Why is pi important in math?

The expected number of pairs of professors in conflict is given by (m choose 2) * 1/n.

It can be calculated using the principle of linearity of expectation. We can first calculate the probability that any two professors pick the same time slot, which is 1/n. Then, we can count the number of pairs of professors, which is given by the binomial coefficient (m choose 2) = m(m-1)/2. Therefore, the expected number of pairs of professors in conflict is:

Expected number of pairs in conflict = (m choose 2) * 1/n

This formula holds when the selection of time slots by each professor is independent of the choices made by all other professors. Note that this formula assumes that each professor selects only one time slot, and does not consider the possibility of a professor selecting multiple time slots. If professors are allowed to select multiple time slots, then the formula would need to be modified accordingly.

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(a)The distribution of X (the average of the 49 races) follows a normal distribution with mean 146 minutes and standard deviation 11 minutes. (b)The probability of the average of the sample being between 144 and 149 minutes is 0.5854.(c)The 80th percentile for the average of these 49 marathons is 157.2 minutes.(d) The median of the average running times is 146 minutes.

Part(a) The distribution of X (the average of the 49 races) follows a normal distribution with mean 146 minutes and standard deviation 11 minutes. Part (b) The probability that the average of the sample will be between 144 and 149 minutes in these 49 marathons can be calculated using the z-score formula: z = (x - mean)/standard deviation
For x = 144, z = (144 - 146)/11 = -0.18
For x = 149, z = (149 - 146)/11 = 0.27,using the z-score table, the probability of the average of the sample being between 144 and 149 minutes is 0.5854 (0.4026 + 0.1828).

Part (c) The 80th percentile for the average of these 49 marathons can be calculated using the z-score formula: z = (x - mean)/standard deviation, For the 80th percentile, z = 0.84 (from z-score table). Therefore, x = 146 + (0.84 * 11) = 157.2 minutes. Part (d) The median of the average running times is 146 minutes. The median is the midpoint of the data which means half of the data is above the median and half of the data is below the median. Therefore, the median of the average running times is equal to the mean.

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In linear equation, The number of suitcases are 503.

What in mathematics is a linear equation?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.  

                                   Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

Equation

y = 1.98x + 7.97

Where,

x = Number of passengers

y = Number of suitcases

Finding the number of suitcases:

y = 1.98x + 7.97

y = 1.98(250) + 7.97

y = 495 + 7.97

y = 502.97 ≈ 503

Hence,

The number of suitcases are 503.

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Anna should use 10 mL of the 72% salt solution and 20 mL of the 54% salt solution to make 30 mL of a 60% salt solution

Let's assume that Anna will use x mL of the 72% salt solution, and therefore she will use (30 - x) mL of the 54% salt solution (since the total volume is 30 mL).

To find out how much of each solution Anna should use, we can set up an equation based on the amount of salt in each solution.

The amount of salt in x mL of 72% salt solution is

= 0.72x

The amount of salt in (30 - x) mL of 54% salt solution is

= 0.54(30 - x)

To make a 60% salt solution, the total amount of salt in the final solution should be

0.6(30) = 18

So we can set up an equation

0.72x + 0.54(30 - x) = 1

Simplifying the equation

0.72x + 16.2 - 0.54x = 18

0.18x = 1.8

x = 10 ml

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Answer: To plot and connect the points, you can use a graphing software or a graphing calculator. Once the points are plotted, you can determine if the graph is linear or exponential by looking at the slope of the line. If the slope is increasing or decreasing exponentially, the graph is exponential. If the slope is constant, the graph is linear.

Step-by-step explanation:

The derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.

Using the product rule of differentiation, we can find the derivative of the function f(x) = x⋅5x as follows:

f'(x) = (x)'(5x) + x(5x)'

where (x)' and (5x)' are the derivatives of x and 5x with respect to x, respectively.

(x)' = 1 (the derivative of x with respect to x is 1)

(5x)' = 5 (the derivative of 5x with respect to x is 5)

Substituting these values, we get:

f'(x) = 1⋅5x + x⋅5

Simplifying further, we get:

f'(x) = 5x + 5x

Therefore, f'(x) = 10x.

In conclusion, the derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.

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Answer:

2 × 2 × 3 × 7=84

Step-by-step explanation:

The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.

WHAT IS AXIS ?

An axis is a straight line around which an object rotates or is symmetrical. In geometry, it is a reference line that is used to measure distances, angles, and other geometric properties of objects. In a two-dimensional plane, there are two axes: the x-axis and the y-axis. The x-axis is the horizontal line that runs from left to right, while the y-axis is the vertical line that runs from bottom to top. Together, the x-axis and y-axis form a coordinate system that is used to plot points and graph functions. In three-dimensional space, there is also a z-axis that runs perpendicular to the x-axis and y-axis

To perform a dilation with a scale factor of 1/2 around the origin, we need to multiply the coordinates of each point by 1/2. This gives us:

A' = (1/2)A = (1/2)(-2,-5) = (-1,-2.5)

B' = (1/2)B = (1/2)(0,3) = (0,1.5)

C' = (1/2)C = (1/2)(6,-3) = (3,-1.5)

The new coordinates of points A, B, and C after the dilation are (-1,-2.5), (0,1.5), and (3,-1.5), respectively.

To reflect each point over the y-axis, we need to change the sign of its x-coordinate, while leaving the y-coordinate unchanged. This gives us:

A" = (-x,y) = (-(-1),-2.5) = (1,-2.5)

B" = (-x,y) = (-0,1.5) = (0,1.5)

C" = (-x,y) = (-3,-1.5) = (-3,-1.5)

The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.

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After figuring out the given issue, we discovered that the original square's area and the dilated square's area differ by 32 square centimeters.

Square Area = Side x Side. Hence, Side2 square units are equivalent to the area of the square. and four side units make up a square's perimeter.

The formula for calculating the area of a initial square with sides that are 2 centimeters long is:

A = s²

A = 2²

A = 4 square centimeters

The revised side length of this square after dilation by a scale factor of 3 is:

s' = 3s

s' = 3(2)

s' = 6 centimeters

Calculations for the dilated square's area are as follows:

A' = s'²

A' = 6²

A' = 36 square centimeters

The area of a dilated square differs from the size of the original square by:

A' - A = 36 - 4

A' - A = 32 square centimeters

As a result, the original square's area and the dilated square's area differ by 32 square centimeters.

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In the following code, you can add another array declaration that creates an array of 5 doubles called prices and another array of 5 Strings called names and corresponding System.out.println commands.

public class Test1 {

public static void main(String[] args) {

// Array example

int[] highScores = new int[10];

// Add an array of 5 doubles called prices.

double[] prices = new double[5];

// Add an array of 5 Strings called names.

String[] names = new String[5];

System.out.println("Array highScores declared with size " + highScores.length);

// Print out the length of the new arrays.

System.out.println("Array prices declared with size " + prices.length);

System.out.println("Array names declared with size " + names.length);

}

}

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For the first question, the indefinite integral of [tex]16x³ - 15x² + 6[/tex] is [tex]4x⁴ - 5x³ + 6x + C[/tex], where C is the constant of integration.

For the second question, the indefinite integral of 7cos(x) + 5sin(x) is [tex]7sin(x) + 5cos(x) + C,[/tex]where C is the constant of integration.

1)To check the result, differentiate the indefinite integral using the power rule and product rule: [tex](16x³ - 15x² + 6)' = 64x² - 30x + 6.[/tex]

2)To check the result, differentiate the indefinite integral using the sum rule and product rule: [tex](7sin(x) + 5cos(x))' = 7cos(x) - 5sin(x).[/tex]

In summary, to find the indefinite integral of a function, use the power rule and product rule to integrate the individual terms. To check the result, differentiate the indefinite integral.

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Answer:

[tex]\large\boxed{\textsf{The Original Number is 13.}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to identify the original number.}[/tex]

[tex]\textsf{Many changes have happened to the original number that it is 9.}[/tex]

[tex]\textsf{We can identify the original number by using the Inverse Operation.}[/tex]

[tex]\large\underline{\textsf{What are Inverse Operations?}}[/tex]

[tex]\textsf{Inverse Operations are like normal operations, but they are reverse.}[/tex]

[tex]\mathtt{(+ , -, \times, \div)}[/tex]

[tex]\large\underline{\textsf{For Example;}}[/tex]

[tex]\textsf{The Inverse Operation of Addition is Subtraction.}[/tex]

[tex]\textsf{The Inverse Operation of Division is Multiplication.}[/tex]

[tex]\textsf{Basically, we will work backwards to find the original number.}[/tex]

[tex]\large\underline{\textsf{Solve;}}[/tex]

[tex]\textsf{Let's start with 9.}[/tex]

[tex]\mathtt{9 \times 5 = 45.} \ \textsf{(The Inverse Operation is Multiplication.)}[/tex]

[tex]\mathtt{45-10=35} \ \textsf{(The Inverse Operation is Subtraction.)}[/tex]

[tex]\mathtt{35 \div 7 = 5} \ \textsf{(The Inverse Operation is Division.)}[/tex]

[tex]\mathtt{5+8=13} \ \textsf{(The Inverse Operation is Addition.)}[/tex]

[tex]\large\boxed{\textsf{The Original Number is 13.}}[/tex]

The equation of the line y = (5/3)x + (10/3) represents the line passing through the point (3, 5) that cuts off the smallest area from the first quadrant.

To find the equation of the line that passes through the point (3, 5) and cuts off the least area from the first quadrant, we need to consider the slope of the line.

Any line passing through the point (3, 5) can be written in a point-slope form as:

y - 5 = m(x - 3)

where m is the slope of the line. We want to find the slope that minimizes the area cut off by the line.

Consider a line passing through the origin with slope m. The area cut off by this line in the first quadrant is given by:

A(m) = (1/2)(3)(m*3) = (9/2)m

Note- that the area cut off by the line passing through (3, 5) with slope m is equal to the area cut off by the line passing through the origin with slope m plus the area of the triangle formed by the point (3, 5), the origin, and the point where the line intersects the y-axis. The y-intercept of the line passing through (3, 5) with slope m is given by:

y - 5 = m(x - 3)

y = mx - 3m + 5

Setting x = 0, we get:

y = -3m + 5

The coordinates of the point where the line intersects the y-axis are (0, -3m + 5), and the area of the triangle is:

(1/2)(3)(|-3m + 5 - 0|) ⇒ (3/2)|-3m + 5|

Therefore, the total area cut off by the line passing through (3, 5) with slope m is:

A(m) = (9/2)m + (3/2)|-3m + 5|

To find the slope that minimizes this expression, we need to consider two cases:

Case 1: -3m + 5 ≥ 0, i.e., m ≤ 5/3

In this case, the expression simplifies to:

A(m) = (9/2)m + (9/2)m - (3/2)(5)

= (9m - (15/2)

This expression is minimized when m = 5/3, which is within the range of possible slopes.

Case 2: -3m + 5 < 0, i.e., m > 5/3

In this case, the expression simplifies to:

A(m) = (9/2)m - (9/2)m + (3/2)(5)

= (15/2)

This expression is minimized when m = 5/3, which is again within the range of possible slopes.

Therefore, the line passing through (3, 5) with slope m = 5/3 cuts off the least area from the first quadrant. The equation of the line is:

y - 5 = (5/3)(x - 3)

Simplifying, we get:

y = (5/3)x + (10/3)

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The actual errors for Exercise 1 can be computed by subtracting the calculated values from the true values of the functions. For example, the actual error for sin(1.1) can be found by subtracting sin(1.1) = 0.8912 from the calculated value of 0.8890. The actual error in this case is 0.0022.

Error bounds for these functions can be found using the error formulas. For the function f(x) = sin x, the error bound can be found using the formula |E| <= M|x-a|, where M is the maximum value of the first derivative of the function, and a is the value of x at which the error is computed. In this case, M = 1 and a = 1.1, so the error bound is |E| <= 1 * |1.1 - 1.1| = 0.  

For the function f(x) = ex - 2x2 + 3x - 1, the error bound can be found using the formula |E| <= M|x-a|2, where M is the maximum value of the second derivative of the function, and a is the value of x at which the error is computed. In this case, M = e and a = 1.1, so the error bound is |E| <= e * |1.1 - 1.1|2 = 0.

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By using the formula of a geometric sequence, the 14th term will be 125/1024

We have, the Fourth term of geometric sequence T_4= 125 and T_10 = 125/64

Let's find the first term and common ratio of the geometric sequence.

Using the formula of the nth term of a geometric sequence, we get,

T_4 = a * r^3 = 125 ....(1)

and,

T_10 = a * r^9 = 125/64 ...(2)

On dividing eq. (2) by eq. (1), we get,

(r^6) = (125/64) / 125 ⇒ 1/64

Taking the sixth root of both sides, we get:

r = (1/64)^(1/6)

r = 1/2

Now that we know the common ratio, we can use the equation for the nth term of a geometric sequence:

T_n = a * r^(n-1)

To find the 14th term, we substitute n=14 and solve:

T_14 = a * (1/2)^(14-1) ⇒ a * (1/2)^13

We don't know the value of a yet, but we can use the fact that the 4th term is 125 to solve for it:

a * r^3 = 125

a * (1/2)^3 = 125

a = 125 * 2^3

a = 1000

Substituting this value for a, we get:

T_14 = 1000 * (1/2)^13

T_14 = 1000 * 1/8192

T_14= 125/1024

Therefore, the 14th term of the geometric sequence is 125/1024.

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No, the statements are not equivalent as the statement "a > b" simply means that "a" is greater than "b" and there exists a number "c" such that when added to "b", it equals "a". This does not necessarily mean that "a" is greater than "b".

The statements "a>b" and "there is the number c, so that a=b+c" are not equivalent. The statement "a>b" simply means that "a" is greater than "b," while the statement "there is the number c, so that a=b+c" means that "a" can be expressed as the sum of "b" and another number "c."

These statements are not equivalent because even if "a" is greater than "b," it may not be possible to express "a" as the sum of "b" and another number "c."

Additionally, even if "a" can be expressed as the sum of "b" and another number "c," it may not necessarily be true that "a" is greater than "b."

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Answer:

B. Isosceles right triangle

C. Quadrilateral

D. Parallelogram

Step-by-step explanation:

Answer:

The geometric figures that could describe how the garden might look are B. Isosceles right triangle and C. Quadrilateral.

Answer:

Since 1 000 paperclips = 1 kilogram

5,600 paperclips x 0.001 kilograms per paperclip = 5.6 kilograms

Answer: 5.6 kilograms

Step-by-step explanation:

5,600 grams = 5.6 kilograms

Answer:

2=x y=-6

Step-by-step explanation:

x+y=-4

x-y=8

(-)(-)=+

(-)(+)=-

2+-6=-4

2--6=8

hwuw8w92kywjaia01owuqta56a62652629292092827wgbs

Using multiple principle, The different area codes that are possible are 144

The multiplication principle is a counting principle that states the total number of possible outcomes of a series of independent events is equal to the product of the number of outcomes for each event. It is used in probability theory and other areas of mathematics and science.

According to the question

Under the original plan for area codes in 1945, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third digit could be any number except 0.

Therefore, the number of possible area codes can be calculated as follows:

For the first digit, there are 8 possibilities (2, 3, 4, 5, 6, 7, 8, or 9). Here 9 is inclusive.

For the second digit, there are 2 possibilities (0 or 1).

For the third digit, there are 9 possibilities (any digit except 0).

Using the multiplication principle, the total number of possible area codes can be determined by multiplying the number of choices for each digit:

8 × 2 × 9 = 144

So there are 144 different area codes possible according to this plan.

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Answer:

To model -2/3 + (-1 1/6) on a number line, we first need to convert the mixed number (-1 1/6) into an improper fraction.

-1 1/6 = -7/6

Now we can add -2/3 and -7/6 by finding a common denominator. The smallest common multiple of 3 and 6 is 6, so we'll convert both fractions to have a denominator of 6.

-2/3 = -4/6

-7/6 = -7/6

Now we can add them:

-4/6 + (-7/6) = -11/6

So -2/3 + (-1 1/6) = -11/6.

To model this on a number line, we would start at zero and move to the left by 1 and 2/3 units (since -2/3 is less than 1 whole unit to the left of zero). Then, we would move an additional 1 and 1/6 units to the left (since -1 1/6 is one whole unit and 1/6 of another unit to the left of zero). This would bring us to the point represented by -11/6 on the number line.

Step-by-step explanation:

Answer:

(-5/6)

Step-by-step explanation:

Start by marking the point 0 on the number line.

To represent -2/3, you'll need to move 2/3 units to the left of 0, since this value is negative. One way to do this is to divide the space between 0 and -1 into three equal parts and move two of them to the left. This brings you to the point -2/3.

To represent (-1 1/6), you'll first need to represent -1, which is one unit to the left of 0. You'll then need to add 1/6 units to the left of -1. One way to do this is to divide the space between -1 and -2 into six equal parts and move one of them to the left. This brings you to the point (-1 1/6).

To find the sum of -2/3 and (-1 1/6), you'll need to add the distances we moved in steps 2 and 3. The total distance moved to the left is 2/3 + 1/6 = 5/6. Starting from 0, we move 5/6 units to the left to arrive at the final point, which is (-5/6).

Answer: 52.173913%

Step-by-step explanation:

The answer is 52.173913%, you can round it <3

I hope this helped! <333

Answer: I got 52.17391304%

Step-by-step explanation:

If you are looking for a rate, you should use R=P/B where R is the % B is the base (23000) and P is the portion (12000) So, it would be set up like this R = 12000/23000



After I got my answer I double checked it by using the formula B = 52.17% x 12000 to see if I got 23000, but instead I got 11,999.10 because I rounded down the long percentage number from 52.17391304% to 52.17%, which would usually be ok and work just fine, but using 52.17% I got 11,999.10. Which is still extremely close to 12000, but not exact. So even though rounding down is usually a requirement in homework situations, this one seems to be an exact percentage with all the digits. Let me know if I can help further. I'm in accounting and I'm curious about this one and would like to know how it goes.

Answer:

( 9, 1/7 )

Step-by-step explanation:

I/9x- y=6/7                  equation    (1)

1/18x-2y=3/14              equation    (2)

9x(1)-18x(2)              multiply equation (1 ) by 9 and (2 ) by 18 and subtract  

                                                                                                      them                                                                                                                

we get y=1/7

then put the value of y=1/7 in ( 1 ) or (2)

1/9x-1/7=6/7 ⇒x=9

therefore  the right answer is ( 9, 1/7 )

Answer:

(-1,1)

(0,2

(1,3)

(2,4)

Answer:

Step-by-step explanation:

Let's say Jeff ordered x tacos and y burritos.

From the problem, we know that:

- The calorie count of one taco is 150 calories

- The calorie count of one burrito is 380 calories

- Jeff ordered a total of 11 items

- Jeff consumed a total of 3030 calories

We can use the information given to form a system of equations:

x + y = 11 (Jeff ordered a total of 11 items)

150x + 380y = 3030 (Jeff consumed a total of 3030 calories)

To solve this system, we can use substitution.

Rearranging the first equation, we get:

x = 11 - y

Substituting this value of x into the second equation, we get:

150(11 - y) + 380y = 3030

Expanding and simplifying:

1650 - 150y + 380y = 3030

230y = 1380

y = 6

So Jeff ordered 6 burritos.

Substituting this value of y back into the first equation, we get:

x + 6 = 11

x = 5

So Jeff ordered 5 tacos.

Therefore, Jeff ordered 5 tacos and 6 burritos.

Answer:

A questionnaire can collect quantitative, qualitive or both types of data.

Step-by-step explanation:

Answer:

Categorical data

Step-by-step explanation:

Data that relates to certain categories e.g males, females or any types of car

The system of linear equations to find the cost of an apple and the cost of an apricot is as follows:

9x + 6y = 8.50

3x + 2y = 7.40

Jeremiah bought 9 apples and 6 apricots for $8.50 yesterday. He bought 3 apples and 2 apricots for $7.40 today.

The system of linear equation to find the cost of an apple and the cost of an apricot can be represented as follows:

Therefore, system of equation can be solved using different method such as elimination method, substitution method and graphical method.

The linear equation is as follows:

9x + 6y = 8.50

3x + 2y = 7.40

where

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Answer:

Every day, the chef consumes 5443.20 grams of butter, which is calculated using the conversion coefficients 16 oz/1 pound and 28.35 grams/1oz.

A restaurant chef uses 12 pounds of butter every day, as specified in the question.

We need to figure out how much butter the chef uses each day in grams.

Applying the conversion parameters provided, 16 oz/1 lb and 28.35 grams/1oz

According to the data provided, the needed solution is as follows: 12 lb 16 oz/1 lb 28.35 g/1 ounce 5443.20 grams

As a result, the chef consumes 5443.20 grams of butter every day.

Step-by-step explanation:

Brainliest pls

The ratios of the radius of the two circles from their circumferences is 19.09 to 25.11

We have the following readings from the question:

Circumference A = 19.09

Circumference B = 25.11

The formula for the circumference of a circle is:

C = 2πr

where C is the circumference and r is the radius of the circle.

We are given the circumferences of circle A and circle B. Let's use the formula to write two equations:

Circumference A = 2πrA = 19.09

Circumference B = 2πrB = 25.11

We can divide these two equations to eliminate π and get the ratio of the radii:

(2πrA) / (2πrB) = 19.09 / 25.11

Simplifying, we get:

rA / rB = 19.09 / 25.11

So, the ratio is 19.09 to 25.11

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