What is the length of the curve y = 1- cos x from x = 0 to x = 4T ? A) 11.314 B 15.281 с 18.850 19.015

Answers

Answer 1

To find the length of the curve y = 1 - cos x from x = 0 to x = 4π, we can use the arc length formula:

Arc length = ∫[sqrt(1 + (dy/dx)^2)] dx from a to b

First, find the derivative of y with respect to x:
dy/dx = d(1 - cos x)/dx = sin x

Now, substitute into the arc length formula:
Arc length = ∫[sqrt(1 + sin^2(x))] dx from 0 to 4π

This integral doesn't have an elementary function as its antiderivative, but we can use the elliptic integral of the second kind to evaluate it.

Arc length = 4 * E(4π, k) where E is the elliptic integral of the second kind and k = 1 (since k = sin^2(x) in our case).

Using a calculator, we find that E(4π, 1) ≈ 3.8202. So, the arc length is approximate:
Arc length ≈ 4 * 3.8202 ≈ 15.2808

Hence, the correct answer is B) 15.281.

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Related Questions

Please help I need to turn this in now. 30pts

Please help I need to turn this in now. 30pts

Answers

Answer:

Hello! Your answer is BELOW

Step-by-step explanation:

Number 1: D)

Number 2: A)

Hope I helped! Hope you make an 100%

-Amelia♥

Answer:

3rd option and 2nd option

Step-by-step explanation:

(10)

A linear pair sum to 180° , then

x - 30 + x + 76 = 180 , that is

2x + 46 = 180 ( subtract 46 from both sides )

2x = 134 ( divide both sides by 2 )

x = 67

Then

∠ 1 = x - 30 = 67 - 30 = 37°

∠ 2 = x + 76 = 67 + 76 = 143°

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

(11)

Since T is the midpoint of SU , then

ST = TU , that is

5x = 3x + 32 ( subtract 3x from both sides )

2x = 32 ( divide both sides by 2 )

x = 16

Then

ST = 5x = 5(16) = 80

TU = 3x + 32 = 3(16) + 32 = 48 + 32 = 80

SU = 5x + 3x + 32 = 8x + 32 = 8(16) + 32 = 128 + 32 = 160

S. The table below shows the relationship between the number of hours a student studied and their
grade on a certain test.
100
80
Test Grade
888
60
20
0246 8
Hours Studying
a) Find the line of best fit
b) Estimate the grade for a student that did not study at all:

Answers

a) The line of best fit is y = 10x  + 60

b) the grade for a student that did not study at all is 0.

What is the slope?

The slope of a line represents the change in the dependent variable (y) for a unit change in the independent variable (x). In this case, the slope of the line of best fit represents the average change in the test grade for a one hour increase in studying. To find the slope, you can use the formula:

m = (Σxy - (Σx)(Σy)/n) / (Σx² - (Σx)²/n)

a) The line of best fit is a line that best approximates the relationship between two variables. To find the line of best fit for this data, we can use the least squares method to minimize the sum of the squares of the differences between the observed data points and the line. The equation of the line of best fit is y = mx + b, where m is the slope and b is the y-intercept. The slope can be calculated as (Σxy - (Σx)(Σy)/n) / (Σx² - (Σx)²/n), and the y-intercept as (Σy - m(Σx)) / n, where n is the number of data points.

(2, 60) and (5, 90)

The slope is

y2 - y1 =  m(x2 - x1)

90 - 60 = m(5-2)

30 = 3m

m = 10

Slope line is

y - 60 = m(x - 2)

y = 10x - 20 + 80

y = 10x  + 60

b) To estimate the grade for a student that did not study at all, we can use the line of best fit and plug in x=0 for the number of hours studying. The estimated grade for a student that did not study at all would be the y-intercept of the line,

which represents the expected test grade when

x = 0

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Complete question:

S. The table below shows the relationship between the number of hours a student studied and theirgrade

From the information​ given, what is the biggest flaw in the​ study? A health researcher randomly selected 500 high school students from the city of Oak Grove. In a private interview the researcher asked the students whether they had ever used drugs. She concluded that only​ 8% of the high school students of Oak Grove have ever used drugs.

Answers

Answer:

The correct answer is - The setting may discourage honest responses.

Step-by-step explanation:

In this particular setting of research and interview regarding this research may not produce hones results as the question is too direct to ask students if they had ever used drugs.

Research should be anonymous and should avoid the questions that can lead to bias results like this case that may lead to dishonest results.

Thus, the correct answer is : The setting may discourage honest responses.

THISS IS SURGENT PLSSS Form a polynomial whose zeros and degree are given.

​Zeros: ​-7, multiplicity​ 1 3;multiplicity​ 2; degree 3

Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below.

Answers

Answer:

The polynomial is x^3-x^2-33x+63

Step-by-step explanation:

Here, we want to find a polynomial of degree 3 with the given roots

-7 with a multiplicity of 1 and 3 with a multiplicity of 2

if x = -7

x + 7 is a root

if x = 3

x -3 is a root

Since the multiplicity of this is 2, we have x-3 twice

The complete polynomial is;

(x-3)(x-3)(x+7)

= x^2-6x+9(x + 7)

= x(x^2-6x+9) + 7(x^2-6x+9)

= x^3-6x^2 + 9x+7x^2-42x+63

= x^3-x^2-33x+63

guys can you help me with the like terms plz plz dont answer if you dont know it

guys can you help me with the like terms plz plz dont answer if you dont know it

Answers

Answer:

like terms are numbers or letters that are a like.

y² is the only letter with a square in this problem. Whilst xy and 2xy are the same.

y²+ xy + 2xy

y² there is no letter with square again.

xy+2xy is the same as one orange + two orange.

1 is never written against a letter.

y²+ xy + 2xy

y²+3xy

What is the area of the parallelogram?

A parallelogram has a base of 3 feet and height of 2 and one-half feet.
7 and one-half feet squared
9 feet squared
10 and one-half feet squared
12 feet squared

Answers

Answer:

7 and one-half feet squared

Step-by-step explanation:

3 x 2.5 = 7.5

Solve for substitution 2x-3y=-12 x+y=9

Answers

Given data:

The first equation is 2x+3y=-12.

The second equation is x+y=9.

The second equation can be written as,

y=9-x

Substitute the above value in second equation.

2x+3(9-x)=-12

2x+27-3x=-12

-x=-39

x=39

The value of y is,

39+y=9

y=-30.

Thus, the value of x is 39, and the value of y is -30.

9) Calculate the approximate area of the regular pentagon, to the nearest tenth.
11 cm

9) Calculate the approximate area of the regular pentagon, to the nearest tenth.11 cm

Answers

7455 this is wrong don’t use it lol

gustavo is in a contest where he will win one of four possible prizes. he creates a four-part spinner to represent the four prizes that he has an equal chance of winning: a video game system, a bicycle, a watch, and a gift card. he spins the spinner several times to demonstrate the likelihood of winning a certain prize.

Answers

If he spins the four-part spinner 100 times, he can expect to win each prize approximately 25 times. This can be determined by using the concept of probability.

Let's say Gustavo spins the spinner 100 times to demonstrate the likelihood of winning each prize. Since the spinner has four equal parts representing the four prizes, each prize has a 1/4 (or 25%) chance of being won on any given spin.

After spinning the spinner 100 times, we can expect Gustavo to win each prize approximately 25 times. However, it's important to note that these are expected values based on probability, and the actual results may vary due to chance.

Here's a breakdown of the expected number of times Gustavo might win each prize:

1. Video game system: Gustavo can expect to win the video game system around 25 times out of the 100 spins.

2. Bicycle: Similarly, Gustavo can expect to win the bicycle approximately 25 times out of the 100 spins.

3. Watch: Gustavo can also expect to win the watch around 25 times out of the 100 spins.

4. Gift card: Lastly, Gustavo can expect to win the gift card approximately 25 times out of the 100 spins.

Keep in mind that these are average values based on the assumption that each prize has an equal chance of being won. In reality, due to random chance, the actual results might deviate slightly from these expected values.

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Un determinado día, Ana ha pagado 3,6 € por 3 dólares, y Alvaro ha pagado 8,4 € por 7
dólares. A Halla la ecuación de la recta que nos da el precio en euros, y, de x dólares

Answers

La ecuación de la recta es y = 1.2x + 0, o simplemente y = 1.2x. Y el precio en euros de 10 dólares es de 12 €.

La ecuación de la recta que nos da el precio en euros, y, de x dólares se puede hallar utilizando la fórmula de la pendiente de una recta, que es m = (y2 - y1) / (x2 - x1). En este caso, los puntos dados son (3, 3.6) y (7, 8.4).

Primero, calculamos la pendiente de la recta:
m = (8.4 - 3.6) / (7 - 3)

m = 4.8 / 4

m = 1.2

Luego, utilizamos la fórmula de la ecuación de la recta, y = mx + b, para encontrar el valor de b. Podemos utilizar cualquiera de los puntos dados para esto. Usaremos el punto (3, 3.6).

3.6 = 1.2(3) + b
3.6 = 3.6 + b
b = 0

Por lo tanto, la ecuación de la recta es y = 1.2x + 0, o simplemente y = 1.2x.

Esta ecuación nos da el precio en euros, y, de x dólares. Si queremos encontrar el precio en euros de una cierta cantidad de dólares, simplemente sustituimos el valor de x en la ecuación y resolvemos para y. Por ejemplo, si queremos encontrar el precio en euros de 10 dólares, sustituimos x = 10 en la ecuación:

y = 1.2(10) = 12

Por lo tanto, el precio en euros de 10 dólares es de 12 €.

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44-17\(44-17\)

Answers

The difference between the given values as a whole number is 27

Finding the difference between values

To find the difference between two numbers, subtract the number with the smallest value from the number with the largest value.

Given the following expression

44 - 17

This can also be expressed as

44 = 40 + 4

17 = 10 + 7

Substitute

44 - 17 = (40 + 4) - (10+ 7)

Expand

44 - 17 = 40 + 4 -10 - 7

44 - 17 = 40 - 10 + 4 - 7

44 - 17 = 30 - 3

44 - 17 = 27

Hence the difference between the given values as a whole number is 27

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please help with congruence

please help with congruence

Answers

Answer:

 a) cannot tell

  b) congruent

  c) not congruent

Step-by-step explanation:

You want to know if the triangles in the given figures are congruent.

Congruence

In order to prove congruence, we must have information to establish one of SSS, SAS, ASA, AAS congruence.

Congruence only of corresponding angles demonstrates the triangles are similar, which means they may or may not be congruent.

A

These triangles have congruent corresponding angles, so are similar. It is impossible to tell if they are congruent.

B

These triangles meet the conditions for demonstrating AAS congruence. The triangles are congruent.

C

Looking at the figure from an AAS or ASA or SAS point of view, the corresponding sides are not congruent, but the corresponding angles are. These triangles are similar, but are not congruent.

__

Additional comment

In the triangles of (c), for the given long side, the short side shown will be 25.73 in the larger triangle and 16.73 in the smaller one. That is, the angles indicate the triangles should be similar, but the given side ratios are not proportional.

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What is binomial of a number?

Answers

An integer known as a binomial number is created by evaluating a homogeneous polynomial with two terms, also known as a binomial.

What is polynomial?

Polynomials are algebraic expressions that include constants and indeterminates. Polynomials are a type of mathematics. In almost every branch of mathematics, they are used to express numbers, and calculus is one of those branches where they play a particularly significant role.

Polynomials include formulas like 2x + 9 and x² + 3x + 11. You may have noticed that none of these examples include the "=" sign.

Each variable's exponent in an algebraic expression for a polynomial must be a whole number. Polynomial exponents have to be non-negative integers. A polynomial cannot be divided by a variable even though it contains both constants and variables.

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HELP ME PLEASE GIVING OUT BRAINLY

HELP ME PLEASE GIVING OUT BRAINLY

Answers

The measures of the angles of a triangle add up to 180 degrees. The measures of 2 of the angles are already given, which are 25 degrees, and the box means 90 degrees. 90 + 25 = 115 so you need to subtract 115 from 180 to get the measure of angle c (65 degrees)

Answer:

65

Step-by-step explanation:

again, I have answered this in the past.

Brainliest/

What is the equation of the line that has a slope of 3 and goes through the point (-3,-5)?
O A. y=3x+4
OB. y= 3x - 14
O cy=3x-4
OD. y= 3x+12

Answers

Answer:

y = 3x + 4

Step-by-step explanation:

Given:

Coordinates

(-3 , -5)

Slope m = 3

Find;

Equation of slope

Computation:

Given, x1 = -3 and y1 = -5

Equation of slope = y - y1 = m(x - x1)

Equation of slope = y - (-5) = 3(x + 3)

Equation of slope = y + 5 = 3x + 9

Equation of slope = y = 3x + 9 - 5

Equation of slope = y = 3x + 4

y = 3x + 4

On a coordinate plane, a straight red line with a positive slope, labeled g of x, crosses the x-axis at (negative 3, 0) and the y-axis at (0, 3). A straight blue line with a positive slope, labeled f of x, crosses the y-axis at (0, negative 3) and the x-axis at (1, 0). Both lines intersect at (3, 6).

Answers

Answer:

f(6) = g(3)

Step-by-step explanation:

Which statement is true regarding the functions on the graph?

f(6) = g(3)

f(3) = g(3)

f(3) = g(6)

f(6) = g(6)

Solution:

g(x) passes through the points (-3, 0) and the y-axis at (0, 3). The equation of g(x) is given as:

g(x) = y

\(y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\)

\(y-0=\frac{3-0}{0-(-3)}(x-(-3))\\\\y=x+3\\\\g(x)=y= x+3\\\\g(x)=x+3\)

f(x) passes through the points (0, -3) and the y-axis at (1, 0). The equation of f(x) is given as:

f(x) = y

\(y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\)

\(y-(-3)=\frac{0-(-3)}{1-0}(x-0))\\\\y+3=x\\\\f(x)=y= x-3\\\\f(x)=x-3\)

f(6) = x-3 = 6 - 3 = 3

g(3) = x + 3 = 3 + 3 = 3

f(6) = g(3)

Graph: y - 4x >= - 1 then shade in the side that satisfies the inequality.

Answers

The given inequality is:

\(y-4x\ge-1\)

To graph it, first, let's find the equation of the line in slope-intercept form:

\(\begin{gathered} y-4x=-1 \\ \text{ Add 4x to both sides} \\ y-4x+4x=4x-1 \\ y=4x-1 \end{gathered}\)

The slope is 4 and the y-intercept is -1.

So, starting at y=-1 and x=0, we move 4 units up and 1 unit to the right to find the next point on the graph.

It will be located at (1,3).

By joining these two points, we can graph the line of the inequality:

Now, as the inequality says y values greater than or equal to this line, then the side that satisfy the inequality is the side above the line:

Graph: y - 4x &gt;= - 1 then shade in the side that satisfies the inequality.
Graph: y - 4x &gt;= - 1 then shade in the side that satisfies the inequality.

Indicate which property is illustrated in Step 4.
Step 1 9x + 8 + 4x + 3 = (9x + 8) + (4x + 3)
Step 2 = 9x + (8 + 4x) + 3
Step 3 = 9x + (4x + 8) + 3
Step 4 = (9x + 4x) + (8 + 3)
Step 5 = (9 + 4)x + (8 + 3)
Step 6 = 13x + 11
A.
associative
B.
arithmetic fact
C.
distributive
D.
commutative

Answers

The property illustrated in Step 4 is the "associative property of addition". which is the correct answer would be an option (A).

What is the Associative Property of Addition?

The associative property of addition is a rule which states that when adding three or more numbers, we can arrange them in any configuration, and the resultant sum is unaffected by how they are arranged.

(a + b) + c = a + (b + c)

The property illustrated in Step 4 is the "associative property of addition". This property states that the order in which we add numbers does not affect the result of the addition.

In Step 4, we use the associative property to rearrange the terms in the expression so that we can more easily apply the distributive property in the next step.

Hence, the correct answer would be option (A).

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The act of factoring is to reduce numbers down to their____.

Answers

prime numbers is the answer i think

The act of factoring is to reduce numbers down to their prime number.

What is a factorization?

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

A prime number is a positive integer having exactly two factors that are 1 and the number itself. Any number that does not follow this is termed a composite number.

For example:

Let the number 2310, then the factor of the number 2310 will be

2310 = 2 × 3 × 5 × 7 × 11

Where, 2, 3, 5, 7, and 11 are the prime numbers.

The act of factoring is to reduce numbers down to their prime number.

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M (4,6) is the midpoint of AB. If B is (5,7), find the
coordinates of A.

Answers

The coordinates of A are (3, 5).

According to the section formula-

If a point (x , y) divides a line in the ratio of m : n, where the two endpoints of the line are (x1, y1) and (x2, y2), then-

x = (nx1 + mx2)/ (m + n)

and y = (ny1 + my2)/ (m + n)

Here, M (4,6) is the midpoint of AB and B is (5,7)

Let the coordinates of A be (x2, y2)

Therefore,

x1 = 5, y1 = 7

x = 4, y = 6

Since, midpoint divides a line segment into two equal halves, m : n = 1 : 1

Now, substituting the values in the section formula we get-

4 = (5 + x2)/2

4 × 2 = 5 + x2

8 = 5 + x2

5 + x2 = 8

x2 = 8 - 5

x2 = 3

and 6 = (7 + y2)/2

6 × 2 = 7 + y2

12 = 7 + y2

7 + y2 = 12

y2 = 12 - 7

y2 = 5

Thus, the coordinates of A are (3, 5).

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write a product of 2 functions with one x intercept. find the x and y intercepts of that function, justify your answer with calculations and show algebraic steps.

Answers

The x-intercepts of the function h(x) = x^2 - ax are x = 0 and x = a,The y-intercept of the function h(x) is y = 0.These results can be justified by the algebraic steps taken to find the x and y intercepts.

To construct a product of two functions with one x-intercept, we can consider the following:

Let's start with two functions:

f(x) = x

g(x) = (x - a), where 'a' is a constant representing the x-coordinate of the x-intercept.

The product of these two functions is given by:

h(x) = f(x) × g(x)

= x × (x - a)

= x^2 - ax

To find the x-intercept of the function, we set h(x) equal to zero and solve for x:

x^2 - ax = 0

Factoring out an 'x' from the equation:

x(x - a) = 0

Now, we have two possibilities for the x-intercept:

   x = 0

   x - a = 0, which gives x = a

Therefore, the function h(x) has two x-intercepts: x = 0 and x = a.

To find the y-intercept, we set x = 0 in the function h(x):

h(0) = 0^2 - a(0)

= 0

Hence, the y-intercept of the function h(x) is y = 0.

In summary:

   The x-intercepts of the function h(x) = x^2 - ax are x = 0 and x = a.

   The y-intercept of the function h(x) is y = 0.

These results can be justified by the algebraic steps taken to find the x and y intercepts.

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Fair tickets for 2 adults and 3 children cost $34. An adult ticket costs $2 more than a child ticket. What is the price of an adult ticket

Answers

The price of an adult ticket is $8.

An algebraic expression is the combination of numbers and variables in expressing and solving a particular mathematical question.

Let x = price of an adult ticket

If an adult ticket costs $2 more than a child ticket, then

price of a child ticket = x - 2

If fair tickets for 2 adults and 3 children cost $34, then

2x + 3(x - 2) = 34

Simplify the equation and solve for the value of x.

2x + 3(x - 2) = 34

2x + 3x - 6 = 34

5x = 40

x = 8

price of an adult ticket = $8

price of a child ticket = x - 2 = $6

Hence, the price of each ticket is $8 and $6 for adult and child, respectively.

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what is 99 times 99 whoever gets write get brainiest

Answers

Answer:

9,801

Step-by-step explanation:

99*99=9,801

Answer:

9801

Step-by-step explanation:

9801÷99=99

99×99=9801

Let f(x) = 2x2 and g(x) = x2-1. Then f(g(2)) =

Answers

Answer:

18

Step-by-step explanation:

Evaluate g(2), then substitute the value obtained into f(x)

g(2) = 2² - 1 = 4 - 1 = 3 , then

f(3) = 2(3)² = 2(9) = 18

Find Indirect utility function of the following function U = max (X, Y) subject to the budget constraint P₁ X+ P₂ Y = M
a. M/max(P1P2)
b. M²/min(P1P2)
c. M²/P1+P2
d. M/min(P1P2)

Answers

Answer:



To find the indirect utility function, we need to solve the utility maximization problem subject to the budget constraint and express the maximum utility achieved as a function of the prices and income.

Given the utility function U = max(X, Y) and the budget constraint P₁X + P₂Y = M, we can solve for X and Y in terms of prices (P₁, P₂) and income (M).

First, let's consider the different cases:

If P₁ ≤ P₂:

In this case, the individual would choose to consume only good X. Therefore, X = M / P₁ and Y = 0.

If P₂ < P₁:

In this case, the individual would choose to consume only good Y. Therefore, X = 0 and Y = M / P₂.

Now, we can express the indirect utility function in terms of the prices (P₁, P₂) and income (M) for each case:

a) If P₁ ≤ P₂:

In this case, the individual maximizes utility by consuming only good X.

Therefore, the indirect utility function is V(P₁, P₂, M) = U(X, Y) = U(M / P₁, 0) = M / P₁.

b) If P₂ < P₁:

In this case, the individual maximizes utility by consuming only good Y.

Therefore, the indirect utility function is V(P₁, P₂, M) = U(X, Y) = U(0, M / P₂) = M / P₂.

c) and d) do not match any of the cases above.

Therefore, among the given options, the correct answer is:

a) M / max(P₁, P₂).

which of the following subsets of r3x3 are subspaces of r3x3? a.the 3 x 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries) b. the diagonal 3 x 3 matrices c. the 3 x 3 matrices whose entries are all integers d. the 3 x 3 matrices whose entries are all greater than or equal to 0 e. the 3 x 3 matrices a such that the vector 4 is in the kernel of a f. the invertible 3 x 3 matrices

Answers

The subsets that are subspaces of R3x3 are b. the diagonal 3x3 matrices, e. the 3x3 matrices where the vector 4 is in the kernel of A, and f. the invertible 3x3 matrices.

The set of 3x3 matrices with trace 0 is not a subspace of R3x3 because it is not closed under scalar multiplication.The set of diagonal 3x3 matrices is a subspace of R3x3 because it is closed under addition and scalar multiplication.

The set of 3x3 matrices with integer entries is not a subspace of R3x3 because it is not closed under addition. The set of 3x3 matrices with entries greater than or equal to 0 is not a subspace of R3x3 because it is not closed under scalar multiplication.

The set of 3x3 matrices where the vector 4 is in the kernel of A is a subspace of R3x3 because it is closed under addition and scalar multiplication. The set of invertible 3x3 matrices is a subspace of R3x3 because it is closed under addition and scalar multiplication.

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Help I will be marking brainliest!!!

A. 28.8
B. 36
C. 48
D. 38.4

Help I will be marking brainliest!!!A. 28.8B. 36C. 48D. 38.4

Answers

Answer:

A

Step-by-step explanation:

TR=\(\sqrt{60^{2} -48^{2} }\)

TR=36

36/60=AR/36

AR=21.6

TA=\(\sqrt{36^{2} -21.6^{2} }\)

TA=28.8

The concept that a message gives different meanings to different objects is called _____.
a. ​encapsulation
b. ​polymorphism
c. ​linear addressing
d. ​dynamic addressing

Answers

The concept that a message gives different meanings to different objects is called option (b) polymorphism.

Polymorphism is a fundamental concept in object-oriented programming (OOP) that allows different objects to respond to the same message or method invocation in different ways. In other words, it allows objects of different classes to be treated as if they were of the same class, as long as they implement the same method or message.

This can make code more flexible, reusable, and easier to maintain. Polymorphism is achieved through inheritance, interfaces, or overloading methods. For example, a "draw" method could be implemented differently for different shapes, such as circles, rectangles, or triangles.

Therefore, the correct option is (b) polymorphism

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A car company uses the function g(x) = 25,400(0.88)* to predict the value of a car x years from now. What will be the approximate value of
the car 6 years from now?
$7,113
A.
B. $11, 796
C $18,288
D. $22,352

Answers

Answer:

The answer might be B. $11, 796 i'm not sure

Find the critical t-value that corresponds to 50% confidence. Assume 23 degrees of freedom.

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The critical t-value that corresponds to a 50% confidence level with 23 degrees of freedom is approximately 0.685.

To find the critical t-value that corresponds to a 50% confidence level, we need to use the t-distribution table or a statistical calculator. The t-distribution is a probability distribution that is used for hypothesis testing and constructing confidence intervals when the sample size is small or when the population standard deviation is unknown.

In this case, we are given 23 degrees of freedom. Degrees of freedom (df) represent the number of independent observations in a sample. For a t-distribution, the degrees of freedom are typically equal to the sample size minus 1.

To find the critical t-value, we need to determine the desired confidence level and the two-tailed probability associated with it. Since the confidence level is given as 50%, we divide it by 2 to obtain the two-tailed probability.

The two-tailed probability for a 50% confidence level is 0.50 / 2 = 0.25.

Now, using the t-distribution table or a statistical calculator, we can find the critical t-value that corresponds to a two-tailed probability of 0.25 and 23 degrees of freedom.

Looking up the t-distribution table with 23 degrees of freedom and a two-tailed probability of 0.25, we find that the critical t-value is approximately 0.685.

Therefore, the critical t-value that corresponds to a 50% confidence level with 23 degrees of freedom is approximately 0.685.

It's important to note that a 50% confidence level is not commonly used in statistical analysis. Confidence levels are typically chosen to be 90%, 95%, or 99% to provide a higher level of confidence in the results. A 50% confidence level implies a high level of uncertainty and is not widely used in practice.

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