#13Change from standard form to vertex formy= -2x²+12x-21

That depends on the function in question.

If f(x) is integrable and non-decreasing, then the right Riemann sum overestimates the integral of f.

If f is non-increasing, the right Riemann sum underestimates the integral.

The value of k in the function is 2.

Option D is the correct answer.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

The coordinates in the graph are ((0, 5), (2, 3), and (4, 5).

Now,

To find the value of k in the function f(x) = |x - h| + k, we need to use one of the given points (0, 5), (2, 3), or (4, 5) and solve for k.

Let's use the point (2, 3):

f(2) = |2 - h| + k = 3

We can simplify this by considering two cases:

when 2 - h is positive and when it is negative.

Case 1:

2 - h is positive

If 2 - h is positive, then we have:

(2 - h) + k = 3

Simplifying this, we get:

-k + h = -1

Case 2:

2 - h is negative

If 2 - h is negative, then we have:

-(2 - h) + k = 3

Simplifying this, we get:

k + h = 5

Now we can solve for k by eliminating h.

Adding the two equations we obtained from the two cases, we get:

k = 2

Therefore,

The value of k is 2.

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Answer:

Please check the explanation.

Step-by-step explanation:

Given the expression

3x-12

1)

Writing the expression in an equivalent form

3x-12

Factor out common term 3

3x-12 = 3(x-4)

Thus, 3(x-4) is equivalent to 3x-12.

2)

Writing the expression in an equivalent form

3x-12

Multiply the expression by 2/2

\(3x-12\:=\frac{2}{2}\:\times \:\left[3x-12\right]\)

Therefore, \(\frac{2}{2}\:\times \:\left[3x-12\right]\) is equivalent to 3x-12.

Answer:

1 1/7

Step-by-step explanation:

positive 10 3/7- 9 2/7 is equal to 1 1/7

Subtracting a negative number is the same as adding a positive number.

-9 2/7 - (-10 3/7)

-9 2/7 + 10 3/7

Turn both into improper fractions.

-9 2/7 = -65/7

10 3/7 = 73/7

Add.

-65/7 + 73/7

8/7 or 1 1/7

Best of Luck!

The number of days would be 10 and the maximum percent of the population infected would be 25.752%.

What are exponential functions?

The exponential function, denoted by \(e^x\), is a mathematical function. Unless otherwise specified, the term refers to a positive-valued function of a real variable, though it can be extended to complex numbers or generalized to other mathematical objects such as matrices or Lie algebras.

\(p(t) = 7te^{-\frac{t}{10}}\)

percentage of infected people a maximum when p '(t) = 0

                    \(p '(t) = 7(1)e^\frac{-1}{10} +7te^\frac{-t}{10}(\frac{-1}{10})\\\\p'(t)=e^{-\frac{t}{10}}(7 -\frac{7t}{10})\\\\e^{-\frac{t}{10}}(7 -\frac{7t}{10})=0\\\\7 -\frac{7t}{10}=0\\\\t = 10\)

Hence percentage of infected people reaches a maximum after 10 days

maximum percent of the population infected = p(10)

    \(p(10) = 7(10)e^{-\frac{10}{10}}\\\\p(10)=\frac{70}{e}\\\\P(10)=25.752\%\)

Hence, the number of days would be 10 and the maximum percent of the population infected would be 25.752%.

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Answer:

The common ratio is -2

Next three terms are 18, -36, and 72

Step-by-step explanation:

The common ratio is -2 since each consecutive term is being multiplied by -2

The next three terms are -9(-2) = 18, 18(-2) = -36, and -36(-2) = 72

Answer:

A -0.5

B 4.5, -2.25, 1.125

Step-by-step explanation:

A: 72× -0.5= -36

-36× -0.5= 18

18× -0.5= -9

B multiple everything by -0.5

The volume of the candle is approximately 94 cubic inches.

A circular cylinder is a three-dimensional solid object made up of two parallel and congruent circular bases and a curving surface connecting the bases.

The volume of a right circular cylinder is given by the formula:

V = πr²h

Where

Substituting the given values into the formula, we get:

V = 3.14 x 2² x 7.5

V = 3.14 x 4 x 7.5

V = 94.2

Rounding to the nearest cubic inch, we get:

V ≈ 94 cubic inches

Therefore, the volume of the candle is approximately 94 cubic inches.

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Answer:

3?

Step-by-step explanation:

If |q| = 5 at an angle of 45° and |r| = 16 at an angle of 300°. then the

|q – r| will be 18.0

To find |q - r|, we need to subtract the complex numbers q and r after which discover the magnitude (or absolute value) of the end result.

First, we want to express q and r in rectangular form, which means that finding their actual and imaginary additives:

For q, we've:

|q| = 5 at an perspective of 45°

Re(q) = |q| cos(45°) = five cos(45°) = 5/√2

Im(q) = |q| sin(45°) = 5 sin(45°) = 5/√2

So q = (5/√2) + (5/√2)i

For r, we have:

|r| = 16 at an angle of 300°

Re(r) = |r| cos(300°) = 16 cos(300°) = sixteen(-√3/2) = -8√three

Im(r) = |r| sin(300°) = 16 sin(300°) = -8

So r = -8√three - 8i

Now we are able to discover q - r by using subtracting the actual and imaginary additives:

q - r = (5/√2) + (5/√2)i - (-8√3 - 8i)

= (5/√2) + 8√3 + (5/√2 + 8)i

To discover |q - r|, we want to take the importance of this complex number:

|q - r| = √[(5/√2 + 8√3)² + (5/√2 + 8)²]

= √[25/2 + 80√3 + 192 + 50/2 + 40 + 64]

= √[125/2 + 80√3 + 256]

= √[625/4 + 320√3 + 1024]

= √[(25/2 + 16√3)²]

= 25/2 + 16√3

≈ 18.0

Hence, |q - r| is about equal to 18.0.

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After answering the presented question, we can conclude that As a result, the line connecting the points A(-7, -1) and B(8, 2) is divided by the line x + y = 2 in the ratio about 65.63:56.63.

In mathematics, ratios demonstrate how frequently one number is contained in another. For example, if there are 8 oranges and 6 lemons in a fruit dish, the ratio of oranges to lemons is 8 to 6. In a similar vein, the orange-to-whole-fruit ratio is 8, whereas the lemon-to-orange ratio is 6:8. A ratio is an ordered pair of numbers a and b represented as a / b, where b is not zero. A ratio is an equation that equates two ratios. For example, if there is one male and three girls (for every boy she has three daughters), 3/4 are girls and 1/4 are boys.

\(m = (y2 - y1)/(x2 - x1) = (2 - (-1))/(8 - (-7)) = 3/15 = 1/5\\y - y1 = m(x - x1)\\y - (-1) = (1/5)(x - (-7))\\y + 1 = (1/5)(x + 7)\\y = (1/5)x + 6/5 - 1\\y = (1/5)x + 1/5\\x + (1/5)x + 1/5 = 2\\(6/5)x = 9/5\\x = 3\\\)

\(y = (1/5)(3) + 1/5\\y = 4/5\\AP = \sqrt[(3 - (-7))^2 + (4/5 - (-1))^2] = \sqrt[10^2 + 9/5^2] = \sqrt[100 + 81/25] = \Sqrt[4301]/5\\PB = \Sqrt[(8 - 3)^2 + (2 - 4/5)^2] = \sqrt[25 + 81/25] = \sqrt[3206]/5\\\)

The ratio in which AB is divided by x + y = 2 is:

\(AP:PB = \sqrt[4301]:\sqrt[3206] = 65.63:56.63 \\\)

As a result, the line connecting the points A(-7, -1) and B(8, 2) is divided by the line x + y = 2 in the ratio about 65.63:56.63.

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These tasks involve analyzing various aspects of cost-volume-profit relationships, sensitivity to changes, and resource constraints to gain insights into the financial performance and operational efficiency of the company.

1) To prepare a contribution margin income statement, you need to classify the costs as variable or fixed and calculate the contribution margin for each product. Subtracting the total variable costs from the total sales revenue will yield the contribution margin, which can be used to determine the operating profit.

2) The weighted average contribution margin ratio can be calculated by dividing the total contribution margin by the total sales revenue. This ratio indicates the average contribution margin earned per dollar of sales.

3) The break-even point in sales dollars can be determined by dividing the total fixed costs by the contribution margin ratio. It represents the level of sales required to cover all costs and achieve a zero-profit position.

4) To earn an annual profit of $400,000, you would need to add this profit amount to the total fixed costs and divide the sum by the contribution margin ratio to find the required sales dollars.

5) By increasing or decreasing the plane sales price by 10 percent while keeping the total sales units constant, you can calculate the impact on the operating profit by multiplying the change in sales price by the total sales units and the contribution margin ratio.

6) If the sales mix shifts more towards the car product, the break-even point in units may decrease. This is because the car product has a lower labor hour requirement per unit compared to the other two products, potentially reducing the total fixed costs and contributing to a lower break-even point.

7) To optimize the use of limited labor hours, you would calculate the contribution margin per labor hour for each product by dividing the contribution margin by the labor hours required per unit. This information can guide decision-making on the allocation of labor hours to maximize profitability.

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Triangle ABC is congruent to Triangle CDA using the SSS rule.

It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.

We have,

SSS rule states that,

If the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.

To prove:

ΔABC ≅ ΔCDA.

AB = DC ( given)

AD = BC (given)

AC = AC (common to both the triangles)

ΔABC ≅ ΔCDA ( By SSS rule )

Thus,

ΔABC ≅ ΔCDA by SSS rule.

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To complete the integer division using the optimized division algorithm, the lowest number of rounds theoretically required depends on the specific algorithm employed. In the given scenario, the specific algorithm is not mentioned. However, we can provide explanations based on common algorithms such as binary division. Additionally, the resulting number in binary representation can be determined by converting the quotient to binary using 8 bits. Lastly, the resulting number in floating-point decimal representation can be determined by converting the quotient to IEEE 754 single precision format.

Part 1: The lowest number of rounds theoretically required to complete the integer division using the optimized division algorithm depends on the algorithm itself.

One common algorithm is binary division, where the dividend is continuously divided by the divisor until the remainder becomes zero or reaches a terminating condition.

The exact number of rounds needed in this case would depend on the values of A (dividend) and B (divisor). Without knowing the specific algorithm being used, it is not possible to determine the exact number of rounds.

Part 2: To represent the resulting quotient in binary format using 8 bits, we need to convert the quotient of A divided by B to binary. In this case, A = +54 and B = -3.

Performing the division, we get a quotient of -18. Representing -18 in 8-bit binary format, we have: 10010010. The most significant bit (MSB) represents the sign, where 1 indicates a negative value.

Part 3: To represent the resulting quotient in FP decimal representation using the IEEE 754 single precision standard, we need to convert the quotient to binary and then apply the specified format. Considering the quotient of -18, in binary it is represented as 10010.

Using IEEE 754 single precision format, the sign bit would be 1 (negative), the true exponent would be biased by 127, and the fraction would be normalized. The IEEE-754 exponent in binary would be 10000101, and in decimal (base 10) it would be 133. The resulting representation in IEEE 754 single precision format would be: 1 10000101 10010000000000000000000.

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A) For the first question, we will use the first and second derivative criteria. First, we will compute the first and second derivatives of the given function:

Now, we set the first derivative equals to zero and solve for q:

Evaluating q=75 in the second derivative, we get a negative value since it is a constant, therefore there is a maximum for q=75.

B) We know the maximum is reached for q=75 therefore to find the maximum profit we evaluate the function at q=75:

Vertical angulation refers to the angle at which the x-ray beam is directed when taking dental radiographs. It is an important factor in obtaining clear and accurate images.

In both the paralleling and bisecting techniques, the group of answer choices remains the same. However, the vertical angulation is generally greater for images taken with the paralleling technique compared to the bisecting technique.

This is because the paralleling technique requires the x-ray beam to be directed more vertically in order to capture the entire tooth structure on the film. On the other hand, the bisecting technique involves angling the x-ray beam downward to intersect the imaginary bisector between the long axis of the tooth and the film.

Therefore, the vertical angulation differs depending on which technique is being used.

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A. The area of the rectangle is 29.16 square inches.

B. The width of the rectangle is 2.7 inches and the height (or length) is 10.8 inches

A parallelogram in which each pair of adjacent sides is perpendicular is referred to as a rectangle.

We should utilize "w" to address the width and "l" to address the length of the square shape.

A. Since the width equals one quarter of the length, we are able to write:

w = (1/4)l

We also know that the perimeter of the rectangle is 27 inches, and

the formula for the perimeter of a rectangle is:

P = 2l + 2w

Substituting the first equation into the second equation, we get:

P = 2l + 2(1/4)l

27 = 2.5l

l = 10.8

Now that we know the length, we can use the first equation to find the width:

w = (1/4)l

w = (1/4)(10.8)

w = 2.7

As a result, the rectangle has a width of 2.7 inches and a height of 10.8 inches.

B. To find the area of the rectangle, we can use the formula:

Area = length *width

Substituting the values we found above, we get:

A = (2.7)*(10.8)

A = 29.16

Therefore, the area of the rectangle is 29.16 square inches.

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The sets A and B, defined as A = {x ∈ Z | x = at + bs for some t, s ∈ Z} and B = {x ∈ Z | x = ct for some t ∈ Z}, are equal. This is because every element in A can be expressed as an element in B and vice versa, based on the properties of the greatest common divisor.

To prove that the sets A = B, where A = {x ∈ Z | x = at + bs for some t, s ∈ Z} and B = {x ∈ Z | x = ct for some t ∈ Z}, we need to show that every element in A is also in B, and vice versa.

First, let's show that every element in A is in B:

Let x be an element in A. This means there exist integers t and s such that x = at + bs.

Since c = gcd(a, b), we can express a and b as a = c * a' and b = c * b', where a' and b' are integers.

Substituting these expressions into x, we have:

x = (c * a')t + (c * b')s

x = c(a't + b's)

Since a't + b's is an integer (as it is the sum of two integers), we can let t' = a't + b's, where t' is also an integer.

Therefore, x = ct', where t' = a't + b's is an integer.

This shows that every element in A is in B.

Next, let's show that every element in B is in A:

Let x be an element in B. This means there exists an integer t such that x = ct.

Since c = gcd(a, b), we know that c divides both a and b. Therefore, we can express a and b as a = cx and b = cy, where x and y are integers.

Substituting these expressions into x, we have:

x = c * t

x = c * (xt)

Since xt is an integer (as it is the product of two integers), we can let s = xt, where s is also an integer.

Therefore, x = at + bs, where t = xt and s = y.

This shows that every element in B is in A.

Since we have shown that every element in A is in B and every element in B is in A, we conclude that A = B.

Hence, the sets A and B are equal, as desired.

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The opposite of 2(3x-8) is 1/(6x-16)

The inverse of an expression is thesame as the opposite of an expression.

Inverse operations are pairs of mathematical manipulations in which one operation undo the action of the other. The inverse of a number usually means its reciprocal.The product of a number and its inverse will be equal to 1.

For example the reciprocal of 5 is 1/5. That is 5×1/5= 1

Also the inverse of a fraction like 5/7 is 7/5. That is 5/7× 7/5 = 1

Similarly, the inverse of 2(3x-8) will be 1/2(3x-8)

= 1/6x- 16.

Therefore the inverse of 2(3x-8) = 1/(6x-16)

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Answer:

75%

Step-by-step explanation:

write it as a fraction first, so 15/20.

make the bottom number 100 to get the mark out of 100.

to do this, multiply the bottom by 5 (20x5=100) and then do the same to the top.

you get 75/100. this equals 75%

Answer:

75%

Step-by-step explanation:

In this question, 20 is the highest score you can get on the test so 20 is 100%. We will now want to divide 100 by 20.

100 ÷ 20 = 5

Each score Amore got on the test was worth 5% of the test.

We know he scored 15 on the text, so now we want to multiply 15 with 5.

15 × 5 = 75

Amore scored 75% on his test.

The solutions to the given quadratic equation are x=[5+13i]/14 or x=[5-13i]/14.

Given that, the quadratic formula is x= [-(-5)±√((-5)²-4×7×7)]/2×7.

Here, x= [5±√(25-196)]/14

x= [5±√(-171)]/14

x=[5±13i]/14

x=[5+13i]/14 or x=[5-13i]/14

Now, (x-(5+13i)/14) (x-(5-13i)/14)=0

Therefore, the solutions to the given quadratic equation are x=[5+13i]/14 or x=[5-13i]/14.

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The x-intercept and y-intercept of the equation  -4x + 7y = 3 are x = - 3 / 4 and y = 3 / 7.

The intercept of the equation is the y-intercept and the x-intercept of the equation.

The y-intercept is the point where the line intersects the y-axis. The y-intercept is the value of y when x is equals to 0.

Therefore,

-4x + 7y = 3

7y = 3 + 4x

y = 3 / 7 + 4 / 7 x

when x = 0

y = 3 / 7 + 4 / 7 (0)

y = 3 / 7

The y-intercept is 3 / 7.

The x-intercept is the point at which the graph of an equation crosses the x-axis. The x-intercept is the value of x when y equals zero,

Therefore,

-4x + 7y = 3

- 4x + 7(0) = 3

-4x  = 3

x = - 3 / 4

The x-intercept is - 3 / 4

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Answer:

Step-by-step explanation:

You want the x- and y-intercept coordinates for the line -4x +7y = 3.

The x-intercept is the value of x that corresponds to a y-value of 0. Substituting y=0 into the equation, we have ...

  -4x +0 = 3

  x = -3/4

The coordinates of the x-intercept point are (-3/4, 0).

The y-intercept is the value of y that corresponds to an x-value of 0. Substituting x=0 into the equation, we have ...

  0 +7y = 3

  y = 3/7

The coordinates of the y-intercept point are (0, 3/7).

The intercepts are ...

__

Additional comment

We often express the intercept using only the coordinate on the respective axis. The other coordinate is always zero.

The attached graph shows decimal approximations of the intercept points. The value 0.429 corresponds to the decimal value 0.428571...(repeating), or 3/7. Of course -0.75 = -3/4.

<95141404393>

Cheri collected 145 number of 5 peso coin and 156 number of 10 peso coin during her family's annual outreach

An equation is used to show the relationship between numbers and variables.

Let x represent the number of 5 peso coins and y represent the number of 10 peso coin

At the end of the year, she has saved a total of Php 2285, hence:

5x + 10y = 2285    (1)

Also, she has 11 more 10 pesos than 5 peso coins, hence:

y = x + 11    (2)

Solving both equations using elimination method gives:

x = 145, y = 156

Cheri had 145 number of 5 peso coin and 156 number of 10 peso coin.  

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The probability that Liz has two boys given that she has at least one boy who is taller is approximately 0.2841

Let's first consider all possible gender combinations of Liz's two children:

Boy, boy (BB)

Boy, girl (BG)

Girl, boy (GB)

Girl, girl (GG)

We know that Liz has at least one boy, which rules out the GG combination. That leaves us with three possible combinations: BB, BG, and GB.

From the given information, we know that in 76% of families consisting of one son and one daughter, the son is taller than the daughter. This means that in the BB combination, the probability that the taller child is a boy is 1 (since both children are boys), and in the BG and GB combinations, the probability is 0.76 (since there is one boy and one girl, and we know the boy is taller).

So, let's calculate the probability that Liz has two boys (BB) given that she has at least one boy who is taller. We can use Bayes' theorem for this

P(BB | taller child is a boy) = P(taller child is a boy | BB) × P(BB) / P(taller child is a boy)

where P(taller child is a boy | BB) = 1 (as both children are boys), P(BB) = 1/4 (since there are four possible gender combinations), and P(taller child is a boy) = P(taller child is a boy | BB) × P(BB) + P(taller child is a boy | BG) × P(BG) + P(taller child is a boy | GB) × P(GB) = 1 × 1/4 + 0.76 × 1/2 + 0.76 × 1/2 = 0.88.

Substituting these values into Bayes' theorem, we get

P(BB | taller child is a boy) = 1 × 1/4 / 0.88 = 0.2841

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Answer:

A is the correct answer. x = 3.

Step-by-step explanation:

\( {49}^{x + 4} = {7}^{5x - 1} \)

\( {7}^{2(x + 4)} = {7}^{5x - 1} \)

\(2x + 8 = 5x - 1\)

\(3x = 9\)

\(x = 3\)

Answer:

x = 6

y = 13

Step-by-step explanation:

since DE and JK are congruent, they are both 18

to find 'y':

3y - 21 = 18

3y = 39

y = 13

DF and JL are diagonals that are congruent so:

9x - 23 = 7x - 11

2x = 12

x = 6

Based on the given information, there are no possible values of ∠m to the nearest degree that make sense. It's possible that there is a typo or error in the problem statement.

In ΔMNO, given m = 50 cm, o = 35 cm, and ∠O = 83°, we can find all possible values of ∠M using the Law of Sines.
First, let's set up the equation:
sin(∠M) / m = sin(∠O) / o
Now, plug in the given values:
sin(∠M) / 50 = sin(83°) / 35
Solve for sin(∠M):
sin(∠M) = (50 * sin(83°)) / 35
Calculate the value of sin(∠M):
sin(∠M) ≈ 0.964
Now, find the angle:
∠M = arcsin(0.964)
∠M ≈ 75° (to the nearest degree)
So, the possible value for ∠M is approximately 75°.

To find the possible values of ∠m, we can use the fact that the sum of angles in a triangle is 180 degrees. First, we can find the measure of ∠n by subtracting the given angle from 180:
∠n = 180 - ∠o
∠n = 180 - 83
∠n = 97 degrees
Now we can use the fact that the sum of angles in a triangle is 180 degrees to find the measure of ∠m:
∠m + ∠n + ∠o = 180
Substituting in the given values:
∠m + 97 + 83 = 180
Simplifying:
∠m = 180 - 97 - 83
∠m = 0 degrees
This doesn't make sense - a triangle cannot have an angle with a measure of 0 degrees.

However, we can also use the fact that the sum of angles in a triangle is 180 degrees to find an inequality for ∠m:
∠m + ∠n + ∠o = 180
Substituting in the given values:
∠m + 97 + 83 = 180
Simplifying:
∠m = 0 degrees
This tells us that if ∠m is 0 degrees, then the other two angles must add up to 180 degrees. But we also know that ∠m and ∠n must be acute angles (less than 90 degrees) since the opposite sides of the triangle are longer than the adjacent sides.
Therefore, the only possible value for ∠m is less than 90 degrees. We can estimate this value by subtracting the sum of the other two angles (180 - 97 - 83 = 0 degrees) from 180:
∠m < 180 - 97 - 83
∠m < 0 degrees
Again, this doesn't make sense.
So, based on the given information, there are no possible values of ∠m to the nearest degree that make sense. It's possible that there is a typo or error in the problem statement.

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The number of times the spinner will land on an odd number in 200 trials is 80 times

The sections on the spinner are given as

Sections = 1 to 7

From these sections, we have

Odd sections = 1, 3, 5 and 7

From the dot plot given, the total frequency of the odd sections is

Frequency of odd sections = 3 + 1 + 2 + 2

Evaluate

Frequency of odd sections = 8

Also, we have

Total frequency = 20

The number of times is then calculated as

Number of times = Frequency of odd sections/Total frequency * 200

So, we have

Number of times = 8/20 * 200

Evaluate

Number of times = 80

Hence, the number of times is 80 times

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Answer:

80 times

Step-by-step explanation:

Explanation:

We were given the equation:

The slope-intercept is represented by:

Rewriting the equation into its slope-intercept form, we have:

Therefore, the equation in slope-intercept is: y = -(8/3)x + 4

We will now input assumed values for "x" to obtain corresponding y-values. We have:

We will proceed to plot these points on the graph, we have:

(a) (i) 0.3174 = 31.74% probability of a defect

(ii) The expected number of defects for a 1,000-unit production run is 317.

(b) (i) 0.0026 = 0.26% probability of a defect

(ii) The expected number of defects for a 1,000-unit production run is 3.

(C) Reduces the process standard deviation and causes no change in the number of defects.

When the distribution is normal, we use the z-score formula.

In a set with mean  and standard deviation, the score of a measure X is given by:

         Z = X-μ /σ

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Question a:

We have that: μ = 10 and σ = 0.12

(i). Calculate the probability of a defect.

Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.

Probability of less than 9.88:

This is the p-value of Z when X = 9.88. Thus,

  Z = X-μ /σ

⇒ Z = 9.88 - 10/0.12

⇒ Z = -1, has a p-value of 0.1587

⇒ 2× 0.1587 = 0.3174

This means 0.3174 = 31.74% probability of a defect

(ii)  Calculate the expected number of defects for a 1,000-unit production run.

The expected number of defects is 31.74% of 1000. So

0.3174*1000 = 317.4

Rounding to the nearest integer

The expected number of defects for a 1,000-unit production run is 317.

Question (b):

The mean remains the same, but the standard deviation is now

(i) Calculate the probability of a defect.

Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.

Probability of less than 9.88:

This is the p-value of Z when X = 9.88. Thus,

   Z = X-μ /σ

⇒ Z = 9.88 -10/0.04

⇒ Z = -3

⇒ Z = -3, has a p-value of 0.0013

which means 2× 0.0013 = 0.0026

from which 0.0026 = 0.26% probability of a defect

(ii) Calculate the expected number of defects for a 1,000-unit production run. The expected number of defects is 31.74% of 1000. Thus,

⇒ 0.0026*1000 = 2.6

Rounding to the nearest integer

The expected number of defects for a 1,000-unit production run is 3.

(C) The advantage of reducing process variation, thereby causing problem limits to be at a greater number of standard deviations from the mean Reduces the process standard deviation and causes no change in the number of defects.

Learn more about Probability:

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