Find the equation of the line.
Use exact numbers.

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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The correct proof of the equation is cos3θ =cos(2θ + θ) =  cos²θ - 3cosθsin²θ (Proved)

Given the trigonometry identity below:

cos3∅=cos²∅-3cos∅sin²∅

We are to prove that both sides of the equation are equal.

cos 3θ = cos(2θ + θ)

Using the double angle formula for cosine, we can expand the first term:

cos(2θ + θ) = cos2θ cos θ - sin2θ sinθ

Since cos2θ = cos²θ - sin²θ

cos(2θ + θ) =  cos²θ - sin²θ - 2cosθsinθsinθ

cos(2θ + θ) = cos²θ - sin²θ - 2cosθsin²θ

On simplifying, we can see that;

cos3θ =cos(2θ + θ) =  cos²θ - 3cosθsin²θ (Proved)

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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

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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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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People in this country own approximately 111 million dogs, 123 million cats, and 258 million fish as pets.

Let's solve the problem step by step to determine how many of each type of pet people in this country own.

Let's assume the number of dogs as D.

According to the information given, the number of cats would be D + 12 million since there are 12 million more cats than dogs.

The number of fish would be the sum of the number of cats and dogs plus 12 million, as there are 12 million more fish than the total number of cats and dogs combined.

So, the equation representing the total number of pets is:

D + (D + 12 million) + (D + 12 million) = 356 million

Simplifying the equation, we have:

3D + 24 million = 356 million

Subtracting 24 million from both sides, we get:

3D = 332 million

Dividing both sides by 3, we find:

D = 110.67 million

Since the number of pets cannot be in fractions, we can round D to the nearest whole number. In this case, D ≈ 111 million.

Now, we can calculate the number of cats:

Cats = D + 12 million

= 111 million + 12 million

= 123 million

And the number of fish:

Fish = (D + Cats) + 12 million

= (111 million + 123 million) + 12 million

= 246 million + 12 million

= 258 million

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Answer:

Step-by-step explanation:

The areas of the triangles are listed below:

Triangle A: A = 30 cm²

Triangle B: A = 14 cm²

Triangle C: A = 15 cm²

Triangle D: A = 40.997 cm²

In this problem we have a composite figure in the form of a square and formed by three right triangles (A, B, C) and a non-right triangle (D). The area of each triangle must be found, the areas of the each right triangle can be determined by this formula:

A = 0.5 · b · h

Where:

And the area of the non-right triangle can be found by Heron's formula:

A = √[s · (s - a) · (s - b) · (s - c)]

s = 0.5 · (a + b + c)

Where:

First, determine the areas of the right triangles:

Triangle A

A = 0.5 · (10 cm) · (6 cm)

A = 30 cm²

Triangle B

A = 0.5 · (7 cm) · (4 cm)

A = 14 cm²

Triangle C

A = 0.5 · (3 cm) · (10 cm)

A = 15 cm²

Second, find the side lengths of the non-right triangle by Pythagorean theorem:

a = √[(10 cm)² + (6 cm)²]

a = 2√34 cm

b = √[(7 cm)² + (4 cm)²]

b = √65 cm

c = √[(3 cm)² + (10 cm)²]

c = √109 cm

Third, calculate the semiparameter of the non-right triangle:

s = 0.5 · (2√34 cm + √65 cm + √109 cm)

s = 15.082 cm

Fourth, calculate the area of the non-right triangle:

A = √[(15.082 cm) · (15.082 cm - 2√34 cm) · (15.082 cm - √65 cm) · (15.082 cm - √109 cm)]

A = 40.997 cm²

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If the value of the truck is at $23,000 which is 23% off the original price. Then the original price of the truck will be $ 29,870.13.

What is the percentage?

The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.

A 4th of July sale at your local truck dealership is advertising big discounts on pickup trucks.

The truck you've had your eye on is now listed at $23,000 which is 23% off the original price.

Then the original price of the truck will be given as

Let the original price of the truck will be x. Then we have

0.77 × x = $ 23,000

         x = $ 29,87

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 is : 22 meters would be saved if it were possible to walk through the pond.

Here, we have,

There is a pond between two points A and B.

To avoid the pond, one must walk to the south from point A to point C (say) by 34 meters and then to the east from point C to point B by 41 meters.

Now, it is clear that Δ ABC is a right triangle with AB as the hypotenuse that is the minimum distance from A to B through the pond.

The two legs of the right triangle are AC and CB.

Applying Pythagoras Theorem,

AB² = AC² + CB²

      = 34² + 41²

      = 2837

⇒ AB = 53 meters (Rounded to the nearest meter)

Therefore, (34 + 41) - 53 = 22 meters will be saved if it were possible to walk through the pond.

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complete question:

To get from point A to point B you must avoid walking through a pond. To

avoid the pond, you must walk 34 meters south and 41 meters east. To the

nearest meter, how many meters would be saved if it were possible to walk

through the pond?

Using the z-score, mean and sample mean, the standard deviation of the sample is 7.57

A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

The formula is given as;

z = x - μ / σ

Substituting the values into the formula;

-2.13 = 19.3 - 29 / σ

Cross multiply both sides;

-2.13σ = 19.3 - 29

-2.13σ = -9.7

Divide both sides by the coefficient;

-2.13σ / -2.13 = -9.7 / -2.13

σ = 7.57

The standard deviation is 7.57

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The data-set in this problem is classified as follows:

D) Negatively skewed.

Negative skew refers to a longer or fatter tail on the left side of the distribution, that is, the majority of the values are on the lower end of the distribution.

Positive skew refers to a longer or fatter tail on the right side of the distribution, that is, the majority of the values are on the right end of the distribution.

For a symmetric distribution, we have that the majority of the values are at the center of the distribution, with an equal proportion at both tails.

For this problem, we have that the larger percentage is less than $4.00, which is at the left tail of the distribution, hence the distribution is negatively skewed.

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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.

Answer:

error

Step-by-step explanation:

I'm sorry, but I need more information to calculate the average budget across the four quarters. Can you please provide me with the values of the budget for each quarter? Once I have this information, I can calculate the average and then determine the percentage difference between the average and the estimated budget for next year.

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!

The  probability of rolling two dice that have a sum of 7 is: 1/6.

When rolling two dice each die will have 6 possible outcomes

Total number of possible outcomes= 6 * 6

Total number of possible outcomes = 36

Now let's roll a sum of 7 dice to determine the amount of favourable results. There are 6 various ways that the number 7 may be combined, and they are as follows:
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1)

The number of favorable outcomes is 6.

Let find the  probability of rolling two dice that have a sum of 7 :

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Probability = 6 / 36

Probability = 1 / 6

Therefore the probability  is 1/6.

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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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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

Answer:

Solution is in the attached photo.

Step-by-step explanation:

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

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 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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Based on the information, we can infer that the area of the floor is 130 square inches. Besides, we can infer that 2 tiles are needed to cover the entire floor area.

To calculate the area of the floor we must divide the figure into fragments to find the area more easily. First of all we can find the area of the left zone that measures 10 base by 9 height.

10 * 9 = 90

For the right part the area would be 10 * 4. So:

10 * 4 = 4040 + 90 = 130

According to the above, the area of the place would be 130. On the other hand, we can infer that 2 tiles are needed to cover the entire floor area.

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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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The cost of each pound of grass seed is $6.

A graph of the solution for the cost of each pound of grass seed is shown below.

In order to determine the cost of each pound of grass seed, we would assign a variable to the cost of each pound of grass seed and then translate the word problem into an algebraic linear equation.

Let the variable x represent the cost of each pound of grass seed.

Since Lori bought 3.6 pounds of grass seed and she was charged $22.90, including a tax of $1.30, an algebraic linear equation that best represent this situation is given by;

3.6x + 1.30 = 22.90

3.6x = 22.90 - 1.30

3.6x = 21.60

x = 21.60/3.6

x = $6.

In conclusion, we would sue an online graphing calculator to plot the solution for the cost of each pound of grass seed as shown in the image attached below.

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Answer:

Set your calculator to Degree mode.

[tex]x = {sin}^{ - 1} \frac{73}{83} = 61.6 \: degrees[/tex]

Angle x measures about 61.6°.

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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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)

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.