Find F'(x): F(x) = Sx² 1 (t³ - 4t² + 2)dt

To investigate whether there is convincing evidence, at a 5 percent level of significance,

that a longer hand length is associated with a decrease in the time to complete the 100-meter freestyle, the researcher should use a two-sample t-test.

The independent variable is hand length, and the dependent variable is time to complete the 100-meter freestyle.

The t-test compares the mean time to complete the 100-meter freestyle for swimmers with longer hand lengths to the mean time for swimmers with shorter hand lengths.

The t-test is appropriate because the sample size is small, and the researcher is comparing two groups.

By conducting a two-sample t-test, the researcher can determine whether the observed difference in mean times is statistically significant or due to chance.

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Answer:

The answer to your problem is, 149.5

Step-by-step explanation:

How to find our number from the problem, 115% of 130.

Calculate:

[tex]\frac{115}{100}[/tex] of 130 = [tex]\frac{115}{300}[/tex] × 130

= 149.5

The new snowboard is 149.5 centimeters now.

Thus the answer to your problem is, 149.5

The length of CB is 4√(3), and we used the Pythagorean theorem and the perpendicular bisector theorem to solve for it.

By the perpendicular bisector theorem, if a point lies on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the segment. Therefore, in triangle ABC, since ray DB is the perpendicular bisector of AC, it follows that BD = DC.

Let x be the length of CB. Then, by the Pythagorean theorem in triangle ABC, we have:

[tex]AB^2 + BC^2 = AC^2[/tex]

Substituting AB = 12 and BD = DC = x/2, we get:

[tex]12^2 + x^2 = (2x)^2[/tex]

[tex]144 + x^2 = 4x^2[/tex]

[tex]3x^2 = 144[/tex]

[tex]x^2[/tex]= 48

x = √(48) = 4√(3)

Therefore, the length of CB is 4√(3), and we used the Pythagorean theorem and the perpendicular bisector theorem to solve for it.

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The equivalent expression to the -49y - 14 are as follow,

7(-7y -2) , -7(7y + 2) ,-1(49y + 14), -14(3.5y + 1), -98/7 - 49y and -7(7y + k₁) + k₂ where -14 = -k₁ + k₂.

Expression is equal to

-49y - 14

The equivalent expressions are ,

By taking -1 as common factor

-(49y + 14)

By taking -7 as common factor

-7(7y + 2)

By taking 7 as common factor .

7(-7y -2)

By taking -14 as common factor

-14(3.5y + 1)

By replacing -14  as -98/7

-98/7 - 49y

By replacing -14 = -21 + 7

-7(7y + 3) + 7

By replacing -14 = -28 + 14

-7(7y + 4) + 14

By replacing -14 = -35 + 21

-7(7y + 5) + 21

By replacing -14 = -42 + 28

-7(7y + 6) + 4

and many more.

All of these expressions are equivalent to -49y -14.

Therefore, the expression which are equivalent to the given expression are 7(-7y -2) , -7(7y + 2) ,-1(49y + 14), -14(3.5y + 1), -98/7 - 49y and -7(7y + k₁) + k₂ where -14 = -k₁ + k₂.

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The radius of convergence is given as [tex]i = {x | 0 \leq x < \frac{1}{3} }[/tex]

To find the radius of convergence, we can use the ratio test:

[tex]\frac{lim |(n+1)!(6x-1)^n^+^1|}{|n!(6x-1)^n|} \\\\[/tex]

[tex]= lim |6x-1| \\=|6x-1|[/tex]

The series will converge if this limit is less than 1. So we solve the inequality:

[tex]|6x-1| < 1[/tex]

which gives us:

-1 < 6x - 1 < 1

0 < 6x < 2

[tex]0 < x < \frac{1}{3}[/tex]

Therefore, the radius of convergence is [tex]\frac{1}{6}[/tex]

To find the interval of convergence, we check the endpoints of the interval [0, [tex]\frac{1}{3}[/tex]]:

When x = 0, the series becomes:

∑(n=1 to infinity) n!(6x-1)^n = ∑(n=1 to infinity) n!(-1)^n

which diverges by the test for divergence.

When x = 1/3, the series converges by the ratio test. Therefore, the interval of convergence is:

i = [0, 1/3)

or in set-builder notation:

[tex]i = {x | 0 \leq x < 1/3}[/tex]

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If area of cattle's "square-field" is 40000 ft², and Mr. Blackburn is constructing a fence diagonal in the field, then length of cattle field's fence should be 282.84 ft.

The area of the Blackburn's family cattle square-field is 40000 ft²,

So, We First, equate this area with area formula,

On Equating ,We get,

⇒ (side × side) = 40000,,

⇒ side is 200 ft,

Now, we substitute the "side-length" of field as 200 ft, in the formula for diagonal of a square,

we get,

⇒ Length of cattle-field's diagonal is = (side)√2,

⇒ Length of cattle-field's diagonal is = (200)√2,

⇒ Length of cattle-field's diagonal is ≈ 282.84 ft.

Therefore, the length of cattle field's fence is 282.84 ft.

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The length of the sides are

AC = 5.4

BC = 10.5

To determine the value, we need to know the different trigonometric identities.

They include;

These identities also have their ratios;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

From the information given, we have;

tan 31 = AC/9

cross multiply the values

AC = 5. 4

sin 59 = 9/BC

cross multiply the values

BC = 10. 5

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The probability of P(x < 4) is 0.6922 or approximately 69.22%.

The exponential distribution is often used to model the time between events that occur randomly and independently at a constant rate over time. The probability density function of the exponential distribution with parameter λ is given by f(x) = λe^(-λx) for x ≥ 0.

In this case, X ~ Exp(0.75) means that the parameter λ is 0.75. To find P(x < 4), we need to calculate the area under the curve of the probability density function to the left of 4. This can be done by integrating the function from 0 to 4 as follows

P(x < 4) = ∫₀⁴ λe^(-λx) dx

= [-e^(-λx)]₀⁴

= -e^(-0.75 * 4) + 1

= 0.6922

Therefore, the probability that X is less than 4 is 0.6922 or approximately 69.22%.

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The volume of the sphere B is 14.13 cubic units.

Given that the two similar spheres, the scale factor of the lengths of the radii of Sphere A to Sphere B is 4.

So, we need to find the volume of the sphere B,

So,

Radius A / Radius B = 4/1

6 / Radius B = 4

Radius B = 1.5

Now we know that the volume of a sphere = 4/3 × πr³

= 4/3 × 3.14 × 1.5³

= 42.39/3

= 14.13

Alternatively, you can calculate by =

We know that the ratio of the volumes of two similar solids is equal to the cube of their scale factor, so here, the scale factor is 4/1.

Therefore,

Volume A / Volume B = (4/1)³

288π / Volume B = 64

Volume B = 4.5π

Volume B = 14.13

Hence, the volume of the sphere B is 14.13 cubic units.

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The points that belong to the image of the equation of a line are (4, - 8) and U' = (1, 4).

In this question we find the definition of the equation of a line, whose image must be found by a kind of rigid transformation known as dilation:

U'(x, y) = O(x, y) + k · [U(x, y) - O(x, y)]

Where:

If we know that O(x, y) = (5, - 8), k = 1 / 4 and y = - 4 · x - 4, then the image of the point:

U'(x, y) = (5, - 8) + (1 / 4) · [(x, - 4 · x - 4) - (5, - 8)]

U'(x, y) = (5, - 8) + (1 / 4) · (x - 5, - 4 · x + 4)

U'(x, y) = (5, - 8) + (x / 4 - 5 / 4, - x + 1)

U'(x, y) = (x / 4 + 15 / 4, - x - 7)

Now we evaluate the expression at each x-value:

x = 1

U' = (1 / 4 + 15 / 4, - 1 - 7)

U' = (4, - 8) (YES)

x = - 11

U' = (- 11 / 4 + 15 / 4, - (- 11) - 7)

U' = (1, 4) (YES)

x = - 23

U' = (- 23 / 4 + 15 / 4, - (- 23) - 7)

U' = (- 2, 16) (NO)

x = 17

U' = (17 / 4 + 15 / 4, - 17 - 7)

U' = (8, - 24) (NO)

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The value of angles in inscribed quadrilateral are μ(∠zyx) is 92⁰ and μ(∠yxw) is 65⁰.

An inscribed quadrilateral is a quadrilateral that can be inscribed in a circle, meaning that its vertices all lie on the circumference of a circle. In other words, the four vertices of an inscribed quadrilateral are concyclic.

The opposite angles of an inscribed quadrilateral are supplementary, which means that they add up to 180 degrees. This property is known as the "interior angle sum" of a quadrilateral.

μ(∠zyx) + μ(∠xwz) = 180⁰ (opposite angle of cyclic quadrilateral are supplementary)

μ(∠xwz) = 88⁰

μ(∠zyx) = 180⁰ - 88⁰ = 92⁰

μ(∠yzw) is an inscribed angle that intercepts the arc 112⁰ and 118⁰. Therefore,

μ(∠yzw)

= (112⁰ + 118⁰)/2

= 230⁰/2

= 115⁰

μ(∠yxw) + μ(∠yzw) = 180⁰  (opposite angle of cyclic quadrilateral are supplementary)

μ(∠yxw) = 180⁰ - 115⁰ = 65⁰

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If a model predicting minutes includes "ftmade" as an explanatory variable, the model would have two parameters: the intercept and the slope of "ftmade." The intercept represents the expected minutes when "ftmade" is zero, and the slope represents the expected increase in minutes for every one-unit increase in "ftmade."

The number of parameters in a model that predicts mins by including ftmade as an explanatory variable depends on the type of model being used.

If a simple linear regression model is used, the model would have two parameters: the intercept and the slope coefficient for the ftmade variable.

If a multiple linear regression model is used, which includes more than one explanatory variable, the model would have additional parameters for each additional explanatory variable included.

For example, if the model also included the variables age and gender, the model would have four parameters: the intercept, the slope coefficient for ftmade, the slope coefficient for age, and the coefficient for gender (assuming gender is coded as a binary variable).

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Answer:

1. 75%

2. 0%

3. 0.1

4. one half

5. six tenths

6. 0.75

7. 64%

Step-by-step explanation:Answer 1:

The spinner has a total of 8 sections, out of which there are 2 yellow sections. Therefore, the probability of not getting a yellow section when the spinner is spun once is 6/8 or 3/4, which is equal to 75%.

Answer 2:

The spinner has only three sections and none of them is colored yellow. Therefore, the probability of getting a yellow section when the spinner is spun once is 0%.

Answer 3:

The total number of students surveyed is:

1 + 3 + 2 + 5 + 9 + 7 + 3 = 30

The number of students who studied for 1 hour is 3. Therefore, the probability of a student studying for 1 hour is 3/30, which simplifies to 1/10 or 0.1.

Answer 4:

Since the coin is fair and has one side colored yellow and the other side colored green, the probability of getting a yellow side when the coin is tossed is 1/2 or one half.

Answer 5:

The set of cards has 10 letters, out of which 4 are vowels (A, I, U). Therefore, the number of consonants in the set is 10 - 4 = 6. The probability of drawing a consonant is therefore 6/10, which simplifies to 3/5 or 0.6.

Answer 6:

The number cube has 8 sides, numbered 1 through 8. The probability of getting a number greater than 2 is the same as the probability of getting any number from 3 to 8. There are 6 such numbers out of 8 total numbers, so the probability is 6/8 or 3/4, which is equal to 0.75.

Answer 7:

The total number of marbles in the bag is:

2 + 4 + 10 + 9 = 25

The number of marbles that are not purple is:

2 + 4 + 10 = 16

Therefore, the probability of not getting a purple marble when one marble is chosen from the bag is 16/25, which is equal to 64%.

Answer:

On x < -5, the function is increasing.

Answer:

$2560 is the total amount to be paid back

Step-by-step explanation:

This is Simple interest described in the question

P = Principal amount

  = Amount of money borrowed

  = $1600  

R = Rate of interest

  = 10% per year

  = 10% per annum

T = Time period

  = 6 years

Make sure the base units of R and T are the same:

S.I. = Simple Interest = [tex]\frac{PRT}{100}[/tex]

                                    = [tex]\frac{(1600 Dollars)(10\frac{Percent}{Year})(6Years)}{(100 Percent)}[/tex]

                                    = $960

This means:

Total amount to be paid back = P + S.I.

                                                   = $1600 + $960

                                                   = $2560                

The right response is "the average number of targeted keywords in a website is different than 5 keywords." The website developer asserts that the value of 5 in the article does not accurately reflect the genuine population mean of the number of keywords targeted in a website.

This claim may be one-tailed (if the website developer thinks the true mean is larger or less than 5) or two-tailed (if the website developer thinks the true mean is merely different from 5) in nature. The website developer feels the true mean is different from the value given in the article, without stating whether it is larger or less than 5. As a result, the claim is two-tailed in this instance.

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a) The Venn diagram is as follows:

b) After selecting at random a Foster junior for a survey, the probability this student has taken at least 2 out of 3 courses is 0.61

c) After selecting at random a Foster junior for a survey, the probability this student has taken exactly one of the three courses is 0.06.

d) After selecting at random a Foster junior for a survey, the probability this student has taken at the most one course is 0.14.

b) To find the probability that the student has taken at least 2 out of 3 courses, we add the probabilities of the following three events: taking both Finance and Accounting, taking both Finance and Marketing, and taking both Accounting and Marketing, plus the probability of taking all three courses:

Therefore, the probability that the student has taken at least 2 out of 3 courses is 0.61.

c) To find the probability that the student has taken exactly one of the three courses, we add the probabilities of the following three events: taking Finance only, taking Accounting only, and taking Marketing only:

Therefore, the probability that the student has taken exactly one of the three courses is 0.06.

d) To find the probability that the student has taken at most one course, we add the probabilities of the following two events: taking no courses and taking exactly one course:

Therefore, the probability that the student has taken at most one course is 0.14.

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The expressions are subtracted to give -a⁴ + a²b² + 4b⁴

It is important to note that algebraic expressions are described as expressions that are made up of terms, variables, constants, coefficients and factors.

These expressions are also identified with arithmetic operations. These operations are;

From the information given, we have that;

4a⁴ + 4a²b² - 2b⁴

3a⁴ + 5a²b² + 2b⁴

Now, subtract the expressions;

3a⁴ + 5a²b² + 2b⁴ - (4a⁴ + 4a²b² - 2b⁴)

expand the bracket

3a⁴ + 5a²b² + 2b⁴ - 4a⁴ - 4a²b² + 2b⁴

add  the  values

-a⁴ + a²b² + 4b⁴

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Answer:

The answer is 3x-5y

Step-by-step explanation:

13x+2y-10x-7y

C.L.T.

13x-10x-7y+2y

3x-5y

Answer:

3x - 5y

Step-by-step explanation:

you 1st have to collect like terms which has the same variable

13x + 2y - 10x - 7y

(13x - 10x) + (2y - 7y)

3x - 5y .... is the simplified form of the equation.

(L3) The orthocenter will lie in the exterior of a(n)  obtuse triangle. Contrary to an acute triangle, the orthocenter of an obtuse triangle will lie in the exterior of the triangle.

The orthocenter is the point of intersection of the altitudes of the triangle, which are perpendicular lines drawn from each vertex to the opposite side. In an obtuse triangle, one of the angles measures more than 90 degrees, so when an altitude is drawn from this vertex to the opposite side, it will lie outside of the triangle. Therefore, the three altitudes will intersect outside of the triangle, forming the orthocenter in the exterior. This is in contrast to an acute triangle, where all three altitudes intersect inside the triangle.

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The answer is that the 70% confidence interval for the slope of the regression line using the provided data is between 0.0019 and 0.0037.


To calculate the 70% confidence interval for the slope of the regression line, we need to use the t-distribution with degrees of freedom equal to n - 2, where n is the number of data points. In this case, n = 116, so we have 114 degrees of freedom.

Using a statistical software or calculator, we can find that the t-value for a 70% confidence interval with 114 degrees of freedom is approximately 1.648.

Next, we need to calculate the standard error of the slope, which is given by:

SE =√[ (SS_residuals / (n - 2)) / SS_x ]

where SS_residuals is the sum of squared residuals, SS_x is the sum of squared deviations of x from its mean, and n is the sample size.

Using the regression model provided, we can find that SS_residuals = 6.3574 and SS_x = 1484.9584. Plugging these values into the formula, we get:

SE = √[ (6.3574 / (116 - 2)) / 1484.9584 ] = 0.00044

Finally, we can calculate the confidence interval for the slope using the formula:

slope +/- t * SE

where slope is the estimated slope from the regression model.

Plugging in the values, we get:

slope +/- 1.648 * 0.00044 = 0.0028 +/- 0.0007

Therefore, the 70% confidence interval for the slope of the regression line is between 0.0019 and 0.0037, rounded to 4 decimal places.

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Joan should run 13 times around the park to complete her goal.

Given that, Joan wants to jog in a circular track which 1/4 mile in diameter.

We need to find the circumference of the track and the number of rounds she needs run to complete her goal.

So,

Circumference = π × diameter

= 3.14 × 1/4 = 0.785 miles

Let she runs x rounds to complete her goal,

So,

0.785x = 10

x = 12.73

x ≈ 13

Hence, Joan should run 13 times around the park to complete her goal.

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a) The sample mean years to maturity for corporate bonds = 16.9625

and the sample standard deviation =  8.2232

b) A 95% confidence interval for the population mean years to maturity: (14.4141, 19.5112)

c) The sample mean yield on corporate bonds is 4.5405

and the sample standard deviation = 2.3082

d) A 95% confidence interval for the population mean yield on corporate bonds:  (3.825, 5.256)

a) The mean of the sample would be,

[tex]\bar{x}[/tex] = (10.25 + 28 +  23 + 13.25 + 3, 7.5 + 26.5 + 21.25 + 3.25, 19 + 9.25 + 28.75 + 1.75 + 17 + 8.75 + 24 + 24.5 + 18+ 11.75 + 22 + 22.75 + 27.75 + 16.75 + 12 + 16.5 + 23.75 + 25.25 + 25.75 + 22.5 + 1.25 + 19.5 + 12.5 + 27.25 + 19.5 + 17.75 + 11.5+ 3.5 + 20 + 25.25 + 6.75) / 40

[tex]\bar{x}[/tex] = 678.5 / 40

[tex]\bar{x}[/tex] = 16.9625

And  the sample standard deviation would be,

s = √(67.6203)

s =  8.2232

b)

We know that the formula for the confidence interval is,

CI = [tex]\bar{x}[/tex] ± (z × s/√n)

Here, n = 40, [tex]\bar{x}[/tex] = 16.9625, s = 8.2232 and z = 1.9600

Using above formula the 95% confidence interval for the population mean years to maturity would be,

CI = 16.9625 ± (1.9600 × 8.2232/√40)

CI = (16.9625 ± 2.548)

CI =  (16.9625 - 2.548,  16.9625 + 2.548)

CI = (14.4141, 19.5112)

c) Consider sample yield on corporate bonds.

The mean would be,

[tex]\bar{x}[/tex] = 181.62 / 40

[tex]\bar{x}[/tex] = 4.5405

And the standard deviation would be,

s = √(5.327594)

s = 2.3082

d) Now we construct a 95% confience interval.

Here, n = 40, s = 2.3082, [tex]\bar{x}[/tex] = 4.4505, and z = 1.9600

Using above formula the 95% confidence interval for the population mean years to maturity would be,

CI =  4.4505 ± (1.9600 × 2.3082/√40)

CI = (4.5405 ± 0.716)

CI = (4.5405 - 0.716, 4.5405 + 0.716)

CI = (3.825, 5.256)

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Find the complete question below.

The 5 number summary of the data set. Min=Q1=Median=Q3=Max is calculated as below.

To help you find the 5 number summary of the data set for the survey of 50 college students' weekly income. I'll provide a step-by-step explanation on how to calculate the 5 number summary.

1. First, arrange the data set in ascending order (from smallest to largest).

2. To find the minimum value (Min), simply identify the smallest number in the data set.

3. To find the maximum value (Max), identify the largest number in the data set.

4. To find the median (the middle value), follow these steps:
  a. If the data set has an odd number of values, the median is the middle value.
  b. If the data set has an even number of values, find the average of the two middle values.

5. To find the first quartile (Q1), you'll calculate the median of the lower half of the data set (excluding the overall median if the data set has an odd number of values).

6. To find the third quartile (Q3), calculate the median of the upper half of the data set (again, excluding the overall median if the data set has an odd number of values).

Once you've completed these steps, you'll have the 5 number summary for the data set:
Min, Q1, Median, Q3, Max.

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To model the decrease in the value of the car over time, we can use an exponential function of the form:

V(t) = V(0) * e^(-rt)

where:

V(0) is the initial value of the car (in this case, $20,000).

r is the annual rate of depreciation, expressed as a decimal (in this case, 0.103).

t is the number of years since the car was purchased.

Plugging in the given values, we get:

V(t) = $20,000 * e^(-0.103t)

This is the exponential function that models the value of the car in t years.

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Answer:

Both shots good: .8(.8) = .64 = 64%

Exactly one shot good: 2(.2)(.8) = .32 = 32%

Neither shot good: .2(.2) = .04 = 4%

∫∫∫ x² dv over the region e is (3/40) a⁵ π².

What is volume?

A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder, or a sphere. Different shapes have different volumes.

To use spherical coordinates, we need to express the volume element dv in terms of ρ, θ, and φ, where ρ is the radial distance from the origin, θ is the angle in the xy-plane measured from the positive x-axis, and φ is the angle measured from the positive z-axis.

We have x² = ρ² sin²(φ) cos²(θ). The volume element dv can be expressed as dv = ρ² sin(φ) dρ dφ dθ.

The region e is bounded by the xz-plane and the hemispheres y = ±√(a² - x²), where a > 0.

Since we're only interested in the part of e that lies in the first octant, we can restrict θ to the range [0, π/2] and φ to the range [0, π/2].

Note that since y is always positive in the first octant, we don't need to consider the negative hemisphere.

So, the integral we need to evaluate is:

∫∫∫ x² dv

= ∫[0,π/2]∫[0,π/2]∫[0,a] (ρ² sin³(φ) cos²(θ)) ρ² sin(φ) dρ dφ dθ

= ∫[0,π/2]∫[0,π/2]∫[0,a] ρ⁴ sin⁴(φ) cos²(θ) dρ dφ dθ

= ∫[0,π/2]∫[0,π/2] [(1/5)ρ⁵ sin⁴(φ) cos²(θ)]|[0,a] dφ dθ

= (1/5) a⁵ ∫[0,π/2] cos²(θ) dθ ∫[0,π/2] sin₄(φ) dφ

= (1/5) a⁵ [(1/2)π] [(3/4)(π/2)] = (3/40) a⁵ π².

Therefore, ∫∫∫ x² dv over the region e is (3/40) a⁵ π².

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The distance of the electricity pole from T is 53 km

Given that;

The ST is 54 kilometer long

(180 - 28) + (180 - 70) = 262 degree is the angle PST,

which is the product of the angles at S and T.

S has a 62 degree angle.

The law of sines can be written as:

sin(62)/SP = sin(262/ST)

To find SP, we can rearrange this equation as follows:

SP = sin(62)/sin(262)x(ST)

When we enter the values we are aware of, we obtain:

SP = sin(62)/sin(262)*54 km

We can evaluate this expression to determine that:

SP ≈ 53.5 km Consequently, 53.5 kilometer or so separate T from the electricity pole.

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0. 81818181818 as a fraction in its simplest form is 9/11

Converting the decimal to fraction we have

x = 81818181818/100000000000

Using a calculator we san reduce the fraction to 9/11

Therefore, 0.81818181818... as a fraction in its simplest form is 9/11.

This is a recurrence fraction and a recurring fraction, also called a recurring decimal, is a decimal number with repeating digits after the decimal point. The repeatability of the lines is indicated by a horizontal line or a line above the repetition line.

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Answer:

the answer is symmetry.