100 POINTS
Don and Celine have been approved for a $400,000, 20-year mortgage with an APR of 3.35%. Using the mortgage and interest formulas, set up a two-month amortization table with the headings shown and complete the table for the first two months.

The geometric sequence is:

[tex]\boxed{15,9,\frac{27}{5},\frac{81}{25}, \frac{243}{125} }[/tex]

A geometric sequence, or geometric progression, is a sequence of numbers where each successive number is the product of the previous number and some constant  [tex]\text{r}[/tex].

Now,

We need to find the 2nd, 3rd and 4th term of the geometric sequence. To find these terms, we need to know the common difference. The common difference can be determined using the formula,

Where [tex]\text{a}_1=15[/tex] and [tex]\text{a}_5=\frac{243}{125}[/tex]

For [tex]\text{n}=5[/tex], we have,

[tex]\dfrac{243}{125}=15(\text{r})^4[/tex]

Simplifying, we have,

[tex]\sf r= \dfrac{3}{5}[/tex]

Thus, the common difference is [tex]\sf r= \frac{3}{5}[/tex]

Now, we shall find the 2nd, 3rd and 4th terms by substituting [tex]\sf n=2,3,4[/tex] in the formula [tex]\text{a}_{\text{n}}=\text{a}_1(\text{r})^{\text{n}-1}[/tex]

For [tex]\sf n=2[/tex]

[tex]\text{a}_2=15\huge \text(\dfrac{3}{5}\huge \text)^1[/tex]

   [tex]=\bold{{9}}[/tex]

Thus, the 2nd term of the sequence is 9

For [tex]\sf n=3[/tex], we have,

[tex]\text{a}_3=15\huge \text(\dfrac{3}{5}\huge \text)^2[/tex]

[tex]=15\huge \text(\dfrac{9}{25}\huge \text)[/tex]

[tex]\bold{=\dfrac{27}{5}}[/tex]

Thus, the 3rd term of the sequence is [tex]\bold{{\frac{27}{5}}}}[/tex]

For [tex]\sf n=4[/tex], we have,

[tex]\text{a}_4=15\huge \text(\dfrac{3}{5}\huge \text)^3[/tex]

[tex]=15\huge \text(\dfrac{27}{25}\huge \text)[/tex]

[tex]\bold{=\dfrac{81}{25}}[/tex]

Thus, the 4th term of the sequence is [tex]\bold{\frac{81}{25}}[/tex]

Therefore, the geometric sequence is:

[tex]\boxed{\bold{15,9,\frac{27}{5},\frac{81}{25}, \frac{243}{125}}}[/tex]

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The probability of winning the jackpot on one try is 1/30 or 0.0333 (rounded to four decimal places).

To calculate the probability of winning the jackpot on one try, we need to determine the number of favorable outcomes (winning combinations) and the total number of possible outcomes.

The number of favorable outcomes can be calculated by selecting 4 different numbers from 1 to 45 (without considering the order) and one number from 1 to 30. This can be represented as:

C(45, 4) * C(30, 1)

where C(n, r) represents the number of combinations of n items taken r at a time.

The total number of possible outcomes is the total number of ways to select 4 numbers from 1 to 45 (without considering order) multiplied by the number of ways to select one number from 1 to 30. This can be represented as:

C(45, 4) * 30

To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = (C(45, 4) * C(30, 1)) / (C(45, 4) * 30)

Simplifying the expression:

Probability = C(30, 1) / 30

Probability = 1 / 30

Therefore, the probability of winning the jackpot on one try is 1/30 or 0.0333 (rounded to four decimal places).

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We have successfully proven the identity cos(4A)/sin(2A) + sin(4A)/cos(2A) = cosec(2A) Thus, we have shown that the given trigonometric identity is true.

To prove the trigonometric identity: cos(4A)/sin(2A) + sin(4A)/cos(2A) = cosec(2A), we'll work on simplifying both sides of the equation separately.

Starting with the left-hand side (LHS):

LHS = cos(4A)/sin(2A) + sin(4A)/cos(2A)

To simplify this expression, we'll use trigonometric identities and algebraic manipulations.

First, let's rewrite cos(4A) and sin(4A) using double-angle formulas:

cos(4A) = cos^2(2A) - sin^2(2A)

sin(4A) = 2sin(2A)cos(2A)

Substituting these expressions into LHS, we have:

LHS = (cos^2(2A) - sin^2(2A))/sin(2A) + (2sin(2A)cos(2A))/cos(2A)

Now, let's simplify each term separately:

LHS = cos^2(2A)/sin(2A) - sin^2(2A)/sin(2A) + 2sin(2A)cos(2A)/cos(2A)

Using the identity cos^2(2A) = 1 - sin^2(2A), we can rewrite the first term:

LHS = (1 - sin^2(2A))/sin(2A) - sin^2(2A)/sin(2A) + 2sin(2A)cos(2A)/cos(2A)

Now, let's combine the fractions with a common denominator:

LHS = (1 - sin^2(2A) - sin^2(2A) + 2sin(2A)cos(2A))/sin(2A)

Simplifying further:

LHS = (1 - 2sin^2(2A) + 2sin(2A)cos(2A))/sin(2A)

Using the identity 2sin(2A)cos(2A) = sin(4A), we have:

LHS = (1 - 2sin^2(2A) + sin(4A))/sin(2A)

Now, let's simplify the right-hand side (RHS) using the reciprocal identity of sin(2A):

RHS = 1/sin(2A) = cosec(2A)

Since LHS = RHS, we have successfully proven the identity:

cos(4A)/sin(2A) + sin(4A)/cos(2A) = cosec(2A)

Thus, we have shown that the given trigonometric identity is true.

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The limit (x -> -2) f(x) is equal to -6.

Let's consider the function f(x) = (x² - 4) / (x + 2). We'll calculate the function values for various x-values in the table below and then determine the limit (x -> -2) f(x).

x f(x)

-3 -5

-2.5 -5.25

-2.1 -5.61

-2.01 -5.9602

-2.001 -5.996002

-2.0001 -5.99960002

Now, let's calculate the limit (x -> -2) f(x) using the function values in the table. As x approaches -2, we can observe that the function values approach a specific value:

lim (x -> -2) f(x) ≈ -6

Therefore, the limit (x -> -2) f(x) is equal to -6.

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Answer:

1 2/3

Step-by-step explanation:

If the company has budgeted 6 2/3 hours to complete a project, and 1/4 of that time is spent on research, we can find the amount of time spent on research as follows:

Total time for the project = 6 2/3 hours

Time spent on research = (1/4) * (6 2/3) hours

We can simplify 6 2/3 to an improper fraction as follows:

6 2/3 = (6 x 3 + 2) / 3 = 20/3

Substituting this value into the equation above, we get:

Time spent on research = (1/4) * (20/3) hours

Multiplying the fractions, we get:

Time spent on research = 5/3 hours

We can convert this improper fraction to a mixed number as follows:

5/3 = 1 2/3

Therefore, the company plans to spend 1 2/3 hours on research.

Step-by-step explanation:

P odd = spin of 1 or 3

P greater than 3  = spin of 4  

   three spins out of 4 possible spins   =  3/4

Part: $1.92 (the discount amount)

Percent: 30% (the discount percentage)

Base: The original price of the item before the discount and sale, which we can calculate as follows:

Let x be the original price of the item.

The discount of $1.92 represents 30% of the original price, so we can write:

0.30x = 1.92

Solving for x, we get:

x = 1.92 / 0.30

x = 6.40

Therefore, the base is $6.40 (the original price of the item before the discount and sale).

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Answer: 40 is the arithmetic mean

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

To find the product of -4 and the vector [8, -1, -5, 9], you need to multiply each element of the vector by -4. Here's the calculation:

-4 * 8 = -32

-4 * -1 = 4

-4 * -5 = 20

-4 * 9 = -36

Therefore, the product of -4 and [8, -1, -5, 9] is:

[-32, 4, 20, -36]

So the correct answer is A: [-32, 4, 20, -36].

y=1/4x-2 is a perpendicular line and y=-4x+3 is a parallel line.

The equation y= -2 -4x in the form of slope intercept form y=mx+b is y=-4x-2.

Where m represents the slope and y intercept is -2.

The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line.

The negative reciprocal of -4 is 1/4.

Therefore, the slope of the perpendicular line is 1/4.

The perpendicular line is y=1/4x-2.

We know that the parallel lines have same slope.

y=-4x+3

Hence, y=1/4x-2 is a perpendicular line and y=-4x+3 is a parallel line.

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The correct sum of numbers from 1 to n is 100.

We have,

Let's assume the correct sum of numbers from 1 to n is S.

According to the problem,

Juan made a mistake and added one of the numbers twice.

Let's call this number x.

The wrong sum that Juan obtained is 100.

The correct sum, S, can be expressed as the sum of numbers from 1 to n excluding the number x, plus the number x itself:

S = (1 + 2 + 3 + ... + (x-1) + (x+1) + ... + n) + x.

Since Juan added x twice, the wrong sum can be expressed as the sum of numbers from 1 to n without excluding any number: 100 = 1 + 2 + 3 + ... + (x-1) + x + (x+1) + ... + n.

We can subtract the correct sum equation (step 4) from the wrong sum equation (step 5) to find the value of x:

100 - S = (1 + 2 + 3 + ... + (x-1) + x + (x+1) + ... + n) - ((1 + 2 + 3 + ... + (x-1) + (x+1) + ... + n) + x).

Simplifying further, we get: 100 - S = x - x = 0.

From step 6, we see that 100 - S = 0, which means S = 100.

Therefore,

The correct sum of numbers from 1 to n is 100.

In conclusion, the correct sum of the numbers from 1 to n is 100.

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The complete question:

Juan wants to find the sum of the numbers from 1 to n, but in doing so he makes a mistake and adds one of these numbers twice, obtaining the wrong result 100.

The surface area of the rectangular cuboid is S = 426 feet²

Given data ,

Let the surface area of the cuboid be S

Let the length of the rectangular cuboid be L = 12 feet

Let the width of the rectangular cuboid be W = 9 feet

Let the height of the rectangular cuboid be H = 5 feet

Now , The total surface area of the cuboid is given by the formula

Surface Area = 2 ( LH + LW + HW )

So, S = 2 ( 12 x 5 + 12 x 9 + 5 x 9 )

S = 2 ( 60 + 108 + 45 )

S = 2 x 213

S = 426 feet²

Hence , the surface area is S = 426 feet²

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Answer:

the answer is 5 the probability of each colors is 5

Answer:

Since all three distances are equal to 2√3, we can conclude that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane. This proves the statement.

Step-by-step explanation:

To prove that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane, we need to show that each side of the triangle has the same length, which is equal to the distance between any two of the points.

Let's first find the distance between points 2 and -1+i√3. We can use the distance formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁, y₁) = (2, 0) and (x₂, y₂) = (-1, √3). Substituting in the values, we get:

d₁ = √[(-1 - 2)² + (√3 - 0)²] = √[9 + 3] = √12 = 2√3

Next, let's find the distance between points -1+i√3 and -1-i√3:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁, y₁) = (-1, √3) and (x₂, y₂) = (-1, -√3). Substituting in the values, we get:

d₂ = √[(-1 - (-1))² + (-√3 - √3)²] = √[0 + 12] = 2√3

Finally, let's find the distance between points -1-i√3 and 2:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁, y₁) = (-1, -√3) and (x₂, y₂) = (2, 0). Substituting in the values, we get:

d₃ = √[(2 - (-1))² + (0 - (-√3))²] = √[9 + 3] = √12 = 2√3

Since all three distances are equal to 2√3, we can conclude that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane. This proves the statement.

Units were not specified in the question, so the distances are expressed in arbitrary units.

Answer:

The next whole number after DB36 in the base-fourteen system would be DB37.

Step-by-step explanation:

In the base-fourteen system, each digit position represents a power of 14. The digits range from 0 to 13, where 0 to 9 represent the numbers 0 to 9, and A to D represent the numbers 10 to 13, respectively.

To determine the next whole number after DB36, we increment the digit in the least significant position. In this case, the least significant position is the units place, represented by the digit 6. Since the next number after 6 is 7, the next whole number in the base-fourteen system is DB37.

Scale factor = (10 inches / 5 inches) = 2

D. 2.

Adding the two cases together, we get a total of 360 + 720 = 1080 six-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 once, such that the number is divisible by its unit digit.

To find the number of 6-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 once, such that the 6-digit number is divisible by its unit digit, we can consider the following cases:

Case 1: The unit digit is 2, 4, or 6

In this case, the unit digit is already divisible by itself. We have 3 choices for the unit digit. For the remaining 5 digits, we have 5 choices for the first digit, 4 choices for the second digit, 3 choices for the third digit, 2 choices for the fourth digit, and 1 choice for the fifth digit. Therefore, the total number of 6-digit numbers is 3 * 5 * 4 * 3 * 2 * 1 = 360.

Case 2: The unit digit is 1, 3, or 5

In this case, the unit digit is not divisible by itself. We have 3 choices for the unit digit. For the remaining 5 digits, we have 5 choices for the first digit, 4 choices for the second digit, 3 choices for the third digit, 2 choices for the fourth digit, and 1 choice for the fifth digit. However, for the sixth digit, it cannot be the same as the unit digit since the number needs to be divisible by the unit digit. Therefore, we have 2 choices for the sixth digit. Hence, the total number of 6-digit numbers is 3 * 5 * 4 * 3 * 2 * 2 = 720.

Adding the two cases together, we get a total of 360 + 720 = 1080 six-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 once, such that the number is divisible by its unit digit.\

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The probability of an atom of substance A chosen at random decaying in the next 70 years is 2.07%.

The decay rate of a radioactive substance can be calculated using the formula: N=N0e−λt, where N0 is the initial number of radioactive atoms, N is the remaining number of radioactive atoms,

t is the time in years, and λ is the decay constant expressed in inverse years.

The probability of a given radioactive atom decaying during a specific period can be found using the formula: P = 1 - e^(-λt).

In this case, the decay rate of substance A is 0.03% per year, which can be expressed as a decimal fraction of 0.0003.

Therefore, the decay constant λ = 0.0003.

The time period of interest is 70 years.

Therefore, the probability of an atom decaying during this period can be calculated as:

P = 1 - e^(-0.0003*70)P

= 1 - e^(-0.021)P

= 0.0207 or 2.07%

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[tex] \sf \large \frac{3x + 4}{x + 2} + \frac{ {x}^{2} + 2x}{2x + 4} [/tex]

[tex] \sf \large \frac{3x + 4}{x + 2} + \frac{ x(x + 2)}{2(x + 2)} [/tex]

[tex] \sf \large \frac{3x + 4}{x + 2} + \frac{ x \cancel{(x + 2)}}{2 \cancel{(x + 2)}} [/tex]

[tex] \sf \large \frac{3x + 4}{x + 2} + \frac{ x}{2} [/tex]

[tex] \sf \large \frac{2(3x + 4) + x(x + 2)}{2(x + 2)} [/tex]

[tex] \sf \large \frac{6x + 8 + {x}^{2} + 2x}{2x + 4} [/tex]

[tex] \sf \large \frac{ {x}^{2} + 8x + 8}{2x + 4} [/tex]

Multiplying 0.79 by √2 yields an irrational answer.

To determine which number produces an irrational answer when multiplied by 0.79, we need to identify a number that, when multiplied by 0.79, results in a non-terminating and non-repeating decimal.

If the result of multiplying a number by 0.79 is a rational number, it would have a terminating or repeating decimal representation. However, if the result is an irrational number, it would have a non-repeating and non-terminating decimal representation.

To find such a number, we can look for an irrational number, such as the square root of a non-perfect square. Let's consider √2.

When we multiply 0.79 by √2, the result is:

0.79 * √2 ≈ 1.1141...

The decimal representation of this product does not terminate or repeat, indicating that it is an irrational number. Therefore, multiplying 0.79 by √2 produces an irrational answer.

In summary, multiplying 0.79 by √2 yields an irrational answer.

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The value of x in the triangle is determined as 45.

The value of x in the triangle is calculated by applying the following formula,.

The measure of angle ABD = 0.2x + 52

The measure of angle CBD = 0.2x + 42

From the diagram, we can set-up the following equations;

x + 16 = 0.2x + 52

Simplify the equation above, by collecting similar terms;

x - 0.2x = 52 - 16

0.8x = 36

Divide both sides of the equation by " 0.8 "

0.8x / 0.8 = 36/0.8

x = 45

Thus, the value of x in the triangle is calculated by equating the appropriate values to each other.

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Answer:

C) 2(x+5) = 48.4

Step-by-step explanation:

Here's what happens each step

1) x

2) x+5

3) 2(x+5) = 48.4

Answer:

The correct answer is C) 2(x + 5) = 48.4.

Step-by-step explanation:

Let's break down the problem step by step:

Therefore, the equation that represents this scenario is 2(x + 5) = 48.4.

To solve for x, we can simplify the equation as follows:

2(x + 5) = 48.4

2x + 10 = 48.4

2x = 48.4 - 10

2x = 38.4

x = 19.2

Therefore, the number Kevin was thinking of was 19.2.

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Let's solve this problem step by step.

1. Start by drawing a diagram to visualize the problem. We have a square piece of sheet metal, and we need to remove squares from each corner to create an open box.

Let's assume the side length of the original square sheet metal is "x" feet.

After removing squares of side length 5 feet from each corner, the remaining dimensions of the sheet metal will be (x - 10) feet.

The height of the box will be 5 feet.

Therefore, the dimensions of the open box will be (x - 10) feet (length and width), 5 feet (height).

2. Use the formula for the volume of a rectangular box to set up an equation:

Volume = Length x Width x Height

Given that the volume is 720 cubic feet, we can write:

(x - 10) * (x - 10) * 5 = 720

Simplifying the equation:

5(x - 10)^2 = 720

3. Solve the equation to find the value of x:

Divide both sides of the equation by 5:

(x - 10)^2 = 144

Take the square root of both sides:

x - 10 = ±12

Solve for x:

x = 10 + 12 or x = 10 - 12

x = 22 or x = -2

Since we are dealing with lengths, we discard the negative value of x.

Therefore, the side length of the original square sheet metal is 22 feet.

4. Finally, calculate the dimensions of the sheet metal:

Length = Width = x - 10 = 22 - 10 = 12 feet

So, the dimensions of the sheet metal are 12 feet by 12 feet.

Answer:

a)9. b) 12

Step-by-step explanation:

a) | -6 -3 | .

-6-3 = -9

|-9| =9

b) | 0 - 12 |.

0-12=-12

|-12|=12

Answer:

  C.  KL = 10.14

Step-by-step explanation:

You want the length of segment KL in ∆JKL given it is similar to ∆XYZ with lengths JK=11.31, XY=8.7, YZ=7.8.

Corresponding sides of similar triangles are proportional. It can be useful to identify corresponding sides and their given measures:

  JK = 11.31, XY = 8.7

  KL = ?, YZ = 7.8

This lets us write the proportion ...

  KL/JK = YZ/XY

  KL/11.31 = 7.8/8.7 . . . . . . . . . . use given values

  KL = 11.31(7.8/8.7) = 10.14 . . . multiply by 11.31

The measurement of KL is 10.14 untis.

__

Additional comment

You have to go by the sequence of vertices in the similarity statement, not the appearance of the figure. One triangle is rotated from the other in the figure, so that parallel sides are not corresponding sides.

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The determination of the independent variable depends on the specific environment and design of the study or trial. It's the variable that's manipulated or controlled by the experimenter to observe its impact on the dependent variable.

The independent variable is the variable that's manipulated or controlled by the experimenter in an trial. It's the variable that's believed to have an effect on the dependent variable. The dependent variable, on the other hand, is the variable that's being measured or observed to determine any changes or goods caused by the independent variable.

To determine which variable represents the independent variable, we need to consider the environment of the trial or study in question. In numerous cases, the independent variable is designedly manipulated or controlled by the experimenter to observe its impact on the independent variable.

For illustration, in an trial testing the effect of different diseases on factory growth, the type of toxin would be the independent variable as it's designedly manipulated by the experimenter. The dependent variable would be the measured factory growth, as it's the variable being observed and affected by the different diseases.

In summary, the determination of the independent variable depends on the specific environment and design of the study or trial. It's the variable that's manipulated or controlled by the experimenter to observe its impact on the dependent variable.

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In this relationship, the arc length is inversely Proportional to the area. As the area increases, the arc length decreases, and vice versa.

The relationship between the arc length (s) and the area (a) of a sector, let's start by understanding the formulas for each.

The arc length of a sector can be calculated using the formula:

s = θr

where s is the arc length, θ is the central angle of the sector in radians, and r is the radius.

The area of a sector can be calculated using the formula:

a = (1/2)θr^2

where a is the area, θ is the central angle of the sector in radians, and r is the radius.

Given that the arc length is equal to one fourth of the radius, we can express this relationship as:

s = (1/4)r

Now, let's express the central angle (θ) in terms of the area (a).

From the formula for the area of a sector, we can rearrange it to solve for θ:

a = (1/2)θr^2

Multiplying both sides by 2/r^2, we get:

2a/r^2 = θ

Substituting this value of θ back into the equation for the arc length (s), we have:

s = (1/4)r = (1/4)r(2a/r^2) = (1/2a)r

Therefore, we have expressed the arc length (s) as a function of the area (a) of the sector:

s = (1/2a)r

In this relationship, the arc length is inversely proportional to the area. As the area increases, the arc length decreases, and vice versa.

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The correct formula to find the area of the surface obtained by rotating the curve, about the x-axis is S = ∫2πy(x) √(1 + (dy/dx)²) dx.

To find the area of the surface obtained by rotating the curve, about the x-axis, we use the formula given below:

S = ∫2πy(x)ds where ds = √(1 + (dy/dx)²) dx

For the curve y = f(x),

we can use the formula given below:

S = ∫2πf(x) √(1 + (dy/dx)²) dx

The student wants to find the area of the surface obtained by rotating the curve, about the x-axis.

So, the correct formula to find the area of the surface is given below:

S = ∫2πy(x)ds = ∫2πy(x) √(1 + (dy/dx)²) dx

Where the curve is y = f(x).

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Answer:

D) 0.203π square units

Step-by-step explanation:

The area of a surface obtained by rotating the curve about the x-axis in the interval [a, b] is given by the formula:

[tex]\large\boxed{\displaystyle S=\int^{b}_{a}2\pi y\sqrt{1+\left(\dfrac{\text{d}y}{\text{d}x}\right)^2}\; \text{d}x}[/tex]

The given interval is 0 ≤ x ≤ 1. Therefore:

The equation of the curve is:

[tex]y=\dfrac{x^3}{3}=\dfrac{1}{3}x^3[/tex]

Differentiate the function using the following differentiation rule:

[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]

Therefore:

[tex]\dfrac{\text{d}y}{\text{d}x}=3 \cdot \dfrac{1}{3}x^{3-1}=x^2[/tex]

Therefore, the integral is:

[tex]\displaystyle S=\int^{1}_{0}2\pi \left(\dfrac{1}{3}x^3\right)\sqrt{1+\left(x^2\right)^2}\; \text{d}x[/tex]

[tex]\displaystyle S=\int^{1}_{0} \left(\dfrac{2\pi}{3}x^3\right)\sqrt{1+x^4}\; \text{d}x[/tex]

[tex]\displaystyle S=\dfrac{2\pi}{3}\int^{1}_{0} x^3\sqrt{1+x^4}\; \text{d}x[/tex]

To evaluate the definite integral, use the method of substitution.

[tex]\textsf{Let}\;u=1+x^4[/tex]

Find du/dx and rewrite it so that dx is on its own:

[tex]\dfrac{\text{d}u}{\text{d}x}=4x^3 \implies \text{d}x=\dfrac{1}{4x^3}\; \text{d}u[/tex]

Find the limits in terms of u:

[tex]\textsf{When}\; x = 0 \implies u=1+0^4=1[/tex]

[tex]\textsf{When}\; x = 1 \implies u=1+1^4=2[/tex]

Rewrite the original integral in terms of u and du:

[tex]\displaystyle S=\dfrac{2\pi}{3}\int^{2}_{1} x^3\sqrt{u} \cdot \dfrac{1}{4x^3}\; \text{d}u[/tex]

[tex]\displaystyle S=\dfrac{2\pi}{3}\int^{2}_{1} \dfrac{1}{4}\sqrt{u}\; \text{d}u[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\int^{2}_{1} \sqrt{u}\; \text{d}u[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\int^{2}_{1} u^{\frac{1}{2}}\; \text{d}u[/tex]

Evaluate the integral using the integration rule:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

Therefore:

[tex]\displaystyle S=\dfrac{\pi}{6}\left[\vphantom{\dfrac12}\dfrac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]^{2}_{1}[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\left[\vphantom{\dfrac12}\dfrac{u^{\frac{3}{2}}}{\frac{3}{2}}\right]^{2}_{1}[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\left[\vphantom{\dfrac12}\dfrac{2u^{\frac{3}{2}}}{3}\right]^{2}_{1}[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\left(\dfrac{2(2)^{\frac{3}{2}}}{3}-\dfrac{2(1)^{\frac{3}{2}}}{3}\right)[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\left(\dfrac{4\sqrt{2}}{3}-\dfrac{2}{3}\right)[/tex]

[tex]\displaystyle S=\dfrac{\pi}{6}\left(\dfrac{-2+4\sqrt{2}}{3}\right)[/tex]

[tex]\displaystyle S=\left(\dfrac{-1+2\sqrt{2}}{9}\right)\pi[/tex]

[tex]\displaystyle S=0.203\pi[/tex]

Therefore, from the given answer options, the correct area is 0.203π square units.

Answer:

1)

A) 0.541 or 0.54(by rounding)

B) 0.675

C)Yes, they are independent because someone doesnt have to attend prom to be a senior and vice versa

2)

a) 0.29

b) 0.04

c) Yes, they are independent because you do not have to live in Long Beach to recommend the provider and vice versa

Step-by-step explanation:

To find the third number in the sequence given that the first number is -4 and the fifth number is 8, we need to determine the pattern or rule that governs the sequence.

One possible pattern is that the sequence is an arithmetic sequence, where each term is obtained by adding a common difference (d) to the previous term.

Let's calculate the common difference using the first and fifth numbers:

Common difference (d) = 8 - (-4)

= 8 + 4

= 12

Now, we can find the third number by adding the common difference to the first number:

Third number = First number + 2 * Common difference

= -4 + 2 * 12

= -4 + 24

= 20

Therefore, the third number in the sequence is 20.