Graph the polar equation and determine about which axis it is symmetric r=2 sin 3 theta

The quotient of  10⁻⁴/10² is 10⁻⁶

Quotient is the result obtained by dividing one quantity by another. It represents the value that is obtained after division.

For example, if you divide 20 by 2, the quotient is 10 because 20 divided by 2 equals 10.

Division law states that:

10ᵃ / 10ᵇ = 10ᵃ⁻ᵇ

In this case have:

10⁻⁴/10² = 10⁻⁴⁻²   (Division law)

10⁻⁴/10² = 10⁻⁶

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Answer:

a. Three students preferred swimming over running and walking.

b. Fourteen students chose running or walking.

c. Alex and Jose preferred swimming and walking, but not running.

Answer:

oo = infinity btw

Decreasing: (-oo, 1)

It is going down in that interval

The ratio of total cholesterol to HDL cholesterol can be found by dividing the total cholesterol by the HDL cholesterol. So, the patient's ratio of total cholesterol to HDL cholesterol is:185 mg/dl ÷ 40 mg/dl = 4.625 mg/dl (dividing both numerator and denominator by 5)⇒ 37/8 (after simplification).

Thus, the patient's ratio of total cholesterol to HDL cholesterol is 37/8

A) The interval where the graph is increasing is: (0,0) to (40, 4)

The  interval where the graph remains constant is from: (120, 3) to (140, 3).

The  interval where the graph is decreasing in depth is from: (140, 3) to (180, 0).

B) The domain is from 0 to 180 minutes.

The range of the graph is from 0 to 4 feet.

C) Yes, the graph represents a function

D) The graph depicts the filling of the pool from 0 to 4 feet, followed by a period of constant depth, and finally, the draining of the pool back to 0 feet.

Part A:

- Increasing interval:

The interval where the graph is increasing is from the starting point coordinate (0,0) to the coordinate (40, 4)

- Constant interval:

The  interval where the graph remains constant at a depth of 4 feet which is from (40, 4) to (80, 4) and then from where the depth is 3 feet which is from the coordinates (120, 3) to (140, 3).

- Decreasing interval:

The  interval where the graph is decreasing  in depth is from the coordinates (140, 3) to (180, 0).

Part B:

- Domain: The domain of the relation is the set of time values represented on the x-axis. The domain here is from 0 to 180 minutes.

- Range: The range of the relation is the set of depth values represented on the y-axis. The range of the graph is from 0 to 4 feet.

Part C:

Yes, the graph represents a function. This is because, each input (x-value) is associated with a unique output (y-value).

Part D:

The graph represents the depth of water in a kids' swimming pool over a certain amount of time between 0 and 180 minutes.

At the beginning, the pool is empty, and then the depth slowly increases with time. From 0 to 40 minutes, the depth increases at a constant rate until it reaches 4 feet. Thereafter, it remains constant at that depth for the next 40 minutes, indicating the pool is full. Afterward, there is a gradual decrease in depth over time. From 140 to 180 minutes, the water is drained from the pool, and the depth eventually reaches 0 feet.

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Answer:

Area of the shaded region= 4 units

Step-by-step explanation:

width (x-axis)= 3-1 = 2

length (y-axis) = 2-0 = 2

Area of rectangle= Length × Width

= 2 × 2 = 4

The coordinates of the vertex of the parabola are (3, -7). The focus of the parabola is located at (3, -3). The equation of the directrix is y = -11.

The given equation of the parabola is in the form (x - h)^2 = 4p(y - k), where (h, k) represents the vertex and p represents the distance between the vertex and the focus/directrix.

Comparing the given equation with the standard form, we can see that the vertex is at (3, -7).

The coefficient 4p in this case is 16, so p = 4. Since the parabola opens upward, the focus will be p units above the vertex. Therefore, the focus is located at (3, -7 + 4) = (3, -3).

To find the directrix, we need to consider the distance p below the vertex. Since the parabola opens upward, the directrix will be p units below the vertex. Hence, the equation of the directrix is y = -7 - 4 = -11.

In summary, the coordinates of the vertex are (3, -7), the focus is located at (3, -3), and the equation of the directrix is y = -11.

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In mathematics, a horizontal asymptote is a horizontal line that a function approaches as the input values (usually x) approach positive or negative infinity. It describes the behavior of a function as its input values become extremely large or small.

For a rational function, which is a function expressed as the ratio of two polynomials, the degree of the numerator and denominator determines the existence and location of horizontal asymptotes. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (y = 0).

If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

For example, consider the function [tex]f(x) = (3x^2 + 2x + 1) / (2x^2 + 5)[/tex]. In this case, both the numerator and denominator have a degree of 2. Therefore, the horizontal asymptote is the ratio of the leading coefficients: y = 3/2.

Horizontal asymptotes help understand the long-term behavior of functions and are useful for graphing and analyzing mathematical models in various fields, including economics, physics, and engineering.

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A. True.

In a two-column proof, the left column consists of statements, which are the facts or assumptions that lead to the proof of the theorem, while the right column consists of the reasons or justifications that explain how each statement logically follows from the previous one. Therefore, the left column states your reasons is a true statement.

The answer to the student's question is True. In a two-column proof, the left column contains the statements (steps) and the right column contains the corresponding reasons (justifications).

The answer to the question, 'True or false? In a two-column proof, the left column states your reasons', is A. True.

A two-column proof is organized into two columns. The left column contains the 'Statements' and the right column contains the corresponding 'Reasons'. The 'Statements' are the steps that lead to the conclusion of the proof, while the 'Reasons' justify each of those steps according to rules or laws of mathematics.

For example, if we want to prove that the opposite angles of a parallelogram are equal. The left column (Statements) could contain the first step 'ABCD is a parallelogram' and the right column (Reasons) would give the explanation 'Given'. Therefore, in a two-column proof, the left column does represent statements, not reasons.

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The graph that represents the solution of y ≤ x² - 1 and x > y² - 3 is option D.The graph that represents the solution of y ≤ x^2 – 1 and x > y^2 – 3 is D.

In order to solve the given inequality y ≤ x² - 1 and x > y² - 3, we need to follow these steps:Firstly, we will rewrite the given inequality in standard form as y² - x + 3 ≤ 0 and x² - y - 1 ≥ 0.Then we need to find the solutions of the above two inequalities. By solving these two inequalities we get the region which is common to both the inequalities and we can plot it on the graph paper.In the first inequality y² - x + 3 ≤ 0, the coefficient of x is negative which means the parabola will be open downwards.

On plotting the parabola we can observe that it passes through (0, -3) and (-2, -1). Also, we can see that it intersects the x-axis at x = 3. Therefore, the solution to the inequality y² - x + 3 ≤ 0 is the region below the parabola including the boundary.In the second inequality x² - y - 1 ≥ 0, the coefficient of y is negative which means the parabola will be open downwards. On plotting the parabola we can observe that it passes through (0, -1) and (-2, 1). Also, we can see that it intersects the y-axis at y = -1.

Therefore, the solution to the inequality x² - y - 1 ≥ 0 is the region above the parabola including the boundary.Now, we need to find the common region between the two solutions obtained above. The common region will be the solution to the inequality y ≤ x² - 1 and x > y² - 3. By observing the graphs we can see that the common region is the shaded region inside the first parabola and outside the second parabola.The correct option is D.

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The y-intercept of the linear function means that her initial distance from the finish line is of 10 kilometers.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

In which:

The graph crosses the y-axis at y = 10, hence the intercept b is given as follows:

b = 10.

The y-values represent the distance in the context of this problem, hence the initial distance is of 10 km.

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Using the ratio of the sides, we have:$x\sqrt{3} = 12\sqrt{3}$ (opposite the $60^{\circ}$ angle is 12$\sqrt{3}$)$x = 2\sqrt{3}\cdot6$ (the hypotenuse is $2x = 12\sqrt{3}$)Simplifying, we have:$x = 12\sqrt{3}$. Therefore $x=y=12\sqrt{3}$, which is our answer

In a 30-60-90 triangle, the sides have the ratio of $1: \sqrt{3}: 2$. Let's apply this to solve for the variables in the given problem.

Instructions: Use the ratio of a 30-60-90 triangle to solve for the variables. Leave your answers as radicals in simplest form. x=y=Let's first find the ratio of the sides in a 30-60-90 triangle.

Since the hypotenuse is always twice as long as the shorter leg, we can let $x$ be the shorter leg and $2x$ be the hypotenuse.

Thus, we have: Shorter leg: $x$Opposite the $60^{\circ}$ angle: $x\sqrt{3}$ Hypotenuse: $2x$

Now, let's apply this ratio to solve for the variables in the given problem. We know that $x = y$ since they are equal in the problem.

Using the ratio of the sides, we have:$x\sqrt{3} = 12\sqrt{3}$ (opposite the $60^{\circ}$ angle is 12$\sqrt{3}$)$x = 2\sqrt{3}\cdot6$ (the hypotenuse is $2x = 12\sqrt{3}$)Simplifying, we have:$x = 12\sqrt{3}$

Therefore, $x=y=12\sqrt{3}$, which is our answer.

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The net price factor for the CDs purchased by Hard Blues Music is approximately 0.91258.

The complement of the trade discounts provided by the wholesaler must be calculated.

The trade discounts of 4/3/2 signify price reductions of 4% for the initial discount, 3% for the subsequent discount, and then 2% for the third discount.

Let's perform a step-by-step analysis of the net price factor:

Second discount: 3% off the purchase price following the first reduction

Discount amount = 3% * $29,760 = $892.80

Price after the second discount = $29,760 - $892.80 = $28,867.20

Third discount: 2% off the price after the second discount

Discount amount = 2% * $28,867.20 = $577.34

Price after the third discount = $28,867.20 - $577.34 = $28,289.86

Now, let's calculate the net price factor by dividing the final price by the list price:

Net Price Factor = Price after discounts / List price

= $28,289.86 / $31,000

≈ 0.91258

So, the net price factor for the CDs purchased by Hard Blues Music is approximately 0.91258.

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B alone can finish the work in 15 days.

Let's assume that B takes x days to finish the work alone.

If A takes 5 days less than B to finish the work, then A takes (x - 5) days to finish the work alone.

We are given that A and B working together can finish the work in 6 days.

Using the concept of work rates, we can set up the following equation:

1/(x - 5) + 1/x = 1/6

To solve this equation, we can find a common denominator:

(x + x - 5)/(x(x - 5)) = 1/6

Simplifying the equation:

[tex](2x - 5)/(x^2 - 5x) = 1/6[/tex]

Cross-multiplying:

[tex]6(2x - 5) = x^2 - 5x[/tex]

[tex]12x - 30 = x^2 - 5x[/tex]

Rearranging the equation:

[tex]x^2 - 17x + 30 = 0[/tex]

Factoring the quadratic equation:

(x - 2)(x - 15) = 0

Setting each factor equal to zero:

x - 2 = 0   or   x - 15 = 0

Solving for x:

x = 2   or   x = 15

We discard the solution x = 2 since it suggests that B can complete the work in two days, which contradicts the fact that A takes five days fewer than B.

Therefore, B alone can finish the work in 15 days.

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Answer:

See below

Step-by-step explanation:

When [tex]-\frac{\pi}{2} < \theta < \frac{\pi}{2}[/tex], the x-coordinate of the rotated point will be positive since they cover Quadrants I and IV, both of which have positive x-values.

When [tex]\frac{\pi}{2} > \theta > \frac{3\pi}{2}[/tex], the x-coordinate of the rotated point will be negative since they cover Quadrants II and III, both of which have negative x-values.

The x-coordinate of the rotated point will be equal to cosθ since both of these values depend on the angle of θ.

cos(90°) = 0, cos(180°) = -1, and cos(270°) = 0

This is why referring to a unit circle is crucial in understanding this material.

Out of the 225 people surveyed, 135 said they shop online, 27 said they always get their order delivered, and 54 said they had loyalty cards.

To calculate the number of people who said they shop online, we can use the percentage given. If 60% of the 225 people surveyed said they prefer to shop online, we can find the number by multiplying the total number of people surveyed by the percentage:

Number of people who shop online = 225 * 0.60

Number of people who shop online = 135

Therefore, 135 people said they shop online.

To find the number of people who have shopping delivered, we need to calculate 20% of the number of people who shop online:

Number of people with shopping delivered = 135 * 0.20

Number of people with shopping delivered = 27

Hence, 27 people said they always get their order delivered.

Now let's calculate the number of people who had loyalty cards. First, we need to find the number of people who said they go in-store to shop. This is 40% of the total surveyed:

Number of people who go in-store to shop = 225 * 0.40

Number of people who go in-store to shop = 90

Out of these 90 people, only 60% said they had loyalty cards:

Number of people with loyalty cards = 90 * 0.60

Number of people with loyalty cards = 54

Therefore, 54 people said they had loyalty cards.

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The percentage of the world population that is in the U.S. is approximately 4.3%.

To find the percentage of the world population that is in the U.S., we need to divide the population of the U.S. by the world population and then multiply the result by 100.

Let's calculate the percentage:

Percentage = (U.S. population / World population) * 100

Given that the U.S. population is approximately 310 million and the world population is approximately 7.2 billion, we can substitute these values into the equation:

Percentage = (310 million / 7.2 billion) * 100

To perform the calculation, we need to convert the populations to the same units. Let's convert 310 million to billion:

310 million = 310 / 1000 billion

          = 0.31 billion

Now, let's substitute the converted values:

Percentage = (0.31 billion / 7.2 billion) * 100

Simplifying the equation:

Percentage = 0.04305555556 * 100

          = 4.305555556

Rounding the result to the nearest tenth of a percent, the percentage of the world population that is in the U.S. is approximately 4.3%.

Therefore, approximately 4.3% of the world population is in the U.S.

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The graph of the function [tex]y = \sqrt{x} - 3[/tex] is given by the image presented at the end of the answer.

The function for this problem is defined as follows:

[tex]y = \sqrt{x} - 3[/tex]

The parent function is defined as follows:

[tex]y = \sqrt{x}[/tex]

The subtraction by 3 means that the function was translated 3 units down.

The function is [tex]y = \sqrt{x} - 3[/tex]

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Answer:

[tex]2y+11x=9[/tex]

Step-by-step explanation:

[tex](1+y)^2-y^2=-11x+10\\1+2y+y^2-y^2=-11x+10\\1+2y=-11x+10\\2y=-11x+9\\2y+11x=9[/tex]

Recall the standard form of a linear equation is [tex]Ax+By=C[/tex].

Answer:

10

Step-by-step explanation:

d = √[(x_2 - x_1)² + (y_2 - y_1)²]

d = √[(2 - 8)² + (10 - 2)²]

d = √[(-6)² + 8²]

d = √(36 + 64)

d = √100

d = 10

Answer: 10

For the ordinal number, this statement means;

1. "α < β if an only if α ∈ β": This is saying that an ordinal number α is less than an ordinal number β if and only if α is a member of β and α is a subset of β.

2.  "α = {β : β < α}" can be interpreted as: "α is the set of all ordinals β such that β is less than α".

1. "α < β if an only if α ∈ β": This is saying that an ordinal number α is less than an ordinal number β if and only if α is a member of β.

In other words, α is less than β if and only if α is contained within β and every element of α is less than or equal to every element of β.

This is a fundamental property of ordinal numbers and is based on the von Neumann definition of ordinals where each ordinal is the well-ordered set of all smaller ordinals.

For an example of how this definition can be used. Let α = 1 and β = 2. Then, α is a member of β because 1 is a natural number and 2 is a natural number.

Additionally, α is a subset of β because every element of α, which is just 1, is less than or equal to every element of β, which is just 2. Therefore, α < β according to the definition.

2.  "α = (β : β < α)" is a definition of ordinal numbers in set theory. It says that an ordinal number α is the set of all ordinal numbers that are less than α. In other words, α is the set of all ordinal numbers that are smaller than α.

For example, when we use the von Neumann definition, the ordinal number 3 would be represented as the set {0, 1, 2} because 0, 1, and 2 are the ordinal numbers less than 3.

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Answer:

Step-by-step explanation:

Rotation symmetry can be also defined as radial symmetry.  This is because no matter how many degrees you turn a figure or shape, it will still be the same figure/shape.

See attachment for an example.

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The best measure of spread to compare data sets depends on the data's nature and the analysis goals. For categorical data, the mode is appropriate. For numerical data, measures like standard deviation, range, or interquartile range (IQR) are commonly used.

In order to determine the best measure of spread to compare the data sets, we need to understand the characteristics of the data and the specific goals of the comparison.

If the data sets are categorical and consist of non-numeric values, such as different types of cakes, using the mode would be appropriate. The mode represents the most frequently occurring value and provides insight into the dominant categories within the data sets.

However, if the data sets are numerical, measures such as standard deviation, range, and interquartile range (IQR) are commonly used to analyze the spread of the data.

The standard deviation measures the dispersion of values around the mean. It provides a measure of the average deviation from the mean, making it useful when the data follows a roughly normal distribution.

The range is the simplest measure of spread, representing the difference between the maximum and minimum values in the data sets. It provides a quick indication of the overall spread, but it is sensitive to outliers.

The interquartile range (IQR) is the range between the first quartile (25th percentile) and the third quartile (75th percentile) of the data. It is resistant to outliers and provides a measure of the spread of the middle 50% of the data.

To determine the best measure of spread, you need to consider the characteristics of your data and the specific context of your analysis. There is no universally "best" measure, as it depends on the nature of the data and the purpose of the comparison.

In summary, the best measure of spread to use for comparing the data sets could be the standard deviation, range, or interquartile range (IQR), depending on the nature of the data and the goals of the comparison.

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Answer:

Step-by-step explanation:

1 oz = $1152

1 oz = 28 grams

0.1 oz = 2.8 grams

21/2.8 = 7.5

0.1 x 7.5 oz = (2.8 x 7.5)grams

0.75 oz = 21 grams

75% of 1152 is 864

So,

0.75 oz = $864

No, his answer is not correct because each of the smaller divisions represents 1/6, not 3.

Mark's answer is not correct.

When dividing a number line into equal sections, the value of each section represents a fraction of the whole.

In this case, Mark divided the number line from 0 to 1 into 3 equal sections and then divided each of those sections into 2 equal parts.

Initially, the whole number line from 0 to 1 represents the number 1. When Mark divided it into 3 equal sections, each section represents 1/3 (one-third) of the whole.

So far, this is correct.

However, when Mark further divided each of the 1/3 sections into 2 equal parts, each part represents 1/2 (one-half) of the 1/3 section.

So, each of these smaller divisions represents 1/2 [tex]\times[/tex] 1/3 = 1/6 (one-sixth) of the whole.

Therefore, the correct representation of each of the smaller divisions is 1/6, not 3.

Mark made an error by mistakenly assuming that each smaller division represents the number 3.

In summary, Mark's answer of 3 is incorrect because each of the smaller divisions represents 1/6, not 3.

The correct representation for dividing the number line as described is 1/6.

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Answer:

7 1/2-(3+7 1/3)= -17/6=-2 5/6 ≅ -2.8333333

Step-by-step explanation:

Step-by-step explanation:

To find the unit rate in kilometers per minute, we need to divide the distance traveled (52/2 kilometers) by the time taken (23/4 minutes).

52/2 kilometers = 26 kilometers

23/4 minutes = 5.75 minutes

So, the unit rate in kilometers per minute is:

26 kilometers / 5.75 minutes = 4.52 kilometers per minute (rounded to two decimal places)

The unit rate of the car that travels 5 2/5 km in 2 3/4 mins is 1.96 km/min.

Distance travelled by a car = 5 2/5 km = 5.4 km

Time travelled = 2 3/4 mins = 2.75 mins

The unit rate in km/min = 5.4/2.75

The unit rate in km/min = 1.96 km/min.

Happy to help, have a great day! :)

The area of triangle YCX is given as follows:

G. 2 units squared.

The area of square WXYZ is 16, hence the side length is given as follows:

s² = 16

s = 4.

Considering the midpoints, we have that the sides of the right triangle are given as follows:

YC = CX = 4/2 = 2

Hence the area of the triangle is given as follows:

0.5 x 2 x 2 = 2 units squared.

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Step-by-step explanation:

Podemos escribir las relaciones como:

a = 7x

b = 5x

c = 8x

Sustituyendo las expresiones anteriores en la ecuación a+b+c=80, obtenemos:

7x + 5x + 8x = 80

20x = 80

x = 4

Sustituyendo el valor de x en las expresiones de a, b y c:

a = 7(4) = 28

b = 5(4) = 20

c = 8(4) = 32

Por lo tanto, c - b = 32 - 20 = 12

Answer:

Step-by-step explanation: