use the paste option to paste the content, including formulas, but none of the formatting into a worksheet.

The x- and y-coordinates of the center of mass are; (20, 0)

The center of mass is the location where the sum of the relative position of the masses in a mass distribution is zero. It is the point where force can be applied on the distributed mass without causing a rotation of the system.

The mas of the steel plate = 800 g = 0.8 kg

The base length of the isosceles triangular plate = 20 cm = 0.2 m

The height of the isosceles triangular plate = 30 cm = 0.3 m

Considering a small strip of height, dx and width, I, we get

Area of small strip, dA = I·dx

Density of the plate for a unit thickness, ρ = M/A

Density of the small strip of mass dm = dm/dA

Considering the density of the plate as uniform, we get;

[tex]\displaystyle {\frac{dm}{dA} =\frac{M}{A}[/tex]

Therefore;

[tex]\displaystyle {dm=\frac{M}{A}\times dA}[/tex]

The area of the triangular plate, A = (1/2) × 0.2 m × 0.3 m = 0.03 m²

Mass of the plate, M = 0.8 kg

Therefore;

[tex]\displaystyle {dm=\frac{0.8}{0.03}\times I\cdot dx}[/tex]

The width of the small strip, I, located at a distance x from the vertex of the triangular plate, using similar triangles, indicates;

[tex]\dfrac{I}{20} = \dfrac{x}{30}[/tex]

[tex]I=20\times \dfrac{x}{30} = \dfrac{2}{3} \cdot x[/tex]

Therefore;

[tex]\displaystyle {dm=\frac{0.8}{0.03}\times I\cdot dx} = \frac{0.8}{0.03}\times \dfrac{2}{3} \cdot x\cdot dx}[/tex]

The center of mass, [tex]x_{cm}[/tex], can be obtained with the formula; [tex]\displaystyle {x_{cm} = \dfrac{1}{M} \cdot \int\limits {x} \, dm }[/tex]

Therefore; [tex]\displaystyle {x_{cm} = \dfrac{1}{0.8} \cdot \int\limits {x} \cdot \frac{0.8}{0.03} \cdot \frac{2}{3}\cdot x\, dx }[/tex]

[tex]\displaystyle {x_{cm} = \dfrac{1}{0.8} \cdot \int\limits {x} \cdot \frac{0.8}{0.03} \cdot \frac{2}{3}\cdot x\, dx } = \dfrac{1}{0.8} \times \frac{0.8}{0.03} \times \frac{2}{3}\cdot \int\limits^3_0 {x^2} \, dx[/tex]

[tex]\displaystyle {x_{cm} = \dfrac{1}{0.8} \times \frac{0.8}{0.03} \times \frac{2}{3}\cdot \int\limits^3_0 {x^2} \, dx = \frac{200}{9} \cdot \left[\frac{x^3}{3} \right]^{0.3}_0[/tex]

[tex]\displaystyle {x_{cm} = \frac{200}{9} \cdot \left[\frac{x^3}{3} \right]^{0.3}_0=\frac{200}{9} \times \frac{0.027}{3} =0.2[/tex]

The location of the x-coordinate center of mass, [tex]x_{cm}[/tex] = 0.2 m = 20 cm from the vertex, which is the same location as the x-coordinate of the centroid of the plate of uniform density.

The plate is symmetrical about the x-axis, therefore, the y-coordinate of the center of mass is along the x-axis, which is [tex]y_{cm}[/tex] = 0

The coordinate of the center of mass = (0.2, 0)

Part of the question requires the location of the coordinate of the center of mass of the triangular plate

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Answer:Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect. Perpendicular lines are lines that intersect at a right (90 degrees) angle.

The Length and width of each pen is 6 feet by 8 feet. The 8 feet side is the adjacent side

An equation is an expression that is used to show the relationship between two or more numbers and variables.

Let x and y represent the length and width of the ship pen (or pig pen) since they have same area and y is the adjacent side. Since both pens are adjacent to each other, he has 48 feet of fencing to make the pens, hence:

Perimeter = x + x + y + x + x + y + y

48 = 4x + 3y

4x = 48 - 3y

x = 12 - (3/4)y

Area = (x + x)y

Area (A) = 2xy

substituting, x = 12 - (3/4)y:

A = 2(12 - (3/4)y)y

A = 24y - (3/2)y²

The maximum area is at A' = 0:

A' = 24 - 3y

24 - 3y = 0

3y = 24

y = 8 feet

x = 12 - (3/4)y = 12 - (3/4)(8) = 6 feet

The Length and width of each pen is 6 feet by 8 feet.

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The values are;

⇒ F ∩ H = Ф

⇒ F ∪ H = (- ∞, 7]

What is Coordinates?

A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

Given that;

F and H are defined as;

F = {v | v < 4 }

H = {v | v ≥ 7 }

Now,

Since, F and H are defined as;

F = {v | v < 4 }

H = {v | v ≥ 7 }

Hence, The values are;

F ∩ H = {v | v < 4 } ∩  {v | v ≥ 7 }

         = Ф

And, F ∪ H = {v | v < 4 } ∪  {v | v ≥ 7 }

                 = (- ∞, 7]

Thus, The values are;

⇒ F ∩ H = Ф

⇒ F ∪ H = (- ∞, 7]

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The company was supposed to finish the order in 9 days.

What 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 preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

Here, we have

The company must have produced 60 items daily if it produced 20 more than the daily average of 40 items.

The company had 120 products to make, which should have taken three days (120 items, 40 items/day), but it finished in just two days (120 items, 60 items/day). They completed this order one day earlier than expected.

However, the issue is that they completed 3 days ahead of schedule. Thus, our estimate is 360 items or three times 120 items.

According to this, they should have completed their order in 9 days (360 things, at a rate of 40 items per day), but they did so in 6, at a rate of 60 items per day. Since 6 is three fewer than 9, our assumption of 360 items was accurate.

We've demonstrated that the business should have completed the order in 9 days.

Let the desired number of days be x.

Then, the company finished its order in x - 3 days.

Since the number of items produced is the number of days times items per day, and it doesn't change:

number of items = 40 items/day * x days

= 60 items/day * (x - 3) days

40x = 60(x - 3) = 60x - 180

180 = 20x

x = 9

Hence, the company was supposed to finish the order in 9 days.

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The inverse function of g is g⁻¹(x) = √(x/2 +  1/2)

From the question, we have the following equations

f(x) = x - 2

g(x) = 2x² - 1

h(x) = 2x³ + 4

To determine the inverse function of g, we start by making the function f(x) as an equation

So, we have

y = 2x² - 1

Swap variables x and y

x = 2y² - 1

Add 2 to both sides

2y² = x +  1

Divide by 2

y² = x/2 +  1/2

Square root both sides

This gives

y = √(x/2 +  1/2)

So, we have

g⁻¹(x) = √(x/2 +  1/2)

Here, we have

f(x) = x - 2

h(x) = 2x³ + 4

This gives

f(h(x)) = h(x) - 2

So, we have

f(h(x)) = 2x³ + 4 - 2

Evaluate

f(h(x)) = 2x³ + 2

So, we have

f(h(x)) = 2(-2)³ + 2

Evaluate

f(h(x)) = -14

Here, we have

f(x) = x - 2

g(x) = 2x² - 1

This gives

g(f(x)) = 2(f(x))² - 1

So, we have

g(f(x)) = 2(x - 2)² - 1

So, we have

g(f(5)) = 2(5 - 2)² - 1

Evaluate

g(f(5)) = 17

Hence, the functions g(f(5)) is 17

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Answer:

a) g^-1(x) = - √2x+1, √2x+1

                        2         2

b) -14

c) 17

Step-by-step explanation:

a) g(x) = 2x² - 1

y= 2x² - 1 becomes x = 2y² - 1

Solve for y: x = 2y² - 1

g^-1(x) = - √2x+1, √2x+1

                    2         2

*answer is: -√2x+1 (over) 2 & √2x+1 (over) 2*

b) Find: f(h(-2))

h(x) = 2x³ +4

h(-2) = 2(-2)³ + 4

= - 12

f(x) = x – 2

f(-12) = (-12) – 2

= -14

c) Find: g(f(5))

f(x) = x – 2

f(5) = (5) – 2

= 3

g(x) = 2x² - 1

g(3) = 2(3)² - 1

= 17

The probability that you take different colors in two"picks" is 31/45.

You could choose one red followed by one blue, one blue followed by one red, one blue followed by one black, one black followed by one blue, one black followed by one red, or one red followed by one black.

P( different colors) = P( one blue followed by one red ) +P( one red followed by one blue) + P( one blue followed by one black ) +P( one black followed by one blue) +P( one black followed by one red ) +P( one red followed by one black)

P(one blue followed by one red) = (10/20)( 6/19) = 3/19

P(one red followed by one blue) = (6/20)( 10/19) = 3/19

P(one blue followed by one black) = (10/20)( 4/19) = 2/19

P(one black followed by one blue) = (4/20)( 10/19) = 2/19

P(one red followed by one black) = (6/20)( 4/19) = 6/95

P(one black followed by one red) = (4/20)( 6/19) = 6/95

P(different colors )= 3/19 + 3/19 + 2/19 + 2/19 + 6/95 + 6/95
                                = 31/45

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Answer:

Step-by-step explanation:

You want to find the distance traveled by light in km for periods of 1–5 seconds at a speed of 299,792,458 m/s.

The speed is given in meters per second, and the table values are wanted in kilometers. There are 1000 meters in 1 km, so it can be convenient to convert the speed to km/s before doing other calculations.

  (299,792,458 m/s) / (1000 m/km) = 299,792.458 km/s

The distance is found by multiplying the speed by time:

  d = 299,792.458t

It is convenient to let a calculator do the multiplication (and rounding). The attachment shows the result of using t = {1, 2, 3, 4, 5}.

The distances are, respectively, {299792, 599585, 899377, 1199170, 1498962} km.

a. [tex]123.1(1.069)^3 \approx 150.4[/tex]

b. [tex]123.1(1.069)^{20} \approx 467.5[/tex]

Answer:the legth is 40

Step-by-step explanation:

We select the option that corresponds to a random number between 1 and 10. We repeat this process until we get 15 good roses.

In the given question, one float decorator found that 10% of the bundles of roses delivered will not open in time for the parade.

20% of the bundles of roses delivered will have bugs on them and be unusable.

60% of the bundles of roses will turn out to be beautiful.

The rest of the bundles of roses delivered will bloom too early and start to discolor before January 1.

We have to conduct a simulation to estimate how many roses the float decorator will need to purchase to have 15 good bundles of roses to place on the float.

We also have to describe how you will use a random number table to conduct this simulation.

The random number table is

Now we select the option that corresponds to a random number between 1 and 10.

We repeat this process until we get 15 good roses.

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Answer:

8+8-8×1

8+8-8

16-8

8 Ans.

Aaron has to read 41 pages on the 6th day to reach his goal.

What is the speed of reading?

Any of the many methods that promise to increase one's reading speed include speed reading. Chunking is one speed-reading technique, as is reducing subvocalization. The many available speed-reading training programs may utilize books, videos, software, and seminars.

We have,

26 pages per day for 6 days:

26×6=156,

So Aaron should read 156 pages by the 6th day to reach his goal,

During the first 5 days, he reads 23 pages per day then,

23 × 5 = 115

That’s how many pages he’s already read.

156-115=41

Hence, Aaron has to read 41 pages on the 6th day to reach his goal.

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The distance travelled per second (i.e speed) of the light is 3×10⁸ m/s

Distance travelled per second is known as speed. It can be expressed as:

Speed = distance / time

With the above formula, we can obtain the distance traveled per second of the light, Details below

From the question given, the following were obtained:

Speed = distance / time

Distance per second = 9.45×10¹⁵ / 3.15×10⁷

Distance per second = 3×10⁸ m/s

From the calculation made above, we can conclude that the distance per second is 3×10⁸ m/s

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Answer:

answers attached int he files below all in order

Answer:

---------------------------

It is assumed the first column represents x- values and the second column represents y- values.

Inverse variation equation:

Substitute one pair of coordinates, x = 3, y = 10:

Correct choice is B.

Answer:

b) k = 30

Step-by-step explanation:

Formula we use,

→ y = k/x

Now the value of k will be,

→ y = k/x

→ 10 = k/3 || → 6 = k/5

→ k/3 = 10 || → k/5 = 6

→ k = 10 × 3 || → k = 6 × 5

→ [ k = 30 ] || → [ k = 30 ]

Hence, the value of k is 30.

Answer:  Choice B) f(x) is a one-to-one function

What this means is that each x input lands on a unique y output. For example, plugging x = 0 leads to y = -3. No other x value will lead to y = -3. This allows us to find the inverse.

The vertical line test is used to see if we have a function or not. The horizontal line test is used to see if an inverse is possible.

The phrasing "The graph of the inverse of f(x) passes the horizontal line test" is close, but we need to erase "the inverse of" to have choice C be useful. In other words, if choice C said "The graph of f(x) passes the horizontal line test", then it would be correct and useful.

Side note: Another term for "one-to-one" is "injective".

Answer:

  A.  n ≥ 11

Step-by-step explanation:

You have a triangle with sides (n), (n-3), and 2(n-2). Its perimeter is at least 37 units, and you want to know the value of n.

The perimeter is the sum of the side lengths:

  P = (n) +(n -3) +2(n -2)

  P = n + n -3 +2n -4

  P = 4n -7

The perimeter is at least 37, so we have ...

  P ≥ 37

  4n -7 ≥ 37

  4n ≥ 44 . . . . . . . add 7

  n ≥ 11 . . . . . . . divide by 4

The value of n is at least 11.

The value of $100 in euros is $87.

Ratio basically compares quantities, that means it show value of one quantity with respect to other quantity.

If a and b are two values, their ratio will be a:b,

Given that,

Worth of 1 United States dollar is $0.87 in euros.

To find the value of $100 in Euros, use ratio property,

1 United States dollar = $0.87 in euros

100 United States dollar = $87 in euros.

The value of $100 in euros is $87.

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Answer:

0.13, 0.16, 0.19, 0.22

Step-by-step explanation:

You are adding 0.03 each time

Answer:

0.13, 0.16, 0.19, 0.22

Step-by-step explanation:

Add 0.03 to each.

The quadratic function, f(x) = 3·x² + 6·x, has the following values;

1. The function has a minimum value

2. The y-intercept of the function is (0, 0)

The axis of symmetry is; x = -1

The x-coordinate of the vertex is; x = -1

3. The table of values can be presented as follows;

x   [tex]{}[/tex]-2    -1    0

f(x) [tex]{}[/tex]0    -3     0

The vertex is (-1, -3)

4. Please find attached the graph of the function

A quadratic function is a function that has 2 as the highest power of the variables.

The specified function is; f(x) = 3·x² + 6·x

1. The coefficient of x² in the specified  quadratic function, which indicates the existence of a minimum or maximum value is positive, therefore, the function has a minimum value

2. The y-intercept is the value of the function when x = 0, therefore;

f(0) = 3 × 0² + 6 × 0 = 0

Therefore, at the y-intercept, x = 0, and f(x) = 0

The y-intercept is (0, 0)

The axis of symmetry is a line that passes through the vertex

A the vertex, we get, f'(x) = 6·x + 6 = 0

Therefore;  6·x + 6 = 0

6·x = -6

x = -6/6 = -1

f(-1) = 3·(-1)² + 6 × (-1) = -3

The vertex is (-1, -3)

Therefore, the  axis of symmetry is the line x = -1

The x-coordinate of the vertex of the function is; x = -1

3. A table of values for the function can be presented as follows;

x    [tex]{}[/tex]           f(x)

-6    [tex]{}[/tex]           72

-5     [tex]{}[/tex]          45

-4     [tex]{}[/tex]          24

-3    [tex]{}[/tex]           9

-2    [tex]{}[/tex]           0

-1    [tex]{}[/tex]           -3

0    [tex]{}[/tex]           0

1    [tex]{}[/tex]            9

2     [tex]{}[/tex]           24

3    [tex]{}[/tex]            45

4    [tex]{}[/tex]            72

5    [tex]{}[/tex]             105

4. . Please find attached the graph of the function created with MS Excel

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c=1/2  which makes fx|y (x, y) a valid conditional probability mass function.

What is conditional probability?

The possibility of an event or outcome occurring based on the existence of a prior event or outcome is known as conditional probability. It is determined by multiplying the likelihood of the earlier event by the increased likelihood of the later, or conditional, event.

Given that;

p(x) = [tex]c (\frac{2}{3} )^n[/tex]   where x = 1,2,3,........

for getting value of c we are using the expression :

[tex]\sum_{n=1}^{\infty} c(\frac{2}{3})^n = 1[/tex]

[tex]c\sum_{n=1}^{\infty} (\frac{2}{3})^n = 1[/tex]

Since we want p(x) to be a Probability Mass Function, your approach is correct, since for any such function

fX:A→[0,1],X:S→A⊆R, it is :

[tex]\sum _{x\in A} \ fX(x)=1[/tex]

As long as you've correctly understood that condition, you can proceed with your specific exercise as follows:

[tex]\sum_{n=1}^{\infty} c(\frac{2}{3})^n = 1[/tex]

⇔ [tex]c\sum_{n=1}^{\infty} (\frac{2}{3})^n = 1[/tex]

⇔2c = 1

⇔c = 1/2

Hence c =  1/2 is the valid conditional probability mass function.

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Hence, the [tex]200th[/tex] term in the sequence is [tex]1613[/tex].

What is the sequence?

A sequence is a list of objects, typically numbers, in which order matters, repetition is allowed, and the same elements can appear multiple times at different positions in the sequence.

Here given that,

The sequence is

[tex]21,29,37,45,53,...[/tex]

The general formula is

[tex]a_n=a_1+d(n-1)[/tex]

[tex]a_1=21[/tex]

[tex]d=a_2-a_1\\d=29-21\\d=8[/tex]

Substitute it in the general formula is

[tex]a_2_0_0=21+8(200-1)\\a_2_0_0=21+8(199)\\a_2_0_0=21+1592\\a_2_0_0=1613[/tex]

Hence, the [tex]200th[/tex] term in the sequence is [tex]1613[/tex].

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The equation of the line that is perpendicular to y = 1/2x - 6 and that passes through the point (-4, -5) is y = -2x + 3

From the question, we are to determine the equation of the line that is perpendicular to the given line and that passes through the given point.

The given equation is y = 1/2x - 6

The given point is (-1, 5)

NOTE: If two lines are perpendicular, then their slopes are negative reciprocals of each other

First, we will determine the slope of the given line

y = 1/2x - 6

Compare to the general form of an equation of a straight line

y = mx + b

Where m is the slope

and b is the y-intercept

Therefore,

The slope of the line is 1/2

But,

The negative reciprocal of 1/2 is -2

Thus, the slope of the equation we are to determine is -2

Now, we will determine the equation of the line that has a slope of -2 and that passes through the point (-1, 5)

Using the point-slope form of the equation of a straight line

y - y₁ = m(x - x₁)

Then,

y - 5= -2(x - (-1))

y - 5 = -2(x + 1)

y - 5 = -2x -2

Add 5 to both sides of the equation

y - 5 + 5 = -2x - 2 + 5

y = -2x + 3

Hence, the equation of the line is y = -2x + 3

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The correct comparison between the given numbers ; √160 and 115 / 9 is; 115 / 9 > √160.

It follows from the task content that the sign which correctly compares the two numbers as given in the task content be determined.

Since the given numbers in the task content can be evaluated as follows;

√160 = 12.64911064.

115 / 9 = 12.7777777778

From the results of the evaluations as represented above, since 12.7777777778 is greater than 12.64911064, it follows that the 115 / 9 > √160.

Therefore, the required correct comparison is;

115 / 9 > √160

The answer choice which is correct is as represented above.

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x under the assumption that x >0 x-70/x<-3, x > -35

WHAT IS INEQUALITY?

Economic inequality comes in many forms, most notably wealth inequality measured by the distribution of wealth and income inequality measured by the distribution of income (the quantity of money people get paid) (the amount of wealth people own). There are significant forms of economic inequality among various groups of people in addition to economic inequality between nations or states. Wealth, income, as well as consumption are the three main focuses of significant economic measurements. There are numerous ways to gauge economic inequality, with the Gini coefficient being among the most popular. The Disparities Human Development Index is a statistic composite index which offers another type of measurement by taking inequality into account.

assuming x > 0,

(x-70/x) < 3

= x-70 < 3x = -70 < 2x

= x > -35

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The cost per ounce in both options is the same. Then the correct option is D.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The rate is the ratio of the amount of something to the unit.

Jorge wants to buy a drink.

Choice 1: 24-ounce cup for $1.92

Choice 2: 12-ounce cups for $0.96 each

The cost per ounce in choice 1 is calculated as,

⇒ $1.92 / 24

⇒ $0.08 per ounce

The cost per ounce in choice 2 is calculated as,

⇒ $0.96 / 12

⇒ $0.08 per ounce

The cost per ounce in both options is the same. Then the correct option is D.

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The probability that the sample mean would be less than 43.4 dollars is 0.9292.

Probability is the occurence of likely events. It is the area of mathematics that deals with numerical estimates of the likelihood that an event will occur or that a statement is true.

From the information, the mean cost of a five pound bag of shrimp is 42 dollars with a variance of 49 and a sample of 54 bags of shrimp is randomly selected.

Therefore, the probability will be:

P(x < 43.4)

Using normal distribution

= P(z < 43.4 - Mean( / 7/✓54)

= P(z < 43.4 - 42( / 7/✓54)

= P(z < 1.4697)

= 0.9292

The probability is 0.9292.

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