what is the scalar product of a vector of length 7 and a vector of length 10 which make an angle of 60∘ with each other?

The value of x that makes the denominators equal to zero are x = -3 and x = 3.

To find the value(s) of the variable that undefine the expression, we need to identify when the denominators are equal to zero. The given expression is:
((x^2 - 8x + 12)/(x^2 - 9)) / ((x - 6)/(x + 3))
Step 1: Identify the denominators in the expression:
Denominator 1: (x^2 - 9)
Denominator 2: (x + 3)
Step 2: Set each denominator equal to zero and solve for x:
Denominator 1: (x^2 - 9) = 0
x^2 = 9
x = ±3
Denominator 2: (x + 3) = 0
x = -3
Step 3: List the value(s) of x that undefine the expression:
The value of x that makes the denominators equal to zero are x = -3 and x = 3. Thus, these are the values that undefine the given expression.

To know more about denominator visit:

https://brainly.com/question/7067665

#SPJ11

The hypothesis referred to in this statement is likely the null hypothesis in a regression analysis, which assumes that the slope coefficient (b1) of the regression line is equal to zero, indicating that there is no relationship between the independent variable and the dependent variable. If the hypothesis is rejected, it means that there is sufficient evidence to suggest that the slope coefficient is not zero and there is a significant relationship between the independent and dependent variables.

In this context, the sample regression coefficient b1 represents the change in the predicted value of the dependent variable for a unit change in the independent variable. In other words, it indicates the slope of the regression line and how much the dependent variable changes for a unit change in the independent variable. If b1 is positive, it means that the dependent variable increases as the independent variable increases, and if b1 is negative, it means that the dependent variable decreases as the independent variable increases. The magnitude of b1 indicates the strength of the relationship between the variables, with larger values indicating a stronger relationship.

To learn more about regression line : brainly.com/question/30243761

#SPJ11

The number of free throws expected basketball player to make by using proportion is equal to 180.

Percent of free throw attempted by basketball player = 80%

Number of free throws basketball player attempts in practice  = 225

If the basketball player makes 80% of the free throws she attempts,

we can expect her to make 80 out of every 100 free throws attempted.

To find out how many free throws she would make if she attempted 225, we can set up a proportion,

80/100 = x/225

We can solve for x by cross-multiplying the expression we get,

⇒ 100x = 80 × 225

⇒ 100x = 18000

⇒ x = 18000/100

⇒ x = 180

Therefore, we can expect the basketball player to make 180 free throws out of 225 attempts in practice using proportion.

learn more about proportion here

brainly.com/question/31548894

#SPJ4

The numbers that would be considered rational are -0.3/2, 0.7 and 3/2. The correct option is B.

We are given that;

The four options

Now,

A rational number is a number that can be written as a fraction of two integers. For example, 3/2 and 0.7 are rational numbers because they can be written as 3/2 and 7/10 respectively. However, √2 and pi are not rational numbers because they cannot be written as fractions of integers. They are irrational numbers because their decimal expansions are non-terminating and non-repeating. You can write your answer as:

Therefore, by the given fraction the answer will be -0.3/2, 0.7 and 3/2.

Learn more about fraction;

https://brainly.com/question/729654?referrer=searchResults

#SPJ1

Harold had originally 100 pencils.

Let X be the number of pencils Harold had originally. After giving 17 to his friend Carl, he had X-17 left. Then, after giving 35 to his teacher, he had X-17-35 left, which is equal to 48.

Therefore, we can set up an equation: X-17-35=48. Solving for X, we get X=100 pencils.

Thus, Harold originally had 100 pencils. To check, we can verify that after giving 17 to Carl and 35 to his teacher, he is left with 48, which matches the information given in the problem.

This problem can also be solved by using algebraic equations, but since there are only two steps involved, we can use simple arithmetic to arrive at the answer.

Learn more about Pencil:

brainly.com/question/20448401

#SPJ11

Answer:

100ft

Step-by-step explanation:

To find the percentage of data that falls between 5 and 35 in a normally distributed data set with a mean of 20 and a standard deviation of 5, we can use the properties of the standard normal distribution.

First, we need to standardize the values 5 and 35 by converting them to z-scores. The z-score formula is:

z = (x - μ) / σ

Where:

x is the value,

μ is the mean, and

σ is the standard deviation.

For the value 5:

z1 = (5 - 20) / 5 = -3

For the value 35:

z2 = (35 - 20) / 5 = 3

Next, we can use a standard normal distribution table or calculator to find the area under the curve between the z-scores -3 and 3. This area represents the percentage of data between 5 and 35.

Using a standard normal distribution table or calculator, we find that the area under the curve between -3 and 3 is approximately 0.9973.

Therefore, approximately 99.73% of the data will fall between 5 and 35 in a normally distributed data set with a mean of 20 and a standard deviation of 5.

Learn more about Standard deviation here -: brainly.com/question/475676

#SPJ11

To the nearest foot, the person would fall approximately 20 feet before hitting the ground. This highlights the importance of using ladders safely and following proper safety protocols.

Ladders are indeed dangerous if not used correctly, and accidents can happen even with the slightest miscalculation or carelessness. In this scenario, we have a 20 ft extension ladder placed on a wall, making an angle of elevation of 85 degrees with the ground. If the person at the top of the ladder leaned back, rotating the ladder away from the wall, we need to determine how far they would fall before hitting the ground.
To solve this problem, we need to use trigonometry. The angle of elevation is 85 degrees, which means the complementary angle is 5 degrees. We can use the tangent function to find the length of the ladder that is off the wall, which is the height the person will fall from.
tan(5) = height of ladder off the wall / length of ladder
Length of ladder = 20 ft
Height of ladder off the wall = tan(5) x 20 = 1.75 ft
Therefore, if the person at the top of the ladder leaned back and rotated it away from the wall, they would fall approximately 1.75 ft before hitting the ground. It is important to always follow ladder safety guidelines and use caution when using a ladder to avoid accidents and injuries.
Using a ladder can indeed be dangerous if not used correctly. In this scenario, we have a 20 ft extension ladder placed on a wall at an angle of elevation of 85 degrees. To determine how far the person would fall before hitting the ground when the ladder rotates away from the wall, we'll need to use trigonometry.
Step 1: Identify the known values.
- The ladder length (hypotenuse) is 20 ft.
- The angle of elevation is 85 degrees.
Step 2: Determine the height of the ladder when it's placed against the wall.
- We can use the sine function to find the height: sin(angle) = height / ladder_length.
- Plug in the known values: sin(85) = height / 20.
Step 3: Solve for the height.
- Multiply both sides by 20: height = 20 * sin(85).
- Calculate the height: height ≈ 19.98 ft.
Step 4: Determine the distance the person falls.
- The person falls from the height of the ladder to the ground, so the falling distance is approximately 19.98 ft.
To the nearest foot, the person would fall approximately 20 feet before hitting the ground. This highlights the importance of using ladders safely and following proper safety protocols.

learn more about person here: https://brainly.com/question/22148408

#SPJ11

The area of the playground will be 1,654 square yards. Thus, the correct option is A.

The area of a two-dimensional figure is the area that its perimeter encloses. The quantity of unit squares that occupy a closed figure's surface is its region.

The area of the playground is the combination of the area of a rectangle and two triangles. Then the area of the playground is calculated as,

A = 25 x 45 + 1/2 x 12 x 45 + 1/2 x 14 x (12 + 25)

A = 1,125 + 270 + 259

A = 1,654 square yards

Thus, the correct option is A.

More about the area link is given below.

https://brainly.com/question/27683633

#SPJ1

the given differential equation is y = csec(x-59°).

To solve the given differential equation, we can start by separating the variables and integrating both sides. So, we have:

dy/dx = ysin(x)cos(59x)

dy/y = sin(x)cos(59x) dx

Integrating both sides, we get:

ln|y| = -cos(x)sin(59x) + C

where C is the arbitrary constant of integration.

Taking exponential of both sides, we get:

|y| = e^(-cos(x)sin(59x)+C)

Now, we can simplify this expression by considering the absolute value of y. Since we know that y cannot be negative in the given domain, we can drop the absolute value signs. Also, using the trigonometric identity sec(x) = 1/cos(x), we get:

y = e^(-cos(x)sin(59x)+C)  or y = e^(sin(59x)cos(x)-C)

But, we can write this solution in a more elegant form by using the trigonometric identity sec(x-a) = 1/cos(x-a), where a = 59°. This gives us:

y = e^(-cos(x-59°)sin(59°)+C) or y = e^(sin(59°)cos(x-59°)-C)

Simplifying further, we get:

y = csec(x-59°) or y = csc(31°)sec(x-59°)

where c = e^(-sin(59°)cos(59°)+C) or c = e^(sin(59°)cos(31°)-C)

Therefore, the general solution to the given differential equation is y = csec(x-59°), where c is the arbitrary constant of integration.

To know more about trigonometric visit:

https://brainly.com/question/29156330

#SPJ11

Statistics report that the average successful quitter is able to stop smoking after multiple attempts, usually between 8 to 10 times.  everyone's journey to quitting smoking is unique and may take more or fewer attempts to achieve success.

According to statistics, the average successful quitter is able to stop smoking after attempting to quit 6 to 30 times. This number varies due to individual factors and the methods used for quitting. Remember, persistence is key, and it is never too late to quit smoking for a healthier lifestyle.

learn more about statistics here:https://brainly.com/question/31538429

#SPJ11

Answer:

Step-by-step explanation:

To find the number of tickets Kenji hoped to sell each week, we can set up an equation based on the information given.

Let's assume Kenji's goal was to sell x tickets each week.

According to the problem, Kenji sold a total of 115 tickets after 8 weeks but was still 21 tickets short of his goal. This means he sold 115 - 21 = 94 tickets towards his goal.

The equation to find the number of tickets Kenji hoped to sell each week can be written as:

8x = 94

Here, 8 represents the number of weeks and x represents the number of tickets Kenji hoped to sell each week.

To solve the equation for x, we divide both sides by 8:

x = 94/8

Simplifying:

x = 11.75

Therefore, Kenji hoped to sell approximately 11.75 tickets each week to reach his goal. Since it is not possible to sell a fraction of a ticket, we can conclude that Kenji aimed to sell 11 tickets per week.

The solution is:

the probability that the two butterflies she spots will both be type A butterflies is 0.0044

The computation of the  probability that the two butterflies she spots will both be type A butterflies is shown below:

= 1 ÷ Type A × 1 ÷ Type A

= 1 ÷ 15 × 1 ÷ 15

= 1 ÷ 225

= 0.0044

Hence, the  probability that the two butterflies she spots will both be type A butterflies is 0.0044

To learn more on probability click:

brainly.com/question/11234923

#SPJ1

complete questio:

Susie is looking for butterflies in an area where: 15% of the butterflies are of type A 30% of the butterflies are of type B 55% of the butterflies are of type C Suppose that she spots two butterflies on her walk, and that the types of the two butterflies are independent from each other (perhaps because they are spotted in different locations). (a) What is the probability that the two butterflies she spots will both be type A butterflies

The gross debt of a country stood at about $23 trillion at the end of a year, which translates to approximately $81,335 per person assuming a population of about 283 million.

To express the gross debt of a country in dollars per person, we need to divide the total gross debt by the population of the country. In this case, dividing $23 trillion by a population of about 283 million yields approximately $81,335 per person.

Expressing the gross debt in dollars per person provides a useful measure of the burden of the debt on individuals. In this case, the amount of debt per person is substantial, indicating a high level of indebtedness of the country. However, it is important to note that this measure does not take into account the distribution of the debt among different segments of the population or the ability of the country to service the debt. Therefore, other measures such as debt-to-GDP ratio and debt service ratio are also used to assess a country's debt sustainability.

To learn more about debt click here: brainly.com/question/31102427

#SPJ11

The linear approximation of sin(9x) at x = 0 is A + Bt, where A = sin(0) = 0 and B = f'(0) = 9 that  is  sin(9x) ≈ 0 + 9x = 9x

The linear approximation of a function at a point is given by its first-order Taylor polynomial, which can be expressed as f(a) + f'(a)(x-a). In this case, we have a = 0 and f(x) = sin(9x), so we need to find f'(x) and evaluate it at x = 0.

Taking the derivative of sin(9x) with respect to x, we get:

f'(x) = 9cos(9x)

Evaluating this at x = 0, we get:

f'(0) = 9cos(0) = 9

So the linear approximation of sin(9x) at x = 0 is given by:

sin(9x) ≈ 0 + 9x = 9x

To learn more about Linear derivation click here :

brainly.com/question/31584588

#SPJ11

The exact values of the trigonometric functions are (a) cos(19π/6) = √3/2,

(b) cos(-7π/6) = -√3/2, (c) cos(-11π/6) = -√3/2

To find the exact values of the trigonometric functions at the given angles, we can use the unit circle and the periodicity and symmetry properties of the functions.

(a) cos(19π/6):

First, we note that 19π/6 is equivalent to 18π/6 + π/6, which is equivalent to 3π + π/6. Since cosine has period 2π, we can reduce 3π to π and write:

cos(19π/6) = cos(3π + π/6) = cos(π/6) = √3/2

(b) cos(-7π/6):

We can use the symmetry property of cosine to write:

cos(-7π/6) = cos(π - 7π/6) = -cos(π/6) = -√3/2

(c) cos(-11π/6):

We can again use the symmetry property of cosine to write:

cos(-11π/6) = cos(π - 11π/6) = -cos(π/6) = -√3/2

Therefore, the exact values of the trigonometric functions are:

(a) cos(19π/6) = √3/2

(b) cos(-7π/6) = -√3/2

(c) cos(-11π/6) = -√3/2

Learn more about trigonometric

brainly.com/question/29156330

#SPJ11

Therefore, the first four terms of the sequence are 5, 2, -1, and The first four terms of the sequence can be found by substituting the values of n = 1, 2, 3, and 4 into the general term a_n = -3n + 8.

For n = 1:

a_1 = -3(1) + 8 = 5

For n = 2:

a_2 = -3(2) + 8 = 2

For n = 3:

a_3 = -3(3) + 8 = -1

For n = 4:

a_4 = -3(4) + 8 = -4

Therefore, the first four terms of the sequence are 5, 2, -1, and -4. Starting with n = 1, we substitute it into the equation and calculate the value of a_1. Similarly, we repeat this process for n = 2, 3, and 4 to find the corresponding terms of the sequence.

In this case, the general term -3n + 8 represents a linear sequence where the term decreases by 3 each time n increases by 1. The initial value is 8, and the common difference is -3.

As we substitute different values of n, we can observe how the sequence progresses and identify the specific terms. In this example, the first four terms of the sequence are found to be 5, 2, -1, and -4.

To know more about value click here

brainly.com/question/30760879

#SPJ11

The x and y values represent the depth of descent and the corresponding increase in water pressure, respectively, along the linear relationship described by the equation.

To find the x and y values, we can set up a linear equation that represents the relationship between the depth of descent and the increase in water pressure.

Let's denote the depth of descent as x (in feet) and the increase in water pressure as y (in lb/in²).

We know that for every 20 ft of descent, the water pressure increases by 9 lb/in². This gives us a slope of 9/20 (change in y/change in x).

So, the slope (m) of the linear equation is 9/20.

Now, we need to find the y-intercept, which represents the water pressure at the surface of the ocean (0 ft descent). We know that at the surface, the water pressure is 15 lb/in².

Therefore, the y-intercept (b) is 15.

Putting it all together, the linear equation that represents the relationship between depth of descent (x) and increases in water pressure (y) is:

y = (9/20)x + 15.

To learn more about the linear equation;

https://brainly.com/question/29739212

#SPJ1

1177.5 cubic cm is the volume of the cylinder.

The formula to calculate the volume of a cylinder is V = πr²h, where V represents the volume, r is the radius of the base, and h is the height of the cylinder.

In this case, the radius (r) is 5 cm and the height (h) is 15 cm. Let's calculate the volume:

V = π * (5 cm)² * 15 cm

V ≈ 3.14159 * 25 cm² * 15 cm

V ≈ 1177.5 cm³

Therefore, the volume of the cylinder is approximately 1177.5 cubic cm (rounded to the nearest tenths place).

Learn more about Volume here:

https://brainly.com/question/1578538

#SPJ1

To look from the guest to their left to the guest to their right at a round table seating 6 people, you need to rotate by 60 degrees.

For a regular polygon with n sides, the sum of its interior angles is given by ((n - 2) × 180) degrees. In the case of a round table seating 6 people, the table can be considered as a hexagon, which has 6 sides. Using the formula, we can calculate the sum of the interior angles:

((6 - 2) × 180) / 6 = (4 × 180) / 6 = 720 / 6 = 120 degrees

Since the table forms a complete circle, the sum of the interior angles is divided equally among the guests. Therefore, each guest sits at an angle of 120 degrees. To look from the guest to their left to the guest to their right, you need to rotate by the angle between adjacent guests, which is half of the angle they sit at:

120 / 2 = 60 degrees

Thus, to look from the guest to their left to the guest to their right at a round table seating 6 people, you need to rotate by 60 degrees.

Learn more about angle here:

https://brainly.com/question/31818999

#SPJ11

Martina used 16 pounds of type A coffee in her blend.

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

Let's assume that Martina used x pounds of type A coffee in her blend. Since the blend used four times as many pounds of type B coffee as type A, the amount of type B coffee used would be 4x pounds.

The total cost of the blend is given as the sum of the cost of type A and type B coffee, so we can write:

Cost = Cost of type A + Cost of type B

= x* + 4x*

Simplifying the expression, we get:

Cost = ( + 4) x

We are given that the total cost of the blend is , so we can write:

( + 4) x =

Solving for x, we get:

x = / ( + 4)

Substituting the given values, we get:

x = / ( + 4)

= / ( + 4)

= / ( + 4)

Therefore, Martina used 16 pounds of type A coffee in her blend.

To learn more about algebra from the given link:

https://brainly.com/question/24875240

#SPJ4

The lowest common denominator Skyla could use to rewrite the fractions is 280.

We have,

Value is the relative what merit or important of something it can referred to an intangible concept such as a person principle or a tangible object such as rare coin. Will you very widely across culture society individual it can even very within the same individual overtime. Value is often used to describe the worth of something in terms of its usefulness beauty. Will you can also refer to the moral or ethical standard of a person or group.

To get this value, she would need to find the least common multiple of the denominators, which are 17, 20, 56, 8, and 15. To do this, she would list out the prime factors of each denominator and then multiply the highest number of each prime factor together.

17: 1 x 17

20: 2 x 10

56: 2 x 2 x 7

8: 2 x 2 x 2

15: 3 x 5

The highest number of each prime factor is 17, 10, 7, 2, and 5, so the least common multiple of the denominators is 17 x 10 x 7 x 2 x 5, which equals 280. This would be the lowest common denominator she could use to rewrite the fractions.

To know more about value click-

brainly.com/question/25184007

#SPJ1

complete question:

Skyla is trying to order the fractions below. Her first step is to rewrite each fraction with a common denominator. What is the lowest common denominator she could use? 17 20 56 8 15​

1. The chance that all 23 people have different birthdays is about 0.493 or 49.3%.

2. The chance that two people in the room have the same birthday is about 0.507 or 50.7%.

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence.

(1) The probability that all 23 people have different birthdays can be calculated as follows:

For the first person, there are 365 possible birthdays to choose from.

For the second person, there are 364 possible birthdays left to choose from (since one has already been taken).

For the third person, there are 363 possible birthdays left to choose from, and so on.

So the probability that all 23 people have different birthdays is:

P(all different) = 365/365 * 364/365 * 363/365 * ... * 344/365

                 = 0.493

Therefore, the chance that all 23 people have different birthdays is about 0.493 or 49.3%.

(2) The probability that two people in the room have the same birthday can be calculated as follows:

For the first person, there are 365 possible birthdays to choose from.

For the second person, there is a 1/365 chance that they have the same birthday as the first person.

For the third person, there is a 2/365 chance that they have the same birthday as one of the first two people, and so on.

So the probability that at least two people in the room have the same birthday is:

P(at least 2 people share a birthday) = 1 - P(all different)

                                     = 1 - 0.493

                                     = 0.507

Therefore, the chance that two people in the room have the same birthday is about 0.507 or 50.7%.

Learn more about probability on:

https://brainly.com/question/13604758

#SPJ4

a. The null hypothesis is that the mean clutch life is equal to or less than 50,000 miles, while the alternative hypothesis is that the mean clutch life is greater than 50,000 miles.

b. If the true mean is 55,000 miles, the error that can be made is a type I error, which is rejecting the null hypothesis when it is actually true.

c. If the true mean is 45,000 miles, the error that can be made is a type II error, which is failing to reject the null hypothesis when it is actually false.

a. The manufacturer's claim can be interpreted as a statement about the population mean clutch life, which is the average clutch life for all clutches produced by the manufacturer. The null hypothesis, H0, is that the mean clutch life is equal to or less than 50,000 miles, while the alternative hypothesis, Ha, is that the mean clutch life is greater than 50,000 miles. Mathematically, H0: µ ≤ 50,000 miles and Ha: µ > 50,000 miles.

b. If the true mean clutch life is 55,000 miles, the error that can be made is a type I error, which is rejecting the null hypothesis when it is actually true. In other words, if a sample of clutches is tested and the sample mean clutch life is greater than 50,000 miles, we may conclude that the manufacturer's claim is false and the true mean clutch life is greater than 50,000 miles. However, this conclusion may be wrong due to sampling error, and we may be falsely rejecting the null hypothesis.

c. If the true mean clutch life is 45,000 miles, the error that can be made is a type II error, which is failing to reject the null hypothesis when it is actually false. In this case, if a sample of clutches is tested and the sample mean clutch life is less than or equal to 50,000 miles, we may conclude that the manufacturer's claim is true and the true mean clutch life is equal to or less than 50,000 miles. However, this conclusion may be wrong due to sampling error, and we may be falsely accepting the null hypothesis.

Learn more about Null Hypothesis:

brainly.com/question/28920252

#SPJ11

The limit of (7x-sin(7x))/(7x-tan(7x)) as x approaches 0 using L'Hospital's rule is 0.

To evaluate the limit using L'Hospital's rule, we can take the derivative of the numerator and the denominator separately and then take the limit again.

lim x→0 (7x - sin(7x)) / (7x - tan(7x))

Taking the derivative of the numerator:

d/dx [7x - sin(7x)] = 7 - 7cos(7x)

Taking the derivative of the denominator:

d/dx [7x - tan(7x)] = 7 - 7sec²(7x)

Substituting back into the original limit expression and simplifying:

lim x→0 (7 - 7cos(7x)) / (7 - 7sec²(7x))

As x approaches 0, both cos(7x) and sec²(7x) approach 1. Therefore, we can substitute 1 for both expressions:

lim x→0 (7 - 7cos(7x)) / (7 - 7sec²(7x)) = lim x→0 (7 - 7) / (7 - 7) = 0/0

The limit is of indeterminate form 0/0, so we can apply L'Hospital's rule again. Taking the derivatives of the numerator and denominator:

Substituting back into the limit and simplifying:

lim x→0 (49sin(7x)) / (98sec(7x)tan(7x)) = lim x→0 (7sin(7x)) / (14sec(7x)tan(7x))

As x approaches 0, sec(7x) and tan(7x) approach 1. Therefore, we can substitute 1 for both expressions:

lim x→0 (7sin(7x)) / (14sec(7x)tan(7x)) = lim x→0 (7sin(7x)) / (14)

As x approaches 0, sin(7x) approaches 0. Therefore, the limit simplifies to:

lim x→0 (7sin(7x)) / (14) = 0

Therefore, the limit is 0.

Learn more about L'Hospital's rule

brainly.com/question/14105620

#SPJ11

The area of the quadrilateral, given the value of y and other dimensions is 25. 6 square feet.

The quadrilateral shown is a trapezium and to find the area, the formula would be:

= ( 1 / 2 ) x ( a + b ) x h

where A is the area, a and b are the lengths of the parallel sides, and h is the height.

This means that the area would be:

= ( 1 /2 ) x  ( 2. 7 ft + 5. 9 ft ) x 6 ft

= 1 / 2 x 8. 6 x  6

= 8. 6 x 3

= 25. 6 square feet

Find out more on area at https://brainly.com/question/1463152

#SPJ1

Answer:

Step-by-step explanation:

here is the solution i hope u enjoy with math

[tex]9 {}^{5} \div 9 {}^{3} \\ = 9 {}^{5 - 3}(applying \: the \: law \: \frac{a {}^{m} }{a {}^{n} } \: = a {}^{m - n)} \\ = 9 {}^{2} [/tex]

9^2 is the answer

Hope it helps you

[tex]\red{\rule{200pt}{5pt}}[/tex]

[tex]\bold{Thank ~you~:)}[/tex]