1 /4 of the schoolyard is used by the fifth graders for recess, as shown by the shaded part of the diagram. Teachers have 3 different activities planned for recess that will take up the same amount of space.

[tex](2^{2})^{4} *5^{4}[/tex][tex](2^{2})^{4} *5^{4}[/tex]The expression is [tex]4^{4} *5^{4}[/tex] is therefore a single power of [tex]2[/tex] and [tex]5[/tex]

[tex]4^{4} *5^{4} =2^{8} *5^{4}[/tex]

A grouping of numbers, symbols, and mathematical operations (such as addition, subtraction, multiplication, and division) is called an expression, and it is used to represent an amount or relationship. Expressions as basic as "[tex]2+3x[/tex]" or as complex as "[tex]3x^{2} +5x-2[/tex]" are both permissible.

Additionally, they might include variables, which are representations of unknown values represented by symbols, such as "[tex]y+2x[/tex]." In algebra, calculus, and other areas of mathematics, expressions are commonly used to discuss problems and describe mathematical properties.

Since [tex]4[/tex] and[tex]5[/tex] may be stated as [tex]2^{2}[/tex] and [tex]5^{1}[/tex] so the [tex](2^{2})^{4} *5^{4}[/tex]

[tex]4^{4} *5^{4} =2^{8} *5^{4}[/tex] [tex]=(2^{2})^{4} *5^{4}[/tex]

So the expression for [tex]4^{4} *5^{4} =2^{8} *5^{4}[/tex] as there are single power of [tex]2[/tex] and [tex]5[/tex]

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Check the picture below.

Therefore, the sales tax rate on the washer and dryer set was 5.25%.

Sales tax rates vary depending on the country, state, and even city or municipality where the sale is taking place. Therefore, it's difficult to provide a single answer without more information about the specific location you're asking about.

For example, in the United States, the sales tax rate can range from 0% in some states to as high as 10.25% in certain cities or counties. In Canada, the federal sales tax is 5%, but provinces and territories may also have their own sales tax rates.

To find the sales tax rate, we need to divide the sales tax by the total cost of the washer and dryer set:

Sales tax rate = sales tax / total cost

Total cost = $2900 (cost of washer and dryer set)

Sales tax = $152.25\2900

Sales tax rate = 152.25 / 2900

Sales tax rate = 0.0525 or 5.25%

Therefore, the sales tax rate on the washer and dryer set was 5.25%.

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The planning value for the population standard deviation ,a sample size of 677 graduates is needed to estimate the population mean starting salary with a margin of error of $200.

We need to use the range of the expected starting salaries, which is between $42,000 and $59,800. The formula for calculating the planning value is:
Planning value = (range of salaries) / (6)
Therefore, the planning value for the population standard deviation is:
Planning value =[tex]($59,800 - $42,000) / (6) = $2,967[/tex]
To determine the sample size needed for a desired margin of error, we need to use the following formula:
Sample size = (Z-score)^2 * (planning value)^2 / (margin of error)^2
For a margin of error of $600 and a 95% confidence level, the Z-score is 1.96. Plugging in the values, we get:
Sample size =[tex](1.96)^2 * ($2,967)^2 / ($600)^2 = 113[/tex]
Therefore, a sample size of 113 graduates is needed to estimate the population mean starting salary with a margin of error of $600.
For a margin of error of $200, the Z-score is still 1.96. Plugging in the new margin of error, we get:
Sample size =[tex](1.96)^2 * ($2,967)^2 / ($200)^2 = 677[/tex]
Therefore, a sample size of 677 graduates is needed to estimate the population mean starting salary with a margin of error of $200
In summary, the planning value for the population standard deviation is $2,967. The sample size needed for a margin of error of $600 is 113, while the sample size needed for a margin of error of $200 is 677. These calculations can help organizations determine the appropriate sample size needed to accurately estimate the starting salaries of business administration graduates.

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The given statement "In the unit method of proportioning, the longest dimension of an object is set equal to one unit." is False because the unit method of proportioning does not require the longest dimension.

In the unit method of proportioning, a chosen dimension of an object is set equal to one unit, and then all other dimensions are expressed in terms of that unit, regardless of whether it is the longest dimension or not.

For example, if a rectangle has a width of 4 cm and a length of 6 cm, we could choose to set the length as the unit and express the width as a fraction of the unit: 4/6 or 2/3 of a unit.

Alternatively, we could choose to set the width as the unit and express the length as a fraction of the unit: 6/4. In either case, the longest dimension of the object is not necessarily the unit.

The unit method of proportioning is commonly used in art and design to help maintain consistent proportions between different elements of a composition. It allows for easy scaling and adjustment of proportions while preserving the overall relationships between the different elements.

Rather, any dimension can be chosen as the unit, and the other dimensions are expressed in terms of that unit.

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The group who has the largest percentage of people who prefer to travel by automobile is given as follows:

Adult.

A bar graph, also known as a bar chart, is a type of graph used to represent and compare data. It consists of rectangular bars of equal width and varying heights, where the height of each bar represents the value of the data being graphed.

Automobiles are represented by the green bars, hence we must observe the input for which the green bar has the largest value, which is for adults, hence adults compose the largest percentage of people who prefer to travel by automobile.

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The value of the technological equipment when it was new according to an exponential decay model is approximately $37,633.52.

It is given to us that the value of a new technological equipment depreciates (decreases) after it is purchased, the value of the technological equipment depreciates according to an exponential decay model.

We know that the value of the equipment is $20000 at the end of 6 years, and its value has been decreasing at the rate of 10% per year.

Let the purchase price of machine be $P

Rate of depreciation = 10% p.a.

Period years.

∴ Present value = $20,000

Depreciated value can be calculated by t(6) ---> [∵ Depriciated value = A]

⇒ $20,000 = P(1−10/100)⁶

⇒ $20,000 = P(9/10)⁶

⇒ P = $20,000 (9/10)⁶ A = P(1-100)t

⇒ P = $20,000 × 10/9 × 10/9 × 10/9 × 10/9 × 10/9 × 10/9

⇒ P = $37,633.528463178 ≈ 37,633.52

The value of the technological equipment when it was new was approximately $37,633.52.

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The sum of the interior angles of the polygon ABCDEF is 720°.

What is the sum of the interior angles of a polygon with n sides?

Sum of interior angles = (n - 2) x 180°

Since polygon ABCDEF is a hexagon (a polygon with six sides), we can substitute n = 6 into the formula to get:

Sum of interior angles = (6 - 2) x 180°

Sum of interior angles = 4 x 180°

Sum of interior angles = 720°

To use triangles to work out the sum of the interior angles of polygon ABCDEF, we can divide the hexagon into four triangles.

Each triangle in the diagram has an interior angle sum of 180°. So, we can find the sum of the interior angles of polygon ABCDEF by adding up the interior angles of the four triangles.

Sum of interior angles = 180° + 180° + 180° + 180°

Sum of interior angles = 720°

Therefore, we get the same answer whether we use the formula or divide the hexagon into triangles to find the sum of the interior angles of polygon ABCDEF, which is 720°.

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Step-by-step explanation:

A   xy = 40   is the same as x = 40/y    an inverse relationship

D   this is the samething as A

Don't know about F...it is truncated

Hence, the volume of the pyramid is 48 cubic inches.

A pyramid is a geometric shape that consists of a polygonal base (usually a square or a triangle) and triangular faces that meet at a single point at the top, called the apex. Pyramids have been built by many ancient civilizations as monumental structures, including the ancient Egyptians, Aztecs, and Mayans. The most famous of these is the Great Pyramid of Giza in Egypt, which was built over 4,500 years ago and is one of the Seven Wonders of the Ancient World. Pyramids have also been used as symbols in many cultures and can represent strength, stability, and spirituality.

The volume of a square pyramid can be calculated using the formula V = (1/3) * B * h, where B is the area of the base and h is the height.

In this case, the base of the pyramid is a square with a side length of 3 in, so the area of the base is:

[tex]B = s^2 = 3^2 = 9 sq. in.[/tex]

The height of the pyramid is given as 16 in.

Therefore, the volume of the pyramid is:

[tex]V = (1/3) * B * h = (1/3) * 9 * 16 = 48 cubic inches.[/tex]

Hence, the volume of the pyramid is 48 cubic inches.

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Answer:

Hery is 3. Jai is 4

Step-by-step explanation:

3×4=12

After answering the presented question, we can conclude that To pay the $19.60 weekly cost for your cell phone Internet access, you would need to save $2.80 (or $2.86 if you round up) per day.

The internet is a vast network of interconnected computers and devices that communicate with each other using a common set of protocols and technologies. It allows people to share information, communicate with each other, and access a wide range of digital services and resources. The internet is a global network that spans across the world, connecting people and businesses across different geographic locations and time zones. It has revolutionized the way people communicate and access information, enabling unprecedented levels of connectivity, collaboration, and innovation.

You can use the following steps to calculate how much you need to save per day to pay the bill:

Divide your weekly bill by the number of days in a week to see how much you need to save per day:

$19.60 ÷ 7 = $2.80

You can round up to the closest cent if you want to be more precise:

$19.60 divided by 7 equals $2.80 per day

$2.80 rounded up to $2.86

To pay the $19.60 weekly cost for your cell phone Internet access, you would need to save $2.80 (or $2.86 if you round up) per day.

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The difference between the perimeter and area of the figure is -20 square units.

In geometry, the perimeter of a shape is defined as the total length of its boundary. The perimeter of a shape is determined by adding the length of all the sides and edges enclosing the shape.

The figure is a rectangle with sides DE and FG measuring 8 units, and sides EF and GD measuring 6 units.

To find the perimeter of the figure, we can add up the lengths of all four sides:

Perimeter = DE + EF + FG + GD

Perimeter = 8 + 6 + 8 + 6

Perimeter = 28

To find the area of the figure, we can use the formula for the area of a rectangle:

Area = length * width

Area = DE * EF

Area = 8 * 6

Area = 48

Therefore, the difference between the perimeter and area of the figure is:

Perimeter - Area = 28 - 48

Perimeter - Area = -20

The difference between the perimeter and area of the figure is -20 square units.

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The differential equation is [tex]-0.625ln|p-1| + 2.3026ln|p+1.5| - 1.6779ln|p+1| = t + K,[/tex] where K is a constant of integration.

We need to find the solution to the differential equation:

[tex]dp/dt = f(p) = -0.4p^3 - 1.6p + 3.6[/tex]

To find the general solution, we first need to separate the variables p and t, and then integrate both sides:

[tex]dp / ( -0.4p^3 - 1.6p + 3.6 ) = dt[/tex]

We can integrate the left-hand side using partial fractions:

[tex]dp / ( -0.4p^3 - 1.6p + 3.6 ) = dp / ( -0.4(p-1)(p+1.5)(p+1) )[/tex]

[tex]dp / ( -0.4(p-1)(p+1.5)(p+1) ) = (A / (p-1)) + (B / (p+1.5)) + (C / (p+1))[/tex]

where A, B, and C are constants to be determined.

Multiplying both sides by the denominator, we get:

[tex]dp = (A(p+1.5)(p+1) + B(p-1)(p+1) + C(p-1)(p+1.5)) / (-0.4(p-1)(p+1.5)(p+1)) dp[/tex]

Expanding the terms on the right-hand side and equating the coefficients of p², p, and the constant term, we get:

A + B + C = 0

1.5A - 1B + 1.5C = -1.6/(-0.4)

A - 1.5B - C = 3.6/(-0.4)

Solving these equations, we get:

A = 1.4286

B = -3.4524

C = 2.0238

Substituting these values back into the partial fractions equation, we get:

[tex]dp / ( -0.4(p-1)(p+1.5)(p+1) ) = (1.4286 / (p-1)) - (3.4524 / (p+1.5)) + (2.0238 / (p+1))[/tex]

Integrating both sides, we get:

[tex]-0.625ln|p-1| + 2.3026ln|p+1.5| - 1.6779ln|p+1| = t + K[/tex]

where K is a constant of integration.

This is the general solution to the differential equation.

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Answer:

The answer to your problem is, D. [tex]-2, \frac{4}{5}[/tex]

Step-by-step explanation:

Using the a-c method to factor the quadratic

The factors of the product 5 x -8 = -40

Which sum to +6 are + 10 and -4

Split the middle term using these factors:

[tex]5x^{2} + 10x - 4x - 8[/tex]

[tex]= 5x(x + 2 ) -4 ( x + 2 )[/tex]

We will then take out the common factor: ( x + 2 )

= ( x + 2 )( 5x - 4 )

[tex]5x^{2} + 6x - 8 = ( x + 2 )( 5x - 4 )[/tex]

Thus the answer to your problem is, [tex]-2, \frac{4}{5}[/tex]

Answer:

Move terms to the left side

Once in standard form, identify a, b, and c from the original equation and plug them into the quadratic formula.

Evaluate the exponent

Multiply the numbers

Add the numbers

Evaluate the square root

Multiply the numbers

Answer:

x= 4/5

x= -2

so the answer is A

To solve the problem, we can use trigonometry and create a right triangle with one leg being the height of building B, the other leg being the distance between building A and B, and the hypotenuse being the line of sight from the top of building A to the top of building B.

Let's call the height of building B "h". Using trigonometry, we can determine the length of the other leg:

tan(30°) = h / x => x = h / tan(30°)

tan(60°) = h / (10√3 - x) => x = 10√3 - h / tan(60°)

Setting these two expressions equal to each other and solving for h, we get:

h / tan(30°) = 10√3 - h / tan(60°)

h (1/tan(30°) + 1/tan(60°)) = 10√3

h = 10√3 / (1/tan(30°) + 1/tan(60°)))

Plugging in the values, we get:

h = 10√3 / (1/(1/√3) + 1/√3)

h = 20√3

Answer:

Step-by-step explanation:

Let's call the height of building A "hA" and the height of building B "hB". We can use trigonometry to solve for hB.

First, let's draw a diagram:

   B

  /|

hB/ |

/  |

/ 60°\

-----

|   /

|  /

| /

|/

A

We know that the distance between building A and B is 10√3. Let's call this distance "d".

Using the angle of depression of 30°, we can form a right triangle with a leg of hA and a hypotenuse of d. The opposite angle is 60°, so the adjacent side is hA/tan(60°) = hA/√3.

Using the angle of depression of 60°, we can form another right triangle with a leg of hB and a hypotenuse of d. The opposite angle is 30°, so the adjacent side is hB/tan(30°) = hB√3.

We know that the sum of the heights of building A and B is equal to the distance between them, so hA + hB = d.

Putting all of this together, we can set up an equation:

hA/√3 + hB√3 = 10√3

Multiplying both sides by √3:

hA + 3hB = 30

But we also know that hA + hB = d = 10√3, so we can substitute:

hB = 10√3 - hA

Substituting into the previous equation:

hA + 3(10√3 - hA) = 30

Simplifying:

-2hA + 30√3 = 30

-2hA = 30 - 30√3

hA = (15√3 - 15)/(-1) = 15 - 15√3

Finally, we can use hA + hB = 10√3 to solve for hB:

hB = 10√3 - hA = 10√3 - (15 - 15√3) = 25√3 - 15

Therefore, the height of building B is 25√3 - 15.

The margin of error for a 90% confidence interval for the proportion of all employees dissatisfied with their jobs, based on a survey of 200 employees where 44% reported dissatisfaction, is  0.068.

We can use the formula for the margin of error for a proportion:

ME = zsqrt((p_hat(1-p_hat))/n)

where z is the z-score associated with the desired level of confidence (90% in this case), p_hat is the sample proportion (0.44), n is the sample size (200), and sqrt is the square root.

From a standard normal distribution table, the z-score for a 90% confidence level is approximately 1.645.

Plugging in the values, we get:

ME = 1.645sqrt((0.44(1-0.44))/200)

ME ≈ 0.068

So the margin of error for a 90% confidence interval is approximately 0.068 or 0.068/1= 0.068 = 0.068 = 6.8% (rounded to 3 decimal places).

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--The given question is incomplete, the complete question is given

"A human resources consulting firm conducted a survey of 200 employees to determine how dissatisfed they were with their jobs. 44% of the employees said they were dissatisfied, what is the margin of error for a 90% confidence interval for the proportion of all employees that are dissatisfied with their jobs? Answer to 3 decimal places"--

These notes provide a comprehensive guide to solving linear equations using the substitution method, and would be a helpful resource for students studying this topic.

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Define these terms in your notes and highlight or underline them for easy reference.
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This will help reinforce your understanding and improve retention of the information.

By following these steps, you'll create organized and effective notes that incorporate vocabulary and core concepts, making it easier to study and understand the material.

The notes provide several examples with detailed explanations on how to apply the substitution method to solve linear equations.

The examples include equations with coefficients, constants, and variables, and demonstrate how to rearrange the equations to solve for the variables.

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Answer:

radius of circle T/smaller circle = 4 units

Diameter of larger circle is 10units

radius of larger circle = D/2

= 10/2 = 5units

CIRCUMFERENCE OF SMALLER CIRCLE =

2πr = 2 × 3.14 × 4

= 25.12 units

Area of smaller Circle =

πr² = 3.14 × 4 × 4

= 50.24 units²

CIRCUMFERENCE OF LARGER CIRCLE =

2πr = 2 × 3.14 × 5

= 31.4 units

Area of larger circle =

πr² = 3.14 × 5 × 5

= 78.5 units²

The approximate probability that x is between 15 and 25 (inclusive) is 0.3026.

To solve this problem, we will use the binomial distribution, since we are dealing with a binary outcome (nonconforming vs. conforming) and a fixed sample size. Let p = 0.11 be the probability of a nonconforming shaft and q = 1 - p = 0.89 be the probability of a conforming shaft. Then, the probability mass function of x is given by:

P(X = x) = (200 choose x) * p^x * q^(200-x)

(a) To find the probability that x is at most 30, we can compute:

P(X ≤ 30) = Σ P(X = x) for x from 0 to 30

However, this is quite tiresome to tally by hand, so we can instead use a normal approximation to the binomial distribution. Specifically, if np and nq are both at least 10, then we can approximate the binomial distribution with a normal distribution with mean μ = np and standard deviation σ = sqrt(npq). In this case, we have np = 22 and nq = 178, both of which are at least 10.

Thus, we can approximate X with a normal distribution:

[tex]X ~ N(mu, σ^2) = N(22, 3.8004)[/tex]

Then, we can compute the probability that X is at most 30 by standardizing and using the standard normal distribution:

[tex]P(X ≤ 30) ≈ P(Z ≤ (30 - mu) / σ) = P(Z ≤ (30 - 22) / 1.9488) = P(Z ≤ 4.1142) = 0.9997[/tex]

(b) To find the probability that x is less than 30, we can use the same normal approximation and compute:

[tex]P(X < 30) ≈ P(Z < (30 - mu) / σ) = P(Z < (30 - 22) / 1.9488) = P(Z < 4.1142) = 0.9997[/tex]

(c) To find the probability that x is between 15 and 25 (inclusive), we can either use the binomial distribution directly or use the normal approximation as before:

P(15 ≤ X ≤ 25) = Σ P(X = x) for x from 15 to 25

However, we can use the normal approximation instead. Using the same approach as before, we get:

[tex]P(15 ≤ X ≤ 25) ≈ P((15 - μ) / σ ≤ Z ≤ (25 - μ) / σ) = P(-3.0806 ≤ Z ≤ -0.5145) = P(Z ≤ -0.5145) - P(Z ≤ -3.0806) = 0.3035 - 0.0009 = 0.3026[/tex]

Therefore, the approximate probability that x is between 15 and 25 (inclusive) is 0.3026.

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The probability of rolling a six when spinning the spinner shown twice, assuming that the outcomes 1, 2, 3, and 4 are equally likely for each spin is 3/16.

A robot spins the spinner shown twice. Assume that the outcomes 1, 2, 3, and 4 are equally likely for each spin.

What is the probability that the sum of the two outcomes will be 6?

The probability that the sum of the two outcomes will be 6 when a robot spins the spinner shown twice, assuming that the outcomes 1, 2, 3, and 4 are equally likely for each spin, is 2/16 or 1/8.However, before determining the probability, we should first determine the possible outcomes of the spinner. When a spinner is spun twice, the possible outcomes are: 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, and 44.There are 16 possible outcomes. Of these 16 possible outcomes, only two add up to six. These two possible outcomes are: 33,24,42. Therefore, the probability of rolling a six is 3/16 .

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The total difference between 4 5/6 and 9 1/6 on a number line using shorter jumps to the nearest integers is 4 1/3.

To find the difference between 4 5/6 and 9 1/6 on a number line using shorter jumps to the nearest integers, we can follow these steps:

Step 1: Convert the mixed numbers to improper fractions.

4 5/6 = (6 x 4 + 5) / 6 = 29/6

9 1/6 = (6 x 9 + 1) / 6 = 55/6

Step 2: Find the distance between the two fractions on the number line by taking the absolute value of their difference.

|29/6 - 55/6| = |-26/6| = 26/6

Step 3: Simplify the fraction and convert it back to a mixed number, if necessary.

26/6 can be simplified to 13/3, which is an improper fraction.

To convert 13/3 back to a mixed number, we divide the numerator (13) by the denominator (3) and write the remainder as a fraction:

13 ÷ 3 = 4 with a remainder of 1

So, 13/3 = 4 1/3

Therefore, on a number line with shorter jumps to the nearest integers, the total distance between 4 5/6 and 9 1/6 is 4 1/3.

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it depends on the shape. There are yt videos you could watch that would explain how to solve the perimeters of that specific shape

OK what are you wanting to find the perimeter to?

Add up all the sides:

EX:

Square: 3+3+3+3

Rectangle: 4+2+4+2

Triangle [Equilateral]: 2+2+2

You get the gist?

Step-by-step explanation:

The total number of pens in the box is:

Total number of pens = 11 black pens + 8 blue pens + 6 red pens

Total number of pens = 25 pens

The probability of Roxy selecting a blue pen can be expressed as the ratio of the number of blue pens to the total number of pens:

Probability of selecting a blue pen = Number of blue pens / Total number of pens

Probability of selecting a blue pen = 8 / 25

So, the probability that the pen Roxy selects is blue is 8/25 or approximately 0.32 (rounded to two decimal places).

Answer:8/25

Step-by-step explanation:

19) Length of AB is a fourth of the circumference of the circle.

C=2πr

Length of AB = [tex]\frac{1}{4} 2\pi r=\frac{1}{2} \pi r[/tex]

=0.5π(12)

Length of AB =1 8.8495559215 in

Area = 1/4*π*r²

Area = 1/4*π*12²

Area = 113.097335529 in²

20) Arc length = θr

[tex]\frac{3 }{4} \pi *10[/tex]

Length = 23.5619449019 in

Area = [tex]\frac{angle}{2\pi } *\pi r^{2}[/tex]

= [tex]\frac{3\pi }{8} *10^2[/tex]

Area = 117.80972451 in²

The measure of angle Q is equal to 40°.

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry. In Geοmetry, triangles are the type οf pοlygοns, which have three sides and three vertices.

This is a twο-dimensiοnal figure with three straight sides. A triangle is cοnsidered a 3-sided pοlygοn. The sum οf all the three angles οf a triangle is equal tο 180°. The triangle is cοntained in a single plane. Based οn its sides and angle measurement, the triangle has six types.

Using Cos θ = adjacent/ Hypotenuse

Cos Q = 720/ 940

Cos Q = 36/47

Q = arccos 36/47

Q = 40.007793734°

Thus, The measure of angle Q is equal to 40°.

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Answer:

The last option/option D/2 over 7/(2/7)

Step-by-step explanation:

There are 3 $10 bills.
There are 4 $1 coins.

3/7=Pull out one $10 dollar bill

However, it is less likely to get the same answer twice.
A Root Graph (or what you like to call it):


As you can see, there are 3 instances where you wound up with 10 BOTH ROUNDS. There are also an additional 11 that DON'T wound up with 10 in both rounds. The ratio is 3/11. 3 divided by 11 is 0.27 (or 0.28). The last option, 2/7 is the one that I think is the answer. It is because 2 divided by 7=0.28...

Therefore our answer is the last option?

The diameter of the lens of a particular telescope is approximately 4 feet.

The circumference of a circle is defined as the linear distance around it. In other words, if a circle is opened to form a straight line, then the length of that line will be the circle's circumference.

The circumference of a circle is given by the formula:

C = πd

where C is the circumference, d is the diameter, and π is approximately equal to 3.14.

In this case, we are given that the circumference of the lens is 12.56 feet. So we can plug in the values:

12.56 = 3.14d

To solve for d, we need to isolate it on one side of the equation. We can do this by dividing both sides by 3.14:

d = 12.56/3.14

Simplifying, we get:

d ≈ 4

Therefore, the diameter of the lens is approximately 4 feet

To know more about circumference, click here

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Answer:

Step-by-step explanation:

To find the equation of a circle that passes through three given points, we can use the fact that the perpendicular bisectors of the chords joining the points intersect at the center of the circle.

Let's first find the midpoint and slope of the chords joining the three points:

The midpoint and slope of the chord joining (-1, 2) and (4, 2):

Midpoint: $((4 - 1)/2, (2 + 2)/2) = (3/2, 2)$

Slope: $(2 - 2)/(4 - (-1)) = 0$

The midpoint and slope of the chord joining (-1, 2) and (-3, 4):

Midpoint: $((-3 - 1)/2, (4 + 2)/2) = (-2, 3)$

Slope: $(4 - 2)/(-3 - (-1)) = 1/2$

The midpoint and slope of the chord joining (4, 2) and (-3, 4):

Midpoint: $((-3 + 4)/2, (4 + 2)/2) = (1/2, 3)$

Slope: $(4 - 2)/(-3 - 4) = -1/2$

Now we can find the equations of the perpendicular bisectors of these chords:

The equation of the perpendicular bisector of the chord joining (-1, 2) and (4, 2) is the horizontal line $y=2$.

The equation of the perpendicular bisector of the chord joining (-1, 2) and (-3, 4) is the line passing through the midpoint $(-2, 3)$ with slope $-2$:

$$y - 3 = -2(x + 2)$$

Simplifying, we get $y = -2x - 1$.

The equation of the perpendicular bisector of the chord joining (4, 2) and (-3, 4) is the line passing through the midpoint $(1/2, 3)$ with slope $2$:

$$y - 3 = 2(x - 1/2)$$

Simplifying, we get $y = 2x + 2$.

The center of the circle is the point where these perpendicular bisectors intersect. Solving the system of equations formed by setting any two of the perpendicular bisectors equal to each other, we get the center of the circle as $(1, 2)$.

Finally, the radius of the circle is the distance from the center to any of the three given points. We can use the distance formula to find that the radius is $\sqrt{10}$.

Putting it all together, the equation of the circle is:

$$(x - 1)^2 + (y - 2)^2 = 10$$

or expanding and simplifying:

$$x^2 + y^2 - 2x - 4y + 5 = 0$$

Therefore, the general equation for the circle that passes through the points (-1, 2), (4, 2), and (-3, 4) is $x^2 + y^2 - 2x - 4y + 5 = 0$.