please help, i don’t understand this at all

The amplitude and the period of the function f(x) = cos(5θ) are 1 and 2π/5

Here, we have,

to determine the amplitude and period of the function

From the question, we have the following parameters that can be used in our computation:

f(x) = cos(5θ)

A sinusoidal function is represented as

f(x) = Acos(B(x + C)) + D

Where

Amplitude = A

Period = 2π/B

So, we have

A = 1

Period = 2π/5

Evaluate

A = 1

Period = 2π/5

Hence, the amplitude is 1 and the period is 2π/5

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complete question:

100 Points! Algebra question. Photo attached. Find the amplitude, if it exists, and period of the function. Then graph the function. Thank you!

Answer:

The distance formula is:

[tex]d = \sqrt{(x_2 - x_1)^2+ (y_2 - y_1)^2}[/tex]

Where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex]are the coordinates of the two points.

(-2, 5) & (0, -4):

[tex]d = \sqrt{(0 - (-2))^2+ ((-4) - 5)^2}=\sqrt{85}[/tex]

So the distance between (-2, 5) and (0, -4) is [tex]\sqrt{85}[/tex].

(0,-4) & (5, 3):

[tex]d = \sqrt{(5 - 0)^2+ ((3) - (-4))^2}=\sqrt{74}[/tex]

So the distance between (0,-4) and (5, 3) is [tex]\sqrt{74}[/tex]

(5, 3) & (-2, 5):

[tex]d = \sqrt{(5 - (-2))^2 + ((3) - 5)^2}=\sqrt{53}[/tex]

So the distance between (5, 3) and (-2, 5) is [tex]\sqrt{53}[/tex].

Answer:

[tex]\sqrt{85}\approx 9.23 \; \sf units\;(3\;s.f.)[/tex]

[tex]\sqrt{74}\approx 8.60\; \sf units\; (3\;s.f.)[/tex]

[tex]\sqrt{53}\approx 7.28\; \sf units \;(3\;s.f.)[/tex]

Step-by-step explanation:

The distance formula is a mathematical equation used to calculate the distance between two points in a coordinate system.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where:\\ \phantom{ww}$\bullet$ $d$ is the distance between two points. \\\phantom{ww}$\bullet$ $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

To find the distance between each set of two points, substitute the coordinates of the points into the distance formula and solve for d.

The distance between points (-2, 5) and (0, -4) is:

[tex]\begin{aligned}d&=\sqrt{(0-(-2))^2+(-4-5)^2}\\&=\sqrt{(2)^2+(-9)^2}\\&=\sqrt{4+81}\\&=\sqrt{85}\\&\approx 9.23 \; \sf units\;(3\;s.f.)\end{aligned}[/tex]

The distance between points (0, -4) and (5, 3) is:

[tex]\begin{aligned}d&=\sqrt{(5-0)^2+(3-(-4))^2}\\&=\sqrt{(5)^2+(7)^2}\\&=\sqrt{25+49}\\&=\sqrt{74}\\&\approx 8.60\; \sf units\; (3\;s.f.)\end{aligned}[/tex]

The distance between points (5, 3) and (-2, 5) is:

[tex]\begin{aligned}d&=\sqrt{(-2-5)^2+(5-3)^2}\\&=\sqrt{(-7)^2+(2)^2}\\&=\sqrt{49+4}\\&=\sqrt{53}\\&\approx 7.28\; \sf units \;(3\;s.f.)\end{aligned}[/tex]

Plan A lasts for 0.5 hours, and Plan B lasts for 0.75 hours based on the given information and the solution to the System of equations.

To determine the duration of each workout plan, let's assume that both Plan A and Plan B have a fixed duration. Let's denote the duration of Plan A as "x" hours and the duration of Plan B as "y" hours.

From the given information, we can form the following equations:

On Wednesday:

5 clients did Plan A, so the total duration for Plan A is 5x hours.

6 clients did Plan B, so the total duration for Plan B is 6y hours.

The total training duration on Wednesday is 7 hours, so we have the equation: 5x + 6y = 7   ... Equation (1)

On Thursday:

3 clients did Plan A, so the total duration for Plan A is 3x hours.

2 clients did Plan B, so the total duration for Plan B is 2y hours.

The total training duration on Thursday is 3 hours, so we have the equation: 3x + 2y = 3   ... Equation (2)

We now have a system of equations (Equation 1 and Equation 2) that we can solve to find the values of x and y.

Multiplying Equation (1) by 2 and Equation (2) by 5 to eliminate the y variable, we get:

10x + 12y = 14   ... Equation (3)

15x + 10y = 15   ... Equation (4)

Multiplying Equation (3) by 3 and Equation (4) by 2 to create a new system of equations:

30x + 36y = 42   ... Equation (5)

30x + 20y = 30   ... Equation (6)

Subtracting Equation (6) from Equation (5), we can eliminate the x variable:

(30x + 36y) - (30x + 20y) = 42 - 30

16y = 12

y = 12/16

y = 0.75

Substituting the value of y into Equation (2), we can solve for x:

3x + 2(0.75) = 3

3x + 1.5 = 3

3x = 3 - 1.5

3x = 1.5

x = 1.5/3

x = 0.5

Therefore, the duration of Plan A is 0.5 hours, and the duration of Plan B is 0.75 hours.

Plan A lasts for 0.5 hours, and Plan B lasts for 0.75 hours based on the given information and the solution to the system of equations.

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The values of the different parameters of the shapes given would be =

5.) = 78.5 in²

6.) = 54cm²

7.) = 50.24 ft²

8.) = 108yd³

10.) = 320mm³

11.)=847.8m³

12.)=366.3ft³

13.)=1004.8in

14.)=113.04in³

For question 5.)

Area of base of cylinder = πr²

radius = 5 in

area = 3.14×5×5 = 78.5 in²

For question 6.)

Area of base of rectangular pyramid = l×w

length = 6 cm

width = 9 cm

Area = 6×9 = 54cm²

For question 7.)

Area of the base of a cone = πr²

radius = 4 ft

area = 3.14×4×4 = 50.24 ft²

For question 8.)

Volume of a square pyramid =1/3 a²h

a = 6 yd

h = 9yd

volume = 1/3× 6×6× 9

= 108yd³

For question 10.)

Volume of the rectangular pyramid;

= 1/3×l×W×h

width= 12mm

height = 8 mm

length = 10

Volume = 1/3× 12×8×10 = 320mm³

For question 11.)

Volume of a cone= ⅓πr²h

height = 10m

radius = 9m

Vol = ⅓×3.14×9×9×10

= 847.8m³

For question 12.)

Volume of cone = ⅓πr²h

height = 14ft

radius = 5ft

Volume = 1/3×3.14×5×5×14

= 366.3ft³

For question 13.)

Volume of cone = ⅓πr²h

height = 15in

radius = 16/2 = 8in

Volume = 1/3× 3.14× 8×8×15

= 1004.8in

For question 14.)

Volume of cone = ⅓πr²h

height = 12in

radius = 3in

volume = 1/3×3.14×3×3×12

= 113.04in³

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Answer:

B. 11

Step-by-step explanation:

√380.13/pi = 10.99, Round: 11

The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is, 3 / 5

We have to given that;

the terminal side of theta in standard position contains the point (6,-8)

Hence, For given condition we get;

The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,

Here, x = 6, y = - 8

Hence, We get;

r = √x² + y²

r  = √6² + 8²

r = √36 + 64

r = √100

r = 10

So, The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,

cos θ = x / r

cos θ = 6 / 10

cos θ =  3 / 5

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Answer:

Solution is in the attached photo.

Step-by-step explanation:

This question tests on the concept on deriving equations and substitution of values.

Triangle ABC is congruent to triangle BAD because all their corresponding angles are equal.

Congruent triangles are triangles with equal corresponding angles and equal corresponding sides. This means that the angles will be equal and their corresponding sides will be equal.

Since AC is equal to BD , then the corresponding angles facing the two lines will be equal and angle A is equal to angle B

Therefore angle C will be equal to angle D, therefore we can say triangle ABC is congruent to triangle BAD.

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Answer:

Step-by-step explanation:

To find the probability of drawing a purple and yellow marble without replacement, we need to determine the number of favorable outcomes (purple and yellow marbles) and the total number of possible outcomes.

Step 1: Determine the number of purple marbles:

The given information states that there are 6 purple marbles.

Step 2: Determine the number of yellow marbles:

The given information states that there are 3 yellow marbles.

Step 3: Determine the total number of marbles:

The given information states that there are 25 marbles in total.

Step 4: Calculate the probability of drawing a purple and yellow marble:

When drawing without replacement, the probability of two events occurring is the product of their individual probabilities.

The probability of drawing a purple marble is: 6 purple marbles / 25 total marbles = 6/25.

After drawing a purple marble, the total number of marbles remaining is 24 (since one purple marble is already drawn).

The probability of drawing a yellow marble from the remaining marbles is: 3 yellow marbles / 24 remaining marbles = 3/24.

To find the probability of both events occurring (drawing a purple and yellow marble), we multiply their individual probabilities:

Probability of drawing a purple and yellow marble = (6/25) * (3/24) = 18/600 = 3/100.

Therefore, the probability of drawing a purple and yellow marble without replacement is 3/100.

The total police officers there altogether given the percentage of police officers who have ride-alongs with teenagers is 430.

Percentage of police officers who have ride-alongs with teenagers = 60%

Number of Percentage of police officers who have ride-alongs = 258

Total police officers = x

So,

(Percentage of police officers who have ride-alongs with teenagers) of (Total police officers) = (Number of Percentage of police officers who have ride-alongs)

60% of x = 258

0.6 × x = 258

0.6x = 258

divide both sides by 0.6

x = 258/0.6

x = 430

Ultimately, there are 430 police officers altogether.

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Candidate A would win the election using the plurality method.

According to the given preference schedule, the number of votes and the corresponding preferences are as follows:

Candidate A: 15 (1st preference: A, 2nd preference: D, 3rd preference: B)

Candidate B: 11 (1st preference: D, 2nd preference: C, 3rd preference: A)

Candidate C: 9 (1st preference: B, 2nd preference: C, 3rd preference: A)

Candidate D: 6 (1st preference: C, 2nd preference: D, 3rd preference: A)

Candidate E: 2 (1st preference: C, 2nd preference: D)

To determine the winner using the plurality method, we look at the candidate with the highest number of first preference votes.

In this case, Candidate A has 15 first preference votes, which is the highest.

Therefore, Candidate A would win the election using the plurality method.

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Step-by-step explanation: there are 3 y's so the answer is 3y

and there are 2 x's so therefore the answer is 2x

and then you just add them together to get 3y + 2x

The mean monthly rainfall is 3.245 inches.

For a set of N numbers {x₁, x₂, ...}, the mean of these N numbers is:

M = {x₁ + x₂ + ...}/N

Here we want to find the mean of the rainfall in inches, then we just need to add the 12 values in the table and then divide by 12, we will get:

M = (2.22+1.51+1.86+2.06+3.48+4.57+ 3.27 + 5.40 + 5.45 + 4.34+2.64+ 2.14)/12

M = 3.245

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The statement you provided refers to a concept known as Russell's paradox in set theory.

The paradox arises when we consider the set of all sets that do not contain themselves as elements. If we assume such a set exists, we can define a paradoxical set, often denoted as R, which contains all sets that do not contain themselves.

The statement you provided is suggesting that the paradox arises from the concept of the set of all sets, rather than from the concept of comprehension itself.

The , in set theory, refers to the idea of defining a set by specifying a property or condition that its elements must satisfy. The paradox does not arise from the idea of defining sets in this way but rather from the specific concept of the set of all sets.

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The value of measure of x in the triangle is,

⇒ x = 4√3 units

We have to given that,

A triangle is shown in image.

Now, WE can formulate by trigonometry formula we get;

⇒ tan 30° = Opposite / Base

⇒ tan 30° = 4 / x

⇒ 1/√3 = 4/x

⇒ x = 4√3 units

Thus, The value of measure of x in the triangle is,

⇒ x = 4√3 units

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The area of triangle ABC is approximately 231.89 square inches.

To solve this problem, we'll use the properties of a triangle and trigonometric functions.

Let's go step by step.

Part A: Determine m ∠A.

To find the measure of angle A, we can use the Law of Cosines.

The Law of Cosines states that for a triangle with sides of lengths a, b, and c, and angle A opposite side a, we have the formula:

c² = a² + b² - 2ab × cos(A)

In this case, c = 60 inches, a = 60 inches, and b = 60√2 inches.

Plugging in these values into the Law of Cosines equation, we get:

(60)² = (60)² + (60√2)² - 2(60)(60√2) × cos(A)

Simplifying:

3600 = 3600 + 7200 - 7200√2 × cos(A)

Now, let's solve for cos(A):

7200√2 × cos(A) = 7200

cos(A) = 7200 / (7200√2)

cos(A) = 1 / √2

cos(A) = √2 / 2

To determine the angle A, we need to find the inverse cosine of (√2 / 2), which is 45 degrees.

Therefore, m ∠A is 45 degrees.

Part B: Explain how to use the unit circle to find the exact values of all six trigonometric functions evaluated at A.

To find the values of the trigonometric functions at angle A, which is 45 degrees or π/4 radians, we can use the unit circle.

The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) in a coordinate plane.

At angle A = 45 degrees or π/4 radians, the coordinates of the point on the unit circle are (√2/2, √2/2).

Using these coordinates, we can find the exact values of the trigonometric functions as follows:

sin(A) = y-coordinate = √2/2

cos(A) = x-coordinate = √2/2

tan(A) = sin(A) / cos(A) = (√2/2) / (√2/2) = 1

csc(A) = 1 / sin(A) = 1 / (√2/2) = √2

sec(A) = 1 / cos(A) = 1 / (√2/2) = √2

cot(A) = 1 / tan(A) = 1 / 1 = 1

Part C: Calculate the area of triangle ABC.

To calculate the area of triangle ABC, we can use Heron's formula, which is based on the lengths of the triangle's sides.

Heron's formula states that for a triangle with side lengths a, b, and c, the area (A) can be calculated as:

A = √(s(s-a)(s-b)(s-c))

where s is the semiperimeter of the triangle, calculated as:

s = (a + b + c) / 2

In this case, a = 60 inches, b = 60√2 inches, and c = 60 inches.

Calculating the semiperimeter:

s = (60 + 60√2 + 60) / 2

s = (120 + 60√2) / 2

s = 60 + 30√2

Now, we can calculate the area using Heron's formula:

A = √((60 + 30√2)((60 + 30√2) - 60)((60 + 30√2) - (60√2))((60 + 30√2) - 60))

Simplifying:

A = √((60 + 30√2)(60)(30√2)(30))

A = √(1800(30√2))

A = √(54,000√2)

A = 231.89 square inches (rounded to the nearest hundredth)

Therefore, the area of triangle ABC is approximately 231.89 square inches.

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Answer: The acceleration of the car is -6 m/s²

Step-by-step explanation:

The acceleration of the car can be found by using the given below equation:

Acceleration = (final velocity - initial velocity) / time

Given:

Initial velocity (u) = 30 m/s

Final velocity (v) = 0 m/s

Time (t) = 5 seconds

Putting the values in the equation:

acceleration = (0 - 30) m/s / 5 s

acceleration = -30 m/s / 5 s

acceleration = -6 m/s²

Therefore, the acceleration of the car is -6 m/s² (negative sign indicates deceleration).

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Answer:

Step-by-step explanation:

To express 6√3 + 6i in polar form, we first need to find the magnitude and argument of the complex number.

The magnitude (or modulus) of the complex number is given by the formula:

|z| = √(a^2 + b^2)

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

In this case, a = 6√3 and b = 6, so:

|6√3 + 6i| = √((6√3)^2 + 6^2) = √(108 + 36) = √144 = 12

The argument (or angle) of the complex number is given by the formula:

arg(z) = tan^-1(b/a)

In this case, arg(z) = tan^-1(6/6√3) = tan^-1(1/√3) = π/6

Therefore, the polar form of 6√3 + 6i is:

12(cos(π/6) + i sin(π/6))

or

12e^(iπ/6)

The given expressions can be identified as:

1. Trinomial

2. Monomial

3. Polynomial

4. Binomial

The given expressions can be identified as:

1. x^3 + y^2 - 3: This is a trinomial since it consists of three terms.

2. 7y^2: This is a monomial since it consists of only one term.

3. x^3 + 2x^2 - x + 10: This is a polynomial since it consists of four terms.

4. x + 7: This is a binomial since it consists of two terms.

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Answer:

The word or phrase that describes the probability that Arianna will roll a number between 1 and 6 (including 1 and 6) is "certain." The reason for this is that a standard six-sided die always has a number between 1 and 6 on each face, and if the die is fair and unbiased, then the probability of rolling a number between 1 and 6 is 1, or 100%. Therefore, option (b) "certain" is the correct choice.

To produce 120 cupcakes, Lihle needs to make 5 batches if one batch produces 24 cupcakes. However, it is important to plan for a little more than needed in case of any failures or discrepancies, so making an extra batch might be helpful. It is also important to consider the amount of frosting needed for 120 cupcakes, which can vary depending on the recipe. As a basic estimate, 1 cup of frosting is suggested for 12 cupcakes.

Answer:

y = 1x + -3

Step-by-step explanation:

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

The y intercept is -3.

Let's calculate the slope. Pick 2 points. I'll choose (0,-3) and (3,0).

The slope should be positive; our line is pointing UP and as x increases, y increases.

Slope = rise/run = (y2-y1)/(x2-x1)

= (-3-0)/(0-3)

= -3/-3 = 1

So y = 1x + -3

We can try another point from the graph to check that we're right. Let's use (1,-2)

y = 1(1) +-3

y = 1-3

y = -2, which is exactly our coordinate from the graph.

844,758,200,030 will be rounded to 845 billion (845,000,000,000).

The rounding digit is 5.

Estimation a is a method used for calculating the approximate value of a quantity (just to get a 'rough answer')

In estimation 1, 2,3, and 4 are rounded down while 5, 6, 7, 8 and 9 are rounded up.

To round 844,758,200,030 to the nearest billion, we look at the digit in the hundred millionths place, which is 7.

Since 7 is greater than or equal to 5, we round 844,758,200,030 up to 845,000,000,000. The rounding digit is therefore 5.

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The volume of the bottle is 25 fluid ounce of iced tea.

from the question, we have the following parameters that can be used in our computation:

Volume of bottle = 750 milliliters of iced tea.

Also, we have

1 fluid ounce = 30 milliliters

This means that

1/30 fluid ounce = 1 milliliters

So, we have

Volume of bottle = 750 * 1/30 fluid ounce of iced tea.

Evaluate

Volume of bottle = 25 fluid ounce of iced tea.

Hence, the volume of bottle is 25 fluid ounce of iced tea.

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Answer:

x = 4

Step-by-step explanation:

a line with an undefined slope is a vertical line with equation

x = c ( c is the value of the x- coordinates the line passes through )

the line passes through the point (4, - 3 ) with x- coordinate 4 , then

x = 4 is the equation of the line

note this equation cannot be expressed in slope- intercept form

The pounds of seafood at the boil was 10 37/40 pounds

A fraction is defined as the part of a whole number, a whole element or a whole variable.

The types of fractions are listed as;

From the information given, we have that;

5 4/5 + 4 1/2 + 5/8, all in pounds

convert the mixed fraction to improper fractions, we have;

29/5 + 9/2 + 5/8

Find the lowest common factor, we have;

232 + 180 + 25  /40

Add the values

437/40

Divide the values

10 37/40 pounds

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Answer:

220 adults and 250 children

Step-by-step explanation:

we require to set up 2 equations and solve simultaneously.

let a be the number of adults and c the number of children , then

a + c = 470 → (1) based on number of people

3a + c = 910 → (2) based on ticket sales

subtract (1) from (2) term by term to eliminate c

(3a - a) + (c - c) = 910 - 470

2a + 0 = 440

2a = 440 ( divide both sides by 2 )

a = 220

substitute a = 220 into (1) and solve for c

220 + c = 470 ( subtract 220 from both sides )

c = 250

that is 220 adults and 250 children attended

The contractor should lower the ramp by approximately 4.62° to meet safety regulations.

We have,

To determine the angle created by the bottom of the ramp and the ground, we can use the trigonometric function of sine.

In the given triangle, the height is 6 ft and the base is 16 ft.

The angle we want to find is opposite the height (let's call it θ).

The sine of an angle is defined as the ratio of the opposite side to the hypotenuse.

sin(θ) = opposite/hypotenuse

sin(θ) = 6/16

sin(θ) = 0.375

To find the angle θ, we can take the inverse sine (also known as arcsine) of 0.375:

θ = arcsin(0.375)

θ ≈ 22.62°

To meet safety regulations, the angle should be no more than 18°. Therefore, the contractor needs to lower the ramp by:

22.62° - 18° ≈ 4.62°

Thus,

The contractor should lower the ramp by approximately 4.62° to meet safety regulations.

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The solution of expression is,

⇒ y = - 39

We have to given that,

An expression is,

⇒ - 5 = (y - 7) / 9

Now, We can simplify as,

An expression is,

⇒ - 5 = (y - 7) / 9

Multiply by 9,

⇒ - 5 × 9 = (y - 7) × 9 / 9

⇒ - 45 =y - 7

Add 7 both side,

⇒ - 45 + 7 = y

⇒ y = - 39

Therefore, The solution of expression is,

⇒ y = - 39

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Explanation:

Each cube has side length 1/2. The volume of each smaller cube is (1/2)^3 = 1/8 cubic units.

The box holds 2 layers and 15 cubes per layer. There are 2*15 = 30 cubes total. The total volume of all 30 cubes is 30*(1/8) = 30/8 = 15/4 = 3.75 cubic units, which also represents the volume of the box.