The circle at each end of the court that surrounds the free throw line is the same size as the jump ball circle. (In other words, they have the same radius. ) What is the area of the rectangle (called the lane) whose length is 19 feet and whose width is the free throw line?

The Pythagorean Theorem with regards to the relationships between the lengths of the sides of a right triangle indicates that we get;

2. x = 51

3. x = 50

4. x = 82

8. x = 2·√(77)

9. x = √(39)

10. x = 2·√(19)

11. x = 2·√(154)

12. x = 3·√3

13. x = 6·√(13)

16. A right triangle

17. A right triangle

18. The triangle is not a right triangle

19. An obtuse triangle

20. An obtuse triangle

The Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the lengths of the other two sides.  

2. x² = 45² + 24² = 2601

x = √(2601) = 51

3. x² = 30² + 40² = 2500

x = √(2500) = 50

x = 50

4. x² = 80² + 18² = 6724

x = √(6724) = 82

x = 82

8. According to the Pythagorean Theorem, in the right triangle we get;

x² = 18² - 4² = 308

x = 2·√(77)

9. x² = 8² - 5² = 39

x = √(39)

10. x² = 20² - 18²

x² = 76

x = √(76) = 2·√(19)

x = 2·√(19)

11. x² = 25² - 3² = 616

x = √(616) = 2·√(154)

x = 2·√(154)

12. x² = 6² - 3² = 27

x = √(27)

x = 3·√3

13. x² = 22² - 4² = 468

x = √(468) = 6·√(13)

x = 6·√(13)

16. A triangle is a right triangle if the square of the side that is the longest is equivalent to the square of the other two sides, therefore;

17² = 289

15² + 8² = 289

Therefore, the triangle is a right triangle

17. 45² = 2025

27² + 36² = 2025

Therefore, the triangle is a right triangle

18. 11² = 121

9² + 4² = 97

Therefore, the triangle is not a right triangle

19. 6² = 36

4² + 3² = 25

The square of the side that is longest is larger than the sum of the squares of the other two sides, which indicates that the angle facing the longest side is lar1ger than 90°, and the triangle is an obtuse triangle.

20. 16² = 256

9² + 11² = 202

16² > 9² + 11²; Therefore, the triangle is an obtuse triangle

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Answer:

Step-by-step explanation:

The length of the hypotenuse is approximately 4.95 cm.

We have,

Since triangle ABC is a 45-45-90 triangle, we know that the measure of angle B is also 45 degrees.

Therefore, we can use the 45-45-90 Triangle Theorem, which states that in a 45-45-90 triangle,

the length of the hypotenuse is √2 times the length of either leg.

In this case,

We know that leg a = 3.5 cm, so we can find the length of the hypotenuse c using the formula:

c = a√2

Substituting the value of a, we get:

c = 3.5√2 ≈ 4.95 cm

Therefore,

The length of the hypotenuse is approximately 4.95 cm.

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complete question:

The triangle above has the following measures. mzC = 45° a = 3.5 cm Use the 45-45-90 Triangle Theorem to find the length of the hypotenuse. Include correct units. Show all your work.​

The total number of possible lock combination using the 40 possible numbers for making lock of 3 numbers is equal to 64,000.

Possible number used for lock combination are 40.

Range is 0 - 39.

Total number chosen for lock combination is equal to 3.

Since there are 40 possible numbers to choose from for each of the three positions on the combination lock.

The total number of possible combinations is equal to ,

40 x 40 x 40

= 40^3

= 64,000

Therefore, there are 64,000 possible lock combinations when choosing 3 numbers out of 40 possible numbers, assuming repeated numbers are allowed.

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The length of the diagonal is c = √269 feet.

To get the length of the diagonal of a rectangular patio, we can use the Pythagorean theorem, which states that for a right triangle with legs of length a and b, and hypotenuse of length c, a² + b² = c². In this case, the legs of the right triangle are the length and width of the rectangular patio, which are 10 feet and 13 feet, respectively. Let's use a and b to represent these lengths.
a = 10 feet
b = 13 feet
We want to find the length of the diagonal, which is the hypotenuse of the right triangle. Let's use c to represent this length.
a² + b² = c²
10² + 13² = c²
100 + 169 = c²
269 = c²
Now we need to find the square root of 269 to get the length of the diagonal.
c = √269
c ≈ 16.4 feet
So the length of the diagonal of the rectangular patio is approximately 16.4 feet. We can also find the ratio of the length, width, and diagonal of the rectangular patio.
length:width = 10:13
width:length = 13:10
length:diagonal = 10:√269
width:diagonal = 13:√269
diagonal:length = √269:10
diagonal:width = √269:13

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The pairs of cities that will not put Jenny back under budget if she drops them are:

a. Moscow and Chelyabinsk

b. Izhevsk and Nizhny Novgorod

To determine which pair of cities Jenny can drop without going over budget, we need to find the total cost of each pair and see if it exceeds her budget of rub 33,000.

a. Moscow and Chelyabinsk:

Total cost = 5,485 + 5,217 = 10,702 rubles

This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Moscow and Chelyabinsk.

b. Izhevsk and Nizhny Novgorod:

Total cost = 4,721 + 6,920 = 11,641 rubles

This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Izhevsk and Nizhny Novgorod.

c. Novosibirsk and Saint Petersburg:

Total cost = 4,870 + 5,960 = 10,830 rubles

This pair of cities is under budget, so Jenny can drop this pair without going over budget.

d. Yaroslavl and Tolyatti:

Total cost = 4,901 + 5,598 = 10,499 rubles

This pair of cities is under budget, so Jenny can drop this pair without going over budget.

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To find the center of a circle when the circumference is not given, you still find it.

1. Determine the coordinates of at least three non-collinear points on the circle. Non-collinear points are points that do not lie on a straight line.
2. Using these points, create two line segments that are chords of the circle. A chord is a line segment connecting two points on the circle.
3. Find the midpoints of each chord. The midpoint formula is given as: Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2).
4. Calculate the slope of each chord using the slope formula: Slope (m) = (y2 - y1) / (x2 - x1).
5. Calculate the slope of the perpendicular bisectors of each chord. Since these lines are perpendicular to the chords, their slopes are the negative reciprocal of the chord slopes: m_perpendicular = -1 / m_chord.
6. Write the equation of each perpendicular bisector using the point-slope formula: y - y_midpoint = m_perpendicular * (x - x_midpoint).
7. Solve the system of equations formed by the two perpendicular bisectors. The solution is the coordinates of the center of the circle.

By following these steps, you can find the center of the circle even when the circumference is not given.

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The probability of rolling two fair six-sided dice and getting a sum of 4 or higher is 11/12

To calculate the probability of rolling two fair six-sided dice and getting a sum of 4 or higher, we first need to calculate the total number of possible outcomes.

The number of possible outcomes when rolling two dice is 6 × 6 = 36, since each die has 6 possible outcomes.

Now, let's find the number of outcomes that result in a sum of 4 or higher. We can do this by listing all the possible outcomes:

Sum of 4: (1, 3), (2, 2), (3, 1) = 3 outcomes

Sum of 5: (1, 4), (2, 3), (3, 2), (4, 1) = 4 outcomes

Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = 5 outcomes

Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 outcomes

Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5 outcomes

Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) = 4 outcomes

Sum of 10: (4, 6), (5, 5), (6, 4) = 3 outcomes

Sum of 11: (5, 6), (6, 5) = 2 outcomes

Sum of 12: (6, 6) = 1 outcome

Therefore, the number of outcomes that result in a sum of 4 or higher is 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 33.

Therefore, the probability of rolling two fair six-sided dice and getting a sum of 4 or higher is 33/36 = 11/12.

To find the probability of getting a sum of 44 or higher, we need to subtract the probability of getting a sum of 43 or lower from 1:

Sum of 2: (1, 1) = 1 outcome

Sum of 3: (1, 2), (2, 1) = 2 outcomes

Sum of 4: (1, 3), (2, 2), (3, 1) = 3 outcomes

Sum of 5: (1, 4), (2, 3), (3, 2), (4, 1) = 4 outcomes

Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = 5 outcomes

Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 outcomes

Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5 outcomes

Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) = 4 outcomes

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you got $45 for 6hours.

one hour=$7.5

two hours=$15

three hours=$7.5*3

calculation=$45/6

Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age.

The problem tells us that Ansley's age is 5 years less than 3 times her cousin's age. We can write this as an equation:

Ansley's age = 3 × Cousin's age - 5

We also know that Ansley is 31 years old. So we can substitute 31 for Ansley's age in the equation:

31 = 3 × Cousin's age - 5

Now we solve for Cousin's age. First, we add 5 to both sides of the equation:

31 + 5 = 3 × Cousin's age

Simplifying:

36 = 3 × Cousin's age

Finally, we divide both sides by 3:

Cousin's age = 12

So Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age in the expression we found earlier:

Ansley's age = 3 × Cousin's age - 5 = 3 × 12 - 5 = 31

So Ansley is indeed 31 years old.

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The  volume of the region U is 16π cubic units.

To find the volume of the region U, we can use cylindrical coordinates. In cylindrical coordinates, a point in space is represented by the coordinates (r, θ, z), where r is the distance from the z-axis, θ is the angle between the x-axis and the projection of the point onto the xy-plane, and z is the height above the xy-plane.

In this case, the region U is defined by 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 12 - r sin(θ).

To find the volume of U, we can integrate over the cylindrical coordinates. The volume of U is given by the integral:

V = ∫∫∫_U dV

where dV = r dz dr dθ is the volume element in cylindrical coordinates.

Substituting in the limits of integration, we have:

V = ∫₀²π ∫₀² ∫₀^(12-rsinθ) r dz dr dθ

Integrating with respect to z, we get:

V = ∫₀²π ∫₀² r(12-rsinθ) dr dθ

Integrating with respect to r, we get:

V = ∫₀²π [(6r² - (1/3)r³sinθ)] from r=0 to r=2 dθ

Simplifying, we get:

V = ∫₀²π [(24 - 16/3 sinθ)] dθ

Integrating, we get:

V = [24θ + 16/3 cosθ] from θ=0 to θ=2π

Simplifying, we get:

V = 48π/3 = 16π

Therefore, the volume of the region U is 16π cubic units.
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Answer:

I believe it is 38

Step-by-step explanation:

The length should be less than or equal to 6 meters in order to have a plot with the maximum area allowed (45 square meters) when the width is 7.5 meters.

To find the length that will give a plot with the maximum area allowed, we can use the formula for the area of a rectangle:

Area = Length × Width

The width is given as 7.5 meters, and the area should not exceed 45 square meters.

Let's denote the length as L.

We want to maximize the area, so we need to find the value of L that satisfies the condition Area ≤ 45 and gives the largest possible area.

Substituting the given values into the area formula, we have:

Area = L × 7.5

Since the area should not exceed 45 square meters, we can write the inequality:

L × 7.5 ≤ 45

To find the maximum value of L, we can divide both sides of the inequality by 7.5:

L ≤ 45 / 7.5

Simplifying the right side:

L ≤ 6

Therefore, the length should be less than or equal to 6 meters in order to have a plot with the maximum area allowed (45 square meters) when the width is 7.5 meters.

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The 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners is $58.06 to $65.94.

To construct a confidence interval for the mean amount of money spent per month on home maintenance by all homeowners, we can use the formula:

CI = [tex]\bar{X}[/tex] ± Zα/2 * (σ/√n)

where [tex]\bar{X}[/tex] is the sample mean, Zα/2 is the critical value from the standard normal distribution corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

In this case, we have:

[tex]\bar{X}[/tex] = $62 (the sample mean)

α = 0.02 (since we want a 98% confidence interval, which means α/2 = 0.01)

Zα/2 = 2.33 (from the standard normal distribution table)

σ = $13 (the population standard deviation)

n = 61 (the sample size)

Substituting these values into the formula, we get:

CI = $62 ± 2.33 * ($13/√61)

Simplifying this expression, we get:

CI = $62 ± $3.94

Therefore, the 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners is $58.06 to $65.94.

This means that we can be 98% confident that the true population mean falls within this range. In other words, if we were to repeat the survey many times and construct confidence intervals in the same way, about 98% of the intervals would contain the true population mean.

It's important to note that this assumes that the sample is representative of the population, and that the population standard deviation is known. If these assumptions are not met, then the confidence interval may not be accurate.

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The Paige walks at a speed of approximately 0.04167 miles per minute to get to the park.

To calculate Paige's speed, we used the formula:

Speed = Distance / Time

Given that Paige walks to the park 2/3 mile away, we substitute Distance with 2/3 mile and Time with 16 minutes. We get:

Speed = 2/3 mile / 16 minutes

Simplifying the expression by converting minutes to hours, we get:

Speed = 2/3 mile / (16/60) hours

Simplifying further by multiplying both the numerator and denominator by 60, we get:

Speed = [tex](2/3) * (60/1)[/tex] mile/hour / (16/1) minutes

Speed = 0.04167 mile/minute (rounded to 5 decimal places)

"Paige walks to the park 2/3 mile away it takes her 16 minutes to get there how many miles per minutes" is that Paige walks at a speed of approximately 0.04167 miles per minute.

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Thus, the new coordinates for the image of translated Vertices of Square RSTU are-  R'(-6, 0), S'(-1, 3), T'(2, -2) and  U'(-3, -5).

In mathematics, a translation moves an object throughout the coordinate plane while preserving its dimensions and shape. After a translation, its area and orientation remain unchanged.

The vertical shift, horizontal shift, or perhaps a combination of the two can be referred to as a translation in mathematics.

Given that:

Vertices of Square RSTU.

R(-2, 1), S(3, 4), T(6, -1), and U(1, -4):

translation: (x, y) → (x-4, y − 1)

New vertices:

R(-2-4, 1 − 1) --> R'(-6, 0)

S(3-4, 4 − 1), ---> S'(-1, 3)

T(6-4, -1 − 1),  -> T'(2, -2)

U(1-4, -4 − 1) --> U'(-3, -5)

Thus, the new coordinates for the image of translated Vertices of Square RSTU are-  R'(-6, 0), S'(-1, 3), T'(2, -2) and  U'(-3, -5).

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21. in the figure, provided line n and line m are parallel the values of x and y are

x = 7

y = 24

21. When line m and line n are parallel then using corresponding angle theorem we have that:

7y - 23 + 8x - 21 = 180

7y - 8x = 180 + 23 - 21

7x - 8x = 182

Also

8x - 21 + 23x - 16 = 180

8x + 23x = 180 + 21 + 16

31x = 217

x = 217 / 31

x = 7

using vertical angle theorem we have:

7y - 23 = 23x - 16

plugging in the value of x

7y - 23 = 23 * 7 - 16

7y - 23 = 145

7y = 145 + 23

7y = 168

y = 24

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Cash outflows and decreases in cash are reported as negative amounts in the operating activities section of the statement of cash flows for Peach Computer using the indirect method.

The operating activities section of the statement of cash flows for Peach Computer using the indirect method would include the following cash inflows and outflows:

Cash inflows:

Cash outflows:

Any decrease in cash would be represented as negative amounts in this section.

It's important to note that the specific amounts and details would vary based on Peach Computer's individual financial transactions and operations. The operating activities section provides a summary of the cash inflows and outflows directly related to the company's core business operations during the specified period.

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Answer: (4)

Step-by-step explanation:

Two lines E and F are same.

5x + 5y = 40

x + y = 8

Deviding both hands of E by 5,

we get F's equation.

So every single point on the line x+y=8

represents the solution of the given system.

The coordinates of triangle X'Y'Z' is X'(-2,-2), Y'(-3,4), Z'(1,2).

What are the coordinates of the triangle?

The triangle's vertices have the coordinates (x1,y1), (x2,y2), and (x3,y3). The line that connects the first two is split in the ratio l:k, and the line that runs from the division point to the opposing angular point is divided in the ratio m:k+l.

Here, we have

Given: Triangle XYZ has coordinates X(-2,2), Y(-3,-4). And Z(1,-2).

The triangle is reflected across the x-axis and we have to find the coordinates of triangle X'Y'Z'.

X(-2,2), Y(-3,-4) and Z(1,-2).

While reflecting any points across the x-axis with coordinates as (x, y) becomes, (x, -y). i.e. sign of y-coordinate changes.

The rule is (x, y) → (x, -y)

After applying the rule, we get

X(-2,2) ⇒ X'(-2,-2)

Y(-3,-4) ⇒ Y'(-3,4)

Z(1,-2) ⇒ Z'(1,2)

Hence, the coordinates of triangle X'Y'Z' is X'(-2,-2), Y'(-3,4), Z'(1,2).

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The height of the right octagonal prism is approximately 10.75 cm.

The given information states that the base of the prism is a regular octagon with each side measuring 2.3 cm, and the area of the base is 25.76 cm². Additionally, the surface area of the prism is 223.56 cm².

First, let's calculate the area of the eight lateral faces. We can do this by subtracting the base's area from the total surface area:

223.56 cm² (total surface area) - 25.76 cm² (base area) = 197.8 cm² (lateral area)

Now, we know that the lateral area is the sum of the areas of all eight rectangular faces. Each rectangle has a base of 2.3 cm (the same as the sides of the octagonal base) and a height equal to the height of the prism (h). Since there are eight faces, the combined area of these rectangles is 8 x 2.3 x h:

8 x 2.3 x h = 197.8 cm²

18.4 x h = 197.8 cm²

To find the height of the prism (h), we can divide both sides of the equation by 18.4:

h = 197.8 cm² / 18.4

h ≈ 10.75 cm

So, the height of the right octagonal prism is approximately 10.75 cm.

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Jerry pays $279.40 this year in property taxes.

To calculate Jerry's property taxes for the year, we need to first decide how many of his month-to-month maintenance fee is going toward property taxes.

The problem states that 20% of the price is for Jerry's assets taxes, which means we will calculate the amount of his belongings taxes with the aid of finding 20% of his monthly charge.

To do this, we multiply the price through 0.20 such as this:

20% of $1,397 = 0.20 x $1,397 = $279.40

Therefore, Jerry pays $279.40 this year in property taxes.

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Answer: The median of the players' heights is not affected.

Step-by-step explanation: B

The median of the players' heights is increased.

we need to consider the current arrangement of heights and the positions

of the new players in relation to the existing players' heights.

If we assume that the heights of the players are sorted in ascending order,

adding two additional players can affect the median in the following ways:

If both new players have heights lower than the current median:

In this case, adding the new players would not change the median.

The median would remain the same because the new players would be

added below the existing median, and the position of the median would

not shift.

If one new player has a height lower than the current median and the other

has a height higher than the current median:

In this case, the median would be increased.

Adding a taller player would shift the median towards the higher end of the data set.

If both new players have heights higher than the current median:

In this case, the  would be increased.

Both new players would be taller than the current median, causing the

median to shift towards the higher end of the data set.

Based on these possibilities, the answer is C.

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Using the Perpendicular Bisector Theorem, the distance from third base to home plate is 90 feet. This means that all the bases are equidistant from home plate, which is a fundamental property of a baseball diamond.

To find the distance from third base to home plate, we need to use the Perpendicular Bisector Theorem, which states that if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

First, we draw a baseball diamond with points A, B, C, and D labeled as described in the problem.

Next, we draw the line segment that joins first base (B) and third base (D), and we construct the perpendicular bisector of this segment by drawing a line through the midpoint of BD and perpendicular to BD. Let's label the point where the perpendicular bisector intersects the line that connects home plate (A) and second base (C) as E.

Since E lies on the perpendicular bisector of BD, it is equidistant from B and D. We know that first base (B) is 90 feet from home plate (A), so the distance from home plate to E must also be 90 feet. Therefore, the distance from third base (D) to home plate (A) is also 90 feet.

In conclusion, using the Perpendicular Bisector Theorem, we determined that the distance from third base to home plate is 90 feet.

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There are 20 different ways the students can be chosen for the positions of editor-in-chief and assistant editor-in-chief.

There are 5 students in the newspaper staff, and two positions to fill i.e. editor-in-chief and assistant editor-in-chief. We need to find the number of different ways the students can be chosen for these positions.

To solve this problem, we can use the formula for permutation

We know the formula for Permutation is

[tex]P(n,r)=\frac{n!}{(n-r)!}[/tex]

Here, n=5 and r=2

So, P(5,2) = 5!/(5-2)!

= 5!/3!

= 120/6

= 20

Therefore, there are 20 different ways the students can be chosen for the positions of editor-in-chief and assistant editor-in-chief.

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Verónica jogged 7/4 or 1 and 3/4 more miles in the first week than in the second week.

Verónica jogged 10 3/16 miles in the first week and 8 7/16 miles in the second week. To find how many more miles she jogged in the first week, we need to subtract the distance she jogged in the second week from the distance she jogged in the first week:

10 3/16 miles - 8 7/16 miles

We need to first convert both mixed numbers to improper fractions:

10 3/16 = (10 x 16 + 3) / 16 = 163 / 16

8 7/16 = (8 x 16 + 7) / 16 = 135 / 16

Now we can subtract the two fractions:

163 / 16 - 135 / 16 = (163 - 135) / 16 = 28 / 16

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor (GCF), which is 4:

28 / 16 = (4 x 7) / (4 x 4) = 7 / 4

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Ranking of the events below in the correct order from least likely to most likely are:

Event B

Event A

Event C

The probability of an event is defined as a number that describes the chance that the event will eventually happen. An event that is sure to happen has a probability of 1. An event that can never possibly happen has a probability of zero. Finally, If there is a chance that an event will happen, then it will have a probability that is between zero and 1.

i) Arrow lands on a section labeled with an odd number: The odd numbers here are 1 and 3.

There are a total of four 1's, and two 3's. This tells us that there are 6 odd numbers on the spinner.

There are 8 numbers in total on the spinner. Thus, 6 out of the 8 numbers are seen as odd numbers. Therefore, the probability that the arrow lands on an odd number would be:

P(odd number) = 6/8 =  75%

ii) Arrow lands on a section labeled with the number 1: There are four 1's on the spinner, and there are seen to be 8 numbers in total on the spinner. Thus, the probability of the arrow landing on a 1 is:

P(Number 1) = 4/8 = 50%.

iii) Arrow lands on a section labeled with a number less than 4:

The numbers that are less than 4 are 3, 2, and 1.

There are two 3's.

There is one 2.

There are four 1's.

2 + 1 + 4 = 7.

The probability of the arrow landing on a number less than 4 is 7/8, which is 88.5%.

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The answer is A. 1. We can use L'Hopital's rule to evaluate the limit:

limit x→9√(f(x))-3/√(x)-3 = limit x→9 (f(x)-9)/(x-9) / (√(f(x))-3)/(√(x)-3)

Now, we know that f(9) = 9 and f'(9) = 4, so we can use the definition of the derivative to write:

f(x) - f(9) = f'(9)(x-9) + o(x-9)

where o(x-9) represents a term that goes to 0 faster than x-9 as x approaches 9. Plugging this into the numerator, we get:

f(x) - 9 = 4(x-9) + o(x-9)

Plugging this into the denominator, we get:

√(f(x)) - 3 = √(4(x-9) + o(x-9)) = 2√(x-9) + o(1)

√(x) - 3 = √(x-9) + o(1)

Therefore, the limit becomes:

limit x→9 (4(x-9) + o(x-9))/(√(x-9) + o(1)) / (2√(x-9) + o(1))/(√(x-9) + o(1))

Simplifying this expression, we get:

limit x→9 2(4(x-9) + o(x-9))/(√(x-9) + o(1))^2

limit x→9 8 + 2o(1)/(x-9)

As x approaches 9, the o(1) term goes to 0, so the limit becomes:

8 + 2*0/0 = 8

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The final result is  ∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C.

To solve the indefinite integral ∫ sin²(x)- cos²(x)/cos(x) dx, we need to use trigonometric identities to simplify the integrand.

First, we use the identity sin²(x) + cos²(x) = 1 to write:

sin²(x) - cos²(x) = sin²(x) + cos²(x) - 2cos²(x) = 2sin²(x) - cos²(x)

Next, we use the identity sin²(x) = 1 - cos²(x) to write:

2sin²(x) - cos²(x) = 2(1-cos²(x)) - cos²(x) = 2 - 3cos²(x)

Substituting this into the original integral, we get:

∫ sin²(x)- cos²(x)/cos(x) dx = ∫ (2 - 3cos²(x))/cos(x) dx

Now, we use the substitution u = cos(x) and du/dx = -sin(x) dx to transform the integral into a simpler form:

∫ (2 - 3cos²(x))/cos(x) dx = ∫ (2 - 3u²)/u (-du/sin(x))

= -∫ (3u² - 2)/u du

= -3∫ u du + 2∫ du/u

= -3u²/2 + 2ln|u| + C

= -3cos²(x)/2 + 2ln|cos(x)| + C

where C is the constant of integration.

Substituting back u = cos(x), we obtain the final result

∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C

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