If there are three different samples of the same size from a set population, is it possible to get three different values for the same statistic?

Answer:

The coordinate of [tex]Z=7.3[/tex].

Step-by-step explanation:

To determine:

What is the coordinate of Z?

Fetching Information and Solution Steps:

The distance [tex]XY[/tex] = [tex]3.8-1.3 = 2.5[/tex]

As

Y is [tex]\frac{5}{12}[/tex] of the way from [tex]X[/tex] to [tex]Z[/tex]

So,

[tex]YZ[/tex] is [tex]\frac{7}{5}[/tex] as big as [tex]XY[/tex]

Therefore,

[tex]Z = 3.8 + (\frac{7}{5} )(2.5)[/tex]

[tex]Z=3.8+\frac{17.5}{5}[/tex]

[tex]Z=7.3[/tex]

Therefore, the coordinate of [tex]Z=7.3[/tex].

Keywords: coordinate, point, length, ratio

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The coordinate of Z is (4.43, 11.57)

The coordinate of X is (1, 3)

The coordinate of Y is (3, 8)

Let the coordinate of z be (x₂, y₂)

The ratio of division is 5:7

m = 5, and n = 7

Let the coordinate of y be (a, b)

a = 3, b = 8

[tex]a=\frac{mx_1+nx_2}{m+n}\\\\3= \frac{5(1)+7x_2}{5+7}\\\\3(12)=5+7x_2\\\\7x_2=36-5\\\\7x_2=31\\\\x_2=4.43[/tex]

[tex]ab=\frac{my_1+ny_2}{m+n}\\\\8= \frac{5(3)+7y_2}{5+7}\\\\8(12)=15+7y_2\\\\7y_2=96-15\\\\7y_2=81\\\\y_2=11.57[/tex]

The coordinate of Z is (4.43, 11.57)

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Answer:

As a fraction, the answer is exactly 86/969

In decimal form, the answer is approximately 0.08875

=====================================================

Work Shown:

The assumption is that no replacements are made for each selection.

5 red, 6 blue, 8 green

5+6+8 = 19 total

A = P(3 red) = (5/19)*(4/18)*(3/17) = 10/969

B = P(3 blue) = (6/19)*(5/18)*(4/17) = 20/969

C = P(3 green) = (8/19)*(7/18)*(6/17) = 56/969

D = P(3 all same color)

D = A+B+C

D = 10/969 + 20/969 + 56/969

D = (10+20+56)/969

D = 86/969

D = 0.08875

Answer:

v(9)=2000

V(15)=8000

Step-by-step explanation:

Answer:

Step-by-step explanation:

You are given a function V(t) and asked to find V(9) and V(15). In each case, the number in parentheses replaces "t" in the function expression. After that, it is just arithmetic. (You may want to use a calculator, but it isn't really necessary.)

Given:

  V(t) = 250·2^(t/3)

Find:

  V(9) = 250·2^(9/3)

  = 250·2^3

  = 250·8

  V(9) = 2000

__

  V(15) = 250·2^(15/3)

  = 250·2^5

  = 250·32

  V(15) = 8000

_____

As you know, an exponent signifies repeated multiplication. The exponent tells you the number of times the base is a factor in the product:

  2^3 = 2·2·2 = 8 . . . . . . the exponent of 3 means 2 is a factor 3 times

Answer:

The answer to your question is below

Step-by-step explanation:

a) x² = 196

   x = ±√196

  x = ± 14       or     x = -14  and x = 14

b) 9x² = 49

      x² = 49/ 9

      x = √(49/9)

      x = ±7/3

      x = ± 2.3      or  x = 2.3 and  x = -2.3

c)  4x² - 8 = 56

    4x² = 56 + 8

    4x² = 64

      x² = 64/4

      x² = 16

      x = ±√16

     x = ±4        or   x = 4 and   x = -4

d) x² = 160

   x = ± √160

Find the prime factors of 160

          160    2

            80    2

             40   2

             20   2

              10   2

               5   5

                1                  160 = 2⁵5      

 x = √2⁴10      

x =  ±4√10

e) x² = 72

    Find the prime factor of 72

    72 = 2²3²6

   x = √2²3²6

   x = ±6√6 or  

   x = ±8.5

Answer:

False

Step-by-step explanation: Are you asking whether or not its false? If so then yes f(x) is not a function because the x's on the left side repeat. Example the six points to two different numbers on the right side. I hope i helped!

Answer:

  your answer is correct

Step-by-step explanation:

When a relation has an input that maps to more than one output, it is not a function.  Both 6 and 12 map to two different outputs, so the relation is not a function. Your answer choice is correct.

Answer:

The population in 2001 was = 23.95 millions

Step-by-step explanation:

Given

Population in 2006 = 30.3 million

Rate of growth since 2001 = 4.7%

To find the population in 2001.

Solution:

The population growth formula is give as:

[tex]P=P_0\times e^{rt}[/tex]

[tex]P\rightarrow[/tex] New population

[tex]P_o\rightarrow[/tex] Initial population

[tex]r\rightarrow[/tex] rate of growth

[tex]t\rightarrow[/tex] time

Data given to us:

New population [tex]P[/tex] (2006) = 30.3 million

Rate = 4.7% = 0.047

Time = [tex]2006-2001[/tex]= 5 years

Plugging in the data in the formula.

[tex]30.3=P_0\times e^{0.047\times 5}[/tex]

[tex]30.3=P_0\times e^{0.235}[/tex]

[tex]30.3=P_0\times 1.265[/tex]

Dividing both sides by 1.265

[tex]\frac{30.3}{1.265}=\frac{P_0\times 1.265}{1.265}[/tex]

[tex]23.95=P_o[/tex]

∴ [tex]P_o=23.95[/tex]

Thus, population in 2001 was = 23.95 millions

The equation for the function with the given characteristics is:

f(x) = (1 * cos((x - pi)/pi)) + (5/2)

We have,

The equation for the function with the given characteristics can be written as follows:

f(x) = (1 * cos((x - pi)/pi)) + (5/2)

Amplitude: The amplitude of the cosine curve is 1, so we multiply the cosine function by 1.

Period: The period of the cosine curve is pi, which means it completes one full cycle over the interval [0, pi]. The period of the cosine function is determined by the argument inside the cosine function, (x - pi)/pi. This shifts the period to start at pi and end at 2pi.

Left phase shift: The left phase shift of pi is achieved by subtracting pi from x inside the argument of the cosine function. This shifts the curve to the left by pi units.

Vertical translation: The vertical translation up 5/2 units is achieved by adding 5/2 to the entire function.

Therefore,

The equation for the function with the given characteristics is:

f(x) = (1 * cos((x - pi)/pi)) + (5/2)

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The equation for the function is y(t) = cos(2t + pi) + 5/2.

To write an equation for the given characteristics, we can start with the general equation for a cosine curve: y(t) = A cos(wt + p).

Given that the period is pi, we know that the angular frequency w is equal to 2pi divided by the period, so w = 2pi/pi = 2. The amplitude is 1, so A = 1.

The left phase shift is pi, so p = pi. Finally, there is a vertical translation up 5/2 units, so we add 5/2 to the equation.

Putting all these values into the equation, we get: y(t) = cos(2t + pi) + 5/2.

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Answer:

(x) free throw = 8

(y) 2-point = 6

(z) 3-point = 2

x+(y*2)+(z*3)

8+(6*2)+(2*3)

8+12+6

=26

Answer:

Side of 22 and height of 11

Step-by-step explanation:

Let s be the side of the square base and h be the height of the tank. Since the tank volume is restricted to 5324 ft cubed we have the following equation:

[tex]V = s^2h = 5324[/tex]

[tex]h = 5324 / s^2[/tex]

As the thickness is already defined, we can minimize the weight by minimizing the surface area of the tank

Base area with open top [tex]s^2[/tex]

Side area 4sh

Total surface area [tex]A = s^2 + 4sh[/tex]

We can substitute [tex]h = 5324 / s^2[/tex]

[tex]A = s^2 + 4s\frac{5324}{s^2}[/tex]

[tex]A = s^2 + 21296/s[/tex]

To find the minimum of this function, we can take the first derivative, and set it to 0

[tex]A' = 2s - 21296/s^2 = 0[/tex]

[tex]2s = 21296/s^2[/tex]

[tex]s^3 = 10648[/tex]

[tex]s = \sqrt[3]{10648} = 22[/tex]

[tex]h = 5324 / s^2 = 5324 / 22^2 = 11[/tex]

To find the dimensions of the rectangular tank with a square base and minimum weight, we set up the volume and surface area formulas in terms of the base's side length and the tank's height, then use calculus to find the dimensions that minimize the surface area.

The problem is a typical unconstrained optimization problem in calculus. The volume of the tank is given as 5324 cubic feet. Since it's a rectangular tank with a square base, we can let x be the length of the side of the base and h be the height of the tank. Then, the volume of the tank is x^2*h = 5324.

The weight of the tank is proportional to the amount of steel used, which, in turn, is proportional to the surface area of the tank. The surface area of the tank is x^2 + 4*x*h. From the volume formula, we can express h in terms of x: h = 5324 / x^2. Substituting it to the surface area formula, we get the surface area is a function of x, such that "A = x^2 + 4x * 5324 / x^2".

To find the dimensions that minimize the weight, we find the derivative of A with respect to x and set it equal to 0. This will give the x that minimizes the surface area. Then substitute this value back into the equation h = 5324 / x^2 to get the optimal height.

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Answer:

y=22000x+40000

Answer:

its a combination

Step-by-step explanation:

because three horses is a combination of horses

Answer:

  y = -2

Step-by-step explanation:

Any asymptotes of a rational function will be described by the quotient of the numerator and denominator (excluding any remainder).

  [tex]f(x)=\dfrac{-2x}{x+1}=-2+\dfrac{2}{x+1}[/tex]

The horizontal asymptote is ...

  y = -2

Answer:

Step-by-step explanation:

The given function is

[tex]f(x)=\frac{-2x}{x+1}[/tex]

An asymptote refers to a restriction in the domain or range set of the function. This happens to rational functions, because there's a scenario where the function is undetermined: when its denominator is zero.

So, in this case, the value that makes the denominator zero is

[tex]x+1=0\\x=-1[/tex]

That means, we need to restrict the domain for [tex]x=-1[/tex]. But this is a vertical asymptote.

To find horizontal asymptotes, we need to find the restrictions for the range of the function, we do that isolating [tex]y[/tex].

[tex]y=\frac{-2x}{x+1}\\yx+y=-2x\\yx+2x=-y\\x(y+2)=-y\\x=-\frac{y}{y+2}[/tex]

So, if we analyse the denominator

[tex]y+2=0\\y=-2[/tex]

Therefore, the restriction is [tex]y=-2[/tex]. In other words, the vertical asymptote is the first choice.

Answer:it will take 0.25 years

Step-by-step explanation:

Let x represent the number of years that it will take for type A and tree B to be the same height.

Let y represent the height of type A after x years.

Let z represent the height of type B after x years.

Type A is 5 feet tall and grows at a rate of 13 inches per year. This means that

y = 5 + 13x

Type B is 3 feet tall and grows at a rate of 21 inches per year. This means that

z = 3 + 21x

To determine the number of years it will take both trees to be of the same height, we would equate equation y to z. It becomes

5 + 13x = 3 + 21x

21x - 13x = 5 - 3

8x = 2

x = 2/8 = 0.25 years

Answer: a) Length of wire to replace old one=9ft

b) Length of wire=3.09ft

Leftover wire=0.91ft

Step-by-step explanation: from the right angle triangle, Tan&=20/15

&=53.13°

Length of wire,x=cos 31.13=x/15

15cos31.13

X=9

b) Tan&=10/6

Tan^-110/6

&=59.05

Length= 6cos59.04

X=3.09

4-3.09=.91= leftover wire

Using the Pythagorean theorem, the length of the guy wire needed for a 20-foot flagpole with a 15-foot distant anchor point is 25 feet, and for a 10-foot flagpole with a 6-foot distant anchor point, a 12-foot guy wire is needed, leaving approximately 4 inches of wire leftover.

The question involves applying the Pythagorean theorem to determine the lengths of guy wires needed to support flagpoles. For the first scenario, a 20-foot tall flagpole with a 15-foot distant anchor point, the guy wire's length can be found using the equation a^2 + b^2 = c^2, where 'c' is the hypotenuse (the guy wire), 'a' is one leg (the height of the flagpole which is 20 feet), and 'b' is the other leg (the distance from the base of the pole to the anchor point which is 15 feet). After calculation, the guy wire's length is found to be 25 feet.

For the second scenario with a 10-foot tall flagpole and a 6-foot distant anchor point, using the same theorem gives us the equation 10^2 + 6^2 = c^2. Estimating using perfect squares, we get c as the square root of 136, which is approximately 11.66 feet. Since guy wires can only be purchased in whole foot lengths, we would purchase a 12-foot guy wire, leaving approximately 0.34 feet (about 4 inches) of wire leftover.

Answer:

A) [tex]2x[/tex]

Step-by-step explanation:

Given:

Number of calculator made by Adnan in this week = [tex]'x'[/tex]

We need to write an algebraic expression for the number of calculators that Maya made this week.

Solution:

Now given:

Maya made twice as many calculators.

So we can say that;

Number of calculators made by Maya this week is equal to 2 times Number of calculator made by Adnan this week.

framing in equation form we get;

Number of calculators made by Maya this week = [tex]2\times x = 2x[/tex]

Hence an algebraic expression for the number of calculators that Maya made this week is [tex]2x[/tex].

Answer:

His diagonal brace need to be 15.55 feet long in the kitchen.

Step-by-step explanation:

Given : The wall of the kitchen is in the shape of a square.

Area of the wall = 121 sq ft

Let us assume the each side of  wall = a ft

Now, AREA OF THE SQUARE = Side x Side

⇒   a x  a = 121 sq units

[tex]\implies a ^2 = 121\\\implies a = 11[/tex]

So, each side of the  kitchen wall is 11 ft.

Now,as  we know in a square : EACH ANGLE IN THE  SQUIRE IS A RIGHT ANGLE.

So, by Pythagoras Theorem, Diagonal D is:

[tex]D^2 = a^2 + a^2\\\implies D^2 = 11^2 + 11^2 = 121 + 121\\\implies D^2 = 242\\\implies D = 15.55[/tex]

Hence,  his diagonal brace need to be 15.55 ft long in the kitchen.

Answer:

True

Step-by-step explanation:

Müller-Lyer illusion - In this illusion two-line having the same length appear to be different from one another.

the line showing in the file attached, the line with arrow moving outward appears to have large length and line having the arrow pointing inward appear to have a short length. But given lines are equal in length, it is just an illusion that might force to believed you that one is shorter or larger than others.

Answer: 15.07°

Step-by-step explanation:

Sin^-1 0.26= 15.07°

Therefore:

Cos15.07= 0.9656

Tan 15.07 = 0.2692

Csc15.07= 1/(Sin15.07)= 3.846

Sec15.07=1/cos15.07=1.0356

Cot15.07=1/tan15.07= 3.7147

We can use the trigonometric identity sin² θ + cos² θ = 1 to find cos θ in a first quadrant triangle where sin θ = 0.26, and get cos θ = 0.966. Then, we calculate tan θ with the formula tan θ = sin θ / cos θ, and get tan θ = 0.269 after rounding.

In trigonometry, there are three primary ratios called trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios can be defined using the sides of a right triangle, and they relate the angles of a triangle to the lengths of its sides. If we know one of these ratios in a right triangle, we can find the others.

In this case, we know that sin θ = 0.26 and θ is in Quadrant I. In the first quadrant, all the trigonometric ratios are positive.

So, firstly we find cos θ. Remember that sin² θ + cos² θ = 1, we can solve for cos θ using the calculator with the square root of (1 - sin² θ), which results in cos θ = 0.966 approximately.

Next, to find tan θ, we use the relationship tan θ = sin θ / cos θ. So, tan θ = 0.26 / 0.966 = 0.269 approximately, after rounding to the nearest hundredth.

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Answer:

There is enough evidence to conclude that the number of minutes spent online per day is related to gender

Step-by-step explanation:

Given that a researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below.

Gender     0-30   30-60   60-90    90+    total

Males         75         45         25       35       180

Females     30         45         45        15       135

H0: there is no association between gender and spending time online

Ha: There is association

(Two tailed test)

Expected for each cell is calculated as row total*grand total/column total

<30 30-60 60-90 >90  Row Totals

Males 75  (60.00)  [3.75] 45  (51.43)  [0.80] 25  (40.00)  [5.62] 35  (28.57)  [1.45]  180

Females 30  (45.00)  [5.00] 45  (38.57)  [1.07] 45  (30.00)  [7.50] 15  (21.43)  [1.93]  135

chi square -= 27.125

p value <0.0001

Since p is less than 1% (significance level) we reject the null hypothesis

There is enough evidence to conclude that the number of minutes spent online per day is related to gender

Answer:

The answer would be 18.146

Step-by-step explanation:

Given that a researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below.Gender     0-30   30-60   60-90    90+    totalMales         75         45         25       35       180Females     30         45         45        15       135H0: there is no association between gender and spending time onlineHa: There is association(Two tailed test)Expected for each cell is calculated as row total*grand total/column total<30 30-60 60-90 >90  Row TotalsMales 75  (60.00)  [3.75] 45  (51.43)  [0.80] 25  (40.00)  [5.62] 35  (28.57)  [1.45]  180Females 30  (45.00)  [5.00] 45  (38.57)  [1.07] 45  (30.00)  [7.50] 15  (21.43)  [1.93]  135chi square -= 27.125p value <0.0001Since p is less than 1% (significance level) we reject the null hypothesisThere is enough evidence to conclude that the number of minutes spent online per day is related to gender

Answer:  The required equation for points P is [tex]4x+2y=17.[/tex]

Step-by-step explanation: We are give two points A(0, 1, 2) and B(6, 4, 2).

To find the equation for points P such that the distance of P from both A and B are equal.

We know that the distance between two points R(a, b, c) and S(d, e, f) is given by

[tex]RS=\sqrt{(d-a)^2+(e-b)^2+(f-c)^2}.[/tex]

Let the point P be represented by (x, y, z).

According to the given information, we have

[tex]PA=PB\\\\\Rightarrow \sqrt{(x-0)^2+(y-1)^2+(z-2)^2}=\sqrt{(x-6)^2+(y-4)^2+(z-2)^2}\\\\\Rightarrow x^2+y^2-2y+1+z^2-4z+4=x^2-12x+36+y^2-8y+16+z^2-4z+4~~~~~~~[\textup{Squaring both sides}]\\\\\Rightarrow -2y+1=-12x-8y+52\\\\\Rightarrow 12x+6y=51\\\\\Rightarrow 4x+2y=17.[/tex]

Thus, the required equation for points P is [tex]4x+2y=17.[/tex]

Answer:

Step-by-step explanation: The brains are very similar insofar that they function in almost the same way when it comes to controlling physical aspects of our existence. Our brains regulate our organs and our body parts just like a sheep's brain.

Answer:

four numbers are 1, 3, 5, 7

Step-by-step explanation:

The sum of four numbers in arithmetic progression is 16

a, a+d, a+2d, a+3d are the four arithmetic series

sum of 4 numbers are

[tex]a+a+d+a+2d+a+3d=4a+6d[/tex]

[tex]4a+6d= 16[/tex]

divide both sides by 2

[tex]2a+3d=8[/tex]

[tex]3d= 8-2a[/tex]

The square of the last number is the square of the first number plus 48.

[tex](a+3d)^2=a^2+48[/tex]

solve for  a  and d

[tex](a+3d)^2=a^2+48\\(a+8-2a)^2=a^2+48\\(8-a)^2=a^2+48\\a^2-16a+64=a^2+48\\-16a=48-64\\-16a=-16\\a=1[/tex]

Now find out 'd'

[tex]3d=8-2a\\3d=8-2\\3d=6\\a=2[/tex]

a, a+d, a+2d, a+3d

four numbers are 1, 3, 5, 7

Answer:

The answer to your question is letter A.

Step-by-step explanation:

Inequality                   x + 2 ≥  5

Solve the inequality, remember that inequalities and equations are solved in similar ways.

- Subtract 2 in both sides

                                   x +2 - 2 ≥ 5 - 2

- Simplify

                                          x ≥ 3      

- Remember that this sign "≥" means that x is greater or equal to a number, and this sign "≤" means that x is lower or equal to a number.

The option that satisfies our inequality is the first option.                    

The answer is

[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]

=======================================================

Work Shown:

[tex](-3+\sqrt{-2})(3-\sqrt{2})[/tex]

[tex](-3+\sqrt{-1*2})(3-\sqrt{2})[/tex]

[tex](-3+\sqrt{-1}*\sqrt{2})(3-\sqrt{2})[/tex]

[tex](-3+i\sqrt{2})(3-\sqrt{2})[/tex]

[tex]x(3-\sqrt{2})[/tex] let [tex]x=-3+i\sqrt{2}[/tex]

[tex]3x-x*\sqrt{2}[/tex] distribute

[tex]3(x)-\sqrt{2}(x)[/tex]

[tex]3(-3+i\sqrt{2})-\sqrt{2}(-3+i\sqrt{2})[/tex]

[tex]-9+3i\sqrt{2}+3\sqrt{2}-i*\sqrt{2}*\sqrt{2}[/tex]

[tex]-9+3i\sqrt{2}+3\sqrt{2}-2i[/tex]

[tex]-9+3\sqrt{2}+3i\sqrt{2}-2i[/tex]

[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]

The use of parenthesis in the last step is to help separate out the terms.

The last expression shown above is in the form a+bi where

[tex]a=-9+3\sqrt{2}[/tex]

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

Answer: Probability that at least two of the triplets will win a medal is19/24

Step-by-step explanation:

Total number of cmpetitors=9

Total number of selecting 3 winners out of 9 competitors=9C3=84

Number of ways of selecting 2 out of 3 triplets=3C2

Number of selecting 1 out of 6 others =6C1

Number of selecting 2 out of 3 triplet and o1 out of 6 others= 3C2 ×6C1=18

Number of selecting 3 out of 3 triplets=3C3=1

Total ways of selecting less than 2 or 2 triplets= 18+1=19

Probability that at least 2 of the triplets will win is 19/84

Final answer:

To find the cost of a youth ticket, you can set up a system of equations using the given information. Solve the system of equations to find the cost of a youth ticket.

Explanation:

To find the cost of a youth ticket, we can set up a system of equations using the given information. Let x be the cost of an adult ticket and y be the cost of a youth ticket. From the problem, we know that 7x + 12y = 525 and x - y = 18. We can solve this system of equations using substitution or elimination.

First, let's solve for x in terms of y using the second equation. Adding y to both sides gives x = y + 18.

Next, substitute this expression for x in the first equation: 7(y + 18) + 12y = 525. Simplifying this equation gives 7y + 126 + 12y = 525. Combining like terms, we get 19y + 126 = 525. Subtract 126 from both sides: 19y = 399. Finally, divide both sides by 19 to solve for y: y = 21.

Therefore, the cost of a youth ticket is $21.

Answer:

[tex]4x+9+4x^2[/tex]

Step-by-step explanation:

The given functions are:

[tex]f(x)=4x+9[/tex] and [tex]g(x)=4x^2[/tex]

We want to find [tex](f+g)(x)[/tex].

Recall that:

[tex](f+g)(x)=f(x)+g(x)[/tex]

We substitute the functions to get:

[tex](f+g)(x)=4x+9+4x^2[/tex]

We now write in standard form to obtain:

[tex](f+g)(x)=4x^2+4x+9[/tex]

The expression for the given sequence of operations is 2(x-4)+8.

The given statement is "subtract 4 from x, double, and add 8".

We need to write an expression for the sequence of operations.

An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.

Now, subtract 4 from x=x-4

Double the difference that is 2(x-4)

Add 8=2(x-4)+8

Therefore, the expression for the given sequence of operations is 2(x-4)+8.

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The distance traveled by the particle is determined as [tex]\int\limits^0_t {v(t)} \, dt=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex].

Given data:

To determine the distance traveled by the particle during the first t seconds, integrate the velocity function with respect to time over the interval [0, t].

Given the velocity function:

[tex]v(t) = t^2 * e^{(-t)}[/tex]

To find the distance traveled, integrate the absolute value of the velocity function over the interval [0, t]:

[tex]\text{distance} = \int\limits^0_t {v(t)} \, dt[/tex]

Since the velocity can be negative (indicating a change in direction), taking the absolute value ensures that we consider the magnitude of the velocity.

Now, calculate the integral:

[tex]D = \int\limits^0_t {t^2 * e^{(-t)}} \, dt[/tex]

To simplify this integral, split it into two parts, considering the cases when the velocity is positive and negative:

[tex]D=[-e^{-t}t^2-\int\limits^0_t {-2e^{-t}} \, dt]_0^t[/tex]

The first integral represents the distance traveled when the velocity is positive, and the second integral represents the distance traveled when the velocity is negative.

Integrating each term separately:

[tex]\int\limits^0_t {v(t)} \, dt=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex]

Hence, the distance is [tex]D=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex].

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The complete question is attached below:

A particle that moves along a straight line has velocity [tex]v(t) = t^2e^{-t}[/tex] meters per second after t seconds. How far will it travel during the first t seconds?

The function v(t) = t2 e−t defines the velocity of a particle. The displacement or distance covered by the particle in the first t seconds is given by the integral from 0 to t of the velocity function.

The given velocity function for the particle is v(t) = t2 e−t. The distance (or displacement) traveled by an object is the integral of the velocity over a given interval. So, to find the distance covered by the particle in the first t seconds, you would set up the following integral:

∫0t t2 e−t dt

We won't compute this integral here, as it requires knowledge of methods beyond basic calculus, specifically the use of integration by parts and possibly a tabular method for successive integrations by parts. However, this setup gives you the right starting point to find the distance covered in the first t seconds.

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