Angelica’s bouquet of a dozen roses contains 5 white roses. The rest of the roses are pink. What fraction of the bouquet is pink roses? There are 12 roses in a dozen.
StartFraction 5 Over 12 EndFraction
StartFraction 7 Over 12 EndFraction
StartFraction 5 Over 7 EndFraction

Answer:

44

Step-by-step explanation:

The sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.

The series is: [tex]\sum_{k=1}^4[/tex] (2k²−4)

Let's find the value of each term for k=1, k=2, k=3, and k=4, and then add them up:

For k=1:

2(1)² - 4 = 2(1) - 4 = 2 - 4 = -2

For k=2:

2(2)² - 4 = 2(4) - 4 = 8 - 4 = 4

For k=3:

2(3)² - 4 = 2(9) - 4 = 18 - 4 = 14

For k=4:

2(4)² - 4 = 2(16) - 4 = 32 - 4 = 28

Now, let's add all the terms:

-2 + 4 + 14 + 28 = 44

So, the sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.

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Answer:

B only

Step-by-step explanation:

Triangles A and C are similar, but not congruent to each other. Similar triangles have proportional sides and congruent angles, while congruent triangles have congruent sides and congruent angles.

Therefore, only B is correct

The question is asking for pairs of congruent triangles but lacks key information needed to accurately answer this, such as lengths or angles. Congruent triangles are identified in geometry based on either side-lengths or angles.

The question is about congruent triangles, which in mathematics means triangles that have the same size and shape. But to accurately answer which pairs show two congruent triangles, we need more information. The options provided include 'A', 'B', and 'C' but without knowing more about these elements (for example their lengths or angles), it is impossible to determine which are congruent. Congruent triangles are identified in geometry based on either side lengths (SSS: Side-Side-Side, SAS: Side-Angle-Side, ASA: Angle-Side-Angle) or angles (AAS: Angle-Angle-Side, HL: Hypotenuse-Leg for right triangles). Without this vital information, we cannot definitively answer the question. Be sure to verify all the properties required to prove two triangles congruent.

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Answer:

Third option

Step-by-step explanation:

-3(2x - 5) < 5(2 - x)

-6x + 15 < 10 - 5x <-- Third option

15 < 10 + x

5 < x

x > 5

There only appears to be one option. The solution to the inequality is x>5, not x<5.

Answer:

4/3

Step-by-step explanation:

1/3:(1/4) = (1/3)/(1/4) = 1/3*4 = 4/3

The amount of medication needed for 30 days is approximately 59.84 tablets of carprofen.

To calculate the amount of carprofen medication needed for 30 days, we need to consider the prescribed dosage, the weight of the patient, and the concentration of the tablets.

The prescribed dosage is 2.2 mg/kg BID x 30d. This means that the patient should take 2.2 milligrams of carprofen per kilogram of body weight, twice a day, for 30 days.

The weight of the patient is 34 kilograms. So, we need to calculate the total amount of carprofen needed for the entire treatment period.

First, we calculate the daily dosage by multiplying the weight of the patient (34 kg) by the prescribed dosage (2.2 mg/kg).

Daily dosage = 34 kg * 2.2 mg/kg = 74.8 mg/day.

Since the medication is prescribed twice a day, we multiply the daily dosage by 2 to get the total dosage per day.

Total dosage per day = 74.8 mg/day * 2 = 149.6 mg/day.

Finally, to find the total amount of medication needed for 30 days, we multiply the total dosage per day by the number of days.

Total medication needed = 149.6 mg/day * 30 days = 4488 mg.

Since the concentration of the tablets is 75 mg, we divide the total medication needed by the tablet concentration to find the number of tablets required.

Number of tablets needed = 4488 mg / 75 mg = 59.84.

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Gabby can get €23.73 for £21 when using the exchange rate of £1 = €1.13.

To calculate how many euros Gabby can get for £21, we need to use the exchange rate of £1 = €1.13.

To find the number of euros, we multiply the amount in pounds (£21) by the exchange rate (€1.13):

£21 * €1.13 = €23.73

Therefore, Gabby can get €23.73 for £21 when using the exchange rate of £1 = €1.13.

This means that for every pound exchanged, Gabby will receive €1.13. So, by multiplying the amount in pounds (£21) by the exchange rate (£1 = €1.13), we can determine the equivalent amount in euros.

It's important to note that exchange rates can fluctuate, and the rate provided here is just an example. When exchanging currency, it's always advisable to check the current exchange rates as they may vary. Additionally, some currency exchange services may charge fees or have different rates, so it's essential to consider those factors as well.

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The strength of the correlation as shown in the scatter plot can be concluded to be: b) no correlation.

To estimate the strength of the correlation in a scatter plot, visually assess the clustering and linearity of the data points. If the points cluster tightly around a linear trend, it indicates a strong correlation, whereas dispersed and non-linear points suggest a weak correlation.

However, if the points on a scatter plot do not exhibit a discernible pattern (as shown in the image given above), it can be concluded that there is no correlation between the variables being plotted.

Therefore, the answer is: b. no correlation.

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Answer:

Part A: Answer: D.

Part B:

true

false

true

Step-by-step explanation:

Part A:

Use a tree.

Round 1                 Round 2                   Round 3

W                               W                                W

W                               W                                L

W                               L                                 W

W                                L                                 L

L                                 W                               W

L                                 W                               L

L                                 L                                W

L                                 L                                L

All possibilities are:

WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL

Answer: d.

Part B:

Winning 3 games is WWW.

WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL

Out of the 8 possible outcomes shown above, only 1 of them is WWW.

p(WWW) = 1/8

Answer: true

Winning exactly 2 games happens when there are exactly 2 W's:

WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL

p(exactly 2 wins) = 3/8

Answer: false

Wining at least 2 games means winning exactly 2 or exactly 3 times.

WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL

p(winning at least 2 games) = 4/8 = 1/2

Answer: true

11. Yes, the information provided can be used to conclude that C is on the perpendicular bisector of AB because CE bisects AB.

12. Yes, the information provided can be used to conclude that C is on the perpendicular bisector of AB because C is equidistant from AB.

In Mathematics and Geometry, a perpendicular bisector divides or bisects a line segment exactly into two (2) equal halves, in order to form a right angle with a magnitude of 90° at the point of intersection.

Question 11.

By critically observing the geometric shape, we can logically deduce that there is enough information to conclude that C is on the perpendicular bisector of AB because CE bisects AB:

AC ≅ BC

CD ≅ CD

AE ⊥ EC

Question 12.

By critically observing the geometric shape, we can logically deduce that there is enough information to conclude that C is on the perpendicular bisector because C is equidistant from line segment AB.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The fraction of the container needed to fill up the gas tank of the lawn mower four times is 6/12. Option D.

The shaded part of the circle represents the fraction of gas needed to fill up the gas tank of the lawn mower once. To determine the fraction needed to fill it up four times, we can simply multiply the fraction by 4.

Let's assume that the shaded part of the circle represents x part of the whole container. Therefore, the fraction needed to fill up the gas tank of the lawn mower once is x.

To fill up the gas tank four times, we need to multiply x by 4. Therefore, the fraction needed to fill up the gas tank four times is 4x.

Now, we can compare this to the answer choices given:

8/48: This can be simplified to 1/6, which is not equal to 4x.

8/12: This can be simplified to 2/3, which is not equal to 4x.

6/16: This can be simplified to 3/8, which is not equal to 4x.

6/12: This can be simplified to 1/2, which is equal to 4x.

Therefore, the correct answer is 6/12, which represents the fraction of the container needed to fill up the gas tank of the lawn mower four times. so Option D is correct.

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Note the complete question is

Answer:

31.  m∠E = 56.1°

32.  c = 24.9 inches

Step-by-step explanation:

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all three sides of the triangle.

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

Given values of triangle DEF:

To find m∠E, substitute the values into the Law of Sines formula and solve for E:

[tex]\dfrac{\sin D}{d}=\dfrac{\sin E}{e}[/tex]

[tex]\dfrac{\sin 81^{\circ}}{25}=\dfrac{\sin E}{21}[/tex]

[tex]\sin E=\dfrac{21\sin 81^{\circ}}{25}[/tex]

[tex]E=\sin^{-1}\left(\dfrac{21\sin 81^{\circ}}{25}\right)[/tex]

[tex]E=56.1^{\circ}\; \sf(nearest\;tenth)[/tex]

Therefore, the measure of angle E is 56.1°, to the nearest tenth.

See the attachment for the accurate drawing of triangle DEF.

[tex]\hrulefill[/tex]

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of triangle ABC:

To find the length of side c, substitute the values into the Law of Cosines formula and solve for c:

[tex]c^2=a^2+b^2-2ab \cos C[/tex]

[tex]c^2=13^2+15^2-2(13)(15) \cos 125^{\circ}[/tex]

[tex]c^2=169+225-390 \cos 125^{\circ}[/tex]

[tex]c^2=394-390 \cos 125^{\circ}[/tex]

[tex]c=\sqrt{394-390 \cos 125^{\circ}}[/tex]

[tex]c=24.8534667...[/tex]

[tex]c=24.9\; \sf inches\;(nearest\;tenth)[/tex]

Therefore, the length of side c is 24.9 inches, to the nearest tenth.

The journal entries for the given transactions are as follows:

Cash A/c Dr. 40,000

Furniture A/c Dr. 5,000

Motor Car A/c Dr. 12,000

Stock A/c Dr. 20,000

To Capital A/c 77,000

Bank A/c Dr. 15,000

To Cash A/c 15,000

Purchase A/c Dr. 9,000

To Sen's A/c 9,000

Basu's A/c Dr. 6,000

To Sales A/c 6,000

Stationery A/c Dr. 200

To Cash A/c 200

Cash A/c Dr. 2,000

To Dalal's A/c 2,000

Travelling Expenses A/c Dr. 600

To Cash A/c 600

Drawings A/c Dr. 1,000

To Bank A/c 1,000

Cash A/c Dr. 3,000

To Bank A/c 3,000

Sen's A/c Dr. 8,800

To Bank A/c 8,800

Freight and Clearing A/c Dr. 400

To Cash A/c 400

Bank A/c Dr. 6,000

To Basu's A/c 6,000

Journal entries for the given transactions are as follows:

Pater commenced business with 40,000 cash and also brought into business furniture worth ₹5,000; Motor car valued for ₹12,000 and stock worth ₹20,000.

Cash A/c Dr. 40,000

Furniture A/c Dr. 5,000

Motor Car A/c Dr. 12,000

Stock A/c Dr. 20,000

To Capital A/c 77,000

Deposited ₹15,000 into State Bank of India.

Bank A/c Dr. 15,000

To Cash A/c 15,000

Bought goods on credit from Sen for ₹9,000.

Purchase A/c Dr. 9,000

To Sen's A/c 9,000

Sold goods to Basu on Credit for ₹6,000.

Basu's A/c Dr. 6,000

To Sales A/c 6,000

Bought stationery from Ram Bros. for Cash ₹200.

Stationery A/c Dr. 200

To Cash A/c 200

Sold goods to Dalal for ₹2,000 for which cash was received.

Cash A/c Dr. 2,000

To Dalal's A/c 2,000

Paid ₹600 as travelling expenses to Mehta in cash.

Travelling Expenses A/c Dr. 600

To Cash A/c 600

Patel withdrew for personal use ₹1,000 from the Bank.

Drawings A/c Dr. 1,000

To Bank A/c 1,000

Withdrew from the Bank ₹3,000 for office use.

Cash A/c Dr. 3,000

To Bank A/c 3,000

Paid to Sen by cheque ₹8,800 in full settlement of his account.

Sen's A/c Dr. 8,800

To Bank A/c 8,800

Paid ₹400 in cash as freight and clearing charges to Gopal.

Freight and Clearing A/c Dr. 400

To Cash A/c 400

Received a cheque for ₹6,000 from Basu.

Bank A/c Dr. 6,000

To Basu's A/c 6,000

These journal entries represent the various transactions and their effects on different accounts in the accounting system.

They serve as the initial records of the financial activities of the business and provide a basis for further accounting processes such as ledger posting and preparation of financial statements.

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1a. Sum of interior angles of a heptagon:

[tex]\displaystyle \sf \text{Sum of interior angles} = (7 - 2) \times 180^\circ = 900^\circ[/tex]

1b. Sum of interior angles of a 13-gon:

[tex]\displaystyle \sf \text{Sum of interior angles} = (13 - 2) \times 180^\circ = 1980^\circ[/tex]

2. Number of sides for a polygon with a sum of interior angles of 1260°:

[tex]\displaystyle \sf (n - 2) \times 180^\circ = 1260^\circ[/tex]

[tex]\displaystyle \sf n - 2 = \frac{1260^\circ}{180^\circ}[/tex]

[tex]\displaystyle \sf n - 2 = 7[/tex]

[tex]\displaystyle \sf n = 7 + 2 = 9[/tex]

Therefore, the sum of the measures of the interior angles of a heptagon is 900°, the sum of the measures of the interior angles of a 13-gon is 1980°, and the polygon with a sum of interior angles of 1260° is a nonagon (9-gon).

[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]

♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

Answer:

1. a. 900°       b. 1980°

2. Nonagon

Step-by-step explanation:

In order to find the interior angles of a polygon, use the formula,

Sum of interior angles = (n-2)*180°

For

a. Heptagon

no of side =7

The sum of the interior angles of a heptagon:

(7-2)*180 = 900°

b. 13-gon

no. of side =13

The sum of the interior angles of a 13-gon:
(13-2)*180 = 1980°

2.

The sum of the interior angles of a convex polygon is 1260°,

where n is the number of sides.

In this case, we have

1260 = (n-2)*180

1260/180=n-2

n-2=7

n=7+2

n=9

Therefore, the polygon has 9 sides and is classified as a nonagon.

Answer: 6+9x or 9x + 6

Step-by-step explanation:

Multiply 3 by each of the numbers inside the parentheses

The answer is:

Work/explanation:

To simplify this expression, we will use the distributive property:

[tex]\sf{3(2+3x)}[/tex]

Distribute the 3:

[tex]\sf{3\cdot2+3\cdot3x}[/tex]

Simplify

[tex]\sf{6+9x}[/tex]

Answer:

(2, 7) and (-6, -1)

Step-by-step explanation:

y = x + 5

y = x² + 5x − 7

Equatig the above,

x² + 5x − 7 = x + 5

⇒ x² + 4x −12 = 0

⇒ x² + 6x - 2x - 12 = 0

⇒ x(x + 6) - 2(x + 6) = 0

⇒  (x - 2)(x + 6) = 0

⇒ x = 2 or x = -6

Eq(1) : y = x + 5 (given)

When x = 2

y = 2 + 5 = 7

Point : (2, 7)

When x = -6

y = -6 + 5 = -1

Point: (-6, -1)

Answer:

27

Step-by-step explanation:

The correct answer is an option (d): Impossible.

Both the geometric mean and harmonic mean require positive values. However, in the given set of values, we have negative values (-11 and -14), which makes it impossible to calculate the geometric mean and harmonic mean. Therefore, the correct answer is that it is impossible to calculate the geometric mean and harmonic mean for the given values.

First, we need to calculate the area of the room Jenny is tiling, which appears to be in the shape of a rectangle. The area is calculated by multiplying the length and width of the rectangle. If the measurements provided (2.6m and 11m) are for the length and width, then the area of the room would be:

Area = length x width

Area = 2.6m x 11m

Area = 28.6 m²

Next, we need to calculate how many packs of tiles Jenny needs to buy. Given that each pack covers an area of 3m², we divide the total area by the area each pack covers:

Number of packs needed = total area / area covered by one pack

Number of packs needed = 28.6m² / 3m² ≈ 9.53 packs

Since Jenny can't buy a fraction of a pack, she needs to purchase 10 packs of tiles.

The tiles are marked at £18.60, so before the discount, the total cost of the tiles would be:

Total cost = number of packs x price per pack

Total cost = 10 packs x £18.60 = £186

Jenny has a discount card which gives her a 25% discount off the marked price. The discount amount can be calculated as:

Discount = total cost x discount rate

Discount = £186 x 25% = £46.5

So, the cost of the tiles after the discount is:

Discounted price = total cost - discount

Discounted price = £186 - £46.5 = £139.5

Jenny has £100 to spend on the tiles. The extra amount Jenny needs is the difference between the cost of the tiles after the discount and the amount she has:

Extra amount needed = discounted price - amount Jenny has

Extra amount needed = £139.5 - £100 = £39.5

The question asks for the answer in pence, and there are 100 pence in a pound, so:

Extra amount needed in pence = extra amount needed in pounds x 100

Extra amount needed in pence = £39.5 x 100 = 3950 pence

Therefore, Jenny needs an extra 3950 pence to buy the tiles.

Answer:

-21

Step-by-step explanation:

2+7-8-13-4-5

9-30

-21

Hope it's helpful.

Using the concept of scale factor, the value of t in polygon s is:  7.2

The amount by which the shape is expanded or contracted is referred to as the scale factor. This is usually utilized when 2D shapes such as circles, triangles, squares and rectangles need to be enlarged. If y = Kx is an equation, K is the scaling factor for x.

From the given image, since Polygon s is a scaled copy of polygon R, then we can say that using the concept of scale factor, we have:

9/t = 12/9.6

t = (9 * 9.6)/12

t = 7.2

Thus, we can be sure that is the value of t of the polygon S using the concept of scale factor

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The concept of equidistant points or lines is fundamental in geometry and plays a significant role in many geometric constructions and properties.

In geometry, an "equidistant" point is a point that is at the same distance from other points or lines. This concept is often used in various geometric constructions and proofs.

For example, in a circle, the center of the circle is equidistant from all points on the circumference. This property is what defines a circle.

In terms of lines, an equidistant point can be found by drawing perpendicular bisectors. A perpendicular bisector of a line segment is a line that is perpendicular to the segment and divides it into two equal parts. The point where the perpendicular bisectors of a triangle intersect is called the circumcenter, and it is equidistant from the vertices of the triangle.

Another example is the concept of an equidistant curve. In some cases, there may be a curve or path that is equidistant from two fixed points. This curve is called the "locus of points equidistant from two given points" and is often referred to as a "perpendicular bisector" when dealing with line segments.

All things considered, the idea of equidistant points or lines is essential to geometry and is important to many geometrical constructions and features.

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The statement "rank(AB) = n" is not always true for invertible n by n matrices A and B with rank(A) = rank(B) = n.

Counterexample:

Consider the following counterexample:

[tex]A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\\B = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}[/tex]

Both matrices A and B are invertible, as they are both identity matrices. However, when we multiply them together, we get:

[tex]AB = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}[/tex]

The rank of AB is 1, not n. In this case, n = 2, but rank(AB) = 1. Therefore, the statement "rank(AB) = n" is not always true for all invertible n by n matrices A and B with rank(A) = rank(B) = n.

In general, the rank of the product AB is less than or equal to the minimum of the ranks of A and B. So, in order for rank(AB) to be equal to n, both A and B need to have rank n individually, but that does not guarantee the rank of their product to be n as well.

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(a) The amount for total expenditures in 2015 was about $63.2 billion.

(b) The first full year in which expenditures exceeded $110 billion was 2027.

(a) To find the amount for total expenditures in 2015, we need to substitute x = 20 into the function h(x) = [tex]23.4(1.08)^{(x-5)[/tex].

h(x) = 23.4(1.0[tex]8)^{(20-5)[/tex]

    = 23.4(1.0[tex]8)^{15[/tex]

    ≈ 23.4(2.717)

Rounding to the nearest tenth, the total expenditures in 2015 were about $63.2 billion.

(b) To determine the first full year in which expenditures exceeded $110 billion, we need to find the value of x when h(x) is greater than $110 billion.

110 = 23.4(1.0[tex]8)^{(x-5)[/tex]

Dividing both sides by 23.4:

4.7008547... = (1.0[tex]8)^{(x-5)[/tex]

Taking the logarithm base 1.08 of both sides:

log₁.₀₈(4.7008547...) = x - 5

Using a logarithm calculator or software, we find:

x - 5 ≈ 11.75

Adding 5 to both sides:

x ≈ 16.75

Rounding to the nearest whole number, the first full year in which expenditures exceeded $110 billion was 2027.

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The percent increase in recreational boating participation over the sixteen-year period is approximately 8.72%. This means that the number of participants increased by around 8.72% from 52.7 million to 57.3 million.

To determine the percent increase in recreational boating participation over the sixteen-year period, we can use the following formula:

Percent Increase = ((New Value - Old Value) / Old Value) * 100

Using the given information, we have an old value of 52.7 million and a new value of 57.3 million.

Percent Increase = ((57.3 million - 52.7 million) / 52.7 million) * 100

= (4.6 million / 52.7 million) * 100

= 0.0872 * 100

= 8.72%

This increase indicates a positive trend in recreational boating, reflecting a growing interest in this activity over time. Factors such as improved accessibility, marketing efforts, and increasing disposable income may have contributed to this upward trend.

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The money raised by Lucas for the fundraiser is $30.

The bars sold on Saturday are 12 bars. Each bar costs $1.50.

So, the amount will be:  12 * 1.50 = 18

The bars sold on Sunday are  8 bars.

So, the amount will be:  8 * 1.50 = 12

Hence, the total amount: is 18+12= 30

The table that shows the same function as the given graph is: A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 2, 4, 6. Column 2 is labeled y with entries 4, 5, 6, 7.

To determine which table shows the same function as the given graph, we need to find the values of y corresponding to the given values of x on the graph.

Point A: (0, 4)

Point B: (2, 5)

Let's check each table:

A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 2, 4, 6. Column 2 is labeled y with entries negative 8, negative 4, 0, 4.

A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 2, 4, 6. Column 2 is labeled y with entries 0, 4, 0, 4.

A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 2, 4, 6. Column 2 is labeled y with entries 4, 5, 6, 7.

A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 2, 4, 6. Column 2 is labeled y with entries 4, 8, 12, 16.

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The square root of m^6 is always positive.

The square root of m^6 can be calculated by dividing the exponent 6 by 2, since taking the square root is equivalent to raising a number to the power of 1/2.

In this case, m^6 divided by 2 gives us m^3.

Thus, the square root of m^6 is m^3.

This means that if we square m^3, the result will be m^6.

It is important to understand that the square root operation yields the positive root, so the square root of m^6 is always positive.

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Answer:

a. 36.65 in

b. 14.14 km²

Step-by-step explanation:

Solution Given:

a.

Arc Length = 2πr(θ/360)

where,

Here Given: θ=150° and r= 14 in

Substituting value

Arc length=2π*14*(150/360) =36.65 in

b.

Area of the sector of a circle = (θ/360°) * πr².

where,

Here  θ = 45° and r= 6km

Substituting value

Area of the sector of a circle = (45/360)*π*6²=14.14 km²

Answer:

[tex]\textsf{a)} \quad \overset{\frown}{AC}=36.65\; \sf inches[/tex]

[tex]\textsf{b)} \quad \text{Area of sector $ABC$}=14.14 \; \sf km^2[/tex]

Step-by-step explanation:

The formula to find the arc length of a sector of a circle when the central angle is measured in degrees is:

[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Arc length}\\\\Arc length $= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the given values into the formula:

[tex]\begin{aligned}\overset{\frown}{AC}&= \pi (14)\left(\dfrac{150^{\circ}}{180^{\circ}}\right)\\\\\overset{\frown}{AC}&= \pi (14)\left(\dfrac{5}{6}}\right)\\\\\overset{\frown}{AC}&=\dfrac{35}{3}\pi\\\\\overset{\frown}{AC}&=36.65\; \sf inches\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, the arc length of AC is 36.65 inches, rounded to the nearest hundredth.

[tex]\hrulefill[/tex]

The formula to find the area of a sector of a circle when the central angle is measured in degrees is:

[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Area of a sector}\\\\$A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the given values into the formula:

[tex]\begin{aligned}\text{Area of sector $ABC$}&=\left(\dfrac{45^{\circ}}{360^{\circ}}\right) \pi (6)^2\\\\&=\left(\dfrac{1}{8}\right) \pi (36)\\\\&=\dfrac{9}{2}\pi \\\\&=14.14\; \sf km^2\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, the area of sector ABC is 14.14 km², rounded to the nearest hundredth.

A. The distribution of X can be represented as X ~ N(μ, σ). B. 1918 is the sample mean. C. The probability that a randomly selected district had fewer than 2009 votes for President Clinton is approximately 0.5641.

D. The probability is approximately 0.0725.

E. The first quartile for votes for President Clinton is approximately 1544.

a. The distribution of X is normally distributed. Therefore, the distribution of X can be represented as X ~ N(μ, σ), where μ is the mean and σ is the standard deviation.

b. In this context, 1918 is the sample mean. It represents the average number of votes per district in the sample of 40 election districts in Alaska.

c. To find the probability that a randomly selected district had fewer than 2009 votes for President Clinton, we need to calculate the z-score and then use the standard normal distribution table (or a calculator with a normal distribution function). The z-score can be calculated as follows:

z = (x - μ) / σ

where x is the value we're interested in (2009 votes), μ is the population mean (1918 votes), and σ is the standard deviation (554 votes).

z = (2009 - 1918) / 554 ≈ 0.164

Using the standard normal distribution table or a calculator, we can find that the probability corresponding to a z-score of 0.164 is approximately 0.5641.

Therefore, the probability that a randomly selected district had fewer than 2009 votes for President Clinton is approximately 0.5641.

d. To find the probability that a randomly selected district had between 1955 and 2056 votes for President Clinton, we need to calculate the z-scores for both values and then use the standard normal distribution table.

For 1955 votes:

z1 = (1955 - 1918) / 554 ≈ 0.066

For 2056 votes:

z2 = (2056 - 1918) / 554 ≈ 0.248

Using the standard normal distribution table or a calculator, we can find the cumulative probabilities corresponding to z1 and z2. Then, we subtract the cumulative probability for z1 from the cumulative probability for z2 to find the probability between these two values.

P(1955 < X < 2056) = P(X < 2056) - P(X < 1955)

Using the standard normal distribution table or a calculator, we can find these probabilities and subtract them to find the desired probability.

P(1955 < X < 2056) = P(X < 2056) - P(X < 1955)

≈ 0.5989 - 0.5264

≈ 0.0725

So, the probability is approximately 0.0725.

e. The first quartile corresponds to the 25th percentile of the distribution. To find the first quartile for votes for President Clinton, we need to find the value of X such that 25% of the districts have fewer votes.

Using the standard normal distribution table or a calculator, we can find the z-score corresponding to the 25th percentile, which is -0.6745. Then we can calculate the value of X using the z-score formula:

-0.6745 = (X - 1918) / 554

Solving for X:

X - 1918 = -0.6745 × 554

X - 1918 = -374.167

X ≈ 1543.833

Rounding to the nearest whole number, the first quartile for votes for President Clinton is approximately 1544.

learn more about sample mean: https://brainly.com/question/29368683

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