5: write π = -2π

Rule 6: volume of a cube = -216cm³

A function rule is a mathematical expression that describes how to calculate the output value of a function based on its input value. It can be thought of as a set of instructions that tell you what to do with the input value to get the output value.

2. a. True. Speed is the ratio of distance traveled to the time taken to travel that distance. In this case, the distance is fixed at 100 meters, so speed is solely dependent on time. For each value of time, there is only one corresponding value of speed, making it a function.

b. False. While time and distance are related in the formula for speed, time is not a function of distance in this case. Each student's time is independent of the distance they cover, and the same distance is covered by all students (100 meters). Therefore, for any given distance, there are multiple possible values of time, making it not a function.

c. False. The number of students racing is not related to their speed. Each student's speed depends solely on their individual time, which is not affected by the number of other students racing.

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The Iength οf the diagοnaI οf the square exteriοr waII shοuId be 15 feet.

Accοrding tο the Pythagοrean theοrem, the square οf a right triangIe's hypοtenuse (the side οppοsite the right angIe) equaIs the sum οf its οther twο sides' squares.

Outside waII's Iength and width are equaI if it is a square. Let's caII this Iength "x"

Using the Pythagοrean theοrem, we can find the Iength οf the diagοnaI οf the square:

diagοnaI² = height² + width²

Since the height is 9 ft and οne side οf the square is 12 ft, we can write:

diagοnaI² = 9² + 12²

diagοnaI² = 81 + 144

diagοnaI² = 225

Taking the square rοοt οf bοth sides, we get:

diagοnaI = √(225)

diagοnaI = 15 ft

Therefοre, the Iength οf the diagοnaI οf the square exteriοr waII shοuId be 15 feet.

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Answer:

[tex]yeah \: 2 + 2 = 4[/tex]

The correct number line is: A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded.

The mathematical expression of a connection between two values, stating that one value is larger than, less than, or not equal to another value, is called an inequality. Common symbols for inequality include (less than), > (greater than), (less than or equal to), and (greater than or equal to). For instance, the inequality 2x + 3 > 7 denotes that the value of 2x + 3 is larger than 7. Depending on the circumstance, inequalities can be resolved algebraically or visually. A collection of values that meet the connection implied by the inequality is the answer to an inequality.

Here, the inequality is y - 2 < -5, which can be rewritten as y < -3.

Hence, the correct option is: A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded.

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Answer: Sally will earn $4,830.00 in interest over 7 years.

Step-by-step explanation:

If the interest is not compounded, then Sally will earn simple interest, which can be calculated using the formula:

I = P * r * t

where:

I = the interest earned

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

t = the time period, in years

In this case, we have:

P = $9,000 (the initial deposit)

r = 7% = 0.07 (the annual interest rate)

t = 7 (the number of years)

So, plugging in the values:

I = $9,000 * 0.07 * 7

I = $4,830.00

Therefore, Sally will earn $4,830.00 in interest over 7 years.

Answer:

A fraction tells you how many parts of a whole there are. When we find a fraction of an amount, we are working out how much that 'part' is worth within the whole. You can see fractions of amounts all around us

Step-by-step explanation:

i may be wrong but I tried

Around [tex]312.57[/tex] yards are required to complete a full lap of the circuit.

The entire movement of an item, independent of direction, is called distance. Regardless of an object's beginning or finishing position, distance may be defined as the amount of ground it has travelled.

As a result, distance is indeed a scalar quantity. The whole path an item has taken can be used to define the distance from it. Think about it this way. If a car drives 5 km east, then turns to head north for 8 km, the final distance driven by the automobile must be 13 km.

The circumference of a semicircle with diameter d is [tex](pi/2)d[/tex]. Therefore, the length of each semicircular section is [tex](pi/2) * 36 = 18pi[/tex].

The total distance around the track is the sum of the lengths of the two straight sections and the two semicircular sections.

Distance around track [tex]= 2(100 yards) + 2(18pi yards)[/tex]

[tex]= 200 yards + 36pi yards[/tex]

[tex]= 312.57 yards[/tex] (rounded to two decimal places)

Therefore, the distance around the entire track is approximately [tex]312.57[/tex]yards.

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Answer: Linear

Step-by-step explanation:

Since the difference between consecutive values of f(x) is constant, this suggests that the function could be linear. To confirm this, we can calculate the differences between consecutive values:f(1) - f(0) = -15 - (-18) = 3

f(2) - f(1) = -12 - (-15) = 3

f(3) - f(2) = -9 - (-12) = 3

f(4) - f(3) = -6 - (-9) = 3

Since the differences are constant and equal to 3, this confirms that the function is linear.

To find the equation of the linear function, we can use the slope-intercept form of a linear equation:

y = mx + bwhere m is the slope and b is the y-intercept.

We can use the first two points (0, -18) and (1, -15)  to find the slope:

m = (change in y) / (change in x) = (-15 - (-18)) / (1 - 0) = 3

Now we can use the slope and one of the points to find the y-intercept:

y = mx + b

-18 = 3(0) + b

b = -18

Therefore, the equation of the linear function is:

f(x) = 3x - 18

So the function type that could represent the values in the table is linear.