Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. ||u||= 4, θu =6 , ||v|| = 1, θv = 4

Answers

Answer 1

The component of u+v along the length of u and v  is (6.3, 0.66).

A vector is quantity which has both magnitude and direction. A magnitude is the length of the vector.

Here in the question u and v makes angle with the positive x axis.

||u||= 4 and θu =6  and  ||v|| = 1, θv = 4

To determine the component of u = |u||(cosu, sinu)

Component of u = 6(0.9, 0.1) = (5.4, 0.6)

Component of v = ||v||(cosv sinv)

Component of v  along the x axis = 1(0.9, 0.06 ) = (0.9, 0.06)

In the question value of u+v needs to be find.

Value of u + v = (5.4 + 0.9 , 0.6+ 0.06)

Value of the component form of u + v = (6.3, 0.66).

Hence, the component of u+v along the length of u and v is (6.3, 0.66).

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Related Questions

match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. enter the appropriate letter (a,b,c,d, or e) in each blank.

Answers

Trigonometric expressions are mathematical functions that use the ratios of the sides of a right triangle to calculate angles. Non-trigonometric functions are mathematical functions that do not use the sides of a triangle to calculate angles, but rather use other mathematical operations such as exponentials, logarithms, and polynomials.

Trigonometric expressions are mathematical functions that use the ratios of the sides of a right triangle to calculate angles. These functions are often used in applications such as navigation and surveying, as they allow us to calculate angles and distances in the real world. Examples of trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant. Non-trigonometric functions are mathematical functions that do not use the sides of a triangle to calculate angles, but rather use other mathematical operations such as exponentials, logarithms, and polynomials. These functions are often used in engineering and scientific applications, as they allow for more precise calculations than trigonometric functions. Examples of non-trigonometric functions include polynomial functions, exponential functions, logarithmic functions, and hyperbolic functions. In each case, the function being used must be appropriate for the type of problem being solved. By understanding the differences between trigonometric and non-trigonometric functions, it is possible to choose the right function for the right problem.

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What is 11 divided by 2.5 and can you show the work

Answers

If you divide 11 by 2.5 you will end up with 4.4. Answer = 4.4

42. HOW DO YOU SEE IT?
Write an expression in rational
exponent form that represents
the side length of the square.
Area:
x in.²

Answers

The exponent form that represents the side length of the square is √x

How to determine the side length of the square.

From the question, we have the following parameters that can be used in our computation:

Area = x in.²

The area of a square is calculated as

Area = Side length²

substitute the known values in the above equation, so, we have the following representation

Side length² = x

Take the square root of both sides

Side length = √x

Hence, the expression is √x

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find the domain of each expression
1. y ^2+1)/(y^2-2y),
2. 25(y-9),
3. 32/y - (y+1)/Y+7)

Answers

The domain of each expression is given as follows:

1. All real values except y = 0 and y = 2.

2. All real values.

3. All real values except y = 0 and y = -7.

How to obtain the domain of the expressions?

The domain of an expression is composed by the set of all the possible input values that the expression can assume.

For an expression that is a fraction, the denominator cannot be zero, hence:

Item 1: y² - 2y = 0 -> y(y - 2) = 0, hence the domain is all real values except y = 0 and y = 2.Item 3: y = 0 and y + 7 = 0 -> y = -7, hence the domain is all real values except y = 0 and y = -7.

For item 2, the function is a multiplication, which has no restrictions on the domain, and thus the domain is composed by all real values.

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Sasha ran at a constant speed of 3.8 meters per 1/2 hour.Then walked 1.3 meters per second for 1/2 hour. How far did she run and walk in 60 minutes.

Answers

Answer: 2343.8 meters

Step-by-step explanation: If 3.8 meters is walked in 1/2 hours then another 4680 meters per hour is ran for a half hour is 2340 meters add those 2 together

A 20.00 g milk chocolate bar is found to contain 12.00 g of sugar.
A) How many milligrams of sugar does the milk chocolate bar contain?
B) What will be the amount of sugar in milligrams if the size of the milk chocolate bar is reduced from 20.00 g to 3.000 g?

Answers

The amount of sugar present in the milk chocolate bar is, 12000   milligrams.

Given,

Amount of milk chocolate bar = 20.00 grams

Amount of sugar = 12.00 grams

The conversion used for grams to milligram is :

1 gram = 1000 milligram

As per question we are given that, 12 grams of sugar

As, 1 gram of sugar = 1000 milligram of sugar

So, 12.00 grams of sugar = (12/1) x 1000

                                          = 12000 milligrams of sugar

Therefore, the amount of sugar present in the milk chocolate bar is, 12000 milligrams.

(b) THe reduced size is 3 g

3 g bar has 12x3/20 g of sugar

i.e 7.2 g

hence , he amount of sugar in milligrams if the size of the milk chocolate bar is reduced from 20.00 g to 3.000 g is 7200mg.

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Determine the number of lines of symmetry for the figure.
The figure has ✓line(s) of symmetry.
Select the angles of rotation, if any, that map the figure onto itself.
O 30°
O 90°
☐ 180°
dh
O 45°
O 120°

Answers

Answer:

The figure has 4 lines of symmetry.Rotating through angles 90° or 180° will map the figure to itself.

Step-by-step explanation:

You want the number of lines of symmetry of the 4-leaved rose shown in the figure.

Lines of symmetry

The figure is symmetrical about a line through opposite petals. It is also symmetrical about a line through the space between petals. The attachment shows the 4 lines of symmetry.

Rotational symmetry

The petals have the same geometry, but are rotated 90° from each other. That means any rotation that is a multiple of 90° will map the figure to itself. (180° is a multiple of 90°)

The figure maps to itself when rotated 90° or 180°.

Can someone help me?

Answers

Answer: Maybe if you post the question

Step-by-step explanation:

Let f(x)= x^2 + 7 and g(x)= x^2+2. Find the following functions, and simplify each as much as possible: your final answer should be a polynomial with only one term in each power of x. (a) f(x) + g(x) = (b) f(x) - g(x) = (c) f(x).g(x)= (c) f(x)/g(x)= (For part (d), you need not simplify as you did in (a)-(c).)

Answers

For function f(x)= x² + 7 and g(x)= x² + 2,

a) f(x) + g(x) = 2x² + 9

b) f(x) - g(x) = 5

c) f(x) . g(x) = x⁴ + 9x² + 14

d) [tex]\frac{f(x)}{g(x)} =\frac{x^2 + 7 }{ x^2+2}[/tex]

Consider given functions  f(x)= x² + 7 and g(x)= x² + 2

a) To find addition of functions: f(x) + g(x)

f(x) + g(x)

= (x² + 7) + (x² + 2)

= x² + x² + 7 + 2             .............(associative property of addition)

= 2x² + 9                       ................(combine like terms)

b) to find subtraction of functions : f(x) - g(x)

f(x) - g(x)

= ( x² + 7 ) - (x² + 2)

= 7 - 2

= 5

c) To find multiplication of functions: f(x).g(x)

f(x) . g(x)

= ( x² + 7 ).(x² + 2)

= x² (x² + 2) + 7(x² + 2)

= x⁴ + 2x² + 7x² + 14

= x⁴ + 9x² + 14              ................(combine like terms)

d) To find division of functions: f(x)/g(x)

[tex]\frac{f(x)}{g(x)}\\\\ =\frac{x^2 + 7 }{ x^2+2}[/tex]

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HELP!!!!!
Make a table for each equation
y=x-2

Chart:
x -2 -1 0 1 2
y ? ? ? ? ?

I basically just need he y row

Answers

Put the x values into the equation to find the y values.
-4, -3, -2, -1, 0
hope this helps, good luck
When we take a look at the problem and subtract 2 from -2 we get -4 we continue this process with all numbers. We can also simplify the equation to x-2 and plug it in to a system like Desmos (see photo)

Find the slope between the points:
(1,7) and (-5, -11)

Answers

Slope between the points (1,7) and (-5, -11) is m = 18/6 as a decimal m = 3

What is slope?

The slope of a hill describes its steepness. The steepness of a line follows a similar pattern. The rise between two points, which is the vertical change, is divided by the run, which is the horizontal change between those same two points, to determine the slope.

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

step1:

calculate the slope m using the slope formula

slope m=y2-y1/x2-x1

with (x₁, y₁ )=(1,-5) and (x₂, y₂ ) =(-5,-11)

step2:

m=-11-7/-5-1=-18/-6=18/6

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Please help!! It’s due in 10 minutes!!! Also please explain how you got the answer!

Answers

Answer: 22 vaccines

Step-by-step explanation:

first, reduce the amount that doctors leave out from the total amount of a vial

6 - 0.5 = 5.5

next divide 5.5 by the amount that is given to each person, which will give you the number of vaccinations the doctor can give

5.5/ 0.25 = 22 vaccines

Laurie, who is 5 feet tall, notices that she casts an 8-foot shadow. If Laurie is standing 90 feet away from the base of a building, what is the height of the building? Show your work.

Answers

The height of the building that cast the shadow 98 feet will be 61.25 feet.

What is the triangle?

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

The ratio of the matching sides will remain constant if two triangles are comparable to one another.

Laurie, who is 5 feet tall, notices that she casts an 8-foot shadow. If Laurie is standing 90 feet away from the base of a building.

Let 'h' be the height of the building. Then the equation is given as,

h / (90 + 8) = 5 / 8

h / 98 = 5 / 8

h = 98 x 5 / 8

h = 12.25 x 5

h = 61.25 feet

The height of the building that cast the shadow 98 feet will be 61.25 feet.

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Allison can sew three scarves in 75 minutes. How many scarves can she sew in four hours?

Answers

The number of scarves in 4 hours is 9.6

How to determine the number of scarves in 4 hours

From the question, we have the following parameters that can be used in our computation:

Rate = 3 scarves in 75 minutes

Time = 4 hours

The number of scarves is then calculated as

Scarves = Rate * Time

substitute the known values in the above equation, so, we have the following representation

Scarves = (3/75 minutes) * 4 hours

Evaluate the product

Scarves = 9.6

Hence, the number of scarves is 9.6

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because of the enormity of the viewing audience, firms that advertise during the super bowl create special commercials that tend to be quite entertaining. thirty-second commercials cost over $5 million for the 2017 game. a random sample of people who watched the game were asked how many commercials they watch in their entirety. Do these data allow us to inter that the mean number of commercials watch is greater than 15?

Answers

No, these data do not allow us to infer that the mean number of commercials watched is greater than 15.

To make such an inference, we would need to know the population mean number of commercials watched, along with the population standard deviation. We can use the sample data to calculate the sample mean and sample standard deviation. The sample mean (X) is the sum of the data values divided by the number of observations. The sample standard deviation (s) is calculated using the equation s = √(sum of (x-X)² / (n-1)) , where n is the number of observations. With the sample mean and standard deviation we can then use a t-test to determine if the population mean is significantly greater than 15.

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Transistors are used to build computer chips. The size of the earliest transistors was about 1 x 10* micrometers. Today, transistors can measure smaller than 1 × 10-2 micrometer. The size of the earliest transistors is about how many times as great as the size of transistors today?

Answers

IIn a case whereby Transistors are used to build computer chips and the size of the earliest transistors was about 1 x 10* micrometers and today, transistors can measure smaller than 1 × 10-2 micrometer the numbert of times the old is as great as the size of transistors today is 1 × 10^6.

How can the size of transistors be known?

The concept that will be use here is division. One of the four fundamental mathematical operations, along with addition, subtraction, and multiplication, is division. Division is the process of dividing a larger group into smaller groups so that each group contains an equal number of things. It is a mathematical operation used for equal distribution and equal grouping. In this post, let's study more about the division operation in math.

We were told that the side of the earliest form of transistors for computer was 1 × 10⁴.

In another sentence, we were told that the size of the transistor today  is expressed as  1 × 10^-2

We can then know the difference in size will  which will be

= 1 × 10^4 / 1 × 10^-2

= 1 × 10^6

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complete question:

Transistors are used to build computer chips. The size of the earliest transistors was about 1 x 10^4 micrometers. Today, transistors can measure smaller than 1 × 10-2 micrometer. The size of the earliest transistors is about how many times as great as the size of transistors today?

Find the product of the binomial factors using the appropriate special product (difference of two squares, square of a binomial sum, or square of a binomial difference).

(x+8)2

Answers

The product of the binomial factors is x² + 16x + 64

How to determine the product of the binomial factors

from the question, we have the following parameters that can be used in our computation:

(x+8)2

Express properly

So, we have

(x+8)²

Using the square of a binomial sum, we have

(x+8)² = x² + 2 * x * 8 + 8²

Evaluate

(x+8)² = x² + 16x + 64

Hence, the solution is x² + 16x + 64

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is there a value of a that makes the statement |a| = - 2 true ? Explain your reasoning.

Answers

Answer:

No, there is no value of a that makes the statement |a| = -2 true.

Step-by-step explanation:

The absolute value of a number is the distance of that number from zero on the number line. It is always non-negative, so it can't be -2. The absolute value of a number is denoted by two vertical lines on either side of the number like this |a|.

For example, the absolute value of -5 is 5 and the absolute value of 5 is 5. In both cases, the distance from zero is 5.

In the statement |a| = -2, the absolute value of a is -2, which is impossible as the absolute value can't be negative. Therefore, there is no value of a that makes the statement true.

Answer:

  no

Step-by-step explanation:

You want to know if there's any value of the variable 'a' that would make its absolute value be -2.

Absolute value

The absolute value of a number is always positive. There is no such thing as a number whose absolute value is negative, not -2 or any other negative number.

  such a value of 'a' Does Not Exist (DNE)

E.
Carry out the following operations to the proper number of significant figures:
1. 212.45 +5.61-37.9
2. 89.001+2.50-91.50
3. 400+125= 500
4. 73.35-3.65-69.70
5. (81.7-23.456)+(78.44+2.72) =

Answers

By adding (81.7-23.456)+(78.44+2.72), we get 139.404

How does one determine the number of significant figures?

Use the three rules below to calculate the number of significant figures in a number:

Non-zero digits are always meaningful.

Any zero between the first and second significant digits is significant.

Only the last zero or trailing zeros in the decimal section are important.

Numbers Have Rules INCLUDING a Decimal Point

START COUNTING FOR SIGNIFICANT FIGURES. On the very first non-zero digit.

STOP COUNTING FOR SIGNIFICANT FIGURES.

Non-zero numbers are ALWAYS meaningful.

After the first non-zero digit, any zero is still relevant. The zeroes preceding the first non-zero digit are unimportant.

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Use the graphs of f and g to evaluate g(f(2))
I have added the photo of the graphs.
When I answered it the first two times I put -2 and then for the second time I put 4 and I got both of them wrong

Answers

Answer:

1

Step-by-step explanation:

We have g(f(2))

Working from the inside out, we start with f(2)

So we go to the f(x) graph and find where X is 2. We get to the point (2, -2). f(2) is pretty much asking for the Y value at 2. On this graph the y value is -2

Now that we know f(2) is -2, we can plug that into g(f(2)), giving us g(-2)

Now we can go to the other graph, g(x) and find where X is -2.

That's where we find the point (-2,1)

Like last time, g(2) is asking for that Y value at that point of -2. On this graph, it's 1

Now we can conclude g(f(2)) = 1

Using the graph of g, we can see that g(4) is approximately 3. So, g(f(2)) = 3.

What is the graph?

A graph is a visual representation of data or information, often used in mathematics and science to depict relationships between variables. Graphs can take many forms, including line graphs, bar graphs, scatter plots, and pie charts.

To evaluate g(f(2)), we need to first find the value of f(2) using the graph of f. From the graph, we can see that f(2) is approximately 4. So, g(f(2)) = g(4).

Then, using the graph of g, we can see that g(4) is approximately 3. So, g(f(2)) = 3.

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Figure out the polynomial being multiplied on top and then fill in the missing boxes

Answers

In order to factor your quadratic polynomials, the box method, as its name suggests, uses a box. You can divide your quadratic into its factors by using the box, which is essentially a square divided into four equal parts.

What is box method ?Be aware that not all quadratics can be factored. Some just lack relevant components. But if a quadratic polynomial does have components, you can utilize the box approach to simplify the operation.Drawing a square divided into four equal sections or a two-by-two grid is the first step. You then enter your first squared term into the upper left box and your final term into the lower right box. the product of your final term and first term.These new term's contributing components are found. You're seeking for a specific combination of elements that, when added together, give you your middle word. Write these elements in the two remaining empty boxes once you've identified this group of components. Your answer will remain the same no matter which empty box each term is entered into. Find the largest common factor for each row and column to determine the quadratic factors. The factors of your polynomial will then be determined by finding the biggest common factors from your columns and rows, respectively.

Employing the Box Method

For our first quadratic, 6x2 - x - 12, let's utilize the box approach to solve it.

The first step is to create a two by two grid, as shown here:

using a box

Following that, we place the first term in the upper left box and the last term in the lower right box.

The Complete Question.

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pair of equation are parallel, perpendicular, or neither y=-7/8x-1, 32x-28y=-36

Answers

The slope of the given lines are representing that these lines are perpendicular.

What is the 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. Sometimes, the aforementioned is referred to as a "linear equation of two variables," with y and x serving as the variables.

Given, the pair of the equation are parallel, perpendicular, or neither y=-7/8x-1, 32x-28y=-36

Slope intercept form of both equations of a line

1)  y = -7/8x - 1

it is already in a slope intercept form

2) 32x - 28y = -36

-28y = -32x -36

y = 8/7x + 9/7

Since their slope is inverse and had a negative constant.

The slope of a perpendicular line from the slope of a given line is obtained by taking the negative reciprocal of the slope of the given line. If the slope of the given line is m1 and the slope of a perpendicular line is m2 then we have m2 = -1/m1.

Therefore, The slope of the given lines are representing that these lines are perpendicular.

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The vertices of triangle xyz are x=(-2,6), y=(4,10), and z=(14,6). Find the coordinates of the centroid of triangle xyz

Answers

The centroid of the given triangle is =(5.3 , 7.3)

What is Centroid of triangle?

The centroid is the term for the object's geometric centre. We apply the centroid formula to find the triangle's centroid's coordinates. The intersection of a triangle's three medians yields the centroid, or centre, of the triangle. All of the medians are divided by the centroid of a triangle in a 2:1 ratio.

The centroid formula of a given triangle can be expressed as,

C = ((x1+x2+x3)/3 , (y1+y2+y3)3/3)

where,

C denotes the centroid of a triangle

x1,x2,x3  are the x-coordinates of the 3 vertices.

y1,y2,y3  are the y-coordinates of the 3 vertices

To find: Centroid of a triangle.

We are taking the Cordinates of the traiangle XYZ

as x1,y1,x1,y2,x3,andy3

so,Given parameters are,

(x1,y1)=(-2,6)

(x2,y2)=(4,10)

(x3,y3)=(14,6)

Using centroid formula,

The centroid of a triangle = ,

 =-2+4+14/3  6+10+6/3

 =5.3 7.3

The centroid of a triangle is ==(5.3 and 7.3)

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Merle tiene una oferta de un emisor de tarjeta de crédito para una APR del 0
% durante los primeros 60 días y una APR del 23,19 % después, con
capitalización diaria. ¿Qué tasa de interés efectiva se ofrece Merle?
UNA 26.00%

Answers

La tarjeta de crédito tiene una tasa efectiva de interés anual de 21,375 %.

¿Cómo determinar la tasa de interés efectiva anual?

En este problema tenemos que Merle obtiene una tarjeta de crédito bajo la siguientes condiciones: 0 % E.A. para los primeros 60 días, 23,19 % E.A. para los 365 restantes.

Debemos determinar la tasa de interés efectiva anual equivalente, esto es posiblemente mediante una aplicación consecutiva del concepto de interés, descrito abajo:

I = (1 + r / 36500)ⁿ

Donde:

r - Tasa de interés anual.n - Número de días.I - Ganancia por interés.

La situación es descrita por la siguiente expresión matemática:

1 + R / 100 = (1 + 0 / 36500)⁶⁰ + (1 + 23,19 / 36500)³⁰⁵

Donde R es la tasa efectiva de interés anual.

Finalmente, despejamos la variable correspondiente:

R = 21,375

La tasa efectiva de interés anual a sufragar por Merle por el uso de tarjeta de crédito es 21,375 %.

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Time to walk 3 1/3 miles on Saturday and 2 4/5 mile on Sunday how much farther does Tonya walk on Saturday then on Sunday

Answers

The distance that Tonya walk on Saturday more than on Sunday would be = 8/15 miles

What is distance?

Distance is defined as the quantity that describes the space that exists between two locations in a field.

The number of miles Tonya covered on Saturday = 3 ⅓ miles

The number of miles that Tonya covered on Sunday = 2 ⅘

Therefore the distance that Tonya walk on Saturday more than on Sunday would be the difference between the distance covered in Saturday and Sunday.

That is; 3⅓ - 2⅘ ( change to single fraction form)

= 10/3 - 14/5

= 50-42/15

= 8/15

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Translate this polygon six units to the left and two units upward:

Answers

The image that gives the translation of the polygon six units to the left and two units upward is given as follows:

The fourth graph.

(which is the third graph if we consider that the first is the original image).

What is a translation?

A translation is a movement to a graph or figure in which only the position of the figure changes, either left, right, up or down, keeping the inclination, orientation and congruence.

The translations are represented as follows:

Left a units: x -> x - a.Right a units: x -> x + a.Up a units: y -> y + a.Down a units: y -> y - a.

For the translation 5 units left, we have that:

The bottom segment of x = 1 to x = 4 will assume coordinates of x = -5 to x = -2.

For the translation 2 units up, we have that:

The vertical segment from y = 1 to y = 3 will assume coordinates from y = 3 to y = 5.

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Two fractions with the same denominator that have numerators 5 and 7 what would the denominator be?

Answers

The denominator could be any real value

How to determine the possible denominator

From the question, we have the following parameters that can be used in our computation:

Fraction = same denominator

Such that the numerators are 5 and 7

This means that the fractions can be represented as

Fractions = 5/x and 7/x

There is no restriction on the possible value of x

hence, the denominator is any value

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A spinner with repeated colors numbered from 1 to 8 is shown. Sections 1 and 8 are purple. Sections 2 and 3 are yellow. Sections 4, 5, and 6 are blue. Section 7 is red.

Spinner divided evenly into eight sections with three colored blue, one red, two purple, and two yellow.

Determine the theoretical probability of the spinner not landing on yellow, P(not yellow).

0.325
0.625
0.750
0.875

Answers

Answer:

0.750

Step-by-step explanation:

Yellow sections: section 2 and section 3

Number of yellow sections: 2

Number of non yellow sections: 6

Total sections: 8

p(not yellow) = 6/8 = 0.75

Answer:

0.750

Step-by-step explanation:

you take a quarter and then you divide and think  

at a certain high school, the distribution of backpack weight is approximately normal with mean 19.7 pounds and standard deviation 3.1 pounds. a random sample of 5 backpacks will be selected, and the weight, in pounds, of each backpack will be recorded.

Answers

A sample of 5 backpacks will be chosen from a population of backpacks whose weights are normally distributed with mean 19.7 pounds with a 3.1-pound standard deviation. Each backpack's weight will be noted.

A sample of 5 backpacks will be chosen from a population of backpacks whose weights are normally distributed with mean 19.7 pounds with a 3.1-pound standard deviation. Each backpack's weight will be noted.. The weight of each backpack will be recorded in order to calculate the mean and standard deviation of the sample. To calculate the mean, the total weight of the five backpacks will be added together and then divided by the number of backpacks in the sample (5). To calculate the standard deviation, the weight of each backpack will be subtracted from the mean and the resulting difference will be squared. All of the squared differences will then be added together, divided by the number of backpacks in the sample (5), and the square root of this result will be taken. This will give us the standard deviation of the sample. These calculations will provide us with valuable information about the weights of the backpacks in the sample, which can then be compared to the weights of the backpacks in the population.

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row reduce the matrices in exercises 3 and 4 to reduced echelon form. circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns.

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Reduced matrix to its reduced echelon form Original Matrix:

[tex]$$ \begin{bmatrix}1 & -1 & 2 & -1 & 3 \\2 & -2 & 3 & 0 & 4 \\0 & 0 & 0 & 1 & -2\end{bmatrix} $$[/tex]

Pivot Positions (Original): (1,1), (2,2), (3,4)

Pivot Columns (Original): 1, 2, 4

Reduced Echelon Form:

[tex]$$ \begin{bmatrix}1 & 0 & \frac{2}{3} & 0 & \frac{11}{3} \\0 & 1 & \frac{1}{3} & 0 & \frac{2}{3} \\0 & 0 & 0 & 1 & -2\end{bmatrix} $$[/tex]

Pivot Positions (Reduced): (1,1), (2,2), (3,4)

Pivot Columns (Reduced): 1, 2, 4

To reduce this matrix to its reduced echelon form, I used the following series of elementary row operations:

1. Divide row 1 by 1 to get the leading 1 in the first column

2. Subtract two times row 1 from row 2 to get the leading 1 in the second column

3. Subtract three times row 1 from row 3 to get the leading 1 in the fourth column

4. Divide row 2 by 3 to get the coefficient 2/3 in the third column

5. Subtract row 2 from row 1 to get the coefficient 11/3 in the fifth column

6. Subtract row 2 from row 3 to get the coefficient 2/3 in the fifth column

Exercise 4:

Original Matrix:

[tex]$$ \begin{bmatrix}1 & 2 & -3 & 1 & 5 \\2 & 4 & -6 & 2 & 8 \\-1 & -2 & 3 & -1 & -4\end{bmatrix} $$[/tex]

Pivot Positions (Original): (1,1), (2,2), (3,3)

Pivot Columns (Original): 1, 2, 3

Reduced Echelon Form:

[tex]$$ \begin{bmatrix}1 & 0 & 0 & \frac{2}{3} & \frac{11}{3} \\0 & 1 & 0 & \frac{-1}{3} & \frac{1}{3} \\0 & 0 & 1 & \frac{2}{3} & \frac{5}{3}\end{bmatrix} $$[/tex]

Pivot Positions (Reduced): (1,1), (2,2), (3,3)

Pivot Columns (Reduced): 1, 2, 3

To reduce this matrix to its reduced echelon form, I used the following series of elementary row operations:

1. Divide row 1 by 1 to get the leading 1 in the first column

2. Subtract two times row 1 from row 2 to get the leading 1 in the second column

3. Subtract row 1 from row 3 to get the leading 1 in the third column

4. Divide row 2 by 3 to get the coefficient -1/3 in the fourth column

5. Add row 2 to row 1 to get the coefficient 2/3 in the fourth column

6. Divide row 3 by 3 to get the coefficients 2/3 and 5/3 in the fourth and fifth columns respectively

7. Add row 3 to row 1 to get the coefficient 11/3 in the fifth column

8. Add row 3 to row 2 to get the coefficient 1/3 in the fifth column

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