For a cost function C 100+ 10Q+ Q2, the average fixed cost of producing 10 units of output is: Multiple Choice 10. 5. None of the answers are correct.

The value of ssI2 dV ≈ 0.061.

The region W is described as the wedge in the first octant that is cut from the cylindrical solid v2+22 < 1 by the planes y = x and x = 0.

First, we can sketch the region W in the xy-plane:

    |

    | /

    |/_____

    0 1

The region is bounded by the curves y = x and y = √[tex](1 - x^2/2)[/tex], which can be found by setting [tex]v^2 + 2^2 = 1[/tex] and solving for v as a function of x. We can set up the integral as follows:

ssI2 dV = ∫∫∫W 2 dV

We can use cylindrical coordinates, where v = r cosθ and 2 = r sinθ, and the limits of integration are 0 ≤ r ≤ √2, 0 ≤ θ ≤ π/4, and r cosθ ≤ x ≤ √[tex](1 - r^2 sin^2\theta)[/tex]. The integrand is 2, so it is constant and can be factored out of the integral:

ssI2 dV = 2 ∫[tex]_0^{(\pi/4)[/tex] ∫[tex]_0^\sqrt2[/tex] ∫[tex]_(r cos\theta)^\sqrt(1 - r^2 sin^2\theta)[/tex] r dz dr dθ

We can integrate with respect to z first:

ssI2 dV = 2 ∫[tex]_0^{(\pi/4)[/tex] ∫[tex]_0^\sqrt2} r(\sqrt(1 - r^2 sin^2\theta) - r cos\theta)[/tex] dr dθ

Next, we can use the substitution u = 1 - [tex]r^2 sin^2\theta[/tex], du = -2r sinθ cosθ dr, to simplify the inner integral:

ssI2 dV = 2 ∫[tex]_0^{(\pi/4)[/tex] ∫[tex]_0^\sqrt 2 (1 - u)^{(1/2)[/tex] du dθ

We can integrate with respect to u using the substitution u = [tex]sin^2[/tex]φ, du = 2 sinφ cosφ dφ:

ssI2 dV = 2 ∫[tex]_0^{(\pi/4)[/tex] ∫[tex]_0^{(\pi/2)} (1 - sin^2[/tex]φ)[tex]^{(1/2)[/tex] sinφ cosφ dφ dθ

The integrand simplifies using the identity [tex]sin^2[/tex]φ + [tex]cos^2[/tex]φ = 1, so sinφ cosφ = 1/2 sin2φ:

ssI2 dV = ∫[tex]_0^{(\pi/4)[/tex] ∫[tex]_0^{(\pi/2)} sin^2[/tex]φ dφ dθ

We can use the identity [tex]sin^2[/tex]φ = (1 - cos2φ)/2 and integrate with respect to φ and θ:

ssI2 dV = ∫[tex]_0^{(\pi/4)[/tex] [φ/2 - 1/4 sin2φ][tex]_0^{(\pi/2)} d\theta[/tex]

= ∫[tex]_0^{(\pi/4)[/tex] (π/4 - 1/4) dθ

= (π/16 - 1/8) π/4

≈ 0.061

Rounding to three decimal places, we get ssI2 dV ≈ 0.061.

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The area of the rectangular animal pen with fencing of 200 m with a length of 60 m is 2400 sq m

Perimeter refers to the length of the boundary of a given shape.

Perimeter = 2(l + b)

where l is the length

b is the breadth

Given,

Perimeter = 200 m

l = 60 m

200 = 2(60 + b)

100 = 60 + b

b = 100 - 60

b = 40 m

The other side of the rectangle is 40 m.

The area is the expanse covered by a shape

Area = l * b

= 60 * 40

= 2400 sq m

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For the given function : (f o g)(x) = x, (g o f)(x) = x/5, (f o g)(5) = 6, (g o f)(5)  = 1.64.

Now,

a.f(g(x)) = 5((x+4)/5) - 4 = x

b. (g o f)(x) =
g(f(x)) = (5x-4 + 4)/5 = x/5

c. (f o g)(5) =
f(g(5)) = f((5+4)/5) = f(1.8) = 5(1.8) - 4 = 6

d. (g o f)(5) =
g(f(5)) = g(5*5-4) = g(21/5) = (21/5 + 4)/5 = 1.64

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C → B is the Chomsky Normal Form (CNF) of the given grammar.

We have,

To convert the given context-free grammar into Chomsky Normal Form (CNF):

Step 1: Eliminate ε-productions

The given grammar has one ε-production: y → ε.

Replace each occurrence of y in the other productions with ε, obtaining:

s → asb | b | a | sbs

y → b | a

Step 2: Eliminate unit productions

The given grammar has no unit productions.

Step 3: Convert all remaining productions into the form A → BC

The remaining productions are already in form A → BC or A → a.

Step 4: Convert all remaining productions into the form A → a

We need to convert the production y → b into the form y → CB, where C is a new nonterminal symbol.

Then we add the production C → b, and replace each occurrence of y by C in the other productions.

This gives:

s → ASB | B | A | SBS

A → AY | AYB | AYC | B | AYCB | AYBSC | ε

B → BZ | A | AS | ZB | ε

S → BB | ε

Y → C

C → B

Thus,

C → B is the Chomsky Normal Form (CNF) of the given grammar.

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The individual events are dependent. The probability of the combined event is  4.3%.

The events are dependent because the probability of drawing a red candy on the second draw depends on whether a red candy was drawn on the first draw.

Let R1 be the event that a red candy is drawn on the first draw, and R2 be the event that a red candy is drawn on the second draw. The probability of R1 is 13/55 since there are 13 red candies out of 55 total. However, the probability of R2 given that R1 has occurred is 12/54, since there will be one less red candy and one less candy in total.

Therefore, the probability of both events occurring is:

P(R1 and R2) = P(R1) * P(R2 given R1)

= (13/55) * (12/54)

= 0.043 or 0.0432 (rounded to three decimal places)

Therefore, the probability of drawing and immediately eating two red candies in a row from the bag is 0.043 or 4.3% (rounded to three decimal places).

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The individual events are independent. The probability of the combined event is 0.043. The correct answer is A.

For two events to be independent, the occurrence of one event should not affect the probability of the other event. In this case, randomly drawing and immediately eating two red pieces of candy from a bag containing 13 red pieces out of 55 total pieces.

Since the first candy is immediately eaten and removed from the bag before the second candy is drawn, the probability of drawing a red candy on the second draw is still the same as the probability of drawing a red candy on the first draw.

The probability of drawing a red candy on the first draw is 13/55 since there are 13 red candies out of 55 total candies.

The probability of drawing a red candy on the second draw, assuming the first candy was red and removed, is also 13/55. The events are independent because the probability of the second draw is unaffected by the outcome of the first draw.

To find the probability of the combined event (drawing and immediately eating two red candies in a row), we multiply the probabilities of the individual events:

P(Combined Event) = P(Draw Red Candy on 1st Draw) * P(Draw Red Candy on 2nd Draw)

P(Combined Event) = (13/55) * (13/55)

P(Combined Event) ≈ 0.043 (rounded to three decimal places)

Therefore, the individual events are independent, and the probability of the combined event is approximately 0.043.

If the individual events were dependent, it would mean that the probability of the second event is influenced by the outcome of the first event. However, in this scenario, the events are independent as explained in part A. Therefore, the probability of the combined event is 0.043, and the correct answer is A.

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Answer: The answer is (d) 700 miles. 35 inches on the map represents 700 miles in actual distance

Step-by-step explanation:

This is a Unitary method problem.

If 25 inches on the map represents 500 miles in actual distance, then we can write:

25 inches / 500 miles = 35 inches / x miles

where x is the number of miles represented by 35 inches on the map.

To solve for x, we can cross-multiply and simplify:

25 inches * x miles = 500 miles * 35 inches

25x = 17500

x = 700

Therefore, 35 inches on the map represents 700 miles in actual distance.

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The requried for a system of the solution by elimination options B and C is not allowed.

To use the elimination method, you can add or subtract the equations to eliminate one of the variables. This means that you can multiply one or both of the equations by a constant before adding or subtracting them.

Option A is allowed since you can multiply equation 1 by 2 to get 4x - 6y = 24 and multiply equation 2 by 3 to get -3x + 6y = 39, and then add the new equations to eliminate y.

Option B is not allowed since we can cant multiply the left side of equation 2 by .

Option C is also not allowed since we can multiply equation 2 by -2 to get 2x - 4y = -26, but then we cannot add this result to equation 1.

Therefore, Options B and C are not allowed.

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Based on the unit rate, if you buy 2 oranges with £1 coin and get 52p change, the cost of 1 orange is 24p.

The unit rate refers to the ratio of one quantity or value compared to another.

The unit rate is computed as the quotient of the total value divided by the number of items in the data set.

We can also refer to the unit rate as the slope, gradient, or constant of proportionality.

The total amount that you have = £1

The change obtained after the transaction = 52p

The amount spent for 2 oranges = 48p (£1 - 52p)

The unit rate of each orange = 24p (48p ÷ 2)

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If I buy 2 oranges with £1 coin and get 52p change, how much is it for 1 orange?

The average value of the function f(x) = x² - 11 on the interval [0, 6] is -1/3.

The average value of a function on an interval is a useful concept in many areas of mathematics and applied fields. It represents the "center of mass" or "balance point" of the function over the interval, and has applications in physics, engineering, economics, and more. fmin ≤ avg(f) ≤ fmax

This property can be used to prove useful inequalities and approximations in various fields. The formula for the average value of a function on an interval [a,b] is: avg(f) = 1/(b-a) * ∫[a,b] f(x) dx.

Using this formula, we can find the average value of f(x) = x² - 11 on [0,6] as: avg(f) = 1/(6-0) * ∫[0,6] (x² - 11) dx

= 1/6 * [x³/3 - 11x]_0^6

= 1/6 * [(216/3) - (66)]

= -1/3

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The surface area of rectangular prism is 48 square units

The surface area of rectangular prism is 2(lw+wh+hl)

From the figure the height is 2 units

width is 2 units

length is 5 units

Plug in these values in formula

Surface area = 2(5×2 + 2×2 + 2×5)

=2(10+4+10)

=2(24)

=48

Hence, the surface area of rectangular prism is 48 square units

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The solution is, 10.8 kg. does the teams soft balls weigh in total.

Here, we have,

given that,

yalianny and gabby are training for softball. during a break in practice they found that one softball weighs 180 grams. the team has a total of 60 soft balls.

now, we have to find that  how many kilograms does the teams soft balls weigh in total.

so, to get the total weight we have to multiply 60 with 180.

as, we have,

one softball weighs 180 grams

and, the team has a total of 60 soft balls.

so total weight = 180 * 60

                         =10800 gm.

                         =10.8 kg.

Hence, The solution is, 10.8 kg. does the teams soft balls weigh in total.

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The discount is of 54.30 and the sale price is 126.69

If we have a discount of X (a percentage) and an original price P, then the discount is:

D = P*X/100%

And the sale price is:

S = P*(1 - X/100%)

Here the original price is 180.99 and the percentage is 30%.

Replacing that we will get.

D = 180.99*(30%/100%) = 180.99*0.3 = 54.30

S = 180.99*(1 - 30%/100%) = 126.69

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The value of k is given as follows:

k = 82º.

The middle segment of the angle bisects the larger angle. A bisection means that the larger angle is divided into two smaller angles of equal measure.

The angle measures are given as follows:

Hence the value of k is obtained as follows:

k = 82º.

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The first 4 terms in the expansion of (1 + x²)⁸ using the binomial theorem are: (1 + x²)⁸ = 1 + 8x² + 28x⁴ + 56x⁶ + ...

The question asks us to find the first 4 terms in the expansion of (1 + x^2)⁸. To expand this binomial, we can use the binomial theorem, which states that for any positive integer n:

To find the value of (1.01)⁸, we substitute x = 0.01 in the above expression:

(1.01)⁸ = (1 + 0.01²)⁸

= 1 + 8(0.01²) + 28(0.01⁴) + 56(0.01⁶) + ...

Using a calculator, we can evaluate this expression to get:

(1.01)⁸ ≈ 1.0824

Therefore, the value of (1.01)⁸ is approximately 1.0824.

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To compute cross products, you follow the determinant method, which is to take the difference of the products of the non-matching components and subtract them.

Here are the calculations for the given vectors:

1. B × C = (5 - 1, -(-6 - 3), -10 - 1) = (4, 9, -11)
2. C × B = -(B × C) = (-4, -9, 11)
3. (2B) × (3C) = (2*(-2, 5, 1)) × (3*(3, 1, 1)) = (-4, 10, 2) × (9, 3, 3)
  = (30 - 6, -(-12 - 18), -36 - 6) = (24, 30, -42)
4. A × (B × C) = (1, 0, -3) × (4, 9, -11)
  = (27 - 0, -(-33 - 12), 0 - 4) = (27, 45, -4)
5. A ⋅ (B × C) = (1, 0, -3) ⋅ (4, 9, -11) = 1*4 + 0*9 + (-3)*(-11) = 4 + 0 + 33 = 37

If V1 and V2 are perpendicular, their dot product is 0, and |V1 × V2| can be calculated using the formula: |V1 × V2| = |V1| * |V2| * sin(θ), where θ = 90 degrees (as they are perpendicular). sin(90) = 1, so |V1 × V2| = |V1| * |V2|.

If V1 and V2 are parallel, their cross product is 0, so |V1 × V2| = 0.

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The graph that most likely shows a system of equations with one solution is the one that intersects at a single point, which is (4, 4) in this case.

The first two steps in the table show that we can set the two equations equal to each other and solve for x:

x² - 6x + 12 = 2x - 4

Simplifying and rearranging, we get:

x² - 8x + 16 = 0

Factoring, we get:

(x - 4)(x - 4) = 0

So the only solution is x = 4, and we can substitute this back into either equation to find y:

y = x² - 6x + 12

y = 4² - 6(4) + 12

y = 4

Therefore, the system of equations has one solution, (4, 4).

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There are initially 750 bacteria. There are approximately 2.11 x 10^49 bacteria after 15 minutes. It takes approximately 0.111 minutes for the number of bacteria to double.

a. The initial number of bacteria (when t=0) can be found by plugging t=0 into the equation A(t) = 750e^(6.25t). So, A(0) = 750e^(6.25*0) = 750e^0 = 750*1 = 750. Thus, there are initially 750 bacteria.

b. To find the number of bacteria after 15 minutes, plug t=15 into the equation: A(15) = 750e^(6.25*15). A(15) ≈ 2.11 x 10^49. So, there are approximately 2.11 x 10^49 bacteria after 15 minutes.

c. To find the time it takes for the number of bacteria to double, set A(t) equal to twice the initial amount, 2 * 750 = 1500: 1500 = 750e^(6.25t). Solve for t by dividing both sides by 750, then taking the natural logarithm: ln(2) = 6.25t. Finally, divide by 6.25: t ≈ 0.111. Thus, it takes approximately 0.111 minutes for the number of bacteria to double.

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The probability that a light aircraft with an emergency locator, which has disappeared, will be discovered is 0.894.

Let A be the event that the aircraft is discovered, and B be the event that the aircraft has an emergency locator. We are given that P(A|B') = 0.28, P(B|A) = 0.63, and P(B'|A') = 0.84, where B' and A' denote the complements of B and A, respectively.

We want to find P(A|B), the probability that the aircraft is discovered given that it has an emergency locator. By Bayes' theorem,

P(A|B) = P(B|A)P(A) / P(B)

We can find P(A) and P(B) using the law of total probability:

P(A) = P(A|B)P(B) + P(A|B')P(B') = 0.63 * (1 - 0.72) + 0.28 * 0.72 = 0.3264

P(B) = P(B|A)P(A) + P(B|A')P(A') = 0.63 * 0.72 + 0.16 * (1 - 0.72) = 0.4656

Now, we can substitute these values into the first equation to get:

P(A|B) = 0.63 * (1 - 0.72) / 0.4656 = 0.894

Therefore, the probability that a light aircraft with an emergency locator, which has disappeared, will be discovered is 0.894.

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The arc length parameter is a parameterization of a space curve where the parameter t represents the distance traveled along the curve.

Here is these steps:

Step 1: Find the derivative of the space curve r(t) with respect to the parameter t. This is denoted as r'(t).

Step 2: Calculate the magnitude of r'(t). This can be done using the formula ||r'(t)|| = √(x'(t)^2 + y'(t)^2 + z'(t)^2), where x'(t), y'(t), and z'(t) are the derivatives of the x, y, and z components of r(t), respectively.

Step 3: Determine if the magnitude of r'(t) is equal to 1 for all values of t. If ||r'(t)|| = 1 for all t, then t is the arc length parameter for the space curve r(t).

By following these steps, you can check whether t is the arc length parameter for the given space curve. If the condition in Step 3 holds true, then t indeed represents the arc length parameter.

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No, the use of the binomial distribution may not be appropriate for calculating the probability that exactly six 18-20 year olds consumed alcoholic beverages in a random sample of ten.

The binomial distribution assumes that the trials are independent, there are only two possible outcomes (success or failure), and the probability of success remains constant throughout the trials. In the case of consuming alcoholic beverages, the assumption of independence may not hold, as one person's decision to consume alcohol may influence another person's decision. Additionally, the probability of consuming alcohol may not remain constant throughout the sample, as some people may have stronger tendencies or preferences for drinking than others.

A more appropriate distribution for this scenario may be the hypergeometric distribution, which takes into account the finite population size (i.e. the total number of 18-20 year olds from which the sample is drawn) and the varying probabilities of success (i.e. the varying number of individuals in the population who consume alcohol).

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The requreid 3,000 liters of leachate escape from the landfill each year.

The total amount of rain that falls on the landfill each year is:

500 x 150 = 75,000 liters of water

60 percent of the rain is runoff, so the amount of water that enters the landfill is:

75,000 liters x 0.4 = 30,000 liters

If the landfill has a 90 percent effective leachate collection system, then the amount of leachate that escapes is:

30,000 liters x (1 - 0.9) = 3,000 liters

Therefore, 3,000 liters of leachate escape from the landfill each year.

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The indefinite integral of (3x² * e^(6x²) dx) is (1/4)(e^(6x²)) + C. We can calculate it in the following manner.

To solve the indefinite integral of (3x² * e^(6x²) dx) using substitution, follow these steps:

1. Let u = 6x². Then, du/dx = 12x.
2. Rearrange to find dx: dx = du/(12x).
3. Substitute u and dx into the integral: ∫(3x² * e^u * (du/(12x))).
4. Simplify the integral: (1/4)∫(e^u du).
5. Integrate with respect to u: (1/4)(e^u) + C.
6. Substitute back for x: (1/4)(e^(6x²)) + C.

So, the indefinite integral of (3x² * e^(6x²) dx) is (1/4)(e^(6x²)) + C.

An indefinite integral of a function is the antiderivative of that function, which is a function whose derivative is equal to the original function, up to a constant of integration.

The indefinite integral of a function f(x) is denoted by ∫f(x) dx and is read as "the integral of f(x) with respect to x." When we take the indefinite integral of a function, we do not specify any limits of integration, and hence the result is an expression involving an arbitrary constant, which is determined by any additional information provided.

For example, the indefinite integral of f(x) = 3x^2 + 2x is:

∫f(x) dx = ∫(3x^2 + 2x) dx = x^3 + x^2 + C,

where C is the constant of integration. Note that if we differentiate the expression x^3 + x^2 + C with respect to x, we get 3x^2 + 2x, which is the original function f(x).

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Based on the given information, we can conclude that TUVW is a rhombus.

A rhombus is a quadrilateral with all four sides of equal length. Given that TU = UV = VW = WT, we can confirm that all sides of TUVW are equal. Additionally, the fact that the diagonals intersect at right angles (UV IVW, and VW IWT) tells us that TUVW is not just any rhombus, but a special kind of rhombus known as a square.

Therefore, TUVW is a square, which is a special type of rhombus, so it also has all the properties of a rhombus. In addition, it is also a parallelogram and a rectangle, since it has all the properties of those shapes. However, it is not a trapezoid, as a trapezoid has at least one pair of parallel sides, which TUVW does not have.

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The probability of the given event (drawing any marble from the jar) is 1, since you are guaranteed to draw a marble.

The probability of drawing a red marble is 30/62, since there are 30 red marbles out of a total of 62 marbles in the jar. Similarly, the probability of drawing a blue marble is 32/62. Given the jar has 30 red marbles (numbered 1-30) and 32 blue marbles (numbered 1-32), there are a total of 62 marbles in the jar. Since a marble is drawn at random, the probability of each event can be calculated as follows:
If the event is drawing a red marble:
Probability = (Number of red marbles) / (Total number of marbles) = 30/62
If the event is drawing a blue marble:
Probability = (Number of blue marbles) / (Total number of marbles) = 32/62
In both cases, the fractions are already reduced to their simplest form.

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The expression in simplified form is 4(2x - 3y).

Given that, an expression, 9x+(-x)+2y+(-14y), we need to simplify it,

9x+(-x)+2y+(-14y)

Opening the brackets,

= 9x - x + 2y - 14y

combining the like terms,

= 8x - 12y

Take 4 common,

= 4(2x - 3y)

Hence the expression in simplified form is 4(2x - 3y).

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1.The parametric equations of the line are:

2. The symmetric equations of the line are:

1.First, we need to find the direction vector of the line, which is perpendicular to both i + j and j + k. We can take their cross product:

(i + j) × (j + k) = i × j + i × k + j × j + j × k = -k + i

So the direction vector of the line is (-k + i), which is the same as (1, 0, -1).

Next, we need to find the parametric equations of the line. Let (x0, y0, z0) = (3, 2, 0) be a point on the line. Then the parametric equations are:

x(t) = x0 + at = 3 + t

y(t) = y0 + bt = 2 + 0t = 2

z(t) = z0 + ct = 0 - t = -t

where a, b, and c are the direction vector coefficients. So the parametric equations of the line are:

x = 3 + t

y = 2

z = -t

2. To find the symmetric equations, we can eliminate the parameter t. From the parametric equations, we have:

x - 3 = t

y - 2 = 0t = 0

z = -t

We can rearrange the first equation to get t = x - 3, and substitute into the third equation to get z = -(x - 3). Then we have:

x - 3 = x - 3

y - 2 = 0

z + x - 3 = 0

These are the symmetric equations of the line. Alternatively, we can eliminate t by setting x - 3 = -y - 2 and z = 0, which gives:

x + y + 5 = 0

z = 0

These are also symmetric equations of the line.

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The parametric form of the equation is;

The symmetric form of the equation is .

Given

The line through (3, 5, 0) and perpendicular to both i + j and j + k

The symmetric form of the equation of the line is given by;

Where the value of .

To find a, b, c by evaluating the product of ( i + j) and ( j + k ).

The value of a = 1, b = -1 and c = 1.

Substitute all the values in the equation.

Therefore,

The parametric form of the equation is;

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For the given vectors u = -i + 2j + k and v = -2i - 5j - 8k, the solution is:

u = (-1, 2, 1)

v = (-2, -5, -8)

u · v, = -16

Write the vectors in component form:
u = (-1, 2, 1)
v = (-2, -5, -8)

Compute the dot product (u · v) using the formula:
u · v = (u1 * v1) + (u2 * v2) + (u3 * v3)

Substitute the components of u and v into the formula:
u · v = (-1 * -2) + (2 * -5) + (1 * -8)
u · v = 2 - 10 - 8
u · v = -16

So, the given vectors are:
u = -i + 2j + k or (-1, 2, 1)
v = -2i - 5j - 8k or (-2, -5, -8)
and their dot product, u · v, is -16.

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If this portion leads to two segments that come to a point, and from that point, there is a height of 3 meters labeled.  The total area is 45 square meters.

Since the rectangle has a length of 4 meters and a width of 9 meters we need to find the area of rectangle

Area of rectangle  = length × width

Area of rectangle = 4 m × 9 m

Area of rectangle = 36 m^2

Since the  triangle has a base of 6 meters (9 meters - 3 meters o) and a height of 3 meters we need to find the Area of triangle

Area of triangle  = (1/2) × base × height

Area of triangle  = (1/2) × 6 m × 3 m

Area of triangle  = 9 m^2

Now let find the total area of the composite figure

Total area = Area  of rectangle + Area of triangle

Total area = 36 m^2 + 9 m^2

Total area = 45 m^2

Therefore the total area of the figure is 45 square meters.

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The length of the rectangular prism with height and width both of 9 cm and a surface area of 432 sq cm is 7.5 cm

A rectangular prism is also known as a cuboid and it has 6 faces made of rectangles.

S = 2(lb + bh + hl)

where l is the length

b is the breadth

h is the height

S is the surface area

Given,

h = 9 cm

b = 9 cm

S = 432 sq cm

S = 2 (9l + 9l * 81)

432 = 2 (18l + 81)

216 = 18l + 81

18l = 216 - 81

18l = 135

l = 7.5 cm

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Both Lagrange and Newton forms are valid methods to find the interpolating polynomial. Choose the most convenient form based on the problem at hand.

To find the polynomial of least degree that interpolates the given data points, we can use (a) Lagrange form and (b) Newton form.

(a) Lagrange form:

1. Calculate the Lagrange basis polynomials L0(x), L1(x), L2(x), and L3(x).
2. Multiply each basis polynomial by its corresponding y-value.
3. Sum the results to obtain the final Lagrange polynomial.

(b) Newton form:

1. Calculate the divided differences for the given data points.
2. Determine the Newton basis polynomials N0(x), N1(x), N2(x), and N3(x).
3. Multiply each basis polynomial by its corresponding divided difference.
4. Sum the results to obtain the final Newton polynomial.

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