Cassandra opens a savings account with an initial deposit of $500.The account earns 3.25% simple interest annually. She plans to leave the money in the account for 4 years without making additional deposits or withdrawals. How much simple interest will will Cassandra earn at the end of the 4 years?

To determine how many times the spinner will land on blue or green, we need to know the number of sections that are shaded blue or green.

Let's assume the spinner has 8 sections in total, with 3 sections shaded blue, 4 sections shaded green, and the remaining sections shaded red.

Out of 24 spins, the probability of landing on blue or green can be calculated as the sum of the individual probabilities. The probability of landing on blue is 3/8, and the probability of landing on green is 4/8 (since there are 4 green sections out of 8 in total).

Therefore, the expected number of times the spinner will land on blue or green in 24 spins is:

(3/8 + 4/8) * 24 = (7/8) * 24 = 21.

So, the spinner is expected to land on blue or green approximately 21 times out of the 24 spins.

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The solution of the logarithmic equation is x = 2.

Given is a logarithmic equation ㏒ x + ㏒ (x+2) = ㏒ 8, we need to solve for x,

So,

The logarithmic equation is ㏒ x + ㏒ (x+2) = ㏒ 8,

Applying the log rule we get,

x(x+2) = 8

x²+2x = 8

x² + 2x - 8 = 0,

Solving for x,

x = 2 and x = -4,

Checking for extraneous solutions.

Verify solution x = 2 [true]

x = -4 [false]

Hence the solution of the logarithmic equation is x = 2.

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The length of side u in triangle STU i s approximately 6.34 units.

Triangle STU measures: s=8.4, t=6.9, and m∠S=58 degrees

In order to find the length u, we will use the law of cosines .

u² = s²+ t² -2stcos(m∠s)

where m∠s is the measure of angles in degrees.

u² = s²+ t² -2stcos(m∠s)

u² =8.4² +6.9² -2(8.4*6.9cos 58

u² = 118.17 -77.95

u² = 40.22

Taking the square root both sides, we get.

u = 6.34

Therefore, the length of side u in triangle STU i s approximately 6.34 units.

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Answer:

  u ≈ 9.7 units

Step-by-step explanation:

You want side u in triangle STU with s = 8.4, t = 6.9 and S = 58°.

We are given two sides and the angle opposite the larger of them. This means the triangle can be solved using the law of sines, and there will be one solution.

  s/sin(S) = t/sin(T) = u/sin(U)

With the given values, we can find angle T to be ...

  T = arcsin(t/s·sin(S))

  T = arcsin(6.9/8.4·sin(58°)) ≈ 44.156°

Then angle U will be ...

  180° -58° -44.156° = 77.844°

Using the same law of sines relation, we find side u to be ...

  u = s·sin(U)/sin(S)

  u = 8.4·sin(77.844°)/sin(58°) ≈ 9.683

The length of side u is about 9.7 units.

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The value of the measure of the line segment CD is,

⇒ CD = 10

We have to given that;

In circle,

CD = 2 + x

BC = 8

AB = 12

Hence, We can formulate;

AB² = BD × CD

12² = (8 + 2 + x) × 8

144 = 8 (10 + x)

18 = 10 + x

x = 18 - 10

x = 8

Thus, The value of the measure of the line segment CD is,

⇒ CD = 2 + x = 2 + 8 = 10

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The answer is 33.

A number is divisible by 9 if and only if the sum of its digits is divisible by 9. We can use this fact to count the number of a_n that are divisible by 9.

For any n, the sum of the digits in a_n is given by:

S_n = 1 + 2 + 3 + ... + n

This is the sum of an arithmetic sequence, which can be computed using the formula:

S_n = n(n+1)/2

Therefore, the sum of the digits in a_n is:

S_n = n(n+1)/2

We want to find the values of n for which S_n is divisible by 9. Since 9 is a factor of 3, we only need to consider the cases where n(n+1) is divisible by 3.

Case 1: n is divisible by 3

In this case, either n or (n+1) is divisible by 3. Therefore, S_n is divisible by 3, and hence by 9.

Case 2: n is not divisible by 3

In this case, neither n nor (n+1) is divisible by 3. Therefore, S_n is not divisible by 3.

Therefore, the only values of n for which S_n is divisible by 9 are those that are divisible by 3. There are 33 such values of n between 1 and 100 (namely, 3, 6, 9, ..., 99), so the answer is 33.

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The percentage of the men in the sample who have blood type O is 40.38%.

What is the percentage?

A percentage is a figure or ratio stated as a fraction of 100 in mathematics. The acronyms pct., pct., and occasionally pc are also used to indicate it, however, the percent sign is most frequently used. A % is a number without dimensions and without a standard measurement.

Here, we have

Given: The contingency table below shows the blood types of a sample of people cross-classified by sex.

We have to find the percentage of the men in the sample who have blood type o.

Total number of males = 104

Number of males who has blood type O = 42

Percentage of the males in the sample have blood type O:

= (42×100%)/104

= 40.38%

Hence, the percentage of the men in the sample who have blood type O is 40.38%.

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The amount of pizza left for the other three classes is 1/2.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

The total number of pizzas ordered is 7.

The total number of pizzas eaten by the three classes is:

2 1/2 + 1 1/3 + 2 2/3

= (5/2) + (4/3) + (8/3)

= 15/6 + 8/6 + 16/6

= 39/6

= 6 3/6

= 6 1/2

Therefore, the amount of pizza left for the other three classes is:

7 - 6 1/2 = 1/2

So, there is half of a pizza left for the other three classes.

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Answer:                69 cm

To find the perimeter of the triangle, add together the length of all sides.

A right triangle has three sides and we know that:

When we know two sides in a right triangle, we can find the third side using the Pythagorean theorem:

a² + b² = c²

The variables 'a' and 'b' represent the base and height. The variable 'c' represents the hypotenuse, the longest side.

Since we know the base and height, substitute them into the formula and solve for the hypotenuse.

a² + b² = c²                Start with the Pythagorean theorem.

24² + 16² = c²            Substitute the base and height.

576 + 16² = c²            Solve 24².

576 + 256 = c²          Solve 16².

832 = c²                     Add.

Now, let's start to isolate 'c', which means making it alone on one side of the equal sign.

√832 = √c²               Square root both sides.

√832 = c                   Square root is the opposite of ², so it cancels out.

c ≈ 28.8...                  Keep the variable on the left side.

The question says to find the approximate perimeter, so let's round 28.8 to the nearest whole number.

The hypotenuse, 'c', is about 29 cm.

Now, we know the three sides:

Add together all three sides.

Perimeter = base + height + hypotenuse

Perimeter = 24 cm + 16 cm + 29 cm

Perimeter = 69 cm

∴ The perimeter of the right triangle is approximately 69 cm.

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The chance that a randomly selected defective component was made by Alice is approximately 0.33 or 33%.

a. The probability of a randomly selected individual being a male who smokes is 19/100 or 0.19.

b. The probability of a randomly selected individual being a male is 0.6.

c. The probability of a randomly selected individual smoking is (19+12)/100 or 0.31.

d. The probability that a randomly selected female is a smoker is 12/40 or 0.3.

e. The probability that a randomly selected smoker is male is 19/(19+12) or 0.61.

To find the chance that a randomly selected defective component was made by Alice, we can use Bayes' Theorem.

Let A, B, and C represent the events that a component is made by Alice, Betty, and Cleo respectively, and let D represent the event that a component is defective.

We want to find P(A|D), the probability that the defective component was made by Alice. Using Bayes' Theorem, we have:

P(A|D) = P(D|A) * P(A) / [P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)]

We know that P(A) = P(B) = P(C) = 1/3, P(D|A) = 0.05, P(D|B) = 0.03, and P(D|C) = 0.02. Substituting these values, we get:

P(A|D) = 0.05 * (1/3) / [0.05 * (1/3) + 0.03 * (1/3) + 0.02 * (1/3)]

       = 0.333 or approximately 0.33

Therefore, the chance that a randomly selected defective component was made by Alice is approximately 0.33 or 33%.

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Complete Question:

             Yes No Total

Male 19 41 60

Female 12 28 40

Total 31 69 100

a. What is the probability of a randomly selected individual being a male who smokes (or male and smoker)?

b. What is the probability of a randomly selected individual being a male?

c. What is the probability of a randomly selected individual smoking?

d. A person is selected at random and if the person is female, what is the probability that she is a smoker?

e. What is the probability that a randomly selected smoker is male?

A manufacturing firm has three machine operators; Alice, Betty, and Cleo. The operators produce a component. Alice has a 5% defective rate, Betty a 3% defective rate, and Cleo a 2% defective rate. The three operators produce equal numbers of components. Suppose a randomly selected component is found to be defective. What is the chance it was made by Alice?

Answer:

1/8

Step-by-step explanation:

2^2 / 2^5

2^2 = 2 x 2= 4

2^5 = 2 x 2 x 2 x 2 x 2 = 32

4/32 = 1/8

Note :

2 power 5 means you need to multiply 2, 5 times itself

(i.e) 2 x 2 x 2 x 2 x 2

Thus, the extreme points of s are (0,0), (2,2), (2,0), and (0,2), and the extreme directions are [2 -4], [-1 1], [0 1], and [0 -1].

To find the extreme points and extreme directions of the polyhedral set s, we need to first write down the set in standard form. We can rewrite the constraints as:

2x1 - 4x2 <= -4
-x1 + x2 <= 2
x2 <= 4
x2 >= 0

The first two constraints can be written as a matrix inequality:
[2 -4; -1 1][x1; x2] <= [4; 2]

The last two constraints can be written as x2 <= 4 and x2 >= 0. Thus, the polyhedral set s can be written as:
s = {x in R^2 : [2 -4; -1 1][x1; x2] <= [4; 2], x2 <= 4, x2 >= 0}

To find the extreme points, we can solve the linear program:

maximize 0x1 + 0x2
subject to [2 -4; -1 1][x1; x2] <= [4; 2]
x2 <= 4
x2 >= 0

The objective function is just 0x1 + 0x2, so it doesn't matter what the values of x1 and x2 are. The constraints, however, determine the feasible region. The intersection of the constraints is a polygon with vertices at (0,0), (2,2), (2,0), and (0,2). These are the extreme points of s.

To find the extreme directions, we need to look at the gradients of the constraints at each extreme point. If the gradient is non-zero, then that constraint is active at that point and the corresponding direction is extreme. The gradients of the constraints are:

[2 -4] for the first constraint
[-1 1] for the second constraint
[0 1] for the third constraint
[0 -1] for the fourth constraint

At the point (0,0), the first two constraints are active and their gradients are non-zero. Thus, the extreme directions are along [2 -4] and [-1 1].

At the point (2,2), the first two constraints and the third constraint are active. The gradients of the first two constraints are non-zero, as before, and the gradient of the third constraint is [0 1]. Thus, the extreme directions are along [2 -4], [-1 1], and [0 1].

At the point (2,0), the first two constraints and the fourth constraint are active. The gradients of the first two constraints are non-zero, and the gradient of the fourth constraint is [0 -1]. Thus, the extreme directions are along [2 -4], [-1 1], and [0 -1].

At the point (0,2), the second constraint and the third constraint are active. The gradient of the second constraint is non-zero, as before, and the gradient of the third constraint is [0 1]. Thus, the extreme directions are along [-1 1] and [0 1].

Therefore, the extreme points of s are (0,0), (2,2), (2,0), and (0,2), and the extreme directions are [2 -4], [-1 1], [0 1], and [0 -1].

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The characteristic equation of the matrix is given by det(A-λI) = 0, where A is the given matrix and λ is the eigenvalue. Thus, for the matrix A = [8 -2; -4 1], the characteristic equation is:

|8-λ  -2|

|-4  1-λ| = (8-λ)(1-λ)+8 = λ^2 - 9λ + 16 = 0

Solving for λ, we get the eigenvalues λ1 = 1 and λ2 = 8. To find the eigenvectors associated with these eigenvalues, we solve the system of linear equations (A - λI)x = 0.

For λ1 = 1, we get:

|7 -2| |x1|   |0|

|-4  0| |x2| = |0|

Solving the system, we get x1 = 2x2/7, so a basis for the eigenspace corresponding to λ1 is given by {[2/7, 1]}.

For λ2 = 8, we get:

|0 -2| |x1|   |0|

|-4 -7| |x2| = |0|

Solving the system, we get x1 = -x2/4, so a basis for the eigenspace corresponding to λ2 is given by {[-2, 4]}.

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Answer:

its correct

Step-by-step explanation:good job

Thus, the area of surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4 is 36π√3 square units.

To find the area of the surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4, we can use the formula:
A = 2π ∫(1 to 4) x √(1+(dy/dx)^2) dy

First, we need to find dy/dx by taking the derivative of y=x−−√3y=x3:

dy/dx = 1/(2√3x^(1/2))

Substituting this into the formula, we get:

A = 2π ∫(1 to 4) x √(1+1/(12x)) dy

Simplifying the expression under the square root, we get:

A = 2π ∫(1 to 4) x √(12x+1)/12 dy

We can simplify this expression further by using a substitution u = 12x+1:

A = π ∫(13 to 49) √u du

Integrating this, we get:

A = π (2/3)(u^(3/2))|(13 to 49)

A = π (2/3)(49√49-13√13)

A = 36π√3 square units

Therefore, the area of the surface obtained by rotating the curve y=x−−√3y=x3 about the y-axis for 1≤y≤4 is 36π√3 square units.

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The combination of two events consisting of all outcomes that are contained in one event or a second event is:

The Union

The correct option is (c)

The union sets are the sets containing all elements that are in A or in B (possibly both). We write the (A ∪ B)

The event A occurs the outcome is contained in A. For any two events A and B, we define the new event A ∪ B, called the union of events A and B. It also says that: The combination of two events consisting of all outcomes that are contained in one event or a second event is: The Union

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The given question is incomplete, complete question is:

The combination of two events consisting of all outcomes that are contained in one event or a second event is :

a. complement b. intersection c. union d. condition

A true statement for most of the data in this plot is c. Most of the data in the Girl's Heights plot are around 52. Therefore, option c. Most of the data in the Girl's Heights plot are around 52 is correct.

For the Boy's Heights plot, we can see that the majority of the dots are above 48 and below 54, with the most dots being above 52. Therefore, a true statement for most of the data in this plot is:

Most of the data in the Boy's Heights plot are around 52.

For the Girl's Heights plot, we can see that the majority of the dots are also above 48 and below 54, with the most dots being above 52 as well. Therefore, a true statement for most of the data in this plot is:

Most of the data in the Girl's Heights plot are around 52.

So, the correct option is:

Most of the data in each plot are around 52.

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Step-by-step explanation:

14^2 + 8^2 = 260

third side=√260 = 16.12 ~ 16

Part A:

Best graphical representation to display data will be line graph.

Given,

Scores of basketball team during the first 13 games of the season

{85, 94, 101, 118, 107, 110, 114, 96, 117, 105, 121, 88, 125}.

Now,

Part B:

We can create line graph with a very simple technique.

Firstly,

On x - axis take the number of season the team has played. In our case the number is 13 so take 13 distinct points on the x - axis.

Secondly,

On y -axis take the scores of the team in each of the 13 seasons corresponding to their values at x - axis.

Then,

Plot the points on the graph for all 13 seasons .

Last step,

Join all the points in the graph with the help of ruler. This will form the required line graph of the question.

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To find the center of enlargement, we need to determine the scale factor of enlargement between the original quadrilateral ABCD and the image quadrilateral A'B'C'D'.

We can do this by finding the ratio of the corresponding side lengths. Let the scale factor be k. Then,

AB' = k * AB

and

AA' = k * AA'

Using the distance formula, we can find the lengths of AB and AA',

AB = sqrt((6-(-1))^2 + (3-3)^2) = sqrt(50)

AA' = sqrt((11-3)^2 + (-5-(-1))^2) = sqrt(80)

Thus, k = AB' / AB = AA' / AA' = sqrt(80 / 50) = sqrt(8 / 5) = 1.7889 (approx)

Next, we can use the formula for the center of enlargement, which states that the center of enlargement is the point of intersection of the corresponding lines joining the original and the image points.

The line joining A(3,-1) and A'(11,-5) has the equation y = -x/2 + 5/2

The line joining B(3,6) and B'(6,8) has the equation y = (2/3)x + 6

\

Solving for the point of intersection, we get:

x/2 + 5/2 = (2/3)x + 6

=> x = 21.6

Substituting x in either of the equations, we get:

y = -x/2 + 5/2

=> y = -21.6/2 + 5/2

=> y = -5.55 (approx)

Therefore, the center of enlargement is approximately (21.6, -5.55).

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To find the center of enlargement, we need to determine the scale factor of enlargement between the original quadrilateral ABCD and the image quadrilateral A'B'C'D'.

We can do this by finding the ratio of the corresponding side lengths. Let the scale factor be k. Then,

AB' = k * AB

and

AA' = k * AA'

Using the distance formula, we can find the lengths of AB and AA',

AB = sqrt((6-(-1))^2 + (3-3)^2) = sqrt(50)

AA' = sqrt((11-3)^2 + (-5-(-1))^2) = sqrt(80)

Thus, k = AB' / AB = AA' / AA' = sqrt(80 / 50) = sqrt(8 / 5) = 1.7889 (approx)

Next, we can use the formula for the center of enlargement, which states that the center of enlargement is the point of intersection of the corresponding lines joining the original and the image points.

The line joining A(3,-1) and A'(11,-5) has the equation y = -x/2 + 5/2

The line joining B(3,6) and B'(6,8) has the equation y = (2/3)x + 6

\

Solving for the point of intersection, we get:

x/2 + 5/2 = (2/3)x + 6

=> x = 21.6

Substituting x in either of the equations, we get:

y = -x/2 + 5/2

=> y = -21.6/2 + 5/2

=> y = -5.55 (approx)

Therefore, the center of enlargement is approximately (21.6, -5.55).

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To find a basis for the subspace dimension w of R4, we need to find a set of linearly independent vectors that span w.

First, we can rewrite the given condition for w as follows:

w = {(s, 0, 0, 5s) + (0, 4t, 0, -t) + (0, 0, s, 0) + (0, 0, 0, -t) : s, t are real numbers}

Notice that each term in the above expression corresponds to one of the four standard basis vectors in R4. Therefore, the subspace w can be expressed as the span of the following four vectors:

v1 = (1, 0, 0, 5)
v2 = (0, 4, 0, -1)
v3 = (0, 0, 1, 0)
v4 = (0, 0, 0, -1)

To show that these vectors form a basis for w, we need to show that they are linearly independent and that they span w.

To show linear independence, suppose that a linear combination of these vectors is equal to the zero vector:

c1 v1 + c2 v2 + c3 v3 + c4 v4 = (0, 0, 0, 0)

Then we have the following system of equations:

c1 = 0
4c2 = 0
c3 = 0
5c1 - c2 = 0

Solving for the coefficients, we get c1 = c2 = c3 = c4 = 0, which shows that the vectors are linearly independent.

To show that they span w, we need to show that any vector in w can be expressed as a linear combination of these vectors. Let (s, 4t, s, 5s - t) be an arbitrary vector in w. Then we can write:

(s, 4t, s, 5s - t) = (s, 0, 0, 5s) + (0, 4t, 0, -t) + (0, 0, s, 0) + (0, 0, 0, -t)

which is a linear combination of the vectors v1, v2, v3, and v4. Therefore, these vectors span w.

Since we have found a set of four linearly independent vectors that span w, we can conclude that they form a basis for w. Thus, the dimension of the subspace w is 4.

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The given f(x,y) is to be evaluated at 6 different points. At (0,0), the  f(x,y) is 4, at (0,1)=  -1, (2,3)= -37. When evaluated at (1,y), the  f(x,y) simplifies to f(1,y) = 3 - 4y^2,  at (x,0), the  f(x,y) simplifies to f(x,0) = 4 - x^2. Finally, when evaluated at (t,1), the  f(x,y) simplifies to f(t,1) = 3 - t^2 - 4.

Explanation:

To evaluate the function f(x,y) = 4 - x^2 - 4y^2 at point (0,0), we simply substitute x=0 and y=0 in the function. Thus, f(0,0) = 4 - 0^2 - 4(0)^2 = 4.

Similarly, to evaluate the function at point (0,1), we substitute x=0 and y=1 in the function. Thus, f(0,1) = 4 - 0^2 - 4(1)^2 = -1.

To evaluate the function at point (2,3), we substitute x=2 and y=3 in the function. Thus, f(2,3) = 4 - (2)^2 - 4(3)^2 = -37.

When we substitute x=1 in the function f(x,y), we get f(1,y) = 4 - 1^2 - 4y^2 = 3 - 4y^2. Hence, the function simplifies to f(1,y) = 3 - 4y^2 when evaluated at point (1,y).

Similarly, when we substitute y=0 in the function f(x,y), we get f(x,0) = 4 - x^2. Hence, the function simplifies to f(x,0) = 4 - x^2 when evaluated at point (x,0).

Finally, when we substitute y=1 in the function f(x,y), we get f(t,1) = 4 - t^2 - 4(1)^2 = 3 - t^2 - 4. Hence, the function simplifies to f(t,1) = 3 - t^2 - 4 when evaluated at point (t,1).

Therefore, the function values at the given points are 4, -1, -37, 3 - 4y^2, 4 - x^2, and 3 - t^2 - 4 for points (0,0), (0,1), (2,3), (1,y), (x,0), and (t,1), respectively.

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The cylinder x^2 + 5z^2 = 8 is parallel to the y-axis.
- The cylinder x^2 + 5y^2 = 8 is parallel to the z-axis.
- The cylinder z^2 + 5y^2 = 8 is parallel to the x-axis.

Let's analyze each given cylinder equation and determine which coordinate axes they are parallel to in ℝ³:

1. x^2 + 5z^2 = 8:
This cylinder is parallel to the y-axis because the equation does not involve the y-coordinate. The equation is in the form x^2 + cz^2 = k, where c and k are constants.

2. x^2 + 5y^2 = 8:
This cylinder is parallel to the z-axis because the equation does not involve the z-coordinate. The equation is in the form x^2 + cy^2 = k, where c and k are constants.

3. z^2 + 5y^2 = 8:
This cylinder is parallel to the x-axis because the equation does not involve the x-coordinate. The equation is in the form cz^2 + cy^2 = k, where c and k are constants.

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The data set for which the median would be the best measure of center is (b) 67, 72, 75, 74, 68, 69, 70, 65. This is because the data set has no extreme values or outliers, and the values are relatively evenly distributed around the middle.

Therefore, the median, which is the middle value when the data is arranged in order, would accurately represent the typical value in this set.

The median would be the best measure of center for data set d.) 93, 95, 88, 38, 79, 85, 88, 90. This is because the median is less sensitive to extreme values (such as 38 in this case) compared to the mean, and it provides a more accurate representation of the central tendency of the data set.

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The required solution to the system of equations in Question 11 is x = -6 and y = -4, and in Question 12 is x = -7 and y = 9.

11:

3x + 2y = -26

4x - 5y = -4

To eliminate one variable, we can multiply the first equation by 4 and the second equation by 3, so the coefficients of x will be the same:

12x + 8y = -104 (Multiplying the first equation by 4)

12x - 15y = -12 (Multiplying the second equation by 3)

Now, subtract the second equation from the first equation to eliminate x:

(12x + 8y) - (12x - 15y) = -104 - (-12)

12x + 8y - 12x + 15y = -104 + 12

y = -4

Now, substitute the value of y back into one of the original equations, let's use the first equation:

3x + 2(-4) = -26

3x = -18

x = -6

Therefore, the solution to the system of equations in Question 11 is x = -6 and y = -4.

Similarly,

The solution to the system of equations in Question 12 is x = -7 and y = 9.

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The answer is "inelastic". When demand is inelastic, changes in price do not significantly affect the quantity of goods or services demanded.

This means that increases in price result in increases in total revenue, while decreases in price result in decreases in total revenue. Inelastic demand occurs when consumers are not very responsive to changes in price and do not have good alternatives to the product or service being offered. Examples of products with inelastic demand are necessities like food, medicine, and gasoline, where consumers will continue to purchase the product regardless of price changes because they need it for daily living. On the other hand, products with elastic demand, like luxury items or non-essential goods, will see a significant decrease in demand when prices increase.

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Answer:

Each person got 7/10 of a sandwich.

Step-by-step explanation:

Question:

At a family reunion 10 people shared 7 sandwiches equally. How much of a sandwich did each person get to eat?

This is a division problem. You must divide the number of sandwiches by the number of people.

7 ÷ 10 = 7/10

Answer: Each person got 7/10 of a sandwich.

We can use the Maclaurin series expansion of sin(x) and plug in π/2 to get: sum

sin(π/2) = ∑n=0^[infinity] (-1)^n (π/2)^(2n+1)/(2n+1)!

Simplifying the right-hand side:

sin(π/2) = π/2 - π^3/2! + π^5/4! - π^7/6! + ...

Multiplying both sides by π/2 and rearranging:

π^2/4 = π/2 - π^3/3! + π^5/5! - π^7/7! + ...

Now, we can use the Maclaurin series expansion of cos(x) and plug in 0 to get:

cos(0) = ∑n=0^[infinity] (-1)^n x^(2n)/(2n)!

Simplifying the right-hand side:

cos(0) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

Multiplying both sides by x^2/2 and rearranging:

π^2/8 = π^2/4 - π^4/4! + π^6/6! - π^8/8! + ...

Now we can substitute these series expansions into the original sum and simplify:

∑n=0^[infinity] (-1)^n π^2n/(6^2n (2n)!)

= π^2/2 - π^4/4! + π^6/6! - π^8/8! + ...

= 2π^2/4 - π^4/4! + π^6/6! - π^8/8! + ...

= (2π^2 - π^4/3! + π^6/5! - π^8/7! + ...) / 4

= (2π^2 - π^4/6 + π^6/120 - π^8/5040 + ...) / 4

So the sum of the series is (2π^2 - π^4/6 + π^6/120 - π^8/5040 + ...) / 4.

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The Maclaurin series expansion of f(x) is: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...

To find the coefficients of the series, we need to evaluate the derivatives of f(x) at x = 0. Let's find the first few derivatives:

f(x) = (9x)arctan(5x^2)

f'(x) = 9arctan(5x^2) + 18x^2/(1 + 25x^4)

f''(x) = 90x/(1 + 25x^4) - 90x^3(1 - 5x^4)/(1 + 25x^4)^2

f'''(x) = 90(1 - 25x^4)/(1 + 25x^4)^2 - 270x^2(1 - 5x^4)/(1 + 25x^4)^2 - 270x^4(1 - 5x^4)/(1 + 25x^4)^2 + 360x^6(1 - 5x^4)/(1 + 25x^4)^3

By evaluating these derivatives at x = 0, we can find the coefficients of the Maclaurin series expansion of f(x).

However, calculating the derivatives and evaluating them at x = 0 can be quite involved and require significant algebraic manipulation. Therefore, it is best to use computational software or tools to calculate the coefficients of the Maclaurin series expansion accurately.

Once we have the coefficients, we can express the Maclaurin series of f(x) by substituting the coefficients into the general formula.

The Maclaurin series expansion allows us to approximate the function f(x) for values of x close to 0, providing a polynomial representation of the function.

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The length of the arc shown in red is 5π/4 metre

To find the length of the arc, we need to find the circumference of the circle, which we find with the following formula :

C = 2πr

where r  is the radius which is indicated in the image: .

so the circumference  is:

C = 2π(3)

C = 6π

This is the measure of the entire perimeter of the circle, it is the measure of the 360 ° arc.

Because we only want 30° of that 360 °, we divide the value of the circumference by 360 and multiply po 45:

30/360=0.125 of full circle,

L(arc)=0.125L=5π/4

The length of the arc is 5π/4 m

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A: The expression is 3y(2x² - x - 8xy + 4y).

B: Completely factorized expression is 3y{x(2x - 1-8y) + 4y}.

Part A: To factor out the greatest common factor (GCF), we need to find the highest power of each variable that appears in all terms. In this expression, the variables are x and y.

The GCF of the coefficients is 3, and the GCF of the variables is xy.

Factoring out the GCF, we get:

3y(2x² - x - 8xy + 4y)

Part B: To factor the entire expression completely, we look for common factors among the terms and apply factoring techniques.

The given expression is:

6x²y − 3xy − 24xy² + 12y²

First, let's factor out the GCF of the coefficients, which is 3:

3y(2x² - x - 8xy + 4y)

Factor out x from the common terms,

3y{x(2x - 1-8y) + 4y}

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