Let C[infinity](R) be the vector space of all infinitely differentiable functions on R (i.e., functions which can be differentiated infinitely many times), and let D : C[infinity](R) → C[infinity](R) be the differentiation operator Df = f ‘ . Show that every λ ∈ R is an eigenvalue of D, and give a corresponding eigenvector

Answer: (x - y) = 13/6

Step-by-step explanation: To find the value of (x-y), we need to substitute the given values of x and y and then perform the subtraction.

So,

(x - y) = (5/3 - (-1/6))

We can simplify this expression by first converting the negative fraction to its equivalent positive fraction and then finding the common denominator.

(x - y) = (5/3 + 1/6) = ((10+3)/6) = 13/6

Therefore, (x - y) = 13/6.

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Answer:

11/6

Step-by-step explanation:

Use substitution.

x = 5/3

y = -1/6

Sub these values into (x-y):

[(5/3) - (-1/6)]

*Make sure to use brackets when subbing in values especially when there are negative signs or exponents

5/3 + 1/6 ⇒ two negatives become a positive

10/6 + 1/6 ⇒ make a common LCD

= 11/6

The explicit formula is C = 3.14 × d.

What is the explicit formula?

The formal equations for L-functions in mathematics are Riemann's zeta function and links between sums over an L-function's complex number zeroes and sums over prime powers.

Here, we have

Given:

d    C

2    6.28

3    9.42

5    15.70

10   31.40

We have to find the explicit formula that describes the given pattern.

We concluded from the given table that

when d = 2

we get

c = 3.14 × 2 = 6.28

When d = 3

We get

c = 3.14 × 3 = 9.42

When d = 5

We get

c = 3.14 × 5 =  15.70

When d = 10

we get

c = 3.14 × 10 = 31.40

Hence, the explicit formula is C = 3.14 × d.

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The possible zeros of the polynomial are -2, -3/2 and  4.

The zeros of the function is calculated as follows;

The zeros of the function are the values of x that will make the function equal to zero.

let x = -2

f(x) = 2x³ - x² - 22x - 24

f(-2) = 2(-2)³ - (-2)² - 22(-2) - 24

f(-2) = -16 - 4 + 44 - 24

f(-2) = 0

So, x + 2 is a factor of the polynomial, and other zeros of the polynomial is calculated as;

                       2x² - 5x - 12

                    ----------------------------------

         x + 2    √ 2x³ - x² - 22x  - 24

                  - (2x³ +  4x²)

                    ------------------------------------

                              -5x² - 22x -24

                            - (-5x² - 10x)

                       -------------------------------------

                                       -12x - 24

                                    - (-12x - 24)

                               -------------------------

                                              0

 2x² - 5x - 12 , so will factorize this quotient as follows;

= 2x² - 8x + 3x - 12

= 2x(x - 4) + 3(x - 4)

= (2x + 3)(x - 4)

2x + 3 = 0

or

x - 4 = 0

x = -3/2 or 4

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The correct statement regarding the rate of change of the exponential function is given as follows:

The bacterial culture loses 1/2 of it's size every 1/6 seconds.

The exponential function that gives the bacteria's population after t seconds is given as follows:

B(t) = 9300(1/64)^t.

The rate of change of the exponential function is given as follows:

1/64.

Hence the half-life of the population is obtained as follows:

(1/64)^t = 1/2

(1/2^6)^t = (1/2)

2^(-6t) = 2^(-1)

6t = 1

t = 1/6.

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Therefore, s1 and s2 do not span the same subspace of R3.

To determine whether the sets s1 and s2 span the same subspace of R3, we need to check if one set can be obtained as a linear combination of the other set.

We can start by checking if the vectors in s2 can be obtained as a linear combination of the vectors in s1. We can set up the following system of equations:

[tex]a(1, 2, -1) + b (0, 1, 1) + c(2, 5, -1) = (-2, -6, 0)[/tex]

[tex]d(1, 2, -1) + e(0, 1, 1) + f(2, 5, -1) = (1, 1, -2)[/tex]

We can write this system in matrix form as follows:

[tex]\left[\begin{array}{ccc}1&0&2|-2\\2&1&5|-6\\-1&1&-1|0\end{array}\right]*\left[\begin{array}{ccc}1&0&2|1\\2&1&5|1\\-1&1&-1|-2\end{array}\right][/tex]

We can row reduce this augmented matrix to find the solutions for the system of equations:

[tex]\left[\begin{array}{ccc}1&0&2|-2\\0&1&1|2\\0&0&0|0\end{array}\right]*\left[\begin{array}{ccc}1&0&2|1\\2&1&5|1\\0&0&0|0\end{array}\right][/tex]

The matrix on the left represents the coefficients for the linear combinations of the vectors in s1 that would give us the vectors in s2. Since the matrix has a row of zeros, this means that we can't obtain the vectors in s2 as a linear combination of the vectors in s1.

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Answer: Brainliest?

Step-by-step explanation:

To prove the identity 1 + 2 + 3 + ... + n = (n+1)/2 combinatorially, we can use a simple argument involving counting the number of ways to arrange a certain set of objects.

Consider a set of (n+1) objects, consisting of n white balls and 1 black ball. We want to count the number of ways to arrange these objects in a row. Let's call this number N.

On the one hand, we can count N directly by considering the number of choices we have for the first ball, then the number of choices we have for the second ball, and so on, until we have made n choices for the n white balls, leaving only the black ball to be placed in the last position. Using the multiplication principle, we see that N is equal to the product of n consecutive integers, which we can write as:

N = n(n-1)(n-2)...(3)(2)(1)

On the other hand, we can count N indirectly by considering the number of ways to divide the (n+1) objects into two groups: the black ball by itself, and the remaining n white balls. Since there are (n+1) objects in total, there are (n+1) ways to choose which object will be the black ball. Once we have made this choice, the remaining n white balls can be arranged in any order, giving us n! possible arrangements. Thus, the total number of arrangements is:

N = (n+1) n!

Now, these two expressions for N must be equal, since they are both counting the same thing. Equating them, we get:

n(n-1)(n-2)...(3)(2)(1) = (n+1) n!

Simplifying, we obtain:

1 + 2 + 3 + ... + n = n(n+1)/2

which is the desired identity.

At a 95% confidence level, the margin of error is approximately 0.762 (rounded to three decimal places).

To find the margin of error at a 95% confidence level, we need to first find the critical value associated with a sample size of 29 and a confidence level of 95%.

Using a t-distribution with n-1 degrees of freedom, we can find the critical value using a t-table or calculator. For n=29 and a confidence level of 95%, the critical value is approximately 2.045 (rounded to three decimal places).

The formula for the margin of error is:

Margin of error = critical value * (standard deviation / sqrt(sample size))

Plugging in the values we have:

Margin of error = 2.045 * (2 / sqrt(29))
Margin of error ≈ 0.762

Therefore, at a 95% confidence level, the margin of error is approximately 0.762 (rounded to three decimal places).

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Answer:

C

Step-by-step explanation:

K^2 + 4

Answer:k^2+4

Step-by-step explanation:

If the company made a profit of $112000, i first year operation, then the total profit made by the company in 26 years of operation is $8170286.

In order to find the profit after 26 years, we use the formula for "future-value" of investment with compound interest;

⇒ FV = PV × (1 + r)ⁿ,

where FV is = future value, PV is = present value, r is = interest rate per period, and n = time (in years),

In this case, the "present-value" is $112,000, the "interest-rate" per period is 18%, and time is 26 years.

We have to find the total profit, which is = "future-value" - "present-value",

⇒ Total profit = FV - PV,

Substituting the value,

We get,

⇒ FV = $112000 × (1 + 0.18)²⁶,

⇒ FV = $8282285.79 ≈ $8282286,

So, Total profit = $8282286 - $112000,

Total profit = $8170286,

Therefore, the company would make a total-profit of $8170286.

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The CFG has a new start state S and the original start state A is now a non-terminal symbol in the set V.

To convert the given CFG into CNF, we need to follow these steps:
Step 1: Add a new start state S and a new production rule S → A.
So, the modified CFG becomes:
S → A
A → BABB | 11
B → 00

Note that the original CFG had productions with single variables on the right-hand side. These productions do not follow the rules of CNF. By introducing a new start state and a new production rule, we can eliminate such productions and bring the CFG into CNF.

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(a) (i) Another pair of polar coordinates for (1, π/2) with r > 0 and 2π < θ < 4π is (1, 5π/2).
(ii) Another pair of polar coordinates for (1, π/2) with r < 0 and 0 < θ < 2π is (-1, π/2).

(b) (i) Another pair of polar coordinates for (-2, π/4) with r > 0 and 2π < θ < 4π is (2, 9π/4).
(ii) Since there is a typo in the question, I assume you meant 0 < θ < 2π. In this case, another pair of polar coordinates for (-2, π/4) with r < 0 and 0 < θ < 2π is (-2, π/4).

(c) (i) Assuming the correct range for θ is 2π < θ < 4π, another pair of polar coordinates for (3, 2) with r > 0 and 2π < θ < 4π is (3, 2 + 2π).
(ii) Another pair of polar coordinates for (3, 2) with r < 0 and 0 < θ < 2π is (-3, 2 + π).

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(a) (i) Another pair of polar coordinates for (1, π/2) with r > 0 and 2π < θ < 4π is (1, 5π/2).
(ii) Another pair of polar coordinates for (1, π/2) with r < 0 and 0 < θ < 2π is (-1, π/2).

(b) (i) Another pair of polar coordinates for (-2, π/4) with r > 0 and 2π < θ < 4π is (2, 9π/4).
(ii) Since there is a typo in the question, I assume you meant 0 < θ < 2π. In this case, another pair of polar coordinates for (-2, π/4) with r < 0 and 0 < θ < 2π is (-2, π/4).

(c) (i) Assuming the correct range for θ is 2π < θ < 4π, another pair of polar coordinates for (3, 2) with r > 0 and 2π < θ < 4π is (3, 2 + 2π).
(ii) Another pair of polar coordinates for (3, 2) with r < 0 and 0 < θ < 2π is (-3, 2 + π).

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The probability of finding an electron between x = 0.1 nm and x = 0.2 nm is 0.1 nm.

The probability density function for finding an electron between two points in space is given by the square of the absolute value of the wave function, integrated over the given range.

Let's start by finding the normalization constant A for the given wave function:

∫|Ψ|^2 dx = 1

∫(2/√l)sin(2πx/l) dx = 1

Using integration by parts, we get:

A = √(l/2)

Now, the probability of finding the electron between x = 0.1 nm and x = 0.2 nm is given by:

P = ∫0.2nm 0.1nm |Ψ|^2 dx

P = A^2 ∫0.2nm 0.1nm (sin(2πx/l))^2 dx

P = (l/2) ∫0.2nm 0.1nm (sin(2πx/l))^2 dx

P = (10/2) ∫0.2nm 0.1nm (sin(2πx/10))^2 dx

P = 2 ∫0.2nm 0.1nm (sin(πx/5))^2 dx

Using the identity sin^2θ = (1/2)(1 - cos(2θ)), we can simplify this expression:

P = 2 ∫0.2nm 0.1nm (1/2)(1 - cos(2πx/5)) dx

P = ∫0.2nm 0.1nm (1 - cos(2πx/5)) dx

P = (∫0.2nm 0.1nm dx) - (∫0.2nm 0.1nm cos(2πx/5) dx)

The first integral is simply the length of the given interval:

∫0.2nm 0.1nm dx = 0.1nm

For the second integral, we can use the fact that the integral of cos(mx) from 0 to 2π is zero, unless m is equal to zero. In this case, m = 5, so we get:

∫0.2nm 0.1nm cos(2πx/5) dx = 0

Therefore, the probability of finding the electron between x = 0.1 nm and x = 0.2 nm is:

P = 0.1nm

So the probability of finding an electron between x = 0.1 nm and x = 0.2 nm is 0.1 nm.

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The hypotenuse of the given triangle is 8√2 units.

In accordance with the Pythagorean theorem, the square of the length of the hypotenuse (the side that faces the right angle) in a right triangle equals the sum of the squares of the lengths of the other two sides. If you know the lengths of the other two sides of a right triangle, you may apply this theorem to determine the length of the third side. By examining whether the sides of a triangle satisfy the Pythagorean equation, it can also be used to assess whether a triangle is a right triangle. Pythagoras, an ancient Greek mathematician, is credited with discovering the theorem, therefore it bears his name.

The third side of the triangle can be determined using the Pythagoras Theorem as follows:

c² = 8² + 8²

c² = 2(8²)

c = 8√2

Hence, the hypotenuse of the given triangle is 8√2 units.

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The number of terms the series needed so that the remainder is less than 0.0005 is 14. The smallest integer value of n for which this is true is 14.

To find the number of terms needed for the remainder to be less than 0.0005, we need to use the remainder formula for an infinite series:
Rn = Sn - S
where Rn is the remainder after adding n terms, Sn is the sum of the first n terms, and S is the sum of the infinite series.

For this series, S can be found using the formula for the sum of a p-series:
S = Σ[infinity] 2/n^6 n=1 = π^6/945

Now we need to find the smallest value of n for which Rn < 0.0005. We can rewrite the remainder formula as:
Rn = Σ[infinity] 2/n^6 - Σ[n] 2/n^6

Simplifying the first term using the formula for the sum of a p-series, we get:
Σ[infinity] 2/n^6 = π^6/945

Substituting this into the remainder formula, we get:
Rn = π^6/945 - Σ[n] 2/n^6

We want Rn < 0.0005, so we can set up the inequality:
π^6/945 - Σ[n] 2/n^6 < 0.0005

Solving for n using a calculator or computer program, we get:
n ≥ 14

Therefore, we need at least 14 terms of the series Σ[infinity] 2/n^6 n=1 to ensure that the remainder is less than 0.0005, and the smallest integer value of n for which this is true is 14.

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Based on the given information, we can determine that events A and B are mutually exclusive. This is because the probability of their intersection, P(A and B), is equal to 0.

Based on the information provided for events A and B, we can determine if they are mutually exclusive, independent, both, or neither. Here's an analysis using the given probabilities:
1. P(A) = 0
2. P(A and B) = 0
3. P(B) = 0.25
Mutually exclusive events are events that cannot occur at the same time. In other words, if A occurs, then B cannot occur, and vice versa. If events are mutually exclusive, then P(A and B) = 0.
Independent events are events where the occurrence of one event does not affect the probability of the other event. If events A and B are independent, then P(A and B) = P(A) * P(B).

Now let's analyze:
A and B are mutually exclusive because P(A and B) = 0.
To check for independence, we calculate P(A) * P(B) = 0 * 0.25 = 0. Since P(A and B) = 0, A and B are also independent.
Therefore, events A and B are both mutually exclusive and independent. If A and B were independent events, then their intersection probability would be equal to the product of their individual probabilities, i.e. P(A and B) = P(A) * P(B), which is not the case here. Therefore, we can conclude that events A and B are mutually exclusive, but not independent.

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The three critical points of the function f(x, y) = (x² + y²)y² - x² are (0, 0), (0, -1), and (0, 1). The point (0, 0) is a saddle point, while (0, -1) is a local maximum and (0, 1) is a local minimum.

To find the critical points, first compute the partial derivatives fx and fy, then set them to zero and solve for x and y. fx = -2x(1+y²) and fy = 2y(3y²+x²).

Solving fx=0 and fy=0 simultaneously, we get (0, 0), (0, -1), and (0, 1) as critical points.

To determine the nature of each critical point, compute the second-order partial derivatives fx, fyy, and fxy, and then find the determinant of the Hessian matrix, D = fx * fyy - fxy². For (0, 0), D < 0, making it a saddle point.

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The required variance for the question is var (x - y) is 2.

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The mean (average) of the snowfall amounts is 17.9 inches.

In statistics, the mean is a measure of central tendency that represents the average value of a dataset. It is calculated by summing up all the values in the dataset and then dividing the result by the total number of values.

In statistics, the terms "mean" and "average" are often used interchangeably to refer to the same concept. Both terms represent a measure of central tendency that represents the typical or average value of a dataset.

To find the mean (average) of the snowfall amounts, we need to add up all the snowfall amounts and divide by the total number of years.

Adding up the snowfall amounts:

15 + 11 + 18 + 25 + 13 + 20 + 16 + 28 + 15 + 18 = 179

Dividing by the total number of years (10):

179/10 = 17.9

So the mean (average) of the snowfall amounts is 17.9 inches.

Therefore, the correct response to the mean is 17.9 in.

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The image magnification is about 0.133 times the size of the object and is inverted.

Magnification is the ratio of the size of the image to the size of the object. In optics, it is often used to describe how much larger or smaller an image appears compared to the object being viewed. It is calculated by dividing the height or size of the image by the height or size of the object. Magnification can be positive or negative, depending on whether the image is upright or inverted, respectively.

We can use the mirror equation to find the position of the image:

1/f = 1/do + 1/di

where f is the focal length, do is the object distance, and di are the image distance.

Since the ornament is a spherical mirror, the focal length is half the radius of curvature, which is equal to half the diameter of the ornament:

f = R/2 = 6.20 cm/2 = 3.10 cm

The object distance is given as 25.0 cm.

Substituting into the mirror equation and solving for di, we get:

1/3.10 = 1/25.0 + 1/di

di = 3.33 cm

The image is formed 3.33 cm from the center of the ornament, which is 0.23 cm beyond the surface of the ornament (since the ornament has a radius of 3.10 cm).

The magnification of the image can be found using the:

m = -di/do

where the negative sign indicates that the image is inverted.

Substituting the values we found, we get:

m = -(3.33 cm)/(25.0 cm) = -0.133

So the image is about 0.133 times the size of the object and is inverted.

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The measure of the angle in radians is approximately 1.9099 radians and Where radian_angle represents the angle's measure in radians, degree_angle represents the angle's measure in degrees, and π (pi) is approximately 3.1416


a. To convert an angle of 110 degrees to radians, we can use the following conversion formula:

[tex]radians = \frac{(degrees × π) }{180}[/tex]

Step 1: Plug in the given angle (110 degrees) into the formula:
[tex]radians= \frac{110×π}{180}[/tex]

Step 2: Calculate the value:
[tex]radians= \frac{(110)(3.1416)}{180} = \frac{343.7756}{180} = 1.9000[/tex]

So, the measure of the angle in radians is approximately 1.9099 radians.

b. To write a general formula that expresses the radian angle measure of an angle, in terms of the degree measure of that angle, you can use the following formula:
[tex]radian angle= \frac{degree angle x π}{180}[/tex]

Where radian_angle represents the angle's measure in radians, degree_angle represents the angle's measure in degrees, and π (pi) is approximately 3.1416.

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Answer: 0.84375

Step-by-step explanation:

1. x = 5.4n

2. 5.4n + n = 5.4

3. n(6.4) = 5.4

4. n = 0.84375

DOUBLE CHECK

x = 5.4(0.84375)

x = 4.55625

4.55625 + 0.84375 = 5.4

0.84375 = 5.4 - 4.55625

0.84375 = 0.84375

The solution is, : for 0.84375 = N such that x+N=5.4 and x/n=5.4 are equivalent equations.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

Here, we have,

the given equations are:

1. x = 5.4n

2. 5.4n + n = 5.4

3. n(6.4) = 5.4

4. n = 0.84375

DOUBLE CHECK

x = 5.4(0.84375)

x = 4.55625

4.55625 + 0.84375 = 5.4

0.84375 = 5.4 - 4.55625

0.84375 = 0.84375

Hence, The solution is, : for 0.84375 = N such that x+N=5.4 and x/n=5.4 are equivalent equations.

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The simplified expression is [tex]\frac{2x}{x-1}[/tex].

Expression  - An expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them.

To simplify the expression, we can factor out a 4x from the numerator and a 2 from the denominator, which gives:

[tex]=\frac{4x^2+4x}{2x^2-2}\\\\ = \frac{4x(x+1)}{2(x^2-1)} \\\\ = \frac{2\cdot 2x(x+1)}{2(x-1)(x+1)} \\\\ = \frac{2x}{x-1}[/tex]

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The value of function f(1) = 1/16.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is typically represented as y = f(x).

To find f(1), we substitute x = 1 into the given expression for f(x):

f(1) = 4(1/4)⁽¹⁺²⁾ = 4(1/4)³ = 4(1/64) = 1/16

Therefore, f(1) = 1/16.

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The equation to determine p, which is the total number of player at the basketball game, is 12 = p x 3 / 4 .

The number of players at the basketball game is 16 players .

The equation to find the number of players is;

Players who chose water = total players x proportion of players who chose water

12 = p x 3 / 4

This means that solving for p gives :

12 = p x 3 / 4

12 ÷ 3 / 4 = p

p = 12 ÷  3 / 4

p = 12 / 0.75

p = 16 players

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The value of r in terms of l and h is,

⇒ r = √((l/2)² + h²)

Now, We can use the Pythagorean theorem to relate r, l, and h as,

Since, The chord of the circle divides the circle into two segments, each with a height of h.

Let's call the segments are A and B.

Then, the length of the chord (l) is equal to the sum of the bases of segments A and B.

Therefore, the length of each base is,

(l/2).

Hence, We can use the Pythagorean theorem to relate r, l/2, and h for one of the segments as;

⇒ r² = (l/2)² + h²

⇒ r = √((l/2)² + h²)

Thus, The value of r in terms of l and h is,

⇒ r = √((l/2)² + h²)

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The height of the tree to the nearest hundredth is 29.65 ft.

The measure of the distance from the tree and the angle of elevation from the ground to the top of the tree is represented as follows:

Therefore, the height of the tree to the nearest hundredth can ne found as follows:

Therefore, using trigonometric ratios,

tan 56° = opposite / adjacent

tan 56° = h / 20

cross multiply

h = 20 tan 56°

h = 20 × 1.48256096851

h = 29.6512193703

h = 29.65 ft

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Answer:

It takes about 1 hour to read 30 pages at an average reading speed

Step-by-step explanation:

Answer:

16 lbs

Step-by-step explanation:

total of apples = 1/4 + 3/8 = 2/8 + 3/8 = 5/8

then 10 x 8/5 = 16

Answer:207

Step-by-step explanation: circumference = to 2πr or πd
substitute the diameter into the equation πd (22/7)(66) and you get 207.4285...