Construct triangle ABC, in which AB = 5 cm, angle BAC = 95° and
angle ABC = 34°.
Measure the length of BC.
Give your answer to 1 d.p.

Let x₁ and x₂ be two independent random variabIes, each with a mean of 10 and a variance of 5.y has a mean of 203 and a variance of 85.

A function, in mathematics, is a reIationship between a set of possibIe inputs and an equaIIy IikeIy set of outputs, where each input is associated to exactIy one outcome. Functions are commonIy represented as equations or graphs, and they are used to modeI many reaI-worId processes in domains such as physics, engineering, and economics.

Function types incIude Iinear, quadratic, trigonometric, and exponentiaI functions, among others. CaIcuIus, a fieId of mathematics that investigates how quantities change over time or space, heaviIy reIies on functions.

given

The foIIowing is the MacIaurin series for the function g(x) = (1+x)e(-x):

g(x) = ∑[n=0 to ∞] ((-1)ⁿ*xⁿ) / n!

We may simpIify and pIug in the first few vaIues of n to determine the first four nonzero terms of this series:

n = 0: ((-1)⁰*x⁰) / 0! = 1

n = 1: ((-1)¹*x¹) / 1! = -x

n = 2: ((-1)²*x²) / 2! = x²/2

n = 3: ((-1)³*x³) / 3! = -x³/6

The MacIaurin series for g(x) therefore has the foIIowing first four nonzero terms:

1 - x + x²/2 - x³/6

Let x₁ and x₂ be two independent random variabIes, each with a mean of 10 and a variance of 5.  y has a mean of 203 and a variance of 85.

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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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The probability of getting an ace and a three is (4/52) × (3/51) = 12/2652 which simplifies to 1/221.

There are 4 aces and 4 threes in a deck of 52 standard cards.

The probability of getting an ace on your first draw is 4/52.

Once you have the ace, there are 51 cards left in the deck, 3 of which are threes.

Therefore, the probability of drawing a three is 3/51.

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The number of 50 cents in the container is 380 fifty cents

Since Thabang save money by putting coins in a money box. The money box has 600 coins that consist of 20 cents and 50 cents

To calculate how many 50 cents pieces are in the container if there are 220 pieces of 20 cents, we proceed as follows.

Let

Since the total number of cents in the container is 600, we have that

x + y = 600

So, making y subject of the formula, we have that

y = 600 - x

Since x = 220

y = 600 - 220

= 380

So, there are 380 fifty cents

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Answer:51

Step-by-step explanation:

To catch the top 1% of all cars on the freeway, the speed of the car needs to be greater than 99 mph. This is because 99 mph is three standard deviations above the mean, which would put it in the top 1% of speeds.

The question asks for the z-scores and probabilities of cars going 60 mph and 86 mph on the freeway. It also asks for the speed at which the highway patrol woman needs to catch the top 1% of cars.

Let xx be the speed for a randomly selected car on the freeway, with a mean of 66 mph and standard deviation of 10 mph.

The z-score for a car that is caught going 60 mph is -2.0000. This means that the car is two standard deviations below the mean. The probability of randomly choosing a car on the freeway and it going below 60 mph is 0.0228.

The z-score for a car that is caught going 86 mph is 2.0000. This means that the car is two standard deviations above the mean. The probability of randomly choosing a car on the freeway and it going above 86 mph is 0.0228.

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Answer:

yes

Step-by-step explanation:

The 41st term of the sequence is 121.

In mathematics, a sequence is a list of numbers or objects that follow a certain pattern or rule. A sequence's terms are typically identified by subscripts, like a1, a2, a3,..., an, where n denotes the number of terms in the sequence.

Sequences can be arithmetic, geometric, or neither, depending on terms follow a static difference, constant ratio, or neither of these series, respectively. Algebra uses geometric sequences to represent exponential development or decay whereas arithmetic sequences are frequently employed to model linear connections.

The given sequence is 2 , 5 , 8 , ...

The common difference is:

d = 5 - 2 = 3

The nth term of a sequence is given as:

an = a1 + (n-1)d

Substituting the value we have:

an = 2 + (n-1)3

an = 3n - 1

a41 = 3(41) - 1 = 122 - 1 = 121

Hence, the 41st term of the sequence is 121.

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Answer:

Step-by-step explanation:

The point that partitions segment AB in a 1:4 ratio is (-1/2, -1).

What is Segment?

In geometry, a segment is a part of a line that has two endpoints. It can be thought of as a portion of a straight line that is bounded by two distinct points, called endpoints. A segment has a length, which is the distance between its endpoints. It is usually denoted by a line segment between its two endpoints, such as AB, where A and B are the endpoints. A segment is different from a line, which extends infinitely in both directions, while a segment has a finite length between its two endpoints.

To find the point that partitions segment AB in a 1:4 ratio, we need to use the midpoint formula to find the coordinates of the point that is one-fourth of the distance from point A to point B. The midpoint formula is:

((x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the segment.

So, let's first find the coordinates of the midpoint of segment AB:

Midpoint = ((-3 + 7)/2, (2 - 10)/2)

= (2, -4)

Now, to find the point that partitions segment AB in a 1:4 ratio, we need to find the coordinates of a point that is one-fourth of the distance from point A to the midpoint. We can use the midpoint formula again, this time using point A and the midpoint:

((x1 + x2)/2, (y1 + y2)/2) = ((-3 + 2)/2, (2 - 4)/2)

= (-1/2, -1)

So, the point that partitions segment AB in a 1:4 ratio is (-1/2, -1).

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Answer:

Clockwise it would be (3,-5)

Step-by-step explanation:

Counterclockwise it would be (-3,-5)

hope this helps!

When the radius of circle 2 is twice the radius of circle 1, the size of circle 2 is larger than circle 1.

In geometry, a triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always 180 degrees. Triangles are one of the most basic and fundamental shapes in geometry and are used in many mathematical and real-world applications, such as in architecture, engineering, and physics. There are different types of triangles based on the length of their sides and the measures of their angles, such as equilateral triangles, isosceles triangles, scalene triangles, acute triangles, obtuse triangles, and right triangles.

Here,

2. This is because the circumference and area of a circle are directly proportional to the radius.

3. To determine if triangles ABC and DEF are similar, we need to check if their corresponding angles are congruent and if their corresponding sides are in proportion. From the diagram, we can see that angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F. This satisfies the angle-angle (AA) similarity criterion. Additionally, we can use the side-side-side (SSS) similarity criterion to determine if the corresponding sides are in proportion. From the diagram, we can see that side AB is parallel to side DE, side AC is parallel to side DF, and side BC is parallel to side EF. Therefore, we can conclude that triangles ABC and DEF are similar.

4. To find the coordinates of D using the coordinates of A, we need to determine the translation from A to D. From the diagram, we can see that A is translated two units to the right and three units down to get to D. Therefore, we can find the coordinates of D by adding two to the x-coordinate of A and subtracting three from the y-coordinate of A. If the coordinates of A are (x1, y1), then the coordinates of D would be (x1 + 2, y1 - 3).

A= (-0.87,0.5)

D=(-0.87 + 2, 0.5 - 3)

D=(1.13,-2.5)

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Answer:

Let's start by defining our variables:

Now, let's write the system of inequalities to represent the constraints:

The company has 260 boards of mahogany, so x ≤ 260.

The company has 320 boards of black walnut, so y ≤ 320.

The company expects to sell at most 380 boards, so x + y ≤ 380.

We cannot sell a negative number of boards, so x ≥ 0 and y ≥ 0.

Graphically, these constraints represent a feasible region in the first quadrant of the xy-plane bounded by the lines x = 260, y = 320, and x + y = 380, as well as the x and y axes.

To maximize profit, we need to write a function that represents the objective. The profit for selling one board of mahogany is $20, and the profit for selling one board of black walnut is $6. Therefore, the total profit P can be calculated as:

To maximize P, we need to find the values of x and y that satisfy the constraints and make P as large as possible. This is an optimization problem that can be solved using linear programming techniques.

The solution to this problem can be found by graphing the feasible region and identifying the corner point that maximizes the objective function P. However, since we cannot draw a graph here, we will use a table of values to find the maximum profit.

Let's consider the corner points of the feasible region:

Corner point (0, 0):

P = 20(0) + 6(0) = 0

Corner point (260, 0):

P = 20(260) + 6(0) = 5200

Corner point (0, 320):

P = 20(0) + 6(320) = 1920

Corner point (100, 280):

P = 20(100) + 6(280) = 3160

Corner point (200, 180):

P = 20(200) + 6(180) = 5520

Corner point (380, 0):

P = 20(380) + 6(0) = 7600

The maximum profit is $7600, which occurs when the company sells 380 boards of wood, all of which are mahogany.

The mean (also called the arithmetic mean or average) is a measure of central tendency that represents the typical or average value of a set of data. The mean is calculated by summing up all the values in the data set and dividing by the number of values.

The mean of x is calculated by the following formula:

mean of x = ∑(x * P(x))

Where, ∑ = Summation operator

            x = Value of random variable  

            P(x) = Probability of the corresponding value of x.

Let's calculate the mean of x using the formula provided above.

mean of x = (-1 × 0.3) + (0 × 0.4) + (1 × 0.3)

                = -0.3 + 0 + 0.3  

                = 0

Therefore, the mean of x is 0.

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Therefore , the solution of the given problem of percentage comes out to be were tuned in to a specific program or show at a given moment.

A number or figure stated as a fraction of 100 is referred to as "a%" in statistics. The versions that begin with "pct," "pct," and "pc" are also uncommon. The common way to indicate it is with the numeral "%," though. Furthermore, there are no indicators and a flat ratio of every single thing to the total number. Percentages are basically integers because they frequently add up to 100.

Here,

The TV ratings, also known as the TV audience share, are a measurement of the proportion of all television-owning households that are watching the same program at the same moment.

Networks and marketers use it as a gauge of a TV show's popularity to decide how successful a program will be and how much to charge for advertising during it.

Companies like Nielsen, which use a sample of homes with televisions to estimate the audience size for a given program or show, are usually in charge of gathering the TV ratings.

TV ratings are expressed as a proportion of all households with televisions that were tuned in to a specific program or show at a given moment.

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Step-by-step explanation:

x is the number.

the equation is

7x + 6 = 3

One characteristic of all exponential functions is that they change by a constant factor at each step, which means that they exhibit exponential growth or decay.

The constant factor by which an exponential function changes is called the base, which is usually denoted by the symbol "b". If b is greater than 1, the function exhibits exponential growth, and if b is between 0 and 1, the function exhibits exponential decay.

For example, the function f(x) = 2^x is an exponential function with a base of 2. At each step, the function increases by a factor of 2. For instance, f(0) = 1, f(1) = 2, f(2) = 4, f(3) = 8, and so on.

On the other hand, the function g(x) = (1/2)^x is an exponential function with a base of 1/2. At each step, the function decreases by a factor of 1/2. For instance, g(0) = 1, g(1) = 1/2, g(2) = 1/4, g(3) = 1/8, and so on.

Therefore, exponential functions exhibit a characteristic change by a constant factor at each step, which leads to either exponential growth or decay.

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The correct order of steps is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

To find the inverse of 55 modulo 89 using the Euclidean algorithm, the steps are:
1. 89 = 1.55 +34
2. 55 = 1.34 +21
3. 34 = 1.21 + 13
4. 21 = 1.13 + 8
5. 13 = 1.8 +5
6. 8 = 1.5 +3
7. 5 = 1.3 +2
8. 3 = 1.2 +1
9. 2 = 1.2
10. The Bézout coefficient of 55 is 34, and it is an inverse of 55 modulo 89.
11. The Bézout coefficients of 55 and 89 are 34 and -21, respectively.
12. The gcd in terms of 55 and 89 is written as 1 = 34-21.89 = 34-21. (55-1.34) = 13.34-21.55 = 13. (21-1.13) - 3.13 = 5.21-8. 13 = 5. (8-1.5) - 1.5 = 2.8-3.5 = 2. (3-1.2) = 2.3-1.5 = 2. (2-1) = 3-1.2 = 3-1. (1-0) = 34-21.89
The correct order of steps is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

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The given lengths of 3 feet, 3 feet, and 7 feet cannot form a triangle because they do not satisfy the Triangle Inequality Theorem, which is the sum of the lengths of any two sides is greater than the length of the third side.

To form a triangle, the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.

Let's apply this theorem to the given lengths of 3 feet, 3 feet, and 7 feet:

The sum of the first two sides is 3 + 3 = 6 feet, which is less than the length of the third side of 7 feet. So, the first two sides cannot form a triangle.

The sum of the first and third sides is 3 + 7 = 10 feet, which is greater than the length of the second side of 3 feet. However, the sum of the second and third sides is 3 + 7 = 10 feet, which is also greater than the length of the first side of 3 feet.

Therefore, neither of the two combinations of sides satisfy the Triangle Inequality Theorem, and so it is impossible to form a triangle with sides of 3 feet, 3 feet, and 7 feet.

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The company should sell their phones for $250 each to get the maximum revenue.

What do you mean by maximum revenue?

Maximum revenue refers to the highest possible amount of income that can be generated from a particular product or service. In the context of the given problem, it means finding the price at which the company can sell its cell phones to earn the highest amount of revenue.

Finding the price at which the company should sell their phones to get the maximum revenue:

We need to find the vertex of the parabolic function [tex]r(x)=x(1,000-2x)[/tex], which represents the revenue as a function of the selling price.

To find the vertex of the function r(x), we need to first rewrite it in standard form by expanding the product:

[tex]r(x) = 1000x - 2x^2[/tex]

Now we can see that the function is a quadratic polynomial in standard form, with [tex]a=-2, b=1000[/tex], and [tex]c=0[/tex]. To find the x-coordinate of the vertex, we can use the formula:

[tex]x = -b / (2a)[/tex]

Substituting the values of a and b, we get:

[tex]x = -1000 / (2\times(-2)) = 250[/tex]

Therefore, the company should sell their phones for $250 each to get the maximum revenue. To find the maximum revenue, we can substitute this value of x into the function r(x):

[tex]r(250) = 250\times(1000-2\times250) = $125,000[/tex]

So the maximum revenue the company can expect to earn is $125,000 if they sell their phones for $250 each.

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Construct a triangle PQR with sides PQ=8cm, PR=5cm, and QR=6cm, then draw a circle passing through P, Q, and R. This circle is called the circumcircle of triangle PQR.

We draw a line segment PQ = 8 cm long. From point P, we draw a line segment PR = 5 cm long at an angle of 60 degrees to PQ. Then, we draw a line segment QR = 6 cm long joining points Q and R to complete the triangle. Next, we use a compass to draw a circle passing through points P, Q, and R. This circle is called the circumcircle or circumscribed circle of the triangle, which is the unique circle that passes through all three vertices of the triangle. The circumcircle has a special property that its center is equidistant from the three vertices of the triangle.

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Answer: x = 6

Step-by-step explanation:

(7x+3)+85+50 = 180

(7x+3)+135 = 180

7x+3 = 180 - 135 = 45

7x = 45-3 = 42

x = 42 / 7 = 6

x = 6

Answer:

  y = -1/3x +6

Step-by-step explanation:

You want the equation of the line through the point (3, 5) and perpendicular to y = 3x +2.

The slope-intercept form of the equation of a line is ...

  y = mx +b

where m is the slope, and b is the y-intercept.

Comparing this to the given equation, we see that m=3 for the given line.

The slopes of perpendicular lines are opposite reciprocals of one another. This means the slope of the line we want is ...

  desired slope = -1/m = -1/3

The slope-intercept equation above can be solved for b to give ...

  b = y -mx

Then the y-intercept for the line we want is ...

  b = 5 -(-1/3)(3) = 5 +1 = 6

The equation of the desired line is y = -1/3x +6.

__

Additional comment

Once you understand how to find the slope of the given line and of the desired line, you can write down the desired equation in point-slope form.

Given slope = 3; perpendicular slope = -1/3

Point-slope equation: y -k = m(x -h) . . . . line through (h, k) with slope m

  y -5 = -1/3(x -3) . . . . . line through (3, 5) with slope -1/3

The only "work" required is to rearrange this equation to whatever form you may want. In standard form it is x +3y = 18.

Answer:

Step-by-step explanation:

(3x+1)^2 = 3x+3

9x^2 +6x +1=3x+3

9x^2+3x-2=0

finally we got a trinomial quadratic equation solve by factorizing

9x^2 -6x+3x-2=0

3x(3x-2)+(3x-2)=0

3x-2 = 0 or 3x+1=0

x= 2/3 or x= -1/3

the variance of X is 9. b) Determine E [2X² - 20X +5]:

Using linearity of expectation, we can find E [2X² - 20X +5] as:

E [2X² - 20X +5] = 2E[X²] - 20E[X] + 5

by the question.

The number of ways to select the first 2 white chips is given by:

Number of ways = (6C2) (5C0) (3C0) = 15

The number of ways to select the third chip as blue given that the first 2 chips were white is given by:

Number of ways = (3C1) = 3

Therefore, the probability of selecting the third chip as blue given that the first 2 chips were white is:

P(The third chip is blue the first 2 were white) = Number of ways / Total number of ways = 3 / 350 = 0.0086 (rounded to 4 decimal places)

c) P(The fourth chip is blue the first 2 were white):

The number of ways to select the first 2 white chips is given by:

Number of ways = (6C2) (5C0) (3C0) = 15

The number of ways to select the third chip as non-white given that the first 2 chips were white is given by:

Number of ways = (8C1) = 8

The number of ways to select the fourth chip as blue given that the first 2 chips were white, and the third chip was non-white is given by:

Number of ways = (3C1) = 3

Therefore, the probability of selecting the fourth chip as blue given that the first 2 chips were white is:

P(The fourth chip is blue the first 2 were white) = Number of ways / Total number of ways = 8*3 / 350 = 0.0686 (rounded to 4 decimal places)

Suppose we have a random variable X such that E[X]= 7 and E[X²] =58.

a) Determine the variance of X:

The variance of X is given by:

Var[X] = E[X²] - (E[X]) ²

Substituting the given values, we get:

Var[X] = 58 - (7) ² = 9

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A fair coin is flipped 40 times. The probability of getting a head when the coin is tossed is 0.5. If the same coin is flipped 40 times, the probability will remain 0.5. That is, there are equal chances of getting heads and tails when a fair coin is flipped.

Based on the confidence interval, if the actual proportion of heads falls within the range of the confidence interval, it can be said that the coin used was fair. If the actual proportion is outside the confidence interval, it may be an indication that the coin was not fair.

The level of confidence is typically 95% or 99%. If a confidence interval is constructed for the proportion of heads based on a sample of 40 flips and the interval includes the expected proportion of 50%, it can be said that the coin used was fair. If the interval does not include 50%, there is evidence that the coin may not be fair.

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For the first equation with y = 3x + 2 and y = -4x + 2, the lines have the same slope, but a different y-intercept. This means that the lines are parallel and they will never intersect. Therefore, the system of equations has no solution.

For the second equation with y = 2x + 7 and y = 2x - 4, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations has one solution.

For the third equation with y = 6x + 3 and y = -x - 4, the lines have a different slope and a different y-intercept. This means that the lines are not parallel and they will intersect at one point. Therefore, the system of equations has one solution.

For the fourth equation with y = 4x + 3 and y = 4x - 1, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations has one solution.

For the fifth equation with y = 3x + 6 and y = 3x + 6, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations

When a data line on a graph slopes down as it goes to the right, it is depicting that the relationship between the variables on the graph is inverse.

An inverse relationship is a kind of correlation between two variables, in which one variable decreases while the other increases, or vice versa. An inverse relationship happens when one variable increases while the other decreases, or when one variable decreases while the other increases.

On a graph, when a data line slopes down as it goes to the right, this is an indication that the relationship between the variables on the graph is inverse. As the values of x increase, the values of y decrease. Therefore, we can conclude that there is an inverse relationship between x and y.

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The answer of the given question based on the the number of students using the cafeteria's healthy lunch line can be found by solving the inequality 1/5s - 51 < 20, the correct answer is (1) 142 , (1) or more.

An equation is  mathematical statement that shows  equality of two expressions, typically separated by  equal sign. Equations are used to express relationships between variables and to solve problems in various fields of mathematics, science, and engineering. In essence, an equation is representation of  real-world scenario or problem in mathematical terms.

The variables in equation can take on different values, and  equation remains true regardless of the values chosen for  variables. Solving  equation means finding  values of the variables that make  equation true.

Equations can be linear, quadratic, polynomial, exponential, logarithmic, or trigonometric, among other types. They are fundamental tool in mathematics and have numerous applications in fields like physics, engineering, economics, and finance.

There are 155 students or more students that use this lunch line.

The correct answer is 1. or more.

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Answer:

if it is perpendicular to eacha other I e 0