tan nx + tanx
1 - tan nxtanx
=tan(n+1)x

Answer:

the area of triangle OAP is 12 square units.

Step-by-step explanation:

First, we need to find the coordinates of the center of the circle, which is (0, 0), and the radius, which is √40 = 2√10.

Since line I is a tangent to the circle at point A, it is perpendicular to the radius OA at point A. Therefore, OA is perpendicular to line I and forms a right angle with it.

We can find the equation of line I using the point-slope form:

slope of radius OA = (0 - 6) / (0 - 2) = -3

slope of line I = 1 / 3 (because it is perpendicular to OA)

equation of line I: y - 6 = (1/3)(x - 2)

simplifying, we get: y = (1/3)x + 4

To find the x-coordinate of point P, we need to solve for x when y = 0:

0 = (1/3)x + 4

x = -12

Therefore, point P has coordinates (-12, 0).

Now, we can find the coordinates of point O, which is the origin (0, 0), and the coordinates of point A, which are given as (2, 6).

To find the area of triangle OAP, we can use the formula for the area of a triangle:

Area = (1/2) x base x height

We know that OA is the base of the triangle and its length is the radius of the circle, which is 2√10.

To find the height of the triangle, we need to find the distance from point A to line I, which is the perpendicular distance. We can use the formula for the distance between a point and a line:

distance = |ax + by + c| / √(a² + b²)

where a, b, and c are the coefficients of the equation of the line, and x and y are the coordinates of the point.

In this case, the equation of line I is y = (1/3)x + 4, so a = -1, b = 3, and c = -12.

Substituting the values, we get:

distance = |(-1)(2) + (3)(6) - 12| / √((-1)² + 3²)

distance = |6| / √10

distance = 3√10 / 5

Now we can substitute the values into the formula for the area of the triangle:

Area = (1/2) x base x height

Area = (1/2) x 2√10 x (3√10 / 5)

Area = 6/5 x 10

Area = 12

Therefore, the area of triangle OAP is 12 square units.

No, the given value of h=2 is not a solution of the equation 1/3h=6.

The probability that wrist watch chosen is a defective watch is 1 / 10

Total number of watches n(S)=500

Number of watches defective n(A)=50

P(A)= n(A) / n(S)

​

⇒P(A)= 50 / 500

​

Probability of watches to be defective = 1 / 10 or  50 / 500

Probability is a branch of mathematics that deals with the study of the likelihood or chance of an event occurring. It is a way of quantifying uncertainty and making predictions about the outcome of a random process. The probability of an event is represented by a number between 0 and 1, where 0 represents an impossible event, and 1 represents a certain event.

Probability theory provides a set of rules and tools for analyzing and modeling the behavior of random phenomena, such as coin flips, dice rolls, or the occurrence of a disease in a population. These tools include probability distributions, which describe the likelihood of different outcomes, and statistical inference, which allows us to make predictions based on a sample of data.​

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The correct option for question 1 is (c), while the correct option for question 2 is (a).

When answering questions on the platform Brainly, it is important to always be factually accurate, professional, and friendly. In addition, one should be concise and not provide extraneous amounts of detail, ignore any typos or irrelevant parts of the question, repeat the question in their answer, and provide a step-by-step explanation.Using Holtz model, the factor of growth in a time series is accounted for by adding trend exponential smoothing to estimate. This is because Holtz model is a double exponential smoothing method that can account for both level and trend in the time series. The exponential smoothing factor α, which ranges between 0 and 1, is used to control the smoothness of the trend parameter.In addition, the adaptive response rate single exponential smoothing model is termed adaptive because it has the ability to respond to changes in the pattern of data. This is because the smoothing parameter changes each period, allowing the model to adjust to the changing patterns in the time series. Therefore, the correct option for question 1 is (c), while the correct option for question 2 is (a).

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The center of mass of a thin plate of constant density covering the given region is (1.8, 3.6).

To find the center of mass, we must calculate the weighted average of all the points in the region. The region is bounded by the parabola y = 3x - x² and the line y = -3x.

We must calculate the integral of the region and divide by the total mass. The mass is equal to the area times the density, .

The integral of the region is calculated using the limits of the two curves, yielding a final integral of 32/15. Dividing this integral by the density gives the total mass, and multiplying by the density gives us the center of mass, (1.8, 3.6).

We can also find the center of mass by calculating the moments of the plate about the x-axis and y-axis.

The moment about the x-axis is calculated by finding the integral of the parabola and line using the x-coordinate, and the moment about the y-axis is calculated by finding the integral of the parabola and line using the y-coordinate. Once the moments are found, we can divide each moment by the total mass to get the center of mass.

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Answer:

a, c and d

Step-by-step explanation:

The probability of drawing four red balls in succession, with replacement, is 0.04 or 4%.

Since we are replacing the ball after each draw, the probability of drawing a red ball remains the same for each draw. The probability of drawing a red ball on any given draw is:

P(Red) = Number of Red Balls / Total Number of Balls

P(Red) = 6 / (8 + 6)

P(Red) = 0.4286

So, the probability of drawing four red balls in a row is the product of the probability of drawing a red ball four times in a row:

P(4 Red Balls) = P(Red) * P(Red) * P(Red) * P(Red)

P(4 Red Balls) = 0.4286 * 0.4286 * 0.4286 * 0.4286

P(4 Red Balls) = 0.04 or 4%

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According to given equation a. Never true, b. Always true, c. Always true, d. Always true, e. Always true.

In mathematics, a function is a rule that assigns each element in one set, called the domain, to a unique element in another set, called the range. The elements in the domain are the input values, and the elements in the range are the output value.

a. x - 13 = x + 1

This equation is never true because if we subtract x from both sides, we get -13 = 1, which is a contradiction.

b. x + 12/1/2 = x - 21/1/2

This equation is always true because both sides simplify to x + 24 = x - 42. If we subtract x from both sides, we get 24 = -42, which is also a contradiction. Therefore, the equation is true for all values of x, because it has no solutions.

c. 2(x + 3) = 5x + 6 - 3x

This equation is always true. Simplifying both sides, we get 2x + 6 = 2x + 6, which is true for all values of x.

d. x - 3 = 2x - 3 - x

This equation is always true. Simplifying both sides, we get x - 3 = x - 3, which is true for all values of x.

e. 3(x - 5) = 2(x - 5) + x

This equation is true for all values of x. Simplifying both sides, we get 3x - 15 = 2x - 10 + x, which simplifies to 3x - 15 = 3x - 10, and then to -15 = -10, which is a contradiction. Therefore, the equation is true for all values of x, because it has no solutions.

Therefore, according to given equation a. Never true, b. Always true, c. Always true, d. Always true, e. Always true

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The distribution of C) X+Y is a Poisson RV with parameter 9.

This is because the sum of two independent Poisson distributions with parameters λ1 and λ2 is also a Poisson distribution with parameter λ1 + λ2. Therefore, X+Y follows a Poisson distribution with parameter 4+5 = 9.

Option A is incorrect because an exponential distribution cannot arise from the sum of two Poisson distributions. Option B is also incorrect because the parameter of X+Y is not the average of the parameters of X and Y. Option C is the correct answer as explained above.

In summary, the distribution of X+Y is Poisson with parameter 9.

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the x-intercepts are (-3, 0) and (1, 0), and the minimum occurs at the vertex (-1, -4).

What is a function?

In mathematics, a function is a relationship between a set of inputs (called the domain) and a set of possible outputs (called the range), where each input is associated with exactly one output.

To find the x-intercepts, we set the function equal to zero and solve for x:

x² + 2x - 3 = 0

Using the quadratic formula, we get:

x = (-b ± sqrt(b² - 4ac)) / 2a

Where a = 1, b = 2, and c = -3.

Plugging in the values, we get:

x = (-2 ± sqrt(2² - 4(1)(-3))) / 2(1)

Simplifying, we get:

x = (-2 ± sqrt(16)) / 2

x = (-2 ± 4) / 2

x1 = -3, x2 = 1

Therefore, the x-intercepts are (-3, 0) and (1, 0).

To find the maximum or minimum, we can use the vertex form of the equation:

f(x) = a(x - h)² + k

where (h, k) is the vertex.

To get the vertex, we complete the square:

f(x) = x² + 2x - 3

f(x) = (x + 1)² - 4

The vertex is (-1, -4), which is a minimum since the coefficient of the squared term is positive.

Therefore, the x-intercepts are (-3, 0) and (1, 0), and the minimum occurs at the vertex (-1, -4).

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According to the survey the probability is that 30 people like chocolate.

Now we know that 20 people like strawberry, and 10 people like both chocolate and strawberry.

Therefore to determine how many people like chocolate we know that

30 people who like chocolate are made up of the 10 people who like both chocolate and strawberry, plus an additional 20 people who only like chocolate.

Therefore, the probability of an event occurring is calculated by dividing the number of ways the event can occur by the total number of possible outcomes

Probability = 30 / (30+20+10) = 30/60 = 0.5

Therefore, 30 people in total like chocolate.

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The false statement is B. 50% of houses in Ames are smaller than 1,499.69 square feet.

Statistics uses both the mean and median as gauges of central tendency, although their definitions and methods of computation vary. A number's mean is determined by adding together all of the values and dividing by the total number of values. A group of numbers has a median, which is the midpoint value, with half of the values above and below.

The median size of a home in Ames is 1,499.69 square feet, thus this claim is untrue. Therefore, 50% of the homes are smaller and 50% are larger than 1,499.69 square feet. Thus, assertion B is untrue.

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The height of the rectangular prism can be calculated using the following formula:

Volume (V) = Length (L) * Width (W) * Height (H)

Therefore, rearranging the formula, we can calculate the height of the prism:

Height (H) = Volume (V) / (Length (L) * Width (W))

For this problem, plugging in the known values, we get:

Height (H) = 490 cm³ / (10 cm * 7 cm)

Therefore,

Height (H) = 8.14 cm

The numerical expression for the verbal expression "The quotient of thirty-two and four divided by the sum of one and three" is 2.

The verbal expression is "The quotient of thirty-two and four divided by the sum of one and three."

To write this as a numerical expression, we can first evaluate the quotient of thirty-two and four, which is 8. Then we can divide 8 by the sum of one and three, which is 4.

Therefore, the numerical expression for the verbal expression is:

= 8 ÷ (1 + 3)

Add the number

= 8 ÷ 4

Divide the numbers

= 2

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The length of the rectangle is 45 yards and the width is 9 yards whose perimeter is 108yd.

A rectangle is a four-sided flat shape with four right angles (90-degree angles) between the adjacent sides. The perimeter of a rectangle is the sum of the lengths of its sides, and the area of a rectangle is the product of its length and width.

Let L be the length.

Let W be the width.

From the problem, we know that L = 5W (since the length is five times the width).

P = 2L + 2W.

Substituting L = 5W into this formula, we get:

P = 2(5W) + 2W = 10W + 2W = 12W

We're also given that the perimeter of the rectangle is 108 yards, so we can set up the equation:

12W = 108

Solving for W, we get:

W = 9

Now that we know the width is 9 yards, we can use the equation L = 5W to find the length:

L = 5(9) = 45

Therefore, the length of the rectangle is 45 yards and the width is 9 yards.

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The statement "Ten cards are selected out of a 52 card deck without replacement and the number of Jacks is observed. This is an example of a Binomial Experiment" is false.

A binomial experiment is an experiment that is repeated multiple times with each repetition having only two potential outcomes. In a binomial experiment, the probability of success remains constant from trial to trial.

The criteria for a binomial experiment are as follows:

P(x) = (ⁿCₓ)(pˣ)(q^(n-x))

Where p is the probability of success, q is the probability of failure (q = 1 - p), and ⁿCₓ is the combination formula.

Therefore, the statement "Ten cards are selected out of a 52-card deck without replacement and the number of Jacks is observed. This is an example of a "Binomial Experiment" being false. This is because the probability of drawing a jack changes with each trial, as the deck's composition changes after each card is drawn.

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Answer:

136

Step-by-step explanation:

35x2+7x6+12x2

Answer:

x/8 + 6 = 4

Step-by-step explanation:

x / 8 + 6 = 4

x/8 = -2

x = 8*-2 = -16.

The probability that both A and B occur is given by P(A and B) = P(B | A) * P(A) = 0.2 * 0.5 = 0.1.

This can be visualized using a Venn diagram, where the intersection of A and B represents the probability of both events occurring, which is equal to 0.1 in this case.

The probability of B occurring and A not occurring is given by P(B and not A) = P(B) - P(B | A) * P(A') = 0.3 - 0.2 * 0.5 = 0.2. This represents the area of the B circle outside of the A circle.

Given that P(A) = 0.6 and P(A and B) = 0.5, we can use Bayes' theorem to find P(B | A) as follows: P(B | A) = P(A and B) / P(A) = 0.5 / 0.6 = 0.83. This means that the probability of B occurring given that A has occurred is 0.83.

We can also visualize this using a Venn diagram, where the overlap between A and B represents the probability of both events occurring, and the B circle represents the probability of B occurring given that A has occurred.

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The length of the missing side is 4x + 5 units

To find the length of the missing side of the triangle, we need to use the fact that the sum of the lengths of all three sides of a triangle is equal to its perimeter. So, we can write:

Perimeter = (length of first side) + (length of second side) + (length of missing side)

Substituting the given values, we get:

8x-2 = (x-7) + 3x + (length of missing side)

Simplifying and solving for the missing side, we get:

8x-2 = 4x-7 + (length of missing side)

Length of missing side = (8x-2) - (4x-7)

Length of missing side = 4x + 5 units

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The given question is incomplete, the complete question is:

The triangle has a perimeter of 8x-2. what is the length of the missing side?

The lengths of LV and OE are 15cm, and the lengths of LD and EV are 3√7 cm and 9/√7 cm, respectively.

Since LOVE is a kite, LV and OE are perpendicular bisectors of each other. Let the length of LD be x, and the length of EV be y. Then, we can use the Pythagorean theorem and the fact that the diagonals bisect each other to set up two equations:

x² + (LV/2)² = DV²/4

y² + (OE/2)² = LE²/4

Simplifying each equation and substituting the given values, we get:

x² + (LV/2)² = 81/4

y² + (OE/2)² = 225/4

We also know that the diagonals bisect each other, so we can set up another equation:

LV/2 + OE/2 = LO = VE

Substituting the given value for LE, we get:

LV/2 + OE/2 = 15

Solving this equation for one of the variables, we get:

LV = 30 - OE

Substituting this expression into the first equation above, we get:

x² + ((30 - OE)/2)² = 81/4

Simplifying and rearranging, we get:

OE² - 60OE + 675 = 0

Using the quadratic formula, we get:

OE = (60 ± √(3600 - 2700)) / 2

OE = 15 or 45

If OE = 15, then LV = 30 - 15 = 15, and we can solve for x and y:

x² + 7.5² = 81/4

y² + 7.5² = 225/4

Solving these equations, we get:

x = 3√7

y = 9/√7

If OE = 45, then LV = 30 - 45 = -15, which is impossible for a length. Therefore, the solution is:

LV = OE = 15

x = 3√7

y = 9/√7

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Complete question:

LOVE is a kite. LV and OE are diagonals. The segments DV = 9cm and LE = 15cm are given. Find the lengths of the other segments of the diagonals, DV, OE, and LV.

The area of the replica is approximately 1.33 square feet, Rounded to the nearest hundredth.

If the original artwork has an area of 12 square feet, then a replica created by dilating the original by a scale factor of 1/3 will have an area that is (1/3)^2 = 1/9 of the original area. Therefore, the area of the replica will be:

12 square feet × 1/9 = 4/3 square feet ≈ 1.33 square feet

A scale factor is a multiplier that scales or resizes a shape, figure, or object. It is used to determine the size of a new shape or figure when it is scaled up or down by a certain factor. The scale factor can be expressed as a ratio, a fraction, or a decimal. For example, if a rectangle has a length of 10 units and a width of 5 units, and it is scaled up by a factor of 2, the new rectangle will have a length of 20 units and a width of 10 units. The scale factor, in this case, is 2.

Scale factors are commonly used in geometry, especially in transformations such as dilation, which involves scaling a shape by a certain factor without changing its shape. Scale factors can also be used in real-world applications such as maps, where a scale factor is used to represent the ratio between the distance on the map and the actual distance on the ground.

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Complete Question:

an art gallery is selling replicas of some famous artworks. a copy of the artwork is dilated by a scale factor of [tex]\frac{1}{3}[/tex] to create a replica of the original piece. if the area of some original artwork was 12 square feet, what will be the area of the replica? enter the answer in the box, rounded to the nearest hundredth.

According to the graph, the balloon ascends between seconds 0 and 2; it remains stable between seconds 2 and 3; drops rapidly between 3 and 4 seconds; it descends slowly between seconds 4 and 6. Additionally, the natural thing is that it does not ascend again because gravity will not allow it to ascend.

To describe the movement of the balloon we must analyze the relationship between the height of the bomb and time. Based on the above, we see that it ascends, holds, descends rapidly, and then slows its rate of descent as described below:

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13/12 will be long the next airplane model. This will be because her next model is to be 1/4 as long as the first.

Fractions are referred to as the components of a whole in mathematics. A single object or a collection of objects might be entire. When we cut a piece of cake in real life from the entire cake, the part represents the per cent of the cake. The word "fraction" is derived from Roman. "Fractus" means "broken" in Latin. The fraction was expressed verbally in earlier times. It was afterwards presented in numerical form.

Given,

The length of her first model is in the fraction [tex]4\frac{1}{3}[/tex]

As her next model has to be 1/4 as long as the first model

Therefore,

Length of her new or next model

1/4 of [tex]4\frac{1}{3}[/tex]

so, 1/4 * 13 / 3

13/12

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Step-by-step explanation:

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

I think the answer is 10,560 feet

Answer:

2.1875 miles

Step-by-step explanation:

1 mile = 5,280 feet

990 feet ÷ 5,280 feet = 0.1875 in miles

(a) Using the Trapezoidal rule, T8 = (1/16)[cos(0) + 2cos(1/16) + 2cos(2/16) + ... + 2cos(7/16) + cos(1)].

Using the Midpoint rule, M8 = (1/8)[cos(1/16) + cos(3/16) + ... + cos(15/16)].

(b) The error in the Trapezoidal rule is bounded by (1/2880)(1-0)^3(max|f''(x)|), where f''(x) = -4x^2sin(x^2) and 0 <= x <= 1. Therefore, the error in T8 is approximately 0.00014. The error in the Midpoint rule is bounded by (1/1920)(1-0)^3(max|f''(x)|), which gives an approximate error of 0.00011 for M8.

(c) Let n be the number of intervals in the approximation.

Then, the error bound for the Trapezoidal rule is (1/2880)(1-0)^3(max|f''(x)|)(1/n^2), and the error bound for the Midpoint rule is (1/1920)(1-0)^3(max|f''(x)|)(1/n^2).

Setting these equal to 0.0001 and solving for n, we get n >= 129 and n >= 160 for the Trapezoidal and Midpoint rules, respectively. Therefore, we should choose n >= 160 to ensure that both approximations are accurate to within 0.0001.
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Jordy's first error in his proof is option (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

Jordy's first error in his proof is that he used an inappropriate congruence criterion to prove that the two triangles are congruent. He established all necessary conditions, but AAA congruence is only valid for proving congruence of triangles in certain special cases, such as when the triangles are similar.

In general, AAA congruence is not a valid congruence criterion. This highlights the importance of choosing the correct congruence criterion when proving that two triangles are congruent, as using an invalid or inappropriate criterion can lead to an incorrect conclusion.

Therefore, the correct option is (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

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the closest estimate for the experimental probability of rolling a 4 is 0.12 or 12%.

Experimental probability is the probability of an event happening based on the results of an experiment or trial. It is also known as empirical probability.

The experimental probability of an event happening is the ratio of the number of times the event occurred to the total number of trials or attempts.

In this case, the event is rolling a 4 on the cube, and the total number of trials is 500.

So, the experimental probability of rolling a 4 can be calculated as:

experimental probability = number of times 4 landed face-up / total number of trials

experimental probability = 58/500

experimental probability = 0.116 or approximately 0.12 (rounded to two decimal places)

Therefore, the closest estimate for the experimental probability of rolling a 4 is approximately 0.12 or 12%.

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The given figure is made up of a square, rectangle and an arc as shown in the figure. The perimeter will be 38 units and the area will be 95 square units.

Perimeter the the total length of the boundaries. Here we know all the lengths except the length of the arc.

Length of an arc = θ/360 × 2πr

 Here θ = 90

So length = 90/360 × 2× 3.14× 5 = 7.85 units

So perimeter = 10 + 5 + 7.85 + 5 + 10 = 37.85 ≈ 38 units

Area is formed by a rectangle, square and an arc

Area of the given rectangle = l×b = 10× 5 = 50 sq. units

Area of the square = s² = 5² = 25 sq. units

Area of the sector =  θ/360 × πr² = 90/360× 3.14× 5² = 19.64 sq. units

So the total area = 50 + 25 + 19.64 = 94.64 ≈ 95 sq. units

So perimeter is 38 units and area is 95 units².

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The complete question is:

This figure is made from a part of a square and a part of a circle.

What is the perimeter of this figure, to the nearest unit?

What is the area of this figure, to the nearest square unit?

Diagram provided as image.