Consider F and C below. F(x, y, z) = 2xz + y2 i + 2xy j + x2 + 15z2 k C: x = t2, y = t + 1, z = 3t − 1, 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.

Answers

Answer 1

(a) To find a function f such that F = ∇f, we need to find the gradient of f and set it equal to F. So,

∇f = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k

F = 2xz + y^2 i + 2xy j + x^2 + 15z^2 k

Setting the corresponding components equal to each other, we get:

∂f/∂x = x^2

∂f/∂y = 2xy

∂f/∂z = 2xz + 15z^2

Integrating each of these with respect to their respective variables, we get:

f(x, y, z) = (1/3)x^3 + x^2y + 5xz^2 + g(y)

where g(y) is an arbitrary function of y.

(b) Using the result from part (a), we have:

∇f = 3x^2 i + 2xy j + (10z + 6xz) k

C: x = t^2, y = t + 1, z = 3t − 1, 0 ≤ t ≤ 1

dr = (2t) i + j + (3) k

∇f · dr = (9t^4) + (4t^2) + (30t^2 - 18t - 3)

= 9t^4 + 34t^2 - 18t - 3

To evaluate C ∇f · dr, we substitute the values of x, y, z, and dr into the expression above and integrate with respect to t from 0 to 1:

C ∇f · dr = ∫₀¹ (9t^4 + 34t^2 - 18t - 3) (2t) dt

= 161/5

Therefore, C ∇f · dr = 161/5.

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Related Questions

What additional measurement would support Amber's hypothesis?

O The measure of ∠C is 32°
O The measure of ∠C is 40°.
O The measure of ∠C is 50°.
O The measure of ∠C is 90°​

Answers

Answer: B

Step-by-step explanation:

I know I’m late :(

Factor completely x3 − 2x2 − 8x + 16.
(x + 2)(x2 + 8)
(x − 2)(x2 + 8)
(x + 2)(x2 − 8)
(x − 2)(x2 − 8)

Answers

The expression x^3 - 2x^2 - 8x + 16 can be factored as (x - 2)(x^2 - 8).

To factor the given expression x^3 - 2x^2 - 8x + 16, we can look for common factors or factor it using grouping. In this case, we can observe that the expression can be factored by grouping.

First, we can factor out a common factor of (x - 2) from the first two terms:

x^3 - 2x^2 - 8x + 16 = (x - 2)(x^2 - 8x - 8)

Now, we can further factor the quadratic expression (x^2 - 8x - 8) by factoring out a common factor of 8:

(x^2 - 8x - 8) = (x - 2)(x - 8)

Therefore, the complete factorization of the expression x^3 - 2x^2 - 8x + 16 is:

(x - 2)(x - 2)(x - 8) which can also be written as (x - 2)(x^2 - 8).

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find the change-of-coordinates matrix from the basisB = {1-7t^2, -6 + t+43t^2, 1+6t} to the standard basis. Then write t^2 as a linear combination of the polynomials in B.

Answers

To find the change-of-coordinates matrix from basis B to the standard basis, we need to express the standard basis vectors as linear combinations of the vectors in B. Then, to write t^2 as a linear combination of the polynomials in B, we can use the change-of-coordinates matrix to transform t^2 into the coordinates with respect to B.

To find the change-of-coordinates matrix from basis B to the standard basis, we express the standard basis vectors as linear combinations of the vectors in B. Let's denote the standard basis vectors as e1, e2, and e3. We can write:

e1 = 1(1 - 7t^2) + 0(-6 + t + 43t^2) + 0(1 + 6t)

e2 = 0(1 - 7t^2) + 1(-6 + t + 43t^2) + 0(1 + 6t)

e3 = 0(1 - 7t^2) + 0(-6 + t + 43t^2) + 1(1 + 6t)

The coefficients in these equations give us the entries of the change-of-coordinates matrix.

To write t^2 as a linear combination of the polynomials in B, we can use the change-of-coordinates matrix. Let [t^2]_B represent the coordinates of t^2 with respect to B. Then, [t^2]_B = C[t^2]_std, where C is the change-of-coordinates matrix. We can solve this equation to find [t^2]_B.

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what are the values of a and b? a = 2y and b = 3 a = and b = –3y a = 2y and b = –3 a = and b = 3y

Answers

The values of a and b are dependent on the equation. For example, in the equation a = 2y and b = 3, then a = 6 and b = 3. However, in the equation a = 2y and b = -3, then a = -6 and b = -3.

In the given options, the values of a and b are stated as a = 2y and b = –3. This means that the value of a is equal to twice the value of y (a = 2y), and the value of b is equal to -3. The other options do not match these conditions. It is important to note that without further context or information about the variable y, we cannot determine a specific value for a or b. The values provided only establish the relationship between a, b, and y as described in the option a = 2y and b = –3.

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Simplify the following expressions to have fewer terms (MIDDLE SCHOOL)

5x-3+(4x-6)+2

4y+3-(8x-2)

Answers

Answer:

Install math calculator

Thank you

Answer:

x=1\9

Step-by-step explanation:

first part:

5x-3+4x+2

5x+4x-3+2

9x-1  

9x=1

x=1\9 answer.

Solve for x. 1/2x - 1/4 = 1/2

Answers

Answer:

x = 3/2

Step-by-step explanation:

Simplify this equation by dividing all three terms by 1/4:

2x - 1 = 2, or

2x  = 3

Then x = 3/2

Determine the area and circumference of a circle with radius 12 cm.

Answers

The area of the circle is 452.16 cm², and the circumference is 75.36 cm.

To determine the area and circumference of a circle with a radius of 12 cm, we can use the formulas:

Area = π * r²

Circumference = 2 * π * r

The radius (r) is 12 cm, we can substitute this value into the formulas to find the area and circumference.

Area = π * (12 cm)²

      = π * 144 cm²

      ≈ 3.14 * 144 cm²

      ≈ 452.16 cm²

The area of the circle is approximately 452.16 square centimeters.

Circumference = 2 * π * 12 cm

                   = 2 * 3.14 * 12 cm

                   ≈ 75.36 cm

The circumference of the circle is approximately 75.36 centimeters.

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HELPPPPP
Directions: Find the slope of the lines graphed below.
1.
2.
3.
4.
5.
6.
Directions: Find the slope between the given two points.
7.(-1,-11) and (-6, -7)
8. (-7.-13) and (1, -5)
9. (8.3) and (-5,3)
10. (15, 7) and (3,-2)
11. (-5, -1) and (-5, -10)
12. (-12, 16) and (-4,-2)
Directions: Use slope to determine if lines PQ and RS are parallel, perpendicular, or neither.
13.P(-9,-4), (-7, -1), R(-2,5), S(-6, -1)
m(PO)
m(RS)
Types of Lines
PLEASE HELLPPPP

Answers

Answer:

7. -4/5

8. 1

9. 0

10. 3/4

11. undefined/nonlinear

12. -9/4

13. parallel

9. What measure of central tendency is usually the preferred number by researchers for describing a group of scores?

10. Using the following data, calculate the standard deviation: 115, 125, 145, 150, 115, 125

Answers

The measure of central tendency that is usually the preferred number by researchers for describing a group of scores is the mean.

The measure of central tendency that is usually the preferred number by researchers for describing a group of scores is the mean. The mean provides the average score for a given set of data. It is calculated by summing up all the scores and dividing by the total number of scores. While other measures of central tendency, such as the median and mode, are also useful, the mean provides the most comprehensive picture of the data.


10. To calculate the standard deviation for the following data: 115, 125, 145, 150, 115, 125, you can use the following formula:

σ = √[(Σ(x - μ)²) / N]

Where:
σ = standard deviation
Σ = sum of
x = each score
μ = mean of all the scores
N = total number of scores

To find the standard deviation, first find the mean of the scores:

115 + 125 + 145 + 150 + 115 + 125 = 775
Total number of scores = 6
Mean = 775 / 6 = 129.17

Now, subtract the mean from each score, square the result, and add up all the squared differences:

(115 - 129.17)² + (125 - 129.17)² + (145 - 129.17)² + (150 - 129.17)² + (115 - 129.17)² + (125 - 129.17)² = 4666.79

Then, divide the sum of squared differences by the total number of scores and take the square root:

σ = √(4666.79 / 6) = 17.20


Mean = (115 + 125 + 145 + 150 + 115 + 125) / 6 = 129.17

Standard Deviation = √[ {(115 - 129.17)² + (125 - 129.17)² + (145 - 129.17)² + (150 - 129.17)² + (115 - 129.17)² + (125 - 129.17)² } / 6 ]

= √[4666.79 / 6]

= 17.20

Therefore, the standard deviation of the given data is 17.20.

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Find the area of the figure

Answers

Answer:

A=153.94

Step-by-step explanation:

Find the volume of each composite figure.

Answers

Answer:

630 in^3

Step-by-step explanation:

6 x 6 x 14 = 504

(0.5 x 3 x 6) x 14 = 126

504 + 126 = 630

7 ⅓ ÷ 225 =
answer this question plz

Answers

Answer:

0.03259..

Step-by-step explanation:

The J.R. Simplot Company produces frozen French fries that are then sold to customers such as McDonald's. The "prime" line of fries has an average length of 6.00 inches with a standard deviation of 0.50 inch. To make sure that Simplot continues to meet the quality standard for "prime" fries, they plan to select a random sample of n = 100 fries each day. The quality analysts will compute the mean length for the sample. They want to establish limits on either side of the 6.00 inch mean so that the chance of the sample mean falling within the limits is 0.99. What should these limits be?

Answers

Answer:

(5.871 `; 6.127)

Step-by-step explanation:

Given :

Mean = 6

Standard deviation. σ = 0.5

Samplr size, n = 100

Zcritical at 99% confidence = 2.58

The confidence interval :

Mean ± margin of error

Margin of Error = Zcritical * σ / sqrt(n)

Margin of Error = 2.58 * 0.5/sqrt(100) = 0.129

Confidence interval :

Lower boundary = 6.00 - 0.129 = 5.871

Upper boundary = 6.00 + 0.129 = 6.129

(5.871 `; 6.127)

A function, f(x),represents the height of a plant x months after being planted. Students measure and record the height on a
Select the appropriate domain for this situation.
OA.
the set of all real numbers
OB.
the set of all integers
O c.
the set of all positive integers
OD
the set of all positive real numbers

Answers

Answer:

I think the answer would be A or B, the best would be B.

Step-by-step explanation:

K so since it is monthly, it would not be all positive data. So you are left with A or B. Integers are numbers that are positive and negative so since the data can be a range of positive and negative numbers, that is why the data set of all integers (b) is the best choice.

Your welcome :D

At 5 am, the temperature was -10°F. By noon the temperature was 7°F, What Integer
represents the change in temperature from 5 am, to noon?

Answers

Answer:

The integer that represents the change in temperature is 17.

Step-by-step explanation:

Please help.
Is algebra.

Answers

Answer:

1 is C

2 is B

3 is C

Step-by-step explanation:

Answer:

give em brainliest because i have no idea

Step-by-step explanation:

1. A package of sticky notes is in the shape of
a parallelogram. The dimensions of one
sticky note are shown. What is the area of
one sticky note?
A. 248.4 cm?
B. 124.2 cm2
C. 62.1 cm2
D. 22.7 cm2

Answers

Answer:

B. 124.2

Step-by-step explanation:

Area= b*h

Area= 13.5*9.2

Area = 124.2

Please Note* = Multiply

Let p be a prime. Let K = F_p(t), let w = t^p - t and let F = F_p (w).
(a) Find a polynomial of degree p in F[x] for which t is a root. Use this to deduce an upper bound on [K: F].
(b) Show that the automorphism δ of K defined by δ (t) = t + 1 fixes F. Use this to factor the polynomial you wrote down in (a) into linear factors in K[x]
(c) Show that K is a Galois extension of F and determine the Galois group Gal(K/F).

Answers

The degree of the extension [K: F] ≤ p.

Suppose K = F_p(t) has transcendence degree n over F_p.

Then K is an algebraic extension of F_p(t^p).

(a) We need to find a polynomial of degree p in F[x] for which t is a root.

In F_p, we have t^p - t ≡ 0 (mod p).

So, we can write t^p ≡ t (mod p).

Since F_p[t] is a polynomial ring over F_p, we have t^p - t ∈ F_p[t] is an irreducible polynomial.

Hence the degree of the extension [K: F] ≤ p.

Suppose K = F_p(t) has transcendence degree n over F_p.

Then K is an algebraic extension of F_p(t^p).

The minimal polynomial of t over F_p(t^p) is x^p - t^p. Thus, [K: F_p(t^p)] ≤ p.

Since K/F_p is an algebraic extension, we have [K: F_p] = [K: F_p(t^p)][F_p(t^p): F_p].

Thus, [K: F_p] ≤ p².

Therefore, [K: F] ≤ p².

(b) We need to show that the automorphism δ of K defined by δ (t) = t + 1 fixes F.

Let f(x) be the polynomial obtained in part (a). Since f(t) = 0, we have f(t + 1) = 0. This implies δ (t) = t + 1 is a root of f(x) also.

Hence, f(x) is divisible by x - (t + 1). We can writef(x) = (x - (t + 1))g(x)for some g(x) ∈ K[x].

Since [K: F] ≤ p², we have deg(g) ≤ p.

Substituting x = t into the above equation yields 0 = f(t) = (t - (t + 1))g(t) = -g(t).

Therefore, f(x) = (x - (t + 1))g(x) = (x - t - 1)(a_{p-1}x^{p-1} + a_{p-2}x^{p-2} + ··· + a_1 x + a_0)where a_{p-1}, a_{p-2}, ..., a_1, a_0 ∈ F_p are uniquely determined.

(c) To show that K is a Galois extension of F and determine the Galois group Gal(K/F), we need to check that K is a splitting field over F.

That is, we need to show that every element of F_p(t^p) has a root in K.Since K = F_p(t)(t^p - t) = F_p(t)(w), it suffices to show that w has a root in K.

Note that w = t^p - t = t(t^{p-1} - 1).

Since t is a root of f(x) = x^p - x ∈ F_p[t], we have t^p - t = 0 in K. Thus, w = 0 in K.

Therefore, K is a splitting field over F_p(t^p).Since [K : F_p(t^p)] ≤ p, the extension K/F_p(t^p) is separable.

Therefore, the extension K/F_p is also separable. Hence, K/F_p is a Galois extension. The degree of the extension is [K: F_p] = p².

The Galois group is isomorphic to a subgroup of S_p. Since F_p is a finite field of p elements, it contains a subfield isomorphic to Z_p. This subfield is fixed by any automorphism of K that fixes F_p.

Since F_p(t^p) is generated by F_p and t^p, any automorphism of K that fixes F_p(t^p) is uniquely determined by its effect on t.

Since there are p choices for δ(t), the Galois group has order p. It follows that the Galois group is isomorphic to Z_p.

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SOMEONE PLS HELP SHEHSJHSHS

Answers

Answer:

1. Quadrant II

2. Quadrant IV

3. Y-axis

Step-by-step explanation:

what are the answers to a, b, c, d?

Answers

Answer:

Step-by-step explanation:

What Statistic information we can get from a histogram?

A-Standard Deviation

B-Interquartile Range

C-Mean

Answers

The answer should be B

Randomly select a painted rock from a bag containing 4 purple rocks, 3 green rocks, 3 orange rocks, and 2 blue rocks.

Answers

Answer:

i got a orange

Step-by-step explanation:

I would choose blue rocks bestie

My Samsung charger broke so I went to Five Below to get a new one. They were on sale for $15,35 (not so five below). There
was a 6% tax too. How much did the new charger cost?
( I don’t need the explanation, only the right answer)

Answers

Answer:

The answer to your question is 16.27

Explanation:

You would start with the equation,

15.35 + 6% = C (Cost)

15.35 · 6% = 0.921

15.35 + 0.921 = 16.271

Round the one down, since it is less than four, and you get 16.27

Hope this helps.

waffletowne

(q11) Find the center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively.

Answers

The center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively is located at the point (1.6, 2.2).

To find the center of mass of the system of objects, we need to consider the masses and positions of each object. Let's go through the steps to calculate the center of mass:

Assign variables to the masses and coordinates of the objects:

Object 1: Mass = 2, Coordinates = (-1, 2)

Object 2: Mass = 3, Coordinates = (1, 1)

Object 3: Mass = 5, Coordinates = (3, 3)

Calculate the total mass of the system by summing the masses of the objects:

Total mass = 2 + 3 + 5 = 10

Calculate the weighted average of the x-coordinates and y-coordinates to find the center of mass:

Center of mass (x-coordinate) = (m1 * x1 + m2 * x2 + m3 * x3) / Total mass

Center of mass (y-coordinate) = (m1 * y1 + m2 * y2 + m3 * y3) / Total mass

Substituting the values:

Center of mass (x-coordinate) = (2 * -1 + 3 * 1 + 5 * 3) / 10 = ( -2 + 3 + 15) / 10 = 16 / 10 = 1.6

Center of mass (y-coordinate) = (2 * 2 + 3 * 1 + 5 * 3) / 10 = (4 + 3 + 15) / 10 = 22 / 10 = 2.2

Therefore, the center of mass of the system of objects is located at the point (1.6, 2.2).

The center of mass represents the average position of the system, taking  into account the masses and positions of the objects. It is a point where the system can be considered to be balanced. Calculating the center of mass is important in various areas of physics, such as studying the motion, stability, and collisions of objects.

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wayfktgfydfug9ugi0b7ffiobv57f9pogasdfyuiohfdfhjkl

Answers

Answer:

Area of the circle = 572.3 square ft.

Step-by-step explanation:

Area of a circle is given by the formula,

Area of a circle = πr²

Here 'r' = Radius of the circle

Diameter of circle given in the picture = 27 ft

Radius of the circle = [tex]\frac{27}{2}[/tex]

                                = 13.5 ft

Area of the circle = π(13.5)²

                             = 3.14(13.5)²

                             = 572.265

                             ≈ 572.3 square ft

What is the volume of this sphere? Use a ~ 3.14 and round your answer to the nearest hundredth. 20 mm​

Answers

Answer:

4186.67 cubic mm=V

Step-by-step explanation:

The volume of a sphere can be found using the equation, [tex]\frac{4}{3} \pi r^{3}[/tex]. It gives us the diameter but we need the radius. To find the radius just divde the diameter by 2, so the radius of the sphere is 10mm.

[tex]\frac{4}{3} (3.14) (10^{3})[/tex]=V

[tex]\frac{4}{3} (3.14) (1000)[/tex]=V

[tex]\frac{4}{3}[/tex](3140)=V

4186.67 cubic mm=V

The graph of a system of equations is
shown below.

a. (2,0) and (0,3)
b. (-40,-50)
c. the system has no solution
d. the system has infinitely many solutions

Answers

Answer:

The system has no solution.

Step-by-step explanation:

In order for the system to have a solution, both graphs must have an intercept to each others. We see in the picture that both graphs are parallel and do not have any interceptions which we don't know the solution to the system.

That means if graphs are parallel and have no interceptions, there are no solutions. The system of equations are for finding the interceptions of both graphs. But of course! Parallel lines do not intercept.

If you have any questions, feel free to ask.

Write and solve an equation to find the missing dimension of the figure.

Answers

Answer:

15.5769230769

Step-by-step explanation:

8×13 = 104

1620÷104 = 15.5769230769

gh¯¯¯¯¯¯ has endpoints g(−3, 2) and h(3, −2). find the coordinates of the midpoint of gh¯¯¯¯¯¯ . a. (−3, 0) b. (0, 2) c. (0, 0) d. (0, −2)

Answers

The coordinates of the midpoint of the line segment GH with endpoints G(-3, 2) and H(3, -2) are (0, 0). The correct option is (C).

To determine the coordinates of the midpoint of the line segment GH with endpoints G(-3, 2) and H(3, -2), we can use the midpoint formula.

The midpoint formula states that the coordinates of the midpoint (M) are given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2)

For GH, plugging in the coordinates, we have:

Midpoint (M) = ((-3 + 3) / 2, (2 + -2) / 2)

Midpoint (M) = (0, 0)

Therefore, the coordinates of the midpoint of GH are (0, 0), which corresponds to option c. (0, 0).

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Solve for X triangle. ​

Answers

Answer:

x= 12.942

Law of sines :)

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Five years ago, $12, 000 was deposited into a savings account that provides an interest of 6.5% compounded annually. If a uniform amount of $3, 000 was withdrawn yearly starting at the end of the first year after the initial deposit was made, how much money can still be withdrawn now? 9.2 Employing Passive-Voice Verbs (L.O. 3) YOUR TASK Revise the following sentences to present thebad news with passive-voice verbs.Chapter9 NegativeMessagesa. Because management now requires more stringent security, we are postponing indefinitely requests for company tours.b. WecannotofferfreeshippingafterJanuary1.Actnow!c. Our retail stores will no longer be accepting credit cards for purchases under $5.d. As manager, I cannot grant you the three-week vacation period you request because others have already reserved that period.e. We do not examine patients until we have verified their insurance coverage.f. Your car rental insurance coverage does not cover large SUVs.g. 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The salvage life of the wire is 20 years, the cost price of the copper is 100 Riyal/kg. and the salvage value of the material wire is 10 Riyal/kg.The energy cost is 0.10 Riyal per KWH. Assume the cross section of the wire is a square, the interest rate is 15% and the density of the copper is 10000 kg/m^3 when clients are admitted to inpatient psychiatric settings, which member of the health care team is often the first whom they meet? Adela and James are married and file tax returns jointly. Last year Adela earned $48,500 and James earned 549.706 in wages. Additional tax information for the year is as follows: Interest earned: $1,200. State and local income taxes paid 4.200, mortgage interest: $5,200, contributions to charity: $1,400, contributions to retirement plans: $3,350. From this information, calculate their taxable income I need help like asap Listed property includes: a. Ambulances, hearses, taxis, school buses, and marked police and fire vehicles. b. Computers used exclusively at a regular business establishment and owned or leased by the person operating the business. c. Passenger automobiles weighing 6,000 pounds or less. d. Video-recording equipment used exclusively in a taxpayer's business. What is pantun?............... Suppose the price elasticity of demand is inelastic then: A. a 1 percent fall in the price will lead to a higher than 1 percent increase in quantity demanded. B. a 1 percent increase in the price will lead to a lower than 1 percent decrease in quantity demanded. C. a 1 percent fall in the price will lead to a lower than 1 percent decrease in quantity demanded. D. a 1 percent increase in the price will lead to a higher than 1 percent decrease in quantity demanded. E. a 1 percent fall in the price will lead to a 1 percent increase in quantity demanded. To begin answering our original question, test the claim that the proportion of children from the low income group that drew the nickel too large is greater than the proportion of the high income group that drew the nickel too large. Test at the 0.1 significance level.Recall 24 of 40 children in the low income group drew the nickel too large, and 13 of 35 did in the high income group.If we use LL to denote the low income group and HH to denote the high income group, identify the correct alternative hypothesis.H1:pL>pHH1:pL>pHH1:pLH1:LH If you deposit today 11,445.85 in an account earning 8% compound interest, for how long should you invest the money in order to earn 12,488.36 (profit)? Which of the following bonds may be purchased at the cheapest price? O A 5% bond yielding 3% O A 5% bond yielding 7% OA 7% bond yielding 4% OA 6% bond yielding 6% What is interoperability?Group of answer choices:a.) A standard that specifies the format of data as well as the rules to be followed during transmission.b.) Refers to the geometric arrangement of the actual physical organization of the computers and other network devices) in a network.c.) The capability of two or more computer systems to share data and resources, even though they are made by different manufacturers.d.) An intelligent connecting device that examines each packet of data it receives and then decides which way to send it onward toward its destination. of the following statements, which is accurate regarding and its sponsored links? the payment involved is not performance-based. the amount that an advertiser owes for each click-through is set by the iab. as in the case of banner ads, advertisers pay just for being included in the search results. the advertisers pay only when a search engine user clicks on the link and visits the sponsor's site. the amount that an advertiser owes for each click-through is set by . A gas has an initial volume of 24. 6 L at a pressure of 1. 90 atm and a temperature of 335 K. The pressure of the gas increases to 3. 50 atm, and the volume of the gas increases to 31. 3 L There are 12 books on a shelf, all with different heights. If three books are chosen at random what is the probability that one of the books chosen is the tallest book on the shell 12?