12. An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.

a. Write a linear function in the form y = mx + b that represents the total cost, y, in dollars, for a single order of x sweatshirts.

b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.​

we get that the wheel of Kathel's car makes about 146 more revolutions than the wheel of Nguye's lorry to cover a distance of 770 m.

The circumference of a circle is the distance around the outer boundary of the circle. It is the length of the boundary or the "perimeter" of the circle. The circumference is calculated using the formula C = 2πr, where C is the circumference, r is the radius of the circle, and π (pi) is a mathematical constant approximately equal to 3.14159. The circumference is measured in units of length, such as meters or feet.

by the question.

To solve this problem, we need to first find the circumference of each wheel:

[tex]Circumference of Nguyon's lorry wheel = 2πr = 2 x (22/7) x 28 cm = 176[/tex]cm

Circumference of Kathel's car wheel = πd = (22/7) x 42 cm = 132 cm

Next, we can calculate how many revolutions each wheel needs to make to cover a distance of 770 m:

Number of revolutions for Nguyon's lorry wheel = distance ÷ distance covered in one revolution

= 77000 cm ÷ 176 cm ≈ 437.5 revolutions

Number of revolutions for Kathel's car wheel = distance ÷ distance covered in one revolution

= 77000 cm ÷ 132 cm ≈ 583.3 revolutions

Finally, we can find the difference in revolutions between the two wheels:

Difference in revolutions = Number of revolutions for Kathel's car wheel - Number of revolutions for Nguyon's lorry wheel

= 583.3 - 437.5 ≈ 145.8 revolutions

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Answer:

To graph the function f(x) = 2sin(x) - 2 by hand, we can use the following steps:

Step 1: Find the amplitude

The amplitude of the function is the absolute value of the coefficient of the sine function, which is 2 in this case. Therefore, the amplitude is 2.

Step 2: Find the period

The period of the function is given by the formula 2π/b, where b is the coefficient of x in the sine function. In this case, b = 1, so the period is 2π/1 = 2π.

Step 3: Find the midline

The midline is the horizontal line that the graph oscillates around. For the sine function, the midline is the y-value of the function when there is no vertical shift. In this case, the midline is y = -2.

Step 4: Find the y-intercept

The y-intercept is the value of the function when x = 0. Plugging in x = 0, we get f(0) = 2sin(0) - 2 = -2. Therefore, the y-intercept is -2.

Step 5: Sketch the graph

Based on the amplitude, period, midline, and y-intercept we just found, we can sketch the graph of the function as follows:

- The graph oscillates between y = 2 and y = -6, with a midline at y = -2.

- The graph starts at (0, -2), then goes up to (π/2, 0), down to (π, -2), down further to (3π/2, -4), and back up to (2π, -2).

- The graph is symmetric about the midline.

Therefore, the information about the function is:

Amplitude: 2 (The distance from the midline to the maximum or minimum of the graph is 2.)

Period: 2π (The graph completes one full cycle over a horizontal distance of 2π.)

Midline: y = -2 (The horizontal line that the graph oscillates around.)

y-intercept: -2 (The value of the function when x = 0.)

According to the given information, the absolute value is |x - 7/6| = 5/6.

What is an algebraic expression?

A mathematical expression known as an algebraic expression can include variables, integers, and mathematical operations including addition, subtraction, multiplication, and division.

When expressing quantities or values that might change depending on the values given to the expression's variables, algebraic expressions are utilised. Expressions in algebra can include one or more variables and be simple or complex.

Two solutions x=1/2, x=-1/3

|x - 7/6| = 5/6.

We know that the absolute value of (1/2 - 7/6) equals the absolute value of (-1/3 - 7/6) because they are opposites and have the same absolute value. So we need an expression that equals 5/6 when either 1/2 - 7/6 or -1/3 - 7/6 is plugged in for x.

To get an expression that equals 5/6 when 1/2 - 7/6 is plugged in for x, we can subtract 7/6 from 1/2 and take the absolute value of the result:

|1/2 - 7/6| = |-1/6| = 1/6.

To get an expression that equals 5/6 when -1/3 - 7/6 is plugged in for x, we can subtract 7/6 from -1/3 and take the absolute value of the result:

|-1/3 - 7/6| = |-9/6| = 3/2.

Since both expressions have the same absolute value of 5/6, we can use either one as the absolute value expression in the equation. Thus,

|x - 7/6| = 5/6.

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Answer:(0,-1).

Step-by-step explanation:

As the diagonals of a parallelogram bisect each other, the midpoint of AC is the same as the midpoint of BD. Therefore, the fourth vertex D is (0,-1).

Since a 16-ounce bottle of orange juice contains 200 milligrams of vitamin C, we can find the amount of vitamin C in 4 ounces by using a proportion:

16 ounces     200 milligrams

4 ounces       x milligrams

Solving for x, we have:

x = 50 milligrams

So, Yosef drank 50 milligrams of vitamin C.

To find the percent of the daily recommended amount of vitamin C this represents, we can use another proportion:

80 milligrams     100%

50 milligrams      x%

Solving for x, we have:

x = (50/80) * 100 ≈ 62.5%

Therefore, Yosef drank orange juice containing 62.5% of the daily recommended amount of vitamin C.

The general solution of  function  f^(40)(x) = 0 if f(x)=x−4.

The general solution of a function is an expression that includes all possible solutions to a given differential equation or equation. It usually contains arbitrary constants that are determined by applying initial or boundary conditions.

The general solution allows for a range of solutions, rather than a single specific solution. This allows for flexibility in solving problems, as different initial or boundary conditions can result in different specific solutions.

We first find the general solution of f(x) by integrating the function 40 times.

f(x) = x - 4

f'(x) = 1

f''(x) = 0

f'''(x) = 0

f''''(x) = 0

and so on, until we get to:

f^39(x) = 0

Now, we can take the 40th derivative of the function to get the final answer:

f^(40)(x) = 0

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It will take nearly three years for the car's value to be half of its original price.

Resistance bands, loop bands, tube bands with handles, and cable machines can be used for single-arm rows

In a single-arm row exercise, a variety of equipment can be used to add resistance and challenge the muscles. Here are some examples of what can be submitted for bands in a single-arm row

Resistance bands: Resistance bands are a popular option for single-arm rows. They come in different colors to indicate the level of resistance and can be easily adjusted to increase or decrease the difficulty. To perform a single-arm row with a resistance band, step on the center of the band with one foot and grasp the other end with one hand

Loop bands: Loop bands are a type of resistance band that is formed into a loop. They can be placed around the wrists or forearms to provide resistance during a single-arm row. To perform a single-arm row with a loop band, loop the band around your hand and hold onto the other end with your other hand.

Tube bands with handles: Tube bands with handles are another option for single-arm rows. These bands have handles on each end, allowing for a more comfortable grip. To perform a single-arm row with a tube band, attach one end of the band to a stationary object and hold onto the other end with one hand.

Cable machines: Cable machines are a piece of gym equipment that can also be used for single-arm rows. To perform a single-arm row on a cable machine, adjust the weight to the desired resistance and attach a handle to the cable.

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The likelihood of being over 67 inches tall is 0.127978, or 12.8%.

The probability of an event occurring is referred to as probability. We frequently have to make predictions about the future in real life. We may or may not be aware of the outcome of an event.At the point when this happens, we broadcast that there is plausible that the occasion will happen.

In conclusion, probability can be put to great use in business as well as in this rapidly expanding field of artificial intelligence.

By simply dividing the total number of outcomes by the favorable number of possibilities, the probability of an event can be calculated using the probability formula.

5 feet 7 inches equals (5*12 +7) inches, or (60 +7) inches, according to our question.

(67 - 69.1)/(2.65) = -1.136 is the z-score.

The probability of having a height greater than 67 inches is 0.127978, or 12.8%.

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The solution to the inequality 2.4/n < 6 is n > 0.4, and the only value from the given options that satisfies this inequality is 0.5.

What is inequality ?

An inequality is a mathematical statement that compares two values or expressions using inequality symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).

Inequalities express a relationship between two quantities, indicating whether one quantity is less than, greater than, or equal to the other quantity.

We can solve the given inequality as follows:

2.4/n < 6

Multiplying both sides by n (note that n cannot be negative):

2.4 < 6n

Dividing both sides by 6:

0.4 < n

So, any value of n that is greater than 0.4 will be a solution to the inequality.

Out of the given options, only 0.5 is greater than 0.4, so 0.5 is the solution to the inequality.

Therefore, the solution to the inequality 2.4/n < 6 is n > 0.4, and the only value from the given options that satisfies this inequality is 0.5.

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Joni is not correct because she used the simple interest rate formula, not the compound interest rate formula. She will have $231.53.

A loan or investment's principle amount is the sole amount used to compute simple interest. Any interest that has been earned on interest that has accrued over time is not taken into account by simple interest.

Contrarily, compound interest is a sort of interest that takes into account both the original principal and any interest that has accrued over time. As a result, the interest gained during one period is added to the principle for the following month, where it continues to earn interest.

Joni is not correct, because $230 is just the interest she will earn; she will have $430 total.

The compound interest is given by the formula:

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

Substituting the values we have:

[tex]A = 200(1 + 0.05/1)^{(1*3)}\\A = 200(1.05)^3[/tex]

A = 200(1.157625)

A = $231.53

Hence, Joni will have a total balance of $231.53 after 3 years.

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The complete question is:

All three methods give the same solutions. So, the cost of a chocolate truffle is (4C1 - C2) / 18 and the cost of a caramel turtle candy is (9C1 - 20C2).

An equation is a mathematical statement that shows the equality between two expressions. It consists of an equal sign between two expressions, which are usually separated by variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Here,

Let x be the cost of a chocolate truffle and y be the cost of a caramel turtle candy. From the given information, we can write the following system of equations:

5x + 2y = C1 (Cost of Chocolate Lovers Bag)

2x + 8y = C2 (Cost of Combusting Caramel Bag)

To solve the system of equations, we can use various strategies. Here are a few:

Method 1: Substitution

We can solve for x or y in one equation and substitute the result into the other equation to eliminate one variable. For example, we can solve for x in the first equation:

5x + 2y = C1

5x = C1 - 2y

x = (C1 - 2y) / 5

Substituting this into the second equation:

2((C1 - 2y) / 5) + 8y = C2

Simplifying and solving for y:

C1/5 - 4y/5 + 8y = C2

y = (5C2 - C1) / 26

Substituting this value of y into the first equation and solving for x:

5x + 2((5C2 - C1) / 26) = C1

x = (3C1 - 10C2) / 65

Therefore, the cost of a chocolate truffle is (3C1 - 10C2) / 65 and the cost of a caramel turtle candy is (5C2 - C1) / 26.

Method 2: Elimination

We can multiply the first equation by 4 and the second equation by -1, and add them to eliminate y:

20x + 8y = 4C1

-2x - 8y = -C2

Adding these equations gives:

18x = 4C1 - C2

x = (4C1 - C2) / 18

Substituting this into the first equation and solving for y:

5((4C1 - C2) / 18) + 2y = C1

y = (9C1 - 20C2) / 36

Therefore, the cost of a chocolate truffle is (4C1 - C2) / 18 and the cost of a caramel turtle candy is (9C1 - 20C2) / 36.

Method 3: Matrix algebra

We can write the system of equations in matrix form:

[5 2] [x] [C1]

[2 8] [y] = [C2]

Using matrix algebra, we can solve for the variables:

[x] [C1] [8 -2] [y]

[y] = [C2] [-2 5 ] [x]

Multiplying the matrices, we get:

[x] [8x - 2y] [C1]

[y] = [-2x + 5y] * [C2]

Solving for x and y:

x = (4C1 - C2) / 18

y = (9C1 - 20C2) / 36

Therefore, the cost of a chocolate truffle is (4C1 - C2) / 18 and the cost of a caramel turtle candy is (9C1 - 20C2) / 36.

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Answer:

$104.11

Step-by-step explanation:

To calculate the monthly payment for a loan, we can use the following formula:

Monthly payment = (Pr(1+r)^n) / ((1+r)^n-1)

where:

P = the principal amount (the amount borrowed)

r = the monthly interest rate (annual interest rate divided by 12)

n = the number of payments (total number of years multiplied by 12)

In this case, the principal amount is $3825, the annual interest rate is 15%, and the loan term is 4 years. So we have:

P = $3825

r = 0.15/12 = 0.0125

n = 4*12 = 48

Substituting these values into the formula, we get:

Monthly payment = (38250.0125(1+0.0125)^48) / ((1+0.0125)^48-1)

Monthly payment ≈ $104.11

Therefore, the monthly payment for a $3825 loan at 15% annual interest for 4 years is approximately $104.11.

the measure of angle V is approximately 21.87 degrees. To find the measure of angle V, we need to use the fact that the sum of the angles in a triangle is 180 degrees. Therefore, we can write:

m∠V + m∠W + m∠X = 180

where angle X is the third angle in the triangle formed by V, W, and X. We know that m∠W is (3x+4) degrees, and we can see that angle X is adjacent to angle V, so we can use the fact that the sum of adjacent angles is equal to the total angle to write:

m∠V + m∠X = (2x+2) degrees

Now we can substitute this expression for m∠X into the first equation to eliminate m∠X:

m∠V + (2x+2 - m∠V) + (3x+4) = 180

Simplifying and solving for x, we get:

5x + 6 = 180

5x = 174

x = 34.8

Now we can substitute this value of x back into our expression for m∠V + m∠X to find m∠V:

m∠V + (2(34.8)+2 - m∠V) = 71.6

Simplifying and solving for m∠V, we get:

3m∠V = 65.6

m∠V = 21.87 (rounded to two decimal places)

Therefore, the measure of angle V is approximately 21.87 degrees.

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To model the given data points, we can use a linear function of the form y = mx + b.

First, we need to find the slope (m) of the line:

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

m = (-6 - (-2)) / (1 - 0)

m = -4/1

m = -4

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

y = mx + b

-2 = (-4)(0) + b

b = -2

Therefore, the linear function that models the given data is:

y = -4x - 2

Answer:

14

Step-by-step explanation:

Rewrite the equation: 4*4+3-5

Multiply: 16+3-5

Add: 19-5

Subtract: 14

The coordinates of point C are (17, -1).

let say a line have two points A (x1,y1) and B(x2,y2)

slope of line AB = (y2 - y1) / (x2 - x1)

Now given that  points B, C, and D are collinear, then they lie on the same straight line. This means that the slope of the line passing through any two of the points will be the same as the slope of the line passing through the other two points. We can use this fact to find the value of x.

The slope which is between points B and C is:

slope of line BC = (y2 - y1) / (x2 - x1) = (-1 - 7) / (x - (-1)) = -8 / (x + 1)

The slope which is between points C and D is:

slope of line CD = (y2 - y1) / (x2 - x1) = (8 - (-1)) / (1 - x) = (9) / (1 - x)

Since B, C, and D are collinear, slope of BC = slope of CD. Therefore:

-8/(x+1) = 9/(1-x)

by solving the x, we get:

-8(1 - x) = 9(x + 1)

-8+8x=9x + 9

x = 17

So, the coordinates of point C are (17, -1).

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The ratios are not the same for all values of x and y, we can conclude that x and y are not proportional.

Ratio and proportion are two mathematical concepts that are closely related.

A ratio is a way of comparing two or more quantities by dividing one quantity by another. It expresses the relative size or magnitude of two quantities. Proportion, on the other hand, refers to the equality of two ratios. It states that two ratios are equal. Proportions can be used to solve problems involving unknown quantities.

To determine whether x and y are proportional, we need to check if the ratio of corresponding values of x and y is always the same.

Let's find the ratio of y to x for each corresponding value:

y/x for x=1: 7/1 = 7

y/x for x=2: 8/2 = 4

y/x for x=3: 9/3 = 3

Since the ratios are not the same for all values of x and y, we can conclude that x and y are not proportional.

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Answer:

The correct responses are:

A) S(10)=S(5) means that there were the same number of students in 2010 as in 2005.

E) S(0) = 2,000 means that there were no students in the year 2000.

B and D are not plausible statements because they imply that there is an inverse relationship between t and S(t), meaning that the number of students decreases as t increases. Also, C is incorrect as it implies that there were fewer students in 2015 than there were in 2010, which contradicts the nature of the situation.

Answer:

The expression that would be used to find the sum of the products in the area model would be:

C) 600 + 60 + 400 + 40 = 1100

(b) The difference in sample mean is 0.544 ml. (c) Using StatKey, we find the p-value to be 0.046. (d) It is unlikely that this difference is due to random chance alone.

(a) The null hypothesis is that there is no difference in average brain size between football players with no history of concussions (FBNoConcuss) and football players with a history of concussions (FBConcuss). The alternative hypothesis is that the average brain size is larger in FBNoConcuss than in FBConcuss.

H0: μFBNoConcuss = μFBConcuss

HA: μFBNoConcuss > μFBConcuss

(b) The relevant sample statistic is the difference in sample mean between the two groups.

Therefore, the difference in sample mean is 0.544 ml.

(c) Using StatKey, we find the p-value to be 0.046.

(d) With a p-value of 0.046, which is less than the commonly used significance level of 0.05, we reject the null hypothesis and conclude that there is evidence to suggest that the average brain size is larger in FBNoConcuss than in FBConcuss. It is unlikely that this difference is due to random chance alone.

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Given the rectangle's dimensions, its area is 12 cm² and the circle's area is π cm². Thus, the shaded region's area is about 8.57 cm².

The area of the shaded region is equal to the area of the rectangle minus the area of the circle.

The area of the rectangle is given by:

Area of rectangle = length x width = 4 cm x 3 cm = 12 cm²

The area of the circle is given by:

Area of circle = π x radius² = π x (1 cm)² = π cm²

Therefore, the area of the shaded region is:

Area of shaded region = Area of rectangle - Area of circle

Area of shaded region = 12 cm² - π cm²

Area of shaded region ≈ 8.57 cm² (rounded to the nearest hundredth)

Hence, the area of the shaded region is approximately 8.57 cm².

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the missing figure is in the image attached below

Answer:

132 seconds aka 2.2 minutes

Step-by-step explanation:

they met at 132 seconds then converted to minutes which is 2.2 minutes

12,24,36,48,60,72,84,96,108,120,132

11,22,33,44,55,66,77,88,99,110,121,132

Answer:

270 students walked to school

30% students walked to school

Step-by-step explanation:

270 /900 is 30%

Answer:

Step-by-step explanation:

first mark the 30% line. then divide 900 by 0.3 to get 30% of the students (270)

The relationship between A and t is exponential, meaning that as time increases, the number of bacteria grows at an accelerating rate.

An exponential function is a mathematical function of the form:

f(x) = a^x

where "a" is a positive constant called the base, and "x" is the independent variable. The base "a" is usually a number greater than 1, but it can also be a fraction between 0 and 1.

Exponential functions have some key properties that make them useful for modeling many natural phenomena. One of the most important properties is that they exhibit exponential growth or decay, meaning that the rate of change of the function is proportional to the current value of the function.

In the given question,

In the exponential function A=2e^(0.3t), the value of A depends on the value of t. Specifically, A is a function of t, where t is the independent variable and A is the dependent variable. This means that as the value of t changes, the value of A changes as well.

The function describes the growth of some bacteria, where A represents the number of bacteria at time t. The value of t represents the time elapsed since the start of the observation period. The function suggests that the growth rate of the bacteria is proportional to the current number of bacteria, with a growth rate of 0.3 per unit time.

The relationship between A and t is exponential, meaning that as time increases, the number of bacteria grows at an accelerating rate. This relationship can be observed by plotting the function on a graph, where A is plotted on the y-axis and t is plotted on the x-axis. The resulting curve will be a steeply upward-sloping curve that gets steeper and steeper as time goes on.

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84 sq ft area does jamie need to cover.

Describe Area?

The region enclosed by an object's form is referred to as the area. The area of the form is the area that the figure or any other two-dimensional geometric shape occupies in a plane.

                        A two-dimensional shape's area, which is expressed in square units, is the total surface area it may occupy. The square metre (m2), a derived measure, serves as the area unit in the SI system. On a sheet of paper, draw a square using a pencil. It has two dimensions. Its Area refers to the area that the form occupies on the paper.

the area of the walkway is:

Area of walkway = length x width = 8 ft x 3.5 ft = 28 sq ft

Next, let's calculate the area of the patio. The patio measures 7 feet from top to bottom and 8 feet across. Therefore, the area of the patio is:

Area of patio = length x width = 7 ft x 8 ft = 56 sq ft

Now, to find the total area that Desiree needs to cover, we just add the area of the walkway and the area of the patio:

Total area = area of walkway + area of patio

Total area = 28 sq ft + 56 sq ft

Total area = 84 sq ft

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The probability of selecting a striped marble, not replacing it, then selecting a black marble is 4/15.

The probability of selecting a striped marble on the first draw is 6/10, since there are 6 striped marbles out of a total of 10 marbles in the bag.

After the first marble is drawn, it is not replaced, so there are now 9 marbles left in the bag. Out of these, there are 4 solid marbles and 5 striped marbles. Therefore, the probability of selecting a black marble on the second draw is 4/9.

To find the probability of both of these events occurring together, we multiply the probabilities

P(striped, then black) = P(striped) × P(black | striped not replaced)

P(striped, then black) = (6/10) × (4/9)

P(striped, then black) = 24/90

P(striped, then black) = 4/15

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The given question is incomplete, the complete question is:

There are 10 marbles in a bag (4 solid, 6 striped) .what is the probability of selecting a striped marble, not replacing it, then selecting a black marble?

Answer: [-4.5  -3.5]

                 [-4     -3]

Step-by-step explanation:

Music and a matrix calculator online, mwah

Convert the rectangular equation to polar form

3x-y+2=0

To convert the rectangular equation 3x - y + 2 = 0 to polar form, we can substitute x = r cos(theta) and y = r sin(theta), where r is the radius and theta is the angle in polar coordinates. This gives:

3(r cos(theta)) - (r sin(theta)) + 2 = 0

Simplifying the equation, we get:

r(3 cos(theta) - sin(theta)) = -2

Dividing both sides by the expression in the parentheses:

r = -2 / (3 cos(theta) - sin(theta))

This is the polar form of the equation.

Convert the rectangular equation to polar form

12.3x-y+2=0

13. xy=4

14. (x+y)-9(x-2)=0 72-36=0

15. y²-8x-16-0 (4-4)(3-4)=0

For the rectangular equation xy = 4, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:

r cos(theta) * r sin(theta) = 4

r^2 sin(theta) cos(theta) = 4

Using the identity 2 sin(theta) cos(theta) = sin(2theta), we can simplify the equation to:

r^2 sin(2theta) = 8

r = sqrt(8 / sin(2theta))

This is the polar form of the equation.

For the rectangular equation (x+y) - 9(x-2) = 0, we can simplify it to:

x - 8y + 18 = 0

Then, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:

r cos(theta) - 8r sin(theta) + 18 = 0

Simplifying the equation, we get:

r = 18 / (cos(theta) - 8sin(theta))

This is the polar form of the equation.

For the rectangular equation y^2 - 8x - 16 = 0, we can complete the square to get:

y^2 - 8x - 16 = (y - 0)^2 - 16 - 8x

Simplifying the equation, we get:

(y - 0)^2 = 8x + 16

Using the substitution x = r cos(theta) and y = r sin(theta), we get:

r^2 sin^2(theta) = 8r cos(theta) + 16

r^2 sin^2(theta) - 8r cos(theta) - 16 = 0

This equation does not simplify nicely into a standard form of polar equation, but it is still a valid polar form.

16. r = 4 sin θ

17.θ= (π/6)

18. r² = sin 2θ

19. r = 6/(2-3 sin θ)

To convert the polar equation r = 4 sin(theta) to rectangular form, we can use the following trigonometric identities:

sin(theta) = y / r

cos(theta) = x / r

Substituting these into the polar equation, we get:

r = 4 sin(theta)

r = 4 y / r

r^2 = 4 y

x^2 + y^2 = 4 y

This is the rectangular form of the equation.

The equation theta = pi/6 represents a line at an angle of pi/6 radians (30 degrees) from the positive x-axis in the polar coordinate system. In rectangular coordinates, this line has the equation y = x tan(pi/6) = x/sqrt(3).

To convert the polar equation r^2 = sin(2theta) to rectangular form, we can use the following trigonometric identities:

sin(2theta) = 2 sin(theta) cos(theta)

sin(theta) = y / r

cos(theta) = x / r

Substituting these into the polar equation, we get:

r^2 = sin(2theta)

r^2 = 2 sin(theta) cos(theta)

r^2 = 2 (y / r) (x / r)

x^2 + y^2 = 2xy

This is the rectangular form of the equation.

To convert the polar equation r = 6 / (2 - 3 sin(theta)) to rectangular form, we can first substitute sin(theta) = y / r and cos(theta) = x / r, giving:

r = 6 / (2 - 3y / r)

r(2 - 3y / r) = 6

2r - 3y = 6

This is the rectangular form of the equation.

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