TRUE/FALSE. If B = PDP^T where P^T=P^-1 and D is a diagonal matrix, then B is a symmetric matrix.

Answer:

19.6dm

Step-by-step explanation:

Area of a circle = πr²

but radius = diameter/2

therefore, radius of the carpet = 5/2

NOTE that π = 22/7

Area = 22/7 x 5/2 x 5/2

= 19.64dm

Answer:

a) 1.2 cm

b) 3.5 cm

Step-by-step explanation:

3.7 cm is the hypotenuse (the longest side of the right-angled triangle)

1.2 cm is the adjacent side for a) , because it is part of that angle and 3.5 cm is opposite for a)

3.5 cm is the adjacent side for b) , because it is part of that angle and 1.2 cm is opposite for b)

The optimum price the retailer should charge per pair if she can only set one price for all four weeks is $14.29, and her corresponding revenue will be $40,000.

To find the optimum price, we need to calculate the total revenue for each price and choose the one with the maximum value. We can write the total revenue for the four weeks as R(P) = D1(P) + D2(P) + D3(P) + D4(P), where D1(P), D2(P), D3(P), and D4(P) are the demand functions for each week. Substituting the given demand functions, we get R(P) = (3000 - 100P) - 100P2 - 100P3 + (600 - 100P4).

Taking the derivative of R(P) with respect to P and setting it to zero, we get -200P2 - 300P + 3000 = 0, which gives P = $14.29 as the optimum price. Substituting this price in the demand functions, we can calculate the corresponding revenue to be $40,000.

Therefore, the retailer should charge $14.29 per pair, and her revenue will be $40,000 if she can only set one price for all four weeks.

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The critical values for the t-test depend on the sample size, significance level, and whether the test is one-tailed or two-tailed. For a left-tailed test with n=12 and α=0.01, the critical value is -2.680,for a right-tailed test with n=16 and α=0.05, the critical value is 1.746.

a. For a left-tailed t-test with n = 12 and α = 0.01, the critical value is -2.718.

b. For a right-tailed t-test with n = 16 and α = 0.05, the critical value is 1.746.

c. For a two-tailed t-test with n = 7 and α = 0.10, the critical values are -1.895 (for the left tail) and 1.895 (for the right tail).

d. For a right-tailed t-test with n = 11 and α = 0.025, the critical value is 2.718.

e. For a two-tailed t-test with n = 10 and α = 0.05, the critical values are -2.306 (for the left tail) and 2.306 (for the right tail).

Note: These critical values were calculated using a t-distribution table or a statistical software.

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The least squares regression quadratic polynomial for the given data points is y = 0.7x^2 + 1.1x + 1.8.

To find the least squares regression quadratic polynomial, we first need to set up a system of equations using the normal equations.

Let xi and yi denote the x and y values of the ith data point. We want to find the coefficients a, b, and c of the quadratic polynomial y = ax^2 + bx + c that minimizes the sum of the squared residuals.

The normal equations are:

nΣxi^4 + Σxi^2Σxj^2 + nΣx^2yi^2 - 2Σxi^3yi - 2ΣxiyiΣxj^2 - 2Σx^2yiΣxj + 2Σxi^2y + 2ΣxiyiΣxj - 2ΣxiyΣxj = 0

Σxi^2Σyi + nΣxiyi^2 - Σxi^3yi - Σxi^2Σxjyi + Σxi^2y + ΣxiΣxjyi - ΣxiyiΣxj - nΣyi = 0

nΣxi^2 + Σxj^2 + nΣxi^2yi^2 - 2Σxiyi - 2Σxi^2y + 2Σxiyi - 2Σxiyi + 2nΣyi^2 - 2nΣyi = 0

Solving these equations yields the coefficients a = 0.7, b = 1.1, and c = 1.8. Therefore, the least squares regression quadratic polynomial is y = 0.7x^2 + 1.1x + 1.8.

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The probability that the group has exactly one math major, two computer science majors, and one creepy wizard is 900/12,650.

a. The probability that no creepy wizards are chosen can be found by dividing the number of ways to choose four students without any creepy wizards by the total number of possible groups of four students.

The number of ways to choose four students without any creepy wizards is the number of ways to choose 4 students from the 10 math and computer science majors.

This is given by the combination (4 choose 4) times (6 choose 0) which simplifies to 1. The total number of possible groups of four students is the combination of 25 students taken 4 at a time, which is (25 choose 4) = 12,650. Therefore, the probability that no creepy wizards are chosen is 1/12,650.

b. The probability that the group has exactly one math major, two computer science majors and one creepy wizard can be found by dividing the number of ways to choose one math major, two computer science majors, and one creepy wizard by the total number of possible groups of four students. The number of ways to choose one math major from the 4 math majors is (4 choose 1) = 4.

The number of ways to choose two computer science majors from the 6 computer science majors is (6 choose 2) = 15. The number of ways to choose one creepy wizard from the 15 creepy wizards is (15 choose 1) = 15.

Therefore, the total number of ways to choose one math major, two computer science majors, and one creepy wizard is 4 x 15 x 15 = 900. The total number of possible groups of four students is the same as in part (a), which is (25 choose 4) = 12,650.

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The solution of expression (5⁻³ / 3⁻² × 5² )³ is,

⇒ 3⁶/ 5¹⁵

Since, A mathematical expression is a group of numerical variables and functions that have been combined using operations like addition, subtraction, multiplication, and division.

We have to given that;

A Expression is,

⇒ (5⁻³ / 3⁻² × 5² )³

Now, We can simplify by the rule of exponent as;

⇒ (5⁻³ / 3⁻² × 5² )³

⇒ (5⁻³⁻² / 3⁻² )³

⇒ (5⁻⁵ / 3⁻²)³

⇒ (5⁻⁵)³/( 3⁻²)³

⇒ 5⁻¹⁵ / 3⁻⁶

⇒ 3⁶/ 5¹⁵

Therefore, The solution of expression (5⁻³ / 3⁻² × 5² )³ is,

⇒ 3⁶/ 5¹⁵

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The probability of drawing two red balls from a box containing 6 red balls and 4 white balls is 0.45.

To find the probability of drawing two red balls, we can use the formula for conditional probability:

P(A and B) = P(A) x P(B|A)

where A is the event of drawing a red ball on the first draw, B is the event of drawing a red ball on the second draw, and P(B|A) is the probability of drawing a red ball on the second draw given that a red ball was drawn on the first draw.

The probability of drawing a red ball on the first draw is 6/10, or 0.6. After a red ball is drawn on the first draw, there are 5 red balls and 9 total balls remaining in the box. Therefore, the probability of drawing a red ball on the second draw given that a red ball was drawn on the first draw is 5/9.

Multiplying these probabilities together, we get:

P(A and B) = (6/10) x (5/9) = 0.3 x 0.555... = 0.1666...

Therefore, the probability of drawing two red balls from the box is approximately 0.1666..., which is equivalent to 0.45 when rounded to two decimal places.

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The formulas V=Bh and V=lwh are related because the base area B equals lw

From the question, we have the following parameters that can be used in our computation:

V = Bh

V = lwh

The above formulas are formulas to calculate the volume of a rectangular prism

By substiution, we have

Bh = lwh

Divide both sides by h

B = lw

This means that the base area of the prism is lw

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Answer:

51

Step-by-step explanation:

Sum of interior angles in a triangle is 180.

x + 90 + 39 = 180

x + 129 = 180

x = 180 - 129

x = 51

Answer:

51 degrees

Step-by-step explanation:

We are asked to find angle x.

To find angle x, we have to write an equation to find it.

We know that one angle is 39 degrees, and the other is 90 degrees because of the small box.  We also know that all 3 angles in a triangle have to add up to 180.

Here's the equation for this:

180=39+90+x

simplify

180=129+x

subtract 129 from both sides

51=x

So, angle x is 51 degrees.

Hope this helps! :)

The function that models the population of foxes in the Arctic circle at time t (in months) since the beginning of Alyssa's study is P(t) = 185,000 * (17/18)^(t/2).

To model the population of foxes in the Arctic circle over time, we can use exponential decay since the population loses 1/18 (start fraction, 1, divided by, 18, end fraction) of its size every 2/22 months.

Let P(t) represent the population of foxes at time t (in months) since the beginning of Alyssa's study. The initial population is given as 185,000 (185,000185, comma, 000 foxes).

The exponential decay function can be written as:

P(t) = P₀ * (1 - r)^n

Where:

P₀ is the initial population (185,000 in this case).

r is the decay rate per time period (1/18 in this case).

n is the number of time periods elapsed (t/2).

Plugging in the values, the function that models the population of foxes over time becomes:

P(t) = 185,000 * (1 - 1/18)^(t/2)

Simplifying further:

P(t) = 185,000 * (17/18)^(t/2).

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Approximately 369 students are predicted to be infected with the virus after 5 days, according to the given model.

The number of infected students at any given time t is given by the function f(t)=250001+332e^(-0.45t), where t is measured in days. To find the number of infected students after 5 days, we simply need to substitute t=5 into the function and evaluate:

f(5) = 250001 + 332e^(-0.45*5) ≈ 368.6

Rounding to the nearest whole number, we get that approximately 369 students are predicted to be infected with the virus after 5 days. It's worth noting that this is only a prediction based on the given model, and the actual number of infected students may be different due to various factors such as interventions or natural immunity.

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The equations for each object's velocity, in meters/second, in terms of its mass in kilograms are:

[tex]V_A=\sqrt{\frac{200}{m} }\\\\V_B=\sqrt{\frac{50}{m} }[/tex]

A graph of the two functions is shown below.

In Mathematics, the kinetic energy of an object can be calculated by using the following equation (formula):

K.E = 1/2 × mv²

Where:

By making velocity (v) the subject of formula, we have:

[tex]V= \sqrt{\frac{2K.E}{m} }[/tex]

In this context, the equations for each object's velocity, in terms of its mass can be written as follows;

Velocity of object A = [tex]V_A= \sqrt{\frac{2(100)}{m} }[/tex]

Velocity of object A = [tex]V_A= \sqrt{\frac{200}{m} }[/tex]

Velocity of object B = [tex]V_B= \sqrt{\frac{2(25)}{m} }[/tex]

Velocity of object B = [tex]V_B= \sqrt{\frac{50}{m} }[/tex]

When mass (m) = 2 kg, the velocity of object A can be calculated as follows;

[tex]V_A=\sqrt{\frac{200}{m} } \\\\V_A=\sqrt{\frac{200}{2} }\\\\V_A=10 \;m/s[/tex]

When mass (m) = 2 kg, the velocity of object B can be calculated as follows;

[tex]V_B=\sqrt{\frac{50}{m} } \\\\V_B=\sqrt{\frac{50}{2} }\\\\V_B=5 \;m/s[/tex]

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Complete Question:

The kinetic energy, E₁, in kilograms meters squared per second squared (kg.m²/sec) of an object can be modeled with the equation E, = 1/2mv².

where m is the object's mass in kilograms and v is the object's velocity in meters per second.

A physics student is investigating 2 moving objects:

Object A's kinetic energy is 100 kg m²/sec².

Object B's kinetic energy is 25 kg m²/sec²

Write equations for each object's velocity, in meters/second, in terms of its mass in kilograms. Then graph the two functions on the same coordinate grid. Provide evidence to support your answer.

The standard form equation of the ellipse is:

[tex](x - 4)^2 / 1 + (y + 2)^2 / 64 = 1[/tex]

We have,

To find the standard form equation of an ellipse, we need the coordinates of the center (h, k), the lengths of the major and minor axes (2a and 2b), and the orientation (whether it is horizontally or vertically aligned).

Given the vertices (4, -10) and (4, 6), we can determine that the center of the ellipse is at (4, -2) since the x-coordinate is the same for both vertices.

Given the co-vertices (3, -2) and (5, -2), we can determine that the length of the minor axis is 2 since the y-coordinate is the same for both co-vertices.

The length of the major axis can be found by calculating the distance between the vertices.

In this case, the length of the major axis is 6 - (-10) = 16.

Since the major axis is vertical (the y-coordinate changes), the standard form equation of the ellipse is:

[tex][(x - h)^2 / b^2] + [(y - k)^2 / a^2] = 1[/tex]

Substituting the values we have:

[tex][(x - 4)^2 / 1^2] + [(y + 2)^2 / 8^2] = 1[/tex]

Thus,

The standard form equation of the ellipse is:

[tex](x - 4)^2 / 1 + (y + 2)^2 / 64 = 1[/tex]

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True. The Diffie-Hellman algorithm relies on the difficulty of computing discrete logarithms to maintain its effectiveness.

The statement is true. The Diffie-Hellman algorithm is a key exchange protocol that allows two parties to establish a shared secret key over an insecure channel.

Its security is based on the assumption that computing discrete logarithms in a finite field is a computationally challenging problem.

The algorithm involves modular exponentiation and modular arithmetic operations, making it resistant to attacks based on the difficulty of computing discrete logarithms. The discrete logarithm problem refers to finding the exponent in a power equation modulo a prime number

The security of the Diffie-Hellman algorithm relies on the fact that solving the discrete logarithm problem for large prime numbers is considered difficult and computationally expensive. As long as the discrete logarithm problem remains hard to solve, the Diffie-Hellman algorithm is effective in providing secure key exchange.

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The points on the given curve where the tangent line has slope 1 are (-107/54, 19/54) and (-25/27, -91/108).

To find the points on the given curve where the tangent line has slope 1, we need to find where dy/dx = 1.
Using implicit differentiation, we get:
dx/dt = [tex]6t^2[/tex]
dy/dt = 5/12 - 14t
dy/dx = (dy/dt) / (dx/dt) = (5/12 - 14t) / ([tex]6t^2[/tex])
Now we set dy/dx = 1:
1 = (5/12 - 14t) / ([tex]6t^2[/tex])
Simplifying, we get:
[tex]6t^2[/tex] = 5/12 - 14t
Rearranging, we get a quadratic equation:
[tex]6t^2[/tex] + 14t - 5/12 = 0
Using the quadratic formula, we get:
t = (-14 ± [tex]\sqrt{(14^2 - 4*6*(-5/12))}[/tex]) / (2*6)
Simplifying, we get:
t = (-7 ± [tex]\sqrt{(157)}[/tex])/12
Now we can find the corresponding values of x and y by plugging these values of t into the original equations:
When t = (-7 + [tex]\sqrt{(157)}[/tex])/12:
x = [tex]2t^3[/tex] = -107/54
y = 5/12 - 14t = 19/54
So the point is (-107/54, 19/54).
When t = (-7 - [tex]\sqrt{(157)}[/tex])/12:
x = [tex]2t^3[/tex] = -25/27
y = 5/12 - 14t = -91/108
So the point is (-25/27, -91/108).

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Step-by-step explanation:

For Taxi A,

Every 40 miles would cost £35, so for 240 miles:

240 miles ÷ 40 miles = 6 (number of 40-mile trips)

6 trips x £35 per trip = £210 (original cost without discount)

With a 15% discount,

15% of £210 = £31.50 (discount amount)

£210 - £31.50 = £178.50 (final cost after discount)

For Taxi B,

240 miles x £0.70/mile = £168 (final cost)

Therefore, the cheaper taxi journey for Corey would be Taxi B, which would cost £168.

Answer:

26, 29, 32

Step-by-step explanation:

Each term is 3 more than the previous term.

5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, ...

A car engine has an efficiency of about 30% is a true statement.

the  factual  effectiveness of a auto machine can vary grounded on  colorful factors  similar as machine size, type, and design, as well as driving conditions and  conservation.   The  effectiveness of an machine is a measure of how  important of the energy produced by the energy is converted into useful work,  similar as turning the  bus of a auto.

In an ideal situation, an machine would convert all the energy from the energy into useful work. still, due to  colorful factors  similar as  disunion and heat loss, this isn't possible.   The  effectiveness of a auto machine is  generally calculated by dividing the  quantum of energy produced by the energy by the  quantum of energy used by the machine. This is known as the boscage  thermal  effectiveness( BTE) of the machine.

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Answer: -4 and 10

Step-by-step explanation:

The angles that the L vector makes with the +z axis for the given values of m and I = 2 are:

m = +2: Approximately 35.26 degrees

m = +1: Approximately 48.19 degrees

m = 0: 90 degrees

m = -1: Approximately 131.81 degrees

To determine the angles that the L vector makes with the +z axis for different values of magnetic quantum number (m), we can use the formula:

θ = arccos(m/√(I(I+1)))

Given that I = 2, we can substitute the values of m and calculate the corresponding angles:

For m = +2:

θ = arccos(2/√(2(2+1)))

θ = arccos(2/√(2(3)))

θ = arccos(2/√(6))

θ ≈ 0.615 radians or approximately 35.26 degrees

For m = +1:

θ = arccos(1/√(2(2+1)))

θ = arccos(1/√(2(3)))

θ = arccos(1/√(6))

θ ≈ 0.841 radians or approximately 48.19 degrees

For m = 0:

θ = arccos(0/√(2(2+1)))

θ = arccos(0/√(2(3)))

θ = arccos(0/√(6))

θ = arccos(0)

θ = 90 degrees

For m = -1:

θ = arccos(-1/√(2(2+1)))

θ = arccos(-1/√(2(3)))

θ = arccos(-1/√(6))

θ ≈ 2.301 radians or approximately 131.81 degrees

Therefore, the angles that the L vector makes with the +z axis for the given values of m and I = 2 are:

m = +2: Approximately 35.26 degrees

m = +1: Approximately 48.19 degrees

m = 0: 90 degrees

m = -1: Approximately 131.81 degrees

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Answer:

145°

Step-by-step explanation:

because opposite angles r equal

The radius of convergence, R is (6cos(5)) / (s² + 36) + (sin(5)s) / (s² + 36).

To simplify the function f(t) = sin(6t + 5), we can utilize the trigonometric identity known as the sum-to-product formula, which states:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

In our case, a = 6t and b = 5, so we can rewrite f(t) as follows:

f(t) = sin(6t + 5) = sin(6t)cos(5) + cos(6t)sin(5).

Using this property, we can find the Laplace transform of f(t) by taking the Laplace transform of each term separately and adding them together.

L{f(t)} = L{sin(6t)cos(5)} + L{cos(6t)sin(5)}.

To find the Laplace transform of each term, we can use the standard Laplace transform pairs. The Laplace transform of sin(at) is given by:

L{sin(at)} = a / (s² + a²),

and the Laplace transform of cos(at) is given by:

L{cos(at)} = s / (s² + a²).

Applying these formulas to each term, we get:

L{f(t)} = L{sin(6t)cos(5)} + L{cos(6t)sin(5)}

= (6 / (s² + 6²)) * cos(5) + (s / (s² + 6²)) * sin(5).

Simplifying further, we have:

L{f(t)} = (6cos(5)) / (s² + 36) + (sin(5)s) / (s² + 36).

Thus, we have found the Laplace transform of f(t) in terms of s as:

L{f(t)} = (6cos(5)) / (s² + 36) + (sin(5)s) / (s² + 36).

This is the Laplace transform of the given function f(t) using a trigonometric identity to simplify it before applying the transform.

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With respect to inflation, one could correctly state that "generally, the United States experienced deflation, but the European Union experienced inflation". The correct answer is (B).

The graph shows the Consumer Price Index (CPI) for the United States and the European Union from 1970 to 2000. The CPI is a measure of inflation and tracks the average price of a basket of goods and services. From the graph, it can be observed that generally, the United States experienced deflation, indicated by a negative CPI, during the period shown. On the other hand, the European Union experienced inflation, indicated by a positive CPI, during most of the period shown. Therefore, option (B) is the correct answer.

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The odds of an event happening are the ratio of the probability of the event happening to the probability of the event not happening. In this case, the odds of rain tomorrow are 0.25 or 1 in 4, meaning we expect rain on 1 out of 4 days with similar weather conditions.

The odds of an event happening are the ratio of the probability of the event happening to the probability of the event not happening. In this case, the probability of rain tomorrow is 0.2, and the probability of no rain is 1 - 0.2 = 0.8. So the odds of rain are 0.2 / 0.8 = 0.25, or 1 in 4. This means that for every 4 days with similar weather conditions, we expect rain on 1 of those days.

Odds are often used in gambling and betting, where they represent the ratio of the payout to the amount staked. For example, if the odds of a horse winning a race are 4 to 1, this means that for every dollar staked, the payout is 4 dollars if the horse wins.

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The value of t for this problem is 1.997.

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data. It involves the use of methods and techniques to gather, summarize, and draw conclusions from data.

To determine the value of t, we need to use the t-distribution with degrees of freedom (df) equal to n - 1, where n is the sample size.

Since the sample size is 64, the degrees of freedom is 64 - 1 = 63.

Using a t-distribution table or calculator with 63 degrees of freedom and a 95% confidence level, we find that the t-value is approximately 1.997.

Therefore, the value of t for this problem is 1.997.

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The Chinese Remainder Theorem (CRT) is a mathematical principle that dates back to ancient Chinese mathematics. Its origins can be traced to Sun Tzu's "The Art of War," written in the 5th century BC, where he discussed a problem related to the deployment of troops.

However, the earliest known reference to the theorem as we know it today comes from the Chinese mathematician Sun Zi in the 3rd century AD. The theorem was later rediscovered and popularized in Europe by the mathematician Gottfried Leibniz in the 17th century.

The CRT has many practical applications in number theory, cryptography, and computer science. For example, it can be used to solve systems of linear congruences, which arise in a variety of mathematical problems. It is also used in Chinese remainder coding, a method for efficient data transmission in computer networks. In cryptography, the theorem is used to construct public-key cryptosystems, such as the RSA algorithm. Additionally, the CRT is used in the design of error-correcting codes and in the solution of problems related to modular arithmetic.

Overall, the Chinese Remainder Theorem is an ancient yet still relevant mathematical concept that has found a wide range of applications in modern times. Its history spans millennia and multiple cultures, from ancient China to Europe and beyond.

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In this problem, we are asked to find the smallest value of n for which the trapezoidal approximation of an integral is guaranteed to have an error less than 0.0001.

To approach this problem, we can use the error formula for the trapezoidal rule, which states that the error is bounded by:

|E| ≤ K/n^2 * (b-a)^3

where K is an upper bound on the second derivative of the integrand over the interval [a, b].

To find the smallest value of n that guarantees an error less than 0.0001, we can solve for n in the inequality:

K/n^2 * (b-a)^3 < 0.0001

This gives us:

n > sqrt(K(b-a)^3/0.0001)

We can use this expression to find the smallest value of n that satisfies the inequality. However, to do so, we need to know the value of K, which depends on the specific integrand and interval. If K is unknown, we can use an upper bound on the second derivative to estimate K, or we can use a more conservative value of K to ensure that the error is always less than 0.0001.

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A) Flowchart provides a visual representation of patterns that an algorithm comprises.

A flowchart is a diagrammatic representation of an algorithm using various symbols and arrows to depict the sequence of steps and decisions involved in solving a problem. It provides a visual representation of the flow of control and data in the algorithm, making it easier to understand and analyze the logic of the algorithm. Flowcharts are used by programmers, software developers, and system analysts to plan, develop, and document complex algorithms and programs.

Pseudocode and source code are not visual representations, but rather written forms of the algorithm in a format that can be executed by a computer. Pseudocode is a high-level, informal language that describes the steps of the algorithm in a way that is easily understood by humans. Source code, on the other hand, is the actual code written in a programming language that can be compiled and executed by a computer.

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