Find the mass of the solid bounded by the xy-plane, yz-plane, xz-plane, and the plane (x/2)+(y/3)+(z/6)=1, if the density of the solid is given by δ(x,y,z)=x+4y.
mass=?

Moores law that number of transistors per IC Prediction = 4,096,000 and Prediction = 25.6 trillion.

What is probability ?

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that an event is impossible and 1 indicating that an event is certain. For example, the probability of flipping a fair coin and getting heads is 0.5, while the probability of rolling a six on a fair six-sided die is 1/6 or approximately 0.1667.

In mathematical terms, probability is calculated by dividing the number of ways an event can occur by the total number of possible outcomes. Probability theory is widely used in many fields, including statistics, finance, science, engineering, and economics, to help predict and analyze the likelihood of various outcomes and make informed decisions based on the probabilities involved.

According to the question:

Part a: Moore's Law predicts that the number of transistors per IC doubles every 18-24 months. Since 20 years is approximately 10 doublings (20/2), we can estimate the number of transistors in 1992 to be [tex]4,000 * 2^{10} = 4,096,000.[/tex]

Prediction = 4,096,000

Part b: Based on the chart, the number of transistors per IC in 2016 appears to be around 10 billion [tex](1 * 10^{10})[/tex]

Using Moore's Law, we can estimate the number of transistors per IC in 2040 to be [tex]1 * 10^{10} * 2^{24/18} = 25.6 * 10^{12} (or 25.6 trillion)[/tex].

Prediction = 25.6 trillion

Part c: The prediction in Part b may not be entirely believable, as there are physical limits to the number of transistors that can be placed on a single chip. Moore's Law has been slowing down in recent years, with transistor density growth rates dropping below historic trends. Additionally, new technologies beyond traditional silicon-based chips may become necessary to continue improving transistor density at the same pace as in the past. Therefore, while the prediction is technically possible, it may not be achievable without significant breakthroughs in semiconductor technology.


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The coordinates of point X are (7/3, 2√(2)) in Quadrant I.

In coordinate geometry, a plane is divided into four regions called quadrants. These quadrants are numbered counterclockwise starting from the upper right quadrant, which is known as the first quadrant.The four quadrants are defined by the x-axis and y-axis. The x-axis is a horizontal line that runs left and right through the origin, while the y-axis is a vertical line that runs up and down through the origin.

An isosceles triangle is a type of triangle in which two sides are of equal length. In an isosceles triangle, the third side is called the base, and the two equal sides are called legs. The two angles formed by the legs and the base are also equal to each other. The angle opposite the base is called the vertex angle.

In the given question,

Since Triangle XYZ is a 45°- 45°- 90° triangle, the two legs of the triangle are congruent.

Let's call the length of each leg "x".We know that point Z is the right-angle vertex of the triangle and has coordinates (6, 2).

Since the triangle is isosceles, we can find the length of the other leg using the distance formula:x² + x² = (distance between Y and Z)²²x² = (6 - (-1))²²x² = 49x² = 24.5.

Now that we know the length of each leg, we can find the coordinates of point X. Since Triangle XYZ is a 45°- 45°- 90° triangle, we know that the hypotenuse is the square root of 2 times the length of a leg.

Let's call the coordinates of point X (x, y).x² + y² = (distance between X and Y)²x² + y² = x² + (y - 2)²y = 2√(2)Now we can find the x-coordinate of point X:x^2 + (2√(2))^2 = (distance between X and Z)^2x^2 + 8 = (6 - x)^2x^2 + 8 = 36 - 12x + x^212x = 28x = 7/3.

Therefore, the coordinates of point X are (7/3, 2√(2)) in Quadrant I.

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Answer:The specific heat of a substance is defined as the amount of heat required to raise the temperature of one gram of the substance by one degree Celsius (or Kelvin).

To find the specific heat of the unknown substance, we can use the following equation:

Q = m x c x ΔT

where Q is the heat gained or lost, m is the mass of the substance, c is its specific heat, and ΔT is the change in temperature.

In this problem, we know the mass and initial and final temperatures of both the unknown substance and the water, as well as the specific heat of water. We can use this information to calculate the heat gained by the water, which must be equal to the heat lost by the unknown substance:

Heat gained by water = Heat lost by unknown substance

m(water) x c(water) x ΔT(water) = m(substance) x c(substance) x ΔT(substance)

We can plug in the values we know and solve for the specific heat of the unknown substance:

m(water) = 75.0 g

c(water) = 4.18 J/g°C

ΔT(water) = 27.1 °C - 20.00 °C = 7.1 °C

m(substance) = 100.0 g

ΔT(substance) = 200.0 °C - 27.1 °C = 172.9 °C

75.0 g x 4.18 J/g°C x 7.1 °C = 100.0 g x c(substance) x 172.9 °C

Simplifying this equation, we get:

c(substance) = (75.0 g x 4.18 J/g°C x 7.1 °C) / (100.0 g x 172.9 °C)

c(substance) = 0.197 J/g°C

Therefore, the specific heat of the unknown substance is 0.197 J/g°C.

Step-by-step explanation:

Answer:

The specific heat of the unknown substance is 0.39 J/g°C.

Step-by-step explanation:

To solve this problem, we can use the principle of conservation of energy, which states that the heat lost by the unknown substance is equal to the heat gained by the water and the calorimeter. We can express this principle mathematically as:

Q_lost = Q_gained

where Q_lost is the heat lost by the unknown substance, and Q_gained is the heat gained by the water and calorimeter.

We can calculate Q_lost using the formula:

Q_lost = m × c × ΔT

where m is the mass of the unknown substance, c is its specific heat, and ΔT is the change in temperature it undergoes.

We can calculate Q_gained using the formula:

Q_gained = (m_water + m_calorimeter) × c_water × ΔT

where m_water is the mass of the water, m_calorimeter is the mass of the calorimeter, c_water is the specific heat of water, and ΔT is the change in temperature of the water and calorimeter.

Since the system reaches an equilibrium temperature, we can set Q_lost equal to Q_gained and solve for the specific heat of the unknown substance (c).

Here's the calculation:

Q_lost = Q_gained

m × c × ΔT = (m_water + m_calorimeter) × c_water × ΔT

100.0 g × c × (200.0 °C - 27.1 °C) = (75.0 g + 75.0 g) × 4.18 J/g°C × (27.1 °C - 20.00 °C)

Simplifying:

c = [(75.0 g + 75.0 g) × 4.18 J/g°C × (27.1 °C - 20.00 °C)] / (100.0 g × (200.0 °C - 27.1 °C))

c = 0.39 J/g°C

Therefore, the specific heat of the unknown substance is 0.39 J/g°C.

Answer:  Choice D)  [tex]2a^3b[/tex]

Work Shown:

[tex]\frac{18a^6b^2}{9a^3b}=\frac{18a^6b^2}{9a^3b}\\\\\frac{18a^6b^2}{9a^3b}=\frac{18}{9}*\frac{a^6}{a^3}*\frac{b^2}{b}\\\\\frac{18a^6b^2}{9a^3b}=2a^{6-3}b^{2-1}\\\\\frac{18a^6b^2}{9a^3b}=2a^3b\\\\[/tex]

The formula used on the 3rd line was (a^b)/(a^c) = a^(b-c). We subtract the exponents when dividing exponential expressions with the same base.

Answer:

The area of a semicircle with diameter 28 cm is 98π cm², or 307.88 cm² to the nearest tenth.

Step-by-step explanation:

A semicircle is a two-dimensional shape that is exactly half of a circle.

The area of a circle is given by the formula:

[tex]\sf A=\pi r^2[/tex]

where A is the area of the circle, and r is the radius of the circle.

The diameter of a circle is twice its radius.

Given the diameter of the semicircle is 28 cm, the radius is:

[tex]\sf r = \dfrac{28}{2} = 14 \; cm[/tex]

Substituting this into the formula for the area of a circle, we get:

[tex]\sf A = \pi(14)^2[/tex]

[tex]\sf A = 196 \pi[/tex]

Finally, divide this by two to get the area of the semicircle:

[tex]\sf Area\;of\;semicircle = \dfrac{1}{2} \cdot 196 \pi[/tex]

[tex]\sf Area\;of\;semicircle = 98 \pi\; cm^2[/tex]

So the area of a semicircle with diameter 28 cm is 98π cm², or 307.88 cm² to the nearest tenth.

Answer:

g(5)=69

Step-by-step explanation:

g(5)=3(5)^2-2(5)+4

g(5)=75-10+4

g(5)=69

Using simple division we know that the acceleration per second is 11.54 m/s.

Multiplication is the opposite of division.

If 3 groups of 4 add up to 12, then 12 divided into 3 groups of equal size results in 4 in each group.

Creating equal groups or determining how many people are in each group after a fair distribution is the basic objective of division.

The division is a mathematical process that includes dividing a sum into groups of equal size.

For instance, "12 divided by 4" translates to "12 shared into 4 equal groups," which would be 3 in our example.

So, to find the acceleration per second:

We need to perform division as follows:

= 27.7/2.4

= 11.54

Therefore, using simple division we know that the acceleration per second is 11.54 m/s.

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Step-by-step explanation:

1. 2/10

2.3/15

3.4/20

4. 5/25

5.6/30

6.7/35

i. The total height of the tent including the spire is 150 m.

ii. The length of the side of the tent  x is 132.7 m.

Trigonometric functions are required functions in determining either the unknown angle of length of the sides of a triangle.

Considering the given question, we have;

a. To determine the total height of the tent, let its height from the ground to the top of the tent be represented by x. Then:

Tan θ = opposite/ adjacent

Tan 42.7 = h/ 97.5

h = 0.9228*97.5

  = 89.97

h = 90 m

The total height of the tent including the spire = 90 + 60

                                           = 150 m

b. To determine the length of the side of the tent x, we have:

Cos θ = adjacent/ hypotenuse

Cos 42.7 = 97.5/ x

x = 97.5/ 0.7349

  = 132.67

The length of the side of the tent x is 132.7 m.

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A vector x orthogonal to the row space of A, and a vector y orthogonal to column space, and a vector z orthogonal to the null space. The orthogonal vector is :

A = [tex]\left[\begin{array}{ccc}1&2&1\\2&1&0\\1&-2&2\end{array}\right][/tex]

The orthogonal complement of the subspace V contains any vector perpendicular to V. This orthogonal subspace is denoted V⊥. (pronounced "V perp").

By this definition, null space is the orthogonal complement of row space. Every x perpendicular to the line satisfies Ax = 0 and lies in null space.

vice versa. If v is orthogonal to null space, it must be in row space. Otherwise, we can add this v as an extra row of the matrix without changing its null space. The rice space will become larger, breaking the rule of r+(n−r) = n.

The column space extent of A. These two vectors are the basis of col(A) , but they are not normalized.

In this case, the columns of A are already orthonormal, so you don't need to use the Gram-Schmidt procedure. To normalize a vector and then divide it by its norm:

                   [tex]\left[\begin{array}{ccc}1&2&1\\2&4&3\\3&6&4\end{array}\right][/tex]

and the vector after orthogonal process is:

                  [tex]\left[\begin{array}{ccc}1&2&1\\2&1&0\\1&-2&2\end{array}\right][/tex]

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Answer:

11

Step-by-step explanation:

[tex]61 {}^{2} - {60 }^{2} = 121[/tex]

[tex] \sqrt{121} = 11[/tex]

Answer:

The total labor charges for the job are P3,500.

Step-by-step explanation:

To find the total labor charges for a job that takes 2 1/2 hours, we need to look at the labor charges for each hour and a half-hour fraction and add them up.

For the first hour, the charges are P1,200. For the second hour, the charges are P1,400. For the third hour (the half-hour fraction), the charges are P1,800 / 2 = P900.

So, the total labor charges for 2 1/2 hours of work are

P1,200 + P1,400 + P900 = P3,500

Therefore, the total labor charges for the job are P3,500.

Answer:

Step-by-step explanation:

First the max degree is 360

Then multiply by 5/8

360 x 5/8 = 1800/8

1800/8 = 225

Answer: 225

The Taylor's series for [tex]1 + 3e^x + x^2[/tex] at [tex]x=0[/tex] is :

[tex]1 + 3e^x+ x^2 = 5 + 3x + (3/2)x^2 + (1/3)x^3 + ...[/tex]

What do you mean by Taylor's series ?

The Taylor's series is a way to represent a function as a power series, which is a sum of terms involving the variable raised to increasing powers. The series is centered around a specific point, called the center of the series. The Taylor's series approximates the function within a certain interval around the center point.

The general formula for the Taylor's series of a function f(x) centered at [tex]x = a[/tex] is:

[tex]f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...[/tex]

where [tex]f'(a), f''(a), f'''(a),[/tex] etc. are the derivatives of f(x) evaluated at [tex]x = a[/tex].

Finding the Taylor's series for [tex]1 + 3e^x + x^2[/tex] at [tex]x=0[/tex] :

We need to find the derivatives of the function at [tex]x=0[/tex]. We have:

[tex]f(x) = 1 + 3e^x + x^2[/tex]

[tex]f(0) = 1 + 3e^0 + 0^2 = 4[/tex]

[tex]f'(x) = 3e^x+ 2x[/tex]

[tex]f'(0) = 3e^0 + 2(0) = 3[/tex]

[tex]f''(x) = 3e^x + 2[/tex]

[tex]f''(0) = 3e^0 + 2 = 5[/tex]

[tex]f'''(x) = 3e^x[/tex]

[tex]f'''(0) = 3e^0 = 3[/tex]

Substituting these values into the general formula for the Taylor's series, we get:

[tex]f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...[/tex]

[tex]f(x) = 4 + 3x + 5x^2/2 + 3x^3/6 + ...[/tex]

Simplifying, we get:

[tex]f(x) = 5 + 3x + (3/2)x^2 + (1/3)x^3 + ...[/tex]

Therefore, the Taylor's series for [tex]1 + 3e^x + x^2[/tex] at [tex]x=0[/tex] is :

[tex]1 + 3e^x+ x^2 = 5 + 3x + (3/2)x^2 + (1/3)x^3 + ...[/tex]

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Answer:

1 - [tex]\frac{2\sqrt{2} }{3}[/tex]

Step-by-step explanation:

[tex]\frac{\sqrt{3}-\sqrt{6} }{\sqrt{3}+\sqrt{6} }[/tex]

rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

the conjugate of [tex]\sqrt{3}[/tex] + [tex]\sqrt{6}[/tex] is [tex]\sqrt{3}[/tex] - [tex]\sqrt{6}[/tex]

= [tex]\frac{(\sqrt{3}-\sqrt{6})(\sqrt{3}-\sqrt{6}) }{(\sqrt{3}+\sqrt{6})(\sqrt{3}-\sqrt{6}) }[/tex] ← expand numerator/ denominator using FOIL

= [tex]\frac{3-\sqrt{18}-\sqrt{18}+6 }{3-\sqrt{18}+\sqrt{18}+6 }[/tex]

= [tex]\frac{9-2\sqrt{18} }{3+6}[/tex]

= [tex]\frac{9-2(3\sqrt{2}) }{9}[/tex]

= [tex]\frac{9-6\sqrt{2} }{9}[/tex]

= [tex]\frac{9}{9}[/tex] - [tex]\frac{6\sqrt{2} }{9}[/tex]

= 1 - [tex]\frac{2\sqrt{2} }{3}[/tex]

1. To break down 12x² into its prime factors,Start dividing 12 by 2, which gives you 6. Then, break 6 by 2, which gives 3. To write 12x² in terms of its prime factors, you simply need to add the x² term, giving you 2 * 2 * 3 * x * x or option C.

A common factor is a number or expression that divides two or more other numbers or expressions without leaving a remainder.

Since 3 is a prime number So the prime factorization of 12 is 2 * 2 * 3.

2. To find the greatest common factor of 8x³ - 12x² + 16x, you can start by factoring out any common factors that the terms share. In this case, all of the terms have a factor of 4x, so you can factor that out to get 4x(2x² - 3x + 4). Therefore, the greatest common factor is 4x or option B.

3. To rewrite 8x³ - 12x² + 16x, pulling out the greatest common factor, you can use the same method as in question 2. The greatest common factor of the terms is 4x, so you can factor that out to get 4x(2x² - 3x + 4) or option A.

4. To find the greatest common factor of 6x²+ 16x + 10, you can factor the expression using the quadratic formula or by factoring the expression as (3x + 5)(2x + 2). Since there is no other common factor between the terms, the greatest common factor is 1 or option A.

5. To factor x² - 4x - 32, you need to find two numbers that multiply to -32 and add to -4. These numbers are -8 and 4. Therefore, x² - 4x - 32 can be factored as (x - 8)(x + 4) or option C.

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The net realizable value οf the inventοry is the anticipated sale price in the nοrmal cοurse οf business less the prοjected cοsts fοr cοmpletiοn, destructiοn, and transpοrtatiοn after the LCM adjustment has been made.

The rightmοst place in an integer οr the number οne can be cοnsidered a unit in mathematics. The unit number inside the number 6713 in this case is 3. The standard measuring units can alsο be referred tο as a unit.

1. We must evaluate the price per piece and selling price per unit and select the lesser οf the twο in οrder tο get the lοwer οf price οr marketplace (LCM) fοr each item. The calculatiοns lοοk like this:

Speakers: LCM = min($105, $113) = $105 per unit

Stereοs: LCM = min($116, $126) = $116 per unit

Amplifiers: LCM = min($101, $110) = $101 per unit

Subwοοfers: LCM = min($57, $67) = $57 per unit

Alarms: LCM = min($155, $165) = $155 per unit

Lοcks: LCM = min($98, $108) = $98 per unit

Cameras: LCM = min($327, $337) = $327 per unit

Tripοds: LCM = min($89, $99) = $89 per unit

Stabilizers: LCM = min($110, $120) = $110 per unit

2. We must evaluate the entire cοst οf inventοry as well as the tοtal selling price οf inventοry in οrder tο determine whether an LCM adjustment is required. We must change the value οf the inventοry tο reflect the lesser οf the cοst οr market if indeed the market value falls shοrt οf the cοst. The calculatiοns lοοk like this:

Tοtal cοst οf inventοry = (350 x $105) + (265 x $126) + (331 x $101) + (209 x $67) + (485 x $165) + (296 x $108) + (217 x $327) + (190 x $89) + (175 x $110)

= $70,657

Tοtal market value οf inventοry = (350 x $113) + (265 x $116) + (331 x $110) + (209 x $57) + (485 x $155) + (296 x $98) + (217 x $327) + (190 x $99) + (175 x $110)

= $70,273

We must make an LCM mοdificatiοn tο the Merchandise Accοunting system because the market price is lοwer than the cοst. The distinctiοn amοng the tοtal cοst and the tοtal market value is the adjustment amοunt, which is:

$70,657 - $70,273 = $384

The LCM adjustment's jοurnal entry is as fοllοws:

Merchandise Inventοry 384

LCM Adjustment 384

The LCM adjustment reduces the inventοry value tο its net realizable value, which is the estimated selling price in the οrdinary cοurse οf business, less the estimated cοsts οf cοmpletiοn, dispοsal, and transpοrtatiοn.

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The categorical data in the table is column C, Position.

In mathematics and statistics, a table is a way of presenting data in a structured manner, typically with columns and rows. Tables are commonly used to organize and present large amounts of data in a clear and concise way, making it easier to read and analyze. Tables can be used to display numerical data, as well as categorical data, such as names, dates, and labels. They can also be used to summarize data and display relationships between different variables. Tables are often used in scientific research, business, finance, and other fields where data analysis is important.

Here,

In the table given, the only column that contains categories or groups is the "Position" column. It contains categorical data as it lists the positions of the players - Goal, Wing, and Center. On the other hand, "Name" and "Goals" columns contain individual values and numerical data, respectively, and are not considered categorical data.

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The approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month, obtained using the z-score for the proportion of the sample, and the standard error, is about 96.327%

The z-score of a sample proportion, z can be obtained using the formula;

z = (p - π)/√(π·(1 - π)/n)

Where;

p = The sample proportion

π = The proportion of the population

n = The sample size

The percentage of the students out of the 1750 students that experienced extreme levels of stress in the school, p = 15%

The number of students in the sample used by the newspaper, n = 160 students

The number of students in the sample that replied "yes" = 10%

The true proportion of the students that experience stress = 15%

The probability that ,more than 10% of the sample would report that they experienced extreme levels of stress during the past month can be found as follows;

The standard error is; SE = √(p × (1 - p)/n)

Therefore;

SE = √(0.15 × (1 - 0.15)/160) ≈ 0.028

The z-score is therefore;

z = (0.1 - 0.15)/0.028 ≈ -1.79

z = -1.79

The z-score indicates the number of standard deviations the proportion of the sample is from the true proportion

The proportion on the of the sample which is larger than 10% is obtained from the area under the normal curve, to the right of the z-score of -1.79, which is obtained as follows;

The z-value at z = -1.79 is 0.03673, which indicates that the area to the left of the z-value is 0.03673, and the area to the right is; (1 - 0.03673) = 0.96327

The probability observing a sample proportion more than 10% if the actual proportion is 15% is therefore; 0.96327 = 96.327%

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Answer:

Step-by-step explanation:

Assuming you mean [tex]9a^2[/tex]:

   Multiply nine by a squared.

OR

   Times nine by a squared.

If you mean 9a^2:

Multiply nine by a number squared

or

Multiply a number squared by nine

The slope of the graph f(x) = 3x² + 7 at (-2, 19) is -12

The slope of a graph is the derivative of the graph at that point.

Since we have tha graph f(x) = 3x² + 7 and we want to find its slope at the point (-2, 19).

To find the slope of the graph, we differentiate with respect to x, since the derivative is the value of the slope at the point.

So,  f(x) = 3x² + 7

Differentiating with respect to x,we have

df(x)/dx = d(3x² + 7)/dx

= d3x²/dx + d7/dx

= 6x + 0

= 6x

dy/dx = f'(x) = 6x

At (-2, 19), we have x = -2.

So, the slope f'(x) = 6x

f'(-2) = 6(-2)

= -12

So, the slope is -12.

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Answer:

m = 1

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (-1,0) (0,1)

We see the y increase by 1 and the x increase by 1, so the slope is

m = 1

The integral expression for the volume of the solid formed by revolving the region bounded by the graphs of y = x₂ - 3x and y = x about the horizontal line y = 6 is  2πx(6 - x² + 3x)dx, which is integrated from x=0 to x=3, which gives us   81π/2.

To find the integral expression for the volume of the solid formed by revolving the region bounded by the graphs of y=x² - 3x and y=x about the horizontal line y=6, we can use the method of cylindrical shells.

First, we need to find the limits of integration, The graphs of y = x² - 3x  and y=x intersect at x=0 and x=3. Therefore, we integrate from x=0 to x=3.

Next, we consider a vertical strip of width dx at a distance x from the y- boxes. the height of the strip is the difference between the height of the curve y=  x² - 3x and the line y=6, which is   6 - (x² - 3x) = 6 - x² + 3x. the circumference of the shell is 2π times the distance x from the y-axis, and the thickness of the shell is dx. the volume of the shell is the product of the height, circumference, and thickness which is

dV = 2πx(6 - x² + 3x)dx

To find the total volume, we integrate this expression from x=0 to x=3.

V = ∫₀³ 2πx(6 - x² + 3x)dx, after simplifying the integrand we get :

V = 2π ∫₀³ (6x - x³ + 3x²)dx, integrating term by term we get :

V = 2π [(3x²/2) - (x⁴/4) + (x^3)] from 0 to 3, now  evaluation at the limits of integration we get:

V = 2π [(3(3)²/2) - ((3)⁴/4) + (3)³] - 2π [(0)^2/2 - ((0)⁴/4) + (0)^3]=  2π [(27/2) - (27/4) + 27] - 0 = 81π/2

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It involves selecting 35 candies from a bag containing 20 strawberry, 20 cherry, and 10 orange Starburst candies. X is the number of strawberry candies selected. X has a hypergeometric distribution, with possible values from 0 to 20. P(X > 18) is 0.0125, and probability mass function P(X = 3) is 0.0783. The expected value of X is 14, and the variance of X is approximately 5.67.

X has a hypergeometric distribution with parameters M=40 (20+20), N=50 (20+20+10), and n=35.

X can take on values from 0 to 20, since there are only 20 strawberry candies in the bag.

Using the cumulative distribution function for the hypergeometric distribution, we have P(X > 18) = 0.0125.

Using the probability mass function for the hypergeometric distribution, we have P(X = 3) = 0.0783.

The expected value of X is E[X] = np = 35(20/50) = 14.

The variance of X is Var[X] = np(1-p)(N-n)/(N-1) = (35)(20/50)(30/49)(40/49) ≈ 5.67.

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The value of angle that maximizes the cross sectional area is 60°

The cross sectional area of a solid is the two dimensional surface formed when the solid is sectioned by a plane.

The shape of the cross sectional area of the gutter is a trapezoid

Area of a trapezoid = ((Sum of the lengths of the parallel sides)/2) × Height

Let a and b represent the parallel sides of the trapezoid, and let a represent the base of the trapezoid we get;

A = ((a + b)/2) × h

a = 3 inches

b = The longer parallel side

h = The height of the trapezoid

The angle formed by the sides of a trapezium and the horizontal = θ

Therefore;

h = 3 × sin(θ)

b = 3 + 2 × 3·cos(θ)

A(θ) = ((3 + 3 + 2 × 3·cos(θ)) × 3 × sin(θ))/2 = 9·sin(θ)·cos(θ) + 9·sin(θ)

At the extremum value of θ, we get;

A'(θ) = 18·cos²(θ) + 9·cos(θ) - 9 = 0

Solving, we get;

cos(θ) = 0.5 or cos(θ) = -1

Therefore;

θ = arccos(0.5) = 60° and θ = arccos(0.5) = 180°

When θ = 60°, we get;

A(max) = 9·sin(60)·cos(60) + 9·sin(60) ≈ 11.69

The cross sectional area when θ is 60° is about 11.69 square inches which is larger than the 9 square inched when θ = 90°

The angle that maximizes the cross sectional area is θ = 60°

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The statement " the continuity correction must be used because the normal distribution assumes variables whereas the binomial distribution uses discrete variables" is true because  continuity correction is used to adjust for the discrepancy between continuous and discrete variables when approximating a discrete distribution

The continuity correction is used when approximating a discrete distribution, such as the binomial distribution, with a continuous distribution, such as the normal distribution. The normal distribution assumes continuous variables, while the binomial distribution uses discrete variables.

The continuity correction helps to account for the fact that the normal distribution is continuous, whereas the binomial distribution is not. It adjusts the boundaries of the intervals used in the approximation, to better reflect the underlying discrete nature of the data.

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Answer:

Step-by-step explanation:

Sure! Here's a number line showing the distance of 0.75 kilometers:

0 -------------|-------------|------------- 0.75 km

The "0" on the left represents the starting point (such as home), and the "|---|" in the middle represents the distance of 0.75 kilometers to the destination (such as school).