The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0

The pie chart is an important type of data representation. It contains different segments and sectors in which each segment and sector of a pie chart forms a specific portion of the total(percentage). The sum of all the data is equal to 360°.

The "pie chart, also known as circle chart divides the circular statistical graphic into sectors or sections to illustrate numerical problems. Each sector represents a proportionate portion of the total.

The Pie-chart works best for determining the composition of something at that time. In most cases, pie charts take the place of other graphs such as bar graphs, line plots, histograms, and so on.

Note: The monthly budget of the family is represented in pie chart​ that is attached as image.

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The area of the castle's floor increases by 5.06% .

How to calculate area of rectangular floor?

By multiplying length and breadth, the rectangle's area will obtain in a square-unit dimension. In the case of a square, the area will become side2. The main difference between square and rectangle is that the length and breadth are equal for square.

Given :

Let the Length and width of the rectangular floor be x and y respectively

then , the area of rectangular floor be x*y

Since,

Ashley increased the length of the castle's floor : 30%

and width of castle's floor : 20%.

New length = x + 30% of x = x + 3x/100 = 103x/100

New width = y + 20% of y =y + 2y/100 = 102y/100

New area of rectangular floor = (103x/100)*(102y/100)

percent increase in area of the castle's floor

= {[(103x/100)*(102y/100) - x*y]*100} / xy

= 506 / 100

= 5.06

Hence, the area of the castle's floor increases by 5.06% .

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To find the number of terms in a geometric progression, we can use the formula:

n = log[size of last term / size of first term] / log[common ratio]

In this case, the first term is 2, the last term is 256, and the common ratio is 4/2 = 2. Plugging these values into the formula, we get:

n = log[256/2] / log[2] = log[128] / log[2] = 7

Therefore, there are 7 terms in the geometric progression 2, 4, 8, ..., 256.

ChatGPT Feb 13 Version. Free Research Preview. Our goal is to make AI systems more natural and safe to interact with. Y

Answer:

Step-by-step explanation:

Cost of 100 gloves = $7.73

Cost of each glove = 7.73/100= $0.0773 (or 7.73 cents each)

Tyshawn will be required to put down 25% of the total cost of the vehicle.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

Tyshawn wants to spend $19,500 on a car, but the bank will only lend him $14,625 for it. The percentage is then stated as,

P = [($19,500 - $14,625) / $19,500] x 100

P = [($4,875) / $19,500] x 100

P = 0.25 x 100

P = 25%

Tyshawn will be required to put down 25% of the total cost of the vehicle.

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The vectors [8 -2 6] and [12 -3 9] are option (c) linearly independent because neither vector is a multiple of the other.

A vector is an entity that has both magnitude and direction. It is represented by an ordered set of numbers, which correspond to the coordinates of its endpoint in a given coordinate system.

In this problem, we are given two vectors: [8 -2 6] and [12 -3 9]. To determine if they are linearly independent or not, we need to check if one vector can be expressed as a linear combination of the other.

Let us assume that the first vector can be expressed as a linear combination of the second vector. That is, there exist constants a, b, and c such that:

[8 -2 6] = a[12 -3 9] + b

where b is a vector. Simplifying the above equation, we get:

8 = 12a + b1

-2 = -3a + b2

6 = 9a + b3

where b1, b2, and b3 are the respective components of vector b. Now, we can solve for a, b1, b2, and b3 using the system of linear equations. Solving for a, we get a = 2/3. Substituting this value in the remaining equations, we get:

b1 = -4/3, b2 = -2/3, and b3 = 2/3

Therefore, the vector b is [-4/3 -2/3 2/3]. We can verify that the first vector cannot be expressed as a linear combination of the second vector and b. Hence, the given vectors are linearly independent.

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The statements about the Delian League that is true is A. Sparta never joined the league and resented Athens’s power and treatment of allies.

Following the Persian War, the Delian League was established in 478 BCE as a military coalition to protect the Ionian Greeks from any potential threats. Athens served as its principal leader and used its sizable and potent navy to defend all of its members who were unable to defend themselves.

Since Athens dominated the Delian League and Athens and Sparta had historically been adversaries, Sparta decided against joining. Although Athens and Sparta put their disagreements aside to overcome their common foe during the Persian Conflicts, they resumed their rivalry when the wars were over.

Therefore, option A is correct.

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The length of the altitude to EM in triangle GEM is 25 units.

Note that if the altitude of the triangle is at most 10, then the maximum area of the intersection of the triangle and the square is 5 * 10=50. This implies that vertex G must be located outside of square AIME.

Now, suppose GE meet AI at X and let GM meet AI at Y.

Clearly, XY=6 since the area of trapezoid XYME is 80. Also, triangle GXY is similar to triangle GEM.

Let the height of GXY be h.

By the similarity, h/6 = {h + 10}/10,

we get h = 15.

Thus, the height of GEM is

h + 10 = 15 + 10

h = 25 units

So the length of altitude is of triangle GEM is 25 units.

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____The given question is incorrect, the correct question is given below:

Square AIME has sides of length 10 units. Isosceles triangle GEM has

base EM, and the area common to triangle GEM and square AIME is 80

square units. Find the length of the altitude to EM in triangle GEM.

0.90 is value P(A and B) of probability .

What are examples and probability?

The likelihood that something will occur is the foundation of it. The logic underpinning probability serves as the basic foundation for theoretical probability.

                           The theoretical likelihood of receiving a head, for instance, is 12 when a coin is tossed.

P(A) = 0.6 P(B) = 0.5

a) P(A and B) =P(A)P(B) {Independent)

P(A and B) = 0.6×0.5

P(A and B) = 0.30

b) Given P(A | B) = 0.3, this is a conditional probability

If P(A | B) = P(A and B)/P(B)

Since If P(A | B) = 0.3

0.3 = P(A and B)/P(B)

P(A and B) = 0.30(P(B)

P(A and B) = 0.3(0.30)

P(A and B) = 0.90

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The solution to the system of linear equations can be expressed in vector form as:

x = [x1, x2, x3, x4] = [2-3x3-x4, 3+6x3+x4, x3, x4]

Breaking down the vector components:

A = [-3, 0, 1, 0]

a = [-1, 0, 0, -1]

B = [2, 3, 0, 0]

b = [0, 0, 1, 0]

bb = [0, 0, 0, 1]

C = [0, 0, 1, 0]

cc = [0, 0, 1, 0]

D = [0, 0, 0, 1]

dd = [0, 0, 0, 1]

To obtain this solution in vector form, we first rewrite the equations in standard form:

x1 + 3x3 + x4 = 2

x2 - 6x3 - x4 = 3

Then, we can write the solution as a combination of the homogeneous solution (the particular solution) and the non-homogeneous solution (the vector on the right-hand side). The homogeneous solution is obtained by setting the right-hand side to zero and solving for x3 and x4:

x1 + 3x3 + x4 = 0

x2 - 6x3 - x4 = 0

which gives x3 = -1/3 and x4 = 0. Therefore, the homogeneous solution is [0, 0, -1/3, 0].

The non-homogeneous solution is obtained by setting x3 and x4 to zero and solving for x1 and x2:

x1 = 2

x2 = 3

Therefore, the non-homogeneous solution is [2, 3, 0, 0].

Combining the homogeneous and non-homogeneous solutions, we obtain the final solution in vector form:

x = [0, 0, -1/3, 0] + [2, 3, 0, 0] = [2-3x3-x4, 3+6x3+x4, x3, x4]

with the vectors A, a, B, b, bb, C, cc, D, dd given as above.

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The 10-ounce box is the better buy as it has a lower cost per ounce.

In mathematics, divides left-hand operands into right-hand operands in the division operation.

To determine which box of cereal is a better buy, you can calculate the cost per ounce for each option.

For the 10-ounce box, the cost per ounce is:

= 1.86/10

Apply the division operation,

= $0.186 per ounce.

For the 14-ounce box, the cost per ounce is:

= 2.85/14

= $0.203 per ounce.

Therefore, the 10-ounce box is the better buy as it has a lower cost per ounce.

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The expression when simplified is 6x^8y^3

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

(2x5y2)(3x3y)

Express properly

So, we have

(2x^5y^2)(3x^3y)

Evaluate the products

This gives

6x^(5+3)y^(2+1)

So, we have

6x^8y^3

Hence, the expression is 6x^8y^3

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Answer:

B

Step-by-step explanation:

took the test xx

The total number of likes given to the shoulder length. The correct option is B.

A survey in human subjects research is a set of questions designed to elicit specific information from a specific group of people. Surveys can be conducted over the phone, by mail, on the internet, or even on street corners or in shopping malls.

In the given survey 10 liked option A mullet, 11 liked option B is shoulder length, 4 liked the short length and 2 liked bald head.

So from the survey, more points are given to the shoulder-length hairstyle.

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Answer: f = 14.6

Step-by-step explanation:

Since you gave the equation 0.3 = 0.5f - 7, here are the following steps to solve:

1. Combine multiplied terms into a single fraction

[tex]0.3=\frac{1f}{2} -7[/tex]

2. Multiply by 1

[tex]0.3 =\frac{f}{2} -7[/tex]

3. Find Common Denominator

[tex]0.3 = \frac{f}{2} + \frac{2(-7)}{2}[/tex]

4. Combine fractions

[tex]0.3 = \frac{f+2(-7)}{2}[/tex]

5. Multiply

[tex]0.3=\frac{f - 14}{2}[/tex]

6. Multiply all terms by the same number to eliminate denominators

[tex]2(0.3)= 2\frac{f-14}{2}[/tex]

7. Simplify

[tex]0.6 = f-14[/tex]

8. Add 14 to both sides

[tex](0.6 + 14) = f (- 14 + 14)[/tex]

9. Simplify

[tex]f= 14.6[/tex]

Cannot be zero because any exponential number with base 0 is equal to 1.

A lot of 1000 components contains 300 that are defective. Two components are drawn at random and tested.

a) P(A) is 0.3

b) P(B|A) is 299/999

c) P(A ∩ B) is 0.089

d) P(A' ∩ B) is 0.211

e) P(B) is 0.3

f) P(A|B) is 0.297

g) A and B are not independent, therefore, it is not reasonable to treat A and B as though they were independent

a) P(A) is the probability that the first component drawn is defective. Since there are 300 defective components out of 1000, the probability that the first component drawn is defective is:

P(A) = 300/1000 = 0.3

b) P(B|A) is the conditional probability that the second component drawn is defective given that the first component drawn is defective. Since there are now 299 defective components left out of 999, the probability that the second component drawn is defective given that the first component drawn is defective is:

P(B|A) = 299/999

c) P(A ∩ B) is the probability that both the first and second components drawn are defective. Since there are 300 defective components out of 1000, the probability that the first component drawn is defective is 0.3. If the first component drawn is defective, then there are now 299 defective components left out of 999.

So the probability that the second component drawn is defective given that the first component drawn is defective is 299/999. Therefore, the probability that both components are defective is:

P(A ∩ B) = P(A) * P(B|A) = 0.3 * 299/999 = 0.089

d) P(A' ∩ B) is the probability that the first component drawn is not defective (i.e., it is good) and the second component drawn is defective. Since there are 700 good components out of 1000, the probability that the first component drawn is good is:

P(A') = 700/1000 = 0.7

If the first component drawn is good, then there are 299 defective components left out of 999. So the probability that the second component drawn is defective given that the first component drawn is good is:

P(B|A') = 299/999

Therefore, the probability that the first component drawn is good and the second component drawn is defective is:

P(A' ∩ B) = P(A') * P(B|A') = 0.7 * 299/999 = 0.211

e) P(B) is the probability that the second component drawn is defective. There are 300 defective components out of 1000, so the probability that the second component drawn is defective is:

P(B) = 300/1000 = 0.3

f) P(A|B) is the conditional probability that the first component drawn is defective given that the second component drawn is defective. Using Bayes' theorem, we have:

P(A|B) = P(A ∩ B) / P(B) = (0.3 * 299/999) / 0.3 = 0.089 / 0.3 = 0.297

g) A and B are not independent, because the probability of B depends on whether or not A has occurred. In other words, the probability of drawing a defective component for the second draw depends on whether or not the first draw yielded a defective component.

If A has occurred (i.e., the first component drawn is defective), then the probability of B is higher than if A had not occurred. Therefore, it is not reasonable to treat A and B as though they were independent.

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Complete question is:

A lot of 1000 components contains 300 that are defective. Twocomponents are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective.

a) find P(A)

b) find P(B|A)

c) find P(A ∩ B)

d) find P(A' ∩ B)

e) find P(B)

f) find P(A|B)

g) Are A and B independent? It is reasonable to treat A and B as though they were independent? Explain

The linear factors of the given polynomial is (2x+1)(4x+3) and (3x-2).

The factorization method uses basic factorization formula to reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets. The factors of any equation can be an integer, a variable, or an algebraic expression itself.

The given function is f(x)=24x³+14x²-11x-6 and one of the factor is (2x+1).

Here, 24x³+14x²-11x-6 can be written as 24x³+12x²+2x²+1x-12x-6

12x²(2x+1)+1x(2x+1)-6(2x+1)

= (2x+1)(12x²+1x-6)

= (2x+1)(12x²+1x-6)

= (2x+1)(12x²+9x-8x-6)

= (2x+1)[3x(4x+3)-2(4x+3)]

= (2x+1)(4x+3)(3x-2)

Therefore, the linear factors of the given polynomial is (2x+1)(4x+3) and (3x-2).

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The results based on the information provided in the question will be as follows:

a) By using the analysis of variance (ANOVA) table,

MSE= 440.1/10 = 44.01

F = 2717.9/44.01 =61.756

b) To interpret the slope of regression model,

the slope represents that for each $1000 increase in the income of a family on an average sale price increase by 2184.3

c) To calculate the value of coefficient of determination and its interpretation,

0.712 and the lowest that coefficient of determination is 0 and highest is 1. Thus, here the coefficient is high.

d) By using regression equation,

Price = 11.8+2.18×income

Predicted value = (-11.8+2.18×20)×1000 = 31800

e) To compute, 95% confidence slope

= 2.1843/2.228×0.278 = 1.565;2.804

f) The linear relationship between income and price,

All the intervals above have a value of more than 0 therefore we can conclude that the slope is significant. Thus, we have sufficient evidence that at 0.05 level of significance the linear relationship between income and price is inter related.

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Note that the full question is:

The National Association of Home Builders ranks the most and least affordable housing markets based on the proportion of homes that a family earning the median income in that market could afford to buy. Data containing the median income $1000s) and the median sale price($1000s) for a sample of 12 honsing markets appearing on the list of most affordable markets were subjected to a simple linear regression analysis. The following results were obtained. The regression equation is price-11.8 +2.18 income Predictor Coef StDev Constant11.80 12.84 income 2.1843 0.2780 S 6.634 Analysis of Variance Source Regression 1 2717.9 2717 DF SS MS F P Residuni Eror 10 440. Total 1 3158.0 a. Using the snalysis of variance table, find MSE b. Interpret the slope of the regression model. the valus ul and interpret it d. Using the estimated regression equation, the value ofy ifx-$20,000 e. Compute 95% confidence al for the slope 43-1+2.2280T g. At the 0.05 level of significunce, is there evidence of a linear relationship between income and price

The probability she pulls out a purple piece of candy would be 0.22.

Given is Sam's fathers collection.

The probability that tle would be green is -

P{E} = 4/20
P{E} = 1/5

Therefore, the probability she pulls out a purple piece of candy would be 0.22.

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The graph of a radical function in form f(x) = a(x + k)^1/n + c can tell you that the domain of the function is all real numbers greater than or equal to -k, and the range of the function depends on the values of "a" and "c".

The domain is defined as the values of the independent variable for which there is a certain value of the dependent variable exists in the range of the function.

Here,
The domain of a radical function in this form is determined by the expression under the radical. For the function to be defined, this expression must be greater than or equal to zero. Therefore, the domain of the function is all real numbers such that x + k is greater than or equal to zero.

The range of the function is determined by the value of "a" and "c". If "a" is positive, the range of the function is all real numbers greater than or equal to "c". If "a" is negative, the range of the function is all real numbers less than or equal to "c".

In summary, the graph of a radical function in form f(x) = a(x + k)^1/n + c can tell you that the domain of the function is all real numbers greater than or equal to -k, and the range of the function depends on the values of "a" and "c".

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Answer:

B

Step-by-step explanation:

you got polluted water that's the problem the solution is strict er laws about dumping

Answer:

i don't know

Step-by-step explanation:

If the two stations have the same bit rate, they will both have to be on air for the same amount of time to broadcast 4 minutes of news, regardless of the specific bit rate

What is time in maths ?

In mathematics, time is often thought of as a dimension, similar to length, width, and height. However, time is unique in that it only moves in one direction, from the past to the future. Time can be measured in various units such as seconds, minutes, hours, days, weeks, months, and years.

The length of time a station will have to be on air to broadcast 4 minutes of news depends on the bit rate of the broadcast.

The bit rate is the amount of data transmitted per second in a digital signal. The higher the bit rate, the more data can be transmitted per second, resulting in higher audio or video quality but also a larger file size.

Assuming the two stations have different bit rates, the station with the lower bit rate will have to be on air longer to broadcast the same 4 minutes of news than the station with the higher bit rate. This is because the station with the lower bit rate can transmit less data per second, so it will take longer to transmit the same amount of information.

For example, if Station A has a bit rate of 128 kbps (kilobits per second) and Station B has a bit rate of 256 kbps, Station A will have to be on air for twice as long as Station B to transmit the same 4 minutes of news, as it can transmit only half as much data per second as Station B.

Hence, If the two stations have the same bit rate, they will both have to be on air for the same amount of time to broadcast 4 minutes of news, regardless of the specific bit rate

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The x-intercepts of the graph of the equation y = x^2 - 3x + 2 are 1 and 2.

The x-intercepts of the graph of the equation y = x^2 - 1 are -1 and 1.

In Mathematics, the x-intercept can be defined as the point at which the graph of a function crosses the x-coordinate (x-axis) and the value of "y" or y-value is equal to zero (0).

In Mathematics, the y-intercept is sometimes referred to as an initial value and the y-intercept of any graph such as a linear function, generally occur at the point where the value of "x" is equal to zero (x = 0).

In this exercise, we would use an online graphing calculator to plot the linear function in order to determine the x-intercepts and y-intercepts as shown in the image attached below.

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The Slope of the lines made from given points will be For A slope=1 and For B slope=∞.

What exactly is a Slope?

A slope of a line is defined as the change in y coordinate divided by the change in x coordinate.

The net change in y-coordinate is denoted by Δy, whereas the net change in x-coordinate is denoted by Δx.

As a result, the change in y-coordinate in relation to the change in x-coordinate is given by,

m = difference in y/difference in x = Δy/Δx=y2-y1/x2-x1

Where "m" shows the slope of a line.

The line's slope may alternatively be represented by

tan θ = Δy/Δx

In general, the slope of a line indicates its steepness and direction. The slope of a straight line between two locations, (x1,y1) and (x2,y2), is simply calculated by finding the difference of the coordinate points.

Now,

Given Coordinates are in form (x1,y1) and (x2,y2)

A. (5, 7) and (-4,-2)

then slope m=-2-7/-4-5

=-9/-9

=1

B. (1,3) and (1, -10)

slope m=-10-3/1-1

=-13/0

=∞

Hence,

           The Slope of the lines made from given points will be For A slope=1 and For B slope=∞.

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The number of points at which a circle can intersect a triangle is 2 and 6 only. The correct answer is "2 and 6 only".

When a circle intersects a triangle, it can intersect at 0, 1, 2, 3, 4, 5, or 6 points. The maximum number of intersection points occurs when the circle passes through all three vertices of the triangle, which results in 6 intersection points.

However, the minimum number of intersection points is 2, which occurs when the circle intersects the triangle at two distinct points. This can happen when the circle passes through one vertex of the triangle and intersects an opposite side at another point.

It is also possible for a circle to intersect a triangle at 0, 1, 3, 4, or 5 points, but these are not listed as options in the given answer choices. Therefore, the correct answer is "2 and 6 only".

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Step 1 of 7: The class width of the data given s 50.

Step 2 of 7: The frequency in 3rd class is 5

Step 3 of 7: Lower class boundary for the second class is 1228.5.

Step 4 of 7: Upper class boundary for the 3rd class is 1328.5

Step 5 of 7: The midpoint of the first class is 1203.5

Step 6 of 7: The relative frequency of the second class is 0.0476

Step 7 of 7: The cumulative frequency of the 5th class is 15.

As per the data given:

the following data representing the price of plasma televisions (in dollars).

1467, 1281, 1342, 1429, 1313, 1357, 1448, 1385, 1304, 1291, 1406, 1410, 1307, 1422, 1264, 1454, 1191, 1191, 1372, 1461, 1447.

                                Class                        Relative      Cumulative
  Class     Frequency   Boundaries      Midpoint      Frequency  Frequency

1179−1228          2       1178.5 - 1228.5        1203.5   0.0952        2

1229−1278     1       1228.5 - 1278.5       1253.5   0.0476        3

1279−1328     5       1278.5 - 1328.5        1303.5   0.2381        8

1329−1378     3       1328.5 - 1378.5        1353.5   0.1429        11

1379−1428     4       1378.5 - 1428.5        1403.5   0.1905        15

1429−1478     6       1428.5 - 1478.5         1453.5   0.2857        21

Step 1 of 7: Determine the class width of the data given.

The class width is the difference between the upper or lower class limits of consecutive classes. All classes should have the same class width. In this case, class width equals to the difference between the lower limits of the first two classes.

w = 1229 - 1179 = 50

Step 2 of 7: Determine frequency in 3rd class

The 3rd class is 1279−1328 and the frequency in 3rd class is 5

Step 3 of 7: Determine the lower class boundary for the second class

The 2nd class is 1229−1278 and the class boundary is 1228.5 - 1278.5

Lower class boundary for the second class is 1228.5.

Step 4 of 7: Determine the upper class boundary for the 3rd class

The 3rd class is 1279−1328 and the class boundary is 1278.5 - 1328.5

Upper class boundary for the 3rd class is 1328.5

Step 5 of 7: Identify the midpoint of the first class

The 1st  class is 1179−1228 and the midpoint of the first class is 1203.5

Step 6 of 7: Calculate the relative frequency of the second class.

The 2nd class is 1229−1278 and the relative frequency of the second class is 0.0476

Step 7 of 7: Compute the cumulative frequency of the 5th class

The 5th class is 1379−1428 and the cumulative frequency of the 5th class is 15.

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a. The probability of Mikayla drawing black pencil from the bag if each pencil is removed from the bag after it is drawn, is 2/7.

b. The probability of Mikayla drawing blue pencil from the bag if each pencil is removed from the bag after it is drawn, is 15/28.

c. The probability of Mikayla drawing cyan pencil from the bag if each pencil is removed from the bag after it is drawn, 5/28.

Mikayla has a bag of pencils with 8 black, 15 blue, and 5 cyan pencils.

Total Number of Black Pencil = 8

Total Number of Blue Pencil = 15

Total Number of Cyan Pencil = 5

Total Number of pencil = Total Number of Black Pencil + Total Number of Blue Pencil + Total Number of Cyan Pencil

Total Number of pencil = 8 + 15 + 5

Total Number of pencil = 28

a. We have to determine the probability of Mikayla drawing black pencil from the bag if each pencil is removed from the bag after it is drawn.

The Probability = Total Number of black pencil/Total Number of pencil

The Probability = 8/28

The Probability = 2/7

b. We have to determine the probability of Mikayla drawing blue pencil from the bag if each pencil is removed from the bag after it is drawn.

The Probability = Total Number of blue pencil/Total Number of pencil

The Probability = 15/28

c. We have to determine the probability of Mikayla drawing cyan pencil from the bag if each pencil is removed from the bag after it is drawn.

The Probability = Total Number of cyan pencil/Total Number of pencil

The Probability = 5/28

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The complete question is:

Mikayla has a bag of pencils with 8 black, 15 blue, and 5 cyan pencils. find the following probabilities of Mikayla drawing the given pencils from the bag if each pencil is removed from the bag after it is drawn.

a. Only black pencil left

b. Only Blue pencil left

c. Only cyan pencil left

Let n be an odd integer. By definition, n must be an integer not divisible by 2. Hence, when n2 is divided by 8, the remainder must be 1.

Let n be an odd integer. First, we can express n as 2k+1, where k is an integer. Next, when we square n, we get (2k+1)2. Expanding this gives 4k2+4k+1. Dividing this by 8 gives 4k2+k as the quotient and 1 as the remainder. This proves that the square of an odd integer always have a remainder of 1 when divided by 8.

To summarize, when an odd integer n is squared, the remainder when the result is divided by 8 is always 1. This is because an odd integer is not divisible by 2, and so the quotient will not be a whole number when divided by 8.

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