1. cos^2x = sin^2x + 1/2
2. 2cos( π/2 - x) * sin (π/2 + x) - 1 = 0
3. 8sin x/2 * cos x/3 * cosx * cos2x = 1
4. 1 + cosx = cos x/2

Answer:

  y = 6

Step-by-step explanation:

You want the value of y, given the sides of isosceles triangle DEF are marked as (7y +14) and (9y +2).

The sides of an isosceles triangle are congruent, so we have ...

  FD = FE

  7y +14 = 9y +2

  12 = 2y . . . . . . . . . . subtract 7y+2

  6 = y . . . . . . . . . divide by 2

The value of y is 6.

__

Additional comment

We know this triangle is isosceles for a couple of reasons. One is that the altitude is a perpendicular bisector of the base. Another is that the two smaller triangles are congruent by the SAS (or LL) postulate, making FD = FE by CPCTC.

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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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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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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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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 expression that is equivalent to the given variables above would be =y¹²/5. That is option A.

To determine the expression that is equivalent to the given variable above, the following steps needs to be taken as follows;

That is:

= 9x⁵y¹⁶/45x⁵y⁴

Divide the whole numbers using 9 for both the numerator and denominator.

= x⁵y¹⁶/5x⁵y⁴

To solve for x':

= X(5-5 = 0)

To solve for y':

= y(16-4= 12)

Therefore the equivalent = =y¹²/5

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The area of the sector is 30.375 ft squared.

The  sector of a circle has a diameter of 18 feet and an angle of 3/4π radians.

Therefore, the area of the sector can be found as follows:

area of the sector = ∅ / 2 r²

where

∅ = central angle in radian

r = radius of the circle

Therefore,

∅ = 3 / 4 π

r = 18 / 2 = 9 ft

area of the sector = 3 / 4 π ÷ 2 × 9²

area of the sector = 3 / 8π × 81

area of the sector = 243 / 8π

area of the sector = 30.375 ft²

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The area of the sector is 95.43 square feet.

The area of sector when the angle is given in radians is given by the formula:

Area of sector = (1/2) × r²θ

where θ is the angle subtended at the center and r is the radius of the circle.

We have:

θ = 3π/4

r = 18/2 = 9 feet

Substituting into the formula:

Area of sector = (1/2) × 9² × 3π/4

Area of sector = 95.43 square feet

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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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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?

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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The perimeter of the shaded figure is 21.42 units.

Given that, radius of two congruent circles is 3 units.

Dimensions of rectangle are length=6 units and width=3 units.

The formula to find the arc length of a circle is θ/360° ×2πr.

Here, arc length = 90°/360° ×2×3.14×3

= 0.25×2×3.14×3

= 4.71

Arc length of two congruent circles = 2×4.71

= 9.42 units

Perimeter = 9.42+6+6

= 21.42 units

Therefore, the perimeter of the shaded figure is 21.42 units.

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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.

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 number of fluid ounces of iced tea does the bottle hold is 25.

Given that, a bottle holds 750 milliliters of iced tea.

1 fluid ounce has 30 milliliters.

Here, the number of fluid ounces = Total quantity of iced tea/Quantity of fluid in a fluid ounce

= 750/30

= 25

Therefore, the number of fluid ounces of iced tea does the bottle hold is 25.

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Answer:

To find out how many jars Jazmine can fill with her 3 1/3 cups of jam, we need to divide the total amount of jam by the amount of jam that each jar can hold.

First, we need to convert 3 1/3 cups into an improper fraction:

3 + 1/3 = (3 x 3 + 1) / 3 = 10/3

So, Jazmine has 10/3 cups of jam.

Next, we need to convert the jar size of 2/5 cup into a fraction with the same denominator as 10/3:

2/5 = 6/15

Now we can divide the total amount of jam by the amount of jam that each jar can hold:

(10/3) ÷ (6/15)

To divide fractions, we need to multiply the first fraction by the reciprocal of the second fraction:

(10/3) x (15/6)

Simplifying:

(10/3) x (5/2) = (50/6)

So Jazmine can fill (50/6) jars with her 3 1/3 cups of jam.

This is an improper fraction, so we can convert it to a mixed number:

(50/6) = 8 2/6 = 8 1/3

Therefore, Jazmine can fill 8 and 1/3 jars with her 3 1/3 cups of jam.

Step-by-step explanation:

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

The correct answer is:

B. cosθ = √5/3 and tanθ = 2/√5

C. cosθ = -√5/3 and tanθ = 2/√5

To solve this question, we can use the given information that sinθ = 2/3 and determine the values of the other trigonometric functions based on that.

Recall the trigonometric identity: sin^2θ + cos^2θ = 1

Since sinθ = 2/3, we can substitute this value into the identity and solve for cosθ:

(2/3)^2 + cos^2θ = 1

4/9 + cos^2θ = 1

cos^2θ = 1 - 4/9

cos^2θ = 5/9

Taking the square root of both sides, we get:

cosθ = ±√(5/9) = ±(√5/3)

Now, let's analyze the given options:

A. secθ = 3/√5 and tanθ = 2/√5

B. cosθ = √5/3 and tanθ = 2/√5

C. cosθ = -√5/3 and tanθ = 2/√5

D. secθ = -3/2 and tanθ = 2/√5

From our calculations, we found that cosθ can be either √5/3 or -√5/3. So, option B and C are valid.

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Answer:

130° and 310°

Step-by-step explanation:

the bearing of A from B is the measure of the clockwise angle from a north line at B to A

the angle at B and A are same- side interior angles and sum to 180°

angle at B = 180° - 50° = 130°

the bearing of A from B is then 130°

the bearing of B from A is the measure of the clockwise angle from the north line at A to B

the complete angle about A = 360° then clockwise angle from north line at A to B is

360° - 50° = 310°

the bearing of B from A is 310°

The number of square inches or area of gift wrap that the makeup artist will need is 10.8π square inches.

Given that,

A makeup artist purchased some lipsticks and wants to wrap them individually with gift wrap.

Radius of each lipstick = 0.6 inch

Height of the lipstick = 2.4 inches

Now,

Area of the each lipstick = 2πrh + 2πr², where r is the radius and h is the height.

Area = 2π (0.6) (2.4) + 2π (0.6)²

        = 2.88π + 0.72π

        = 3.6π

Area of the wrap for 3 lipsticks = 3 × 3.6π = 10.8π square inches.

Hence the area is 10.8π square inches.

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The statement that describes the type of solutions of the specified  equation is the third option C.;

C. The equation has two valid solutions and no extraneous solutions

A solution to an equation are values of the variables of the equation that make the equation true.

The possible equation obtained from a similar question on the website can be presented as follows;

4/(3·x + 1) = x/(2·x + 10)

The above equation expressions can be combined into a quadratic equation as follows;

4 × (2·x + 10) = x × (3·x + 1)

8·x + 40 = 3·x² + x

3·x² + x - 8·x - 40 = 0

3·x² - 7·x - 40 = 0

3·x² - 15·x + 8·x - 40 = 0

3·x·(x - 5) + 8·(x - 5) = 0

(3·x + 8)·(x - 5) = 0

x = -8/3 and x = 5

Therefore, the equation has two valid solutions and no extraneous solutions

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Answer:

look at explanation

Step-by-step explanation:

it's a 50/50 chance

Summary statistics provide a single number to describe data, while tables and graphs visually represent data to facilitate analysis and answer questions.

We have,

Data can be described by a single number using summary statistics. Summary statistics provide a concise representation of the main characteristics of a dataset.

Commonly used summary statistics include measures such as the mean, median, mode, range, variance, and standard deviation.

These single numbers provide insights into different aspects of the data, such as the central tendency, variability, or distribution.

Tables and graphs are effective tools for representing data and answering questions because they provide a visual representation that helps to organize and understand the information.

Tables present data in a structured format, typically using rows and columns, which allows for easy comparison and analysis.

They can display raw data, summary statistics, or both.

Graphs, on the other hand, use visual elements to represent data in a more intuitive and visual manner.

Different types of graphs can be used depending on the nature of the data and the question at hand.

For example, bar graphs can be used to compare categories or discrete data, line graphs can show trends over time, and scatter plots can display the relationship between two variables.

Tables and graphs help to answer questions about data by providing a clear and organized presentation.

They allow us to observe patterns, trends, and relationships, making it easier to draw conclusions and make informed decisions.

Visually representing data, tables, and graphs enhances our ability to analyze and interpret complex information quickly and effectively.

Thus,

Summary statistics provide a single number to describe data, while tables and graphs visually represent data to facilitate analysis and answer questions.

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Answer:

$224.42

Step-by-step explanation:

P is the principal balance

r is the interest rate (as a decimal)

n is the number of times interest is compounded per year

t is the number of years

Input all of this into the equation you have.

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

[tex]A=160(1+\frac{.07}{1})^{1(5)}[/tex]

Do the math.

[tex]A=160(1+.07)^{5}[/tex]

[tex]A=160(1.07)^{5}[/tex]

[tex]A=160(1.403)[/tex]

[tex]A=224.418[/tex]

Answer: 43.75 miles per day.

Step-by-step explanation: All you have to do is divide 350 by 8, when you divide you basically just split the number evenly 8 times, which is what 350 divided by 8 is. He will have to drive 43.75 miles a day.

The radius of the sphere is 2cm

The circumference of the sphere is about 12.56 cm

cm²

The volume of the sphere is about 33. 5cm³

From the information given, we have that;

We have that the largest cross sectional area of the sphere takes the value

πr²

The circumference of the sphere = 2πr

Equate the expressions;

2πr = πr²

in the ratio of 4: 1

2πr = 4πr²

Divide the values

r =2cm

Substitute the value to determine the circumference of the sphere, we get;

Circumference = 2 × 3.14 × 2

Multiply the values, we get

Circumference = 12.56 cm

The volume of the sphere is represented as;

=4/3 ×  3.14 × (2)³

Multiply the values, we have;

=4/3 × 3.14 ×8

Multiply the values, we get;

= 33. 5cm³

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The export will reach $11.2 billion in 5.4 years or in the year 2013 from the given function.

Given that,

The value of China's exports of automobiles and parts (in billions of dollars) is approximately,

f(x) = 1.8208 e^(0.33872x)

We have to find the value of x, when f(x) = $11.2 billion

11.2 = 1.8208 e^(0.33872x)

6.15 = e^(0.33872x)

Taking logarithms on both sides,

ln (6.15) = 0.33872x ln (e)

ln (6.15) = 0.33872x

x = 5.36 ≈ 5.4 years.

The value will reach 11.2 billion after 5.4 years which is in the year 2013.

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The probability of choosing the expected value of games is

a) P ( G ) = 1/4

b) P ( R ) = 3/8

c) P ( B ) = 1/4

d) P ( Y ) = 1/8

Given data ,

Color Payout

Red $2

Blue $6

Green $6

Yellow $20

Now , game costs $5 to play and you win money based on the color the spinner lands on

On simplifying , we get

The number of yellow spinner = 1

The number of blue spinner = 2

The number of green spinner = 2

The number of red spinner = 3

So , the probabilities are

a) P ( G ) = 1/4

b) P ( R ) = 3/8

c) P ( B ) = 1/4

d) P ( Y ) = 1/8

Hence , the probability is solved

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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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1. The weight of the patient in kilograms is 27.216 kg. (60 lb * 0.4536 kg/lb = 27.216 kg)

2. The total dosage of medication required for the patient is 13.608 mg. [tex](0.5 mg/kg \times 27.216 kg = 13.608 mg)[/tex]

3. The patient should be administered 1.3608 mL of the stock medication. (13.608 mg / 10 mg/mL = 1.3608 mL)

To calculate the necessary values based on the given information, let's follow the steps below:

Determine the weight of the patient in kilograms:

Given that the patient weighs 60 lb, we can convert this to kilograms using the conversion factor of 1 lb = 0.4536 kg.

Weight (in kg)[tex]= 60 lb \times 0.4536 kg/lb = 27.216 kg.[/tex]

Calculate the total dosage of medication required for the patient:

The medication order is 0.5 mg/kg, and the patient weighs 27.216 kg.

Total dosage [tex]= 0.5 mg/kg \times 27.216 kg = 13.608 mg.[/tex]

Determine the amount of stock medication required in milliliters (mL):

The stock medication is available in a concentration of 10 mg/mL.

To find the volume required, we need to divide the total dosage by the concentration of the stock medication.

Volume (in mL) = Total dosage (in mg) / Concentration (in mg/mL) = 13.608 mg / 10 mg/mL = 1.3608 mL.

Therefore, based on the given information, the weight of the patient is 27.216 kilograms, the total dosage of medication required is 13.608 milligrams, and 1.3608 milliliters of the stock medication should be administered to the patient.

Please note that when administering medication, it is crucial to follow the guidance of a healthcare professional and consider other factors such as the specific medication instructions, patient's condition, and any allergies or contraindications.

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Question: A patient weighs 60 lb, and the medication order is 0.5 mg/kg. The stock medication is available in a concentration of 10 mg/mL.

Based on this information, calculate the following

What is the weight of the patient in kilograms? (1 lb = 0.4536 kg)

What is the total dosage of medication required for the patient?

How many milliliters (mL) of the stock medication should be administered to the patient?

Please provide the necessary calculations and steps to find the answers based on the given information.