Find the missing angle measure of the right triangle. Show all work. Round to the nearest tenth.

There were initially 4 million bacteria in the culture.

To determine the initial number of bacteria in the culture, we can follow these steps:

Step 1: Identify the given data.
At the end of 6 hours, there are 5 million bacteria.
At the end of 9 hours, there are 8 million bacteria.

Step 2: Find the rate of bacterial growth.
Since the number of bacteria increased by 3 million (8 million - 5 million) over 3 hours (9 hours - 6 hours), the rate of bacterial growth is 1 million per hour (3 million ÷ 3 hours).

Step 3: Calculate the initial number of bacteria.
The number of bacteria increased by 5 million over the 6 hours. Therefore, if we divide this number by the rate of growth, we can determine the initial number of bacteria. Divide 5 million by 1 million per hour, which results in 5 hours.

Step 4: Subtract the time from the initial time.
Since we know that the culture was growing for 5 hours before reaching 5 million, we can subtract 5 hours from 6 hours, which equals 1 hour.

Step 5: Calculate the initial number of bacteria at 1 hour.
Since the bacteria grow at a rate of 1 million per hour, at the end of the 1st hour, there would be 1 million bacteria. Subtracting this number from the 5 million at the end of 6 hours will give us the initial number of bacteria: 5 million - 1 million = 4 million bacteria.

So, there were initially 4 million bacteria in the culture.

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To check if the line drawn is diameter it should be intersected at centre and it must be equidistant from the centre of the circle.

A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.

To determine if the line Neil drew is an actual diameter of the circle and not just a smaller chord, he can use the following method -

Extend the line on both sides to create two lines that intersect at a point outside the circle.

Use a compass to draw a circle with the same center as the original circle, and with a radius that is greater than half the length of the line Neil drew.

Check if the two lines intersect the circle at two points that are equidistant from the center of the circle.

If they do, then the line Neil drew is an actual diameter of the circle.

If they do not, then the line Neil drew is just a chord.

This method works because a diameter of a circle is the longest chord that passes through the center of the circle.

By extending the line Neil drew and creating two lines that intersect outside the circle, we can compare the distance from the center of the circle to each of the two intersection points with the radius of the circle we drew.

If the distances are equal, then the line Neil drew must be a diameter of the circle.

Therefore, the method to find the actual diameter is explained.

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Answer:

$32.50

Step-by-step explanation:

20% * y = 6.5 =>

y =6.5 ÷ 20% =

6.5 ÷ (20 ÷ 100) =

(100 × 6.5) ÷ 20 =

650 ÷ 20 =

32.5

Percentage of 20% of what number = 6.5?

20% × ? = 6.5

20% of 32.5 = 6.5

Answer:

x:5,x:7

x^2-12x+35

x^2-12x=-35

x^2-12x+6^2=-35+36

(x+6)^2=1

insert a radical anywhere

x+6=1

x=1-6

x=-5. x=5

x=-1-6

x=-7. x=7

The given statement "if we take a simple random sample from a normal distribution, the probability that the sample mean is equal to the population mean is 1 (i.e., 100%)" is false.

The probability that a sample mean is equal to the population mean is actually zero, since any given sample will differ slightly from the population mean due to sampling error. However, as the sample size increases, the sample mean is likely to be close to the population mean, and the probability of it being equal approaches 1 (i.e., 100%).

To explain further, the population mean is the mean of the entire population and is calculated by adding up all the values of the population and dividing them by the total number of individuals in the population. A sample mean, on the other hand, is the mean of a sample taken from the population and can be calculated by adding up the values of the sample and dividing them by the total number of individuals in the sample. Since the sample size is usually smaller than the population size, the sample mean is likely to be slightly different from the population mean due to sampling error.

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The line-plot of the the given data 9 3/4, 10 1/2, 9 1/4, 10 1/4, 11, 10 1/4, 9 1/2 that displays the lengths of the books on the booom shelf is attached.

What is line plot?

A line plot is a type of graph that represents the kind of data representation. It is line plot or a line chart shows the data in a visual form or visual representation. The graph is plotted with help of straight lines and points or dots. The x-axis is represented by straight lines or the it is labelled about the data. The y-axis is represented by the dots or frequency of the data. It is generally used to represent the small data set. This graph uses symbols to show or represent the frequency of the observations.

The data represented here is 9 3/4, 10 1/2, 9 1/4, 10 1/4, 11, 10 1/4, 9 1/2

we can make the frequency table using the above values, after which it can be plotted on the line-plot graph.

1-9 1/4--1 dot

2-9 1/2--1 dot

3-9 1/2--1 dot

4-10 1/4-2 dots

5-10 1/2--1 dot

6-11--1 dot

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Attached is a line-plot of the given data showing the lengths of the books on the book shelf: 9 3/4, 10 1/2, 9 1/4, 10 1/4, 11, 10 1/4, and 9 1/2.

An example of a graph that shows this sort of data representation is a line plot. The data is presented visually in the form of a line plot or a line chart. With the aid of straight lines, points, or dots, the graph is drawn. Straight lines or labels that describe the data are used to depict the x-axis. The dots or frequency of the data are used to depict the y-axis. It often serves as a representation for a tiny group of data. The frequency of the observations is shown or represented by symbols on this graph.

9 3/4, 10 1/2, 9 1/4, 10 1/4, 11, 10 1/4, and 9 1/2 are the numbers shown.

Using the aforementioned variables, we can create the frequency table, which can then be plotted on the line-plot graph.

1-9 1/4--1 dot

2-9 1/2--1 dot

3-9 1/2--1 dot

4-10 1/4-2 dots

5-10 1/2--1 dot

6-11--1 dot

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The sampling method used in the study where the sample is chosen by separating all cars by size, and selecting 10 of each size grouping is known as stratified random sampling.

Stratified random sampling is a method of sampling that involves dividing the population into smaller sub-groups known as strata. In this method, each stratum is composed of elements that have similar characteristics.

The strata could be based on demographic characteristics such as age, sex, education, and so on. Once the strata have been created, the next step is to select a sample from each of the strata to make up the final sample. The selection is done randomly, and the number of elements selected from each stratum is proportional to the size of the stratum. This ensures that the sample is representative of the entire population in terms of the different characteristics of the population.

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Answer:

Let's denote the first term of the sequence as a, and the common difference between consecutive terms as d.

Then, we know that the fifth term is 5, so:

a + 4d = 5

Similarly, we know that the sixth term is 2.5, so:

a + 5d = 2.5

We can solve this system of equations by subtracting the first equation from the second:

(a + 5d) - (a + 4d) = 2.5 - 5

d = -2.5

Now, we can substitute this value of d into either equation to find the value of a:

a + 4d = 5

a + 4(-2.5) = 5

a - 10 = 5

a = 15

Therefore, the first term of the sequence is 15, and the common difference is -2.5. We can use this to find the value of the second term:

a + d = 15 + (-2.5) = 12.5

Therefore, the second term of the sequence is 12.5.

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At exactly 2:00, the distance between the tips of the two hands is changing at the rate of 4 cm/min.Therefore, at exactly 2:00, the distance between the tips of the two hands is changing at the rate of [tex]2\sqrt{13}[/tex] cm/min.

The phrase "at exactly 2:00" means that at 2:00:00, the hour hand and the minute hand overlap. Both hands are together pointing upwards in the vertical direction. At this moment, the angle between the minute hand and the hour hand is 0 degrees.How do we determine the rate of change of the distance between the tips of the two hands?We start by calculating the angle between the two hands. The angle is given by:

θ(t) = 30h - 11/2m

where h is the number of hours past midnight and m is the number of minutes past the most recent hour. At exactly 2:00, h = 2 and m = 0. Thus:

θ(t) = 30(2) - 11/2(0) = 60 degrees

Next, we use the formula for the distance between two points (the tips of the two hands). Let the hour hand be at the point (0,0) and let the minute hand be at the point (x,y).

The distance between the two points is given by:

[tex]d(t) = \sqrt{(x^2 + y^2)}[/tex]

To find x and y, we use the formulas:

x(t) = r cos θ(t)y(t) = r sin θ(t)

where r is the length of each hand. At exactly 2:00:

[tex]r = 4 cmx(t) = 4 cos 60 = 2 my(t) = 4 sin 60 = 2\sqrt{(3) m}[/tex]

Therefore: [tex]d(t) = \sqrt{(x^2 + y^2)}  = \sqrt{(2 m)^2 + (2sqrt(3) m)^2)}  = 2\sqrt{(13) m}[/tex]

Finally, we differentiate with respect to time t to find the rate of change of [tex]d(t):d'(t) = 2\sqrt{13}[/tex] cm/min

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It is important to understand that the force of gravity acting on an object does not violate Newton's second law. It is an external force that acts on an object and produces an acceleration in it. Therefore, the acceleration due to gravity does not violate Newton's second law.

According to the Newton's second law, a force acting on a body of mass m produces an acceleration a in the body which is directly proportional to the force and inversely proportional to the mass. For instance, If a body of mass m is acted upon by a force F, then the acceleration produced in the body is given by F/m.As per the law, a larger force will produce a larger acceleration in the object.

When a body falls under gravity, it is said to experience weight. The weight of an object is the force of gravity acting on the object. According to Newton's law of gravitation, every object attracts every other object with a force proportional to their masses and inversely proportional to the square of their distance. In general, the weight of an object can be given by the formula weight = mass x acceleration due to gravity.

Therefore, as the acceleration due to gravity is the same for all objects, the weight of an object is proportional to its mass. It implies that a heavier object has more weight, while a lighter object has less weight.As per the above facts, a heavier object would require a larger force to move it than a lighter object. However, when they fall under gravity, both objects fall at the same rate. It appears as a contradiction, but the reason for it is the difference between weight and mass.

The mass of an object is a measure of the amount of matter in it, while the weight of an object is the force exerted on it by gravity.The acceleration produced in an object by an external force is not the same as the acceleration due to gravity. A force F produces an acceleration F/m, whereas the acceleration due to gravity is the same for all objects.

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Answer:

331/2

Step-by-step explanation:

5 1/2= 11/2

3 1/2 = 7/2

11/2x7/2x6=115 1/2 = 331/2

Answer:

115.5 cm^3

Step-by-step explanation:

Volume=LengthxWidthxHeight

The values from the diagram are 3.5, 6, and 5.5

Multiply!

3.5x6x5.5=115.5

The value of x in the given linear equation of 4x = 28 is determined as 7.

To find the value of x in the linear equation 4x = 28, we need to isolate x on one side of the equation.

We can do this by dividing both sides of the equation by 4:

4x/4 = 28/4

Simplifying:

x = 7

Thus, identify the equation and the variable: In this case, the equation is 4x = 28, and the variable we want to solve for is x. Also simplify the linear equation by dividing both sides by 4, we get x = 7, which is the solution.

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The complete question is below:

4x = 28

find the value of x

The seller purchased the product at a cost of 7500 and sold it for 7950, which resulted in a gain of 6%. So, the cost price is 7500.

In business, it is essential to keep track of cost prices and profit margins to ensure profitability. If the selling price is 7950 and the gain is 6%, we can use the following formula to calculate the cost price:

Cost price = Selling price / (1 + Gain%)

Substituting the given values, we get:

Cost price = 7950 / (1 + 0.06)

Cost price = 7500

Therefore, the cost price of the product is 7500, the gain percentage is calculated based on the cost price and not the selling price.

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Odd coefficients of the polynomial are obtained by multiplying both odd integers.

Even coefficients of the polynomial are obtained by multiplying both even integers.

Even and odd coefficients of the polynomial are obtained by multiplying both odd integers.

Lets take two integer m and n,

If both 'm' and 'n' are odd integers, then the coefficients of the polynomial obtained by multiplying out will be odd.

If both 'm' and 'n' are even integers, then the coefficients of the polynomial obtained by multiplying out will be even.

If one of 'm' and 'n' is even and the other is odd, then the coefficients of the polynomial obtained by multiplying out will be even and odd.

There are no other cases.

The general form of the polynomial obtained by multiplying out a binomial is:

[tex](a+b)^n = nC_0a^n + nC_1a^{(n-1)}b + nC_2a^{(n-2)}b^2 + .....+ nC_n-1ab^{(n-1)} + nC_nb^n[/tex]

where [tex]nC_k[/tex] is a binomial coefficient.

In general, for a polynomial [tex](a+b)^n[/tex], the coefficient of the term of degree k is [tex]nC_k[/tex].

The binomial coefficients are defined as:  [tex]nC_k = n! / (k!\times(n-k)!)[/tex]

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Therefore , the solution of the given problem of graphs comes out to be a = [P² * G * Ms / [tex](4\pi ^2* Mp)]^(1/3)[/tex]

Graphs are used by theoretical coordinate points physicists to analytically and graphically present assertions rather than values. Typically, a graph point shows the relationship between several distinct objects. A graph is a particular kind of pas de container construction composed of groups and lines. The borders, also known as the channels, should be joined with glue.

Here,

According to Kepler's third law, one can determine the connection between a planet's orbital period and its separation from a star by:

=> P² = (4π² / GM) * a³

where P denotes the orbit's duration in years, a denotes the semi-major axis in astronomical units (AU), G denotes gravity's constant, and M denotes the star's mass in solar masses.

If we rewrite this equation, we obtain:

=> a = [P² * G * Ms / [tex](4\pi ^2* Mp)]^(1/3)[/tex]

We can rewrite this equation as follows since our goal is to estimate the distance from the sun in AU based on the mass of the planet in relation to the sun:

=> a = [P² * G * Ms / [tex](4\pi ^2* Mp)]^(1/3)[/tex]

where Mp is the mass of the planet in relation to the sun and Ms is the mass of the sun.

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The answer of the given question based on the supplementary angle is  the two angles are 58° degrees and 122° degrees.

An angle is a geometric figure that is formed by two rays or lines that share a common endpoint, called the vertex. The rays or lines are called  sides or arms of  angle. Angles are typically measured in the degrees or the radians.

The measure of an angle is determined by the amount of rotation needed to bring one of its sides into coincidence with the other side. A full rotation is 360° degrees, so an angle that is half of a full rotation is 180 degrees, an angle that is a quarter of a full rotation is 90° degrees, and so on.

Let's call the two angles "x" and "y". We know that they are supplementary, which means that their sum is 180° degrees:

x + y = 180

We also know that one angle (let's say "x") is two times the sum of the other angle (y) and 3:

x = 2(y + 3)

Now we can substitute the second equation into the first equation to eliminate "x":

(2(y + 3)) + y = 180

Simplifying the equation, we get:

3y + 6 = 180

Subtracting 6 from both sides, we get:

3y = 174

Dividing both sides by 3, we get:

y = 58

Now that we know "y", we can substitute it back into the equation for "x" to find its value:

x = 2(y + 3) = 2(58 + 3) = 122

Therefore, the two angles are 58° degrees and 122° degrees.

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Answer:

The answer to your problem is, A. 60%

Step-by-step explanation:

54 of 90 can be written as: [tex]\frac{54}{90}[/tex]

To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100

[tex]\frac{54}{90} * \frac{100}{100}[/tex]

= [tex]\frac{( 54 * 100)}{90} * \frac{1}{100} = \frac{60}{100}[/tex]

Thus the answer to your problem is, A. 60%

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Answer:

x = 60%

Step-by-step explanation:

Let the unknown percentage be x.

We can make an expression using the given information.

[tex]\sf90*\frac{x}{100} =54[/tex]

Solve for x to find the percentage.

[tex]\sf\frac{90x}{100} =54\\90x=54*100\\\\90x=5400\\\\\frac{90x}{90}=\frac{5400}{90}\\\\ x=60[/tex]

The given inequality is [tex]|x-3|/11 \geq 1[/tex]and the solution to the inequality is [tex]x \leq -8[/tex] or [tex]x \geq 14.[/tex]

We want to solve for [tex]x[/tex], which means we want to find all possible values of [tex]x[/tex] that satisfy the inequality.

To start, we multiply both sides of the inequality by 11, which gives us:

[tex]| x - 3 | \geq 11[/tex]

Now we have an inequality without any fractions. The absolute value sign means that we need to consider two cases: one where [tex]x-3[/tex] is positive, and one where [tex]x-3[/tex] is negative.

Case 1: [tex]x-3[/tex] is positive

If [tex]x-3[/tex] is positive, then [tex]| x - 3 |[/tex] is equal to [tex]x-3[/tex]. So we can rewrite the inequality as:

[tex]x - 3 \geq 11[/tex]

Adding 3 to both sides, we get:

[tex]x \geq 14[/tex]

This means that if [tex]x[/tex] is greater than or equal to 14, then the inequality is satisfied.

Case 2: [tex]x-3[/tex] is negative

If [tex]x-3[/tex] is negative, then [tex]| x - 3 |[/tex] is equal to [tex]-(x - 3)[/tex]. So we can rewrite the inequality as:

[tex]-(x - 3) \geq 11[/tex]

Multiplying both sides by -1, we get:

[tex]x - 3 \leq -11[/tex]

Adding 3 to both sides, we get:

[tex]x \leq -8[/tex]

This means that if [tex]x[/tex] is less than or equal to -8, then the inequality is satisfied.

Putting these two cases together, we get:

[tex]x \leq -8[/tex] or [tex]x \geq 14[/tex]

These are all the possible values of [tex]x[/tex] that satisfy the inequality.

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The complete question is:

Solve the equation: | x - 3 | / 11 ≥ 11 and determine the type of solution.

Step-by-step explanation:

[tex]volume = {x}^{3 \\} \\ so \: 1000000 = {x}^{3 \\} \\ x = \sqrt[3]{1000000} \\ x = 100[/tex]

The probability that Emme answers exactly 3 questions correctly is 0.1961.

Since each question has four answer choices and only one is correct, the probability of Emme randomly guessing the correct answer for each question is 1/4 = 0.25. Emme needs to answer exactly 3 out of the 9 questions correctly, so we use the binomial distribution with n = 9 and p = 0.25 to calculate the probability:

P(X = 3) = (9 choose 3) * (0.25)^3 * (0.75)^6

Using a calculator, we get:

P(X = 3) = 0.1961

Therefore, the probability that Emme answers exactly 3 questions correctly is 0.1961.

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The year that saw 164 billion dollars being spent on advertising, given the function for the advertising would be c. 1994

The dimensions of the rectangle, given the area of the rectangle would be:

The function is given by y = 0.93x² + 2.2x + 130, where y is the amount of money spent (in billions of dollars) and x is the number of years since 1990. We need to find the value of x when y = 164.

164 = 0.93x² + 2.2x + 130

0.93x² + 2.2x + 130 - 164 = 0

0.93x² + 2.2x - 34 = 0

x = 2.5

Now, we'll find the year by adding the value of x to 1990:

Year = 1990 + 2.5 = 1992.5

= 1994 which is the closest year in options

The area of the rectangle is given by:

Area = width × length

91 = (x + 2)(2x + 3)

91 = 2x² + 3x + 4x + 6

91 = 2x² + 7x + 6

2x² + 7x - 85 = 0

When solved differently:
x = 4.9

x = -3.5

Using the positive values:

Width = x + 2 ≈ 4.9 + 2 = 6.9 meters

Length = 2x + 3 ≈ 2(4.9) + 3 ≈ 9.8 + 3 = 12.8 meters

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The probability of hitting the black circle is closer to 0 and the probability of hitting the white portion of the target is closer to 1.

What is probability?

Probability refers to the numerical measure of the likelihood or possibility of a specific event taking place. It is a mathematical concept that involves calculating the ratio of the number of desired outcomes to the total number of possible outcomes. Probability ranges from 0 (representing an impossible event) to 1 (representing a certain event), with values in between indicating the degree of likelihood. Probability is widely used in various fields, including mathematics, statistics, science, engineering, finance, and social sciences, to make predictions, analyze data, and make decisions under uncertainty.

Explanation of the given questions :

Part A: The black circle has a radius of 1 unit and its center is the same as that of the square. Therefore, its diameter is 2 units, which is much smaller than the side length of the square, which is 11 units. This means that the area of the circle is much smaller than the area of the square.

Therefore, the probability of hitting the black circle is closer to 0.

Part B: The white portion of the target is the area of the square minus the area of the black circle. The area of the square is [tex]11 x 11 = 121[/tex] square units.

The area of the circle is [tex]\pi \times r^2 = \pi\times 1^2 = \pi[/tex] square units, which is approximately 3.14 square units. Therefore, the area of the white portion of the target is [tex]121 - 3.14 = 117.86[/tex] square units.

Since this is much larger than the area of the black circle, the probability of hitting the white portion of the target is closer to 1.

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Answer:

Step-by-step explanation:

To find out how many more hours Jason worked than Sheldon, we need to subtract the number of hours Sheldon worked from the number of hours Jason worked:

Jason's hours = 14/3

Sheldon's hours = 25/6

Jason's hours - Sheldon's hours = (14/3) - (25/6)

To subtract fractions, we need to have a common denominator, which in this case would be 6:

(14/3) - (25/6) = (28/6) - (25/6) = 3/6

Simplifying the result, we get:

3/6 = 1/2

Therefore, Jason worked 1/2 more hours than Sheldon.

Jason worked 1/3 more hours than Sheldon.

To find out how many more hours Jason worked than Sheldon, we need to subtract the number of hours Sheldon worked from the number of hours Jason worked.

Jason worked for 14/3 hours, which can be simplified to 4 and 2/3 hours. Sheldon worked for 25/6 hours, which can be simplified to 4 and 1/6 hours.

Subtracting the hours Sheldon worked from the hours Jason worked, we get: 4 and 2/3 - 4 and 1/6 = 2/3 - 1/6 = 1/3.

Therefore, Jason worked 1/3 more hours than Sheldon.

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The theoretical probability of not drawing an "o" is 3/4.

We have,

Each of the letters of the word Colorado are written on a piece of paper and then put into a bag.

Here, The word "Colorado" has 8 letters, including 2 "o"s.

Therefore, the probability of drawing an "o" is,

P = 2/8

P = 1/4

Hence, The probability of not drawing an "o" is the complement of this probability is,

1 - 1/4 = 3/4.

Therefore, the theoretical probability of not drawing an "o" is 3/4.

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Answer:

[tex]2 \sqrt{2(6 + \sqrt{3} )} [/tex]

Step-by-step explanation:

There's your answer.

Only the final option, which is typical of a longitudinal study, calls for retesting the same set of individuals over a lengthy period of time.

It is a numerical value that runs from 0 to 1, with 0 denoting the impossibility of an event and 1 denoting its certainty. The probability of an event A is expressed mathematically as P(A), where P(A) is the number of possible ways that event A could happen divided by the total number of outcomes.

given

The final choice is the long-term study: "An intelligence test is given to a class of kindergarteners. During 30 years, they are retested every other year."

In a longitudinal study, a group of individuals are followed over a lengthy period of time, frequently years or even decades. A longitudinal study's goal is to look at how people change or grow over time and to pinpoint any potential contributing elements.

Only the final option, which is typical of a longitudinal study, calls for retesting the same set of individuals over a lengthy period of time.

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The point (2,2) may be considered an anomaly and should be investigated further to determine if it is a valid data point or if it should be excluded from the analysis.

The relation of the point (2,2) to the line of best fit can be determined by analyzing the residual value of this point. The residual value is the difference between the actual y-value and the predicted y-value based on the line of best fit. If the residual value is close to zero, then the point lies on the line of best fit. If the residual value is positive, then the actual y-value is higher than the predicted y-value, and if the residual value is negative, then the actual y-value is lower than the predicted y-value.

Unfortunately, the data necessary to determine the residual value of the point (2,2) is not provided in the question. Therefore, it is impossible to determine the exact relation of this point to the line of best fit.

However, we can make some generalizations based on the location of the point relative to the trend of the data.

If the point (2,2) lies close to the line of best fit, with a residual value close to zero, then it can be said that the point follows the trend of the data and is a typical data point. However, if the point lies far away from the line of best fit, with a large positive or negative residual value, then it can be said that the point is an outlier and does not follow the trend of the data.

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The expression 75 + 1.45x represents the total cost of a training session with Terrell

Let x be the distance (in miles) that Terrell must drive to and from the training session.

Then the total cost C of the training session, in dollars, can be expressed as:

C = 75 + 1.45x

Therefore, the expression 75 + 1.45x represents the total cost of a training session with Terrell, including both the base fee and the additional cost for driving.

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Answer:

=(x+9)(x+5)

Step-by-step explanation:

Answer:

[tex]2 \sqrt{61} \: in[/tex]

Step-by-step explanation:

Use the Pythagorean theorem (assume, that the length of hypotenuse is x):

[tex] {x}^{2} = {10}^{2} + {12}^{2} = 100 + 144 = 244[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{244} = 2 \sqrt{61} [/tex]