Dean's savings account started with $100. After one year, it had $110. The
second year, it had $121. The third year it had $133.10.
Is a single linear model a reasonable option for this data? Yes or No?

a) The binomial expansion of (1+12x)^1/2 is given by:

(1+12x)^1/2 = 1^1/2 + (1/2)(1^-1/2)(12x) + (1/2)(1^-1/2)(-1/2)*(12x)^2 + ...

Binomial expression: what is it?

A polynomial with only terms is a binomial. An illustration of a binomial is x + 2, where x and 2 are two distinct terms. Additionally, in this case, x has a coefficient of 1, an exponent of 1, and a constant of 2. As a result, a binomial is a two-term algebraic expression that contains a constant, exponents, a variable, and a coefficient.

The first four terms, in ascending powers of x, are:

1, (1/2)(12x), (1/8)(12x)^2, (1/16)*(12x)^3

b) To use x=1/36 in the expansion, you would substitute x=1/36 into the expansion, and keep only the terms up to x^3 (the fourth term in the expansion). This will give an approximation of the value of (1+12x)^1/2 when x=1/36.

This will be an approximation of √12 because x=1/36 corresponds to 12x = 1 and the initial value of the expansion is 1 + 12x .

We can use this approximation to approximate square root of 12 and also for other square roots by using appropriate x values.

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The UCL and LCL are 1.00434 and 0.99566. The process is not in control and the z-value is 2.17.

We already know that the mean, x = 1.0, and the standard deviation of the process, σ = 0.01.

We already know that according to the central limit theorem, the standard deviation of sample means is equal to the standard deviation of the population divided by the square root of the sample size.

So, it is equal to = 0.01 / √25 ⇒ 0.002

For a sample size of 25. The z value, or the z-score, is about 2.17 for a confidence level of 97 percent.

So, UCL = 1.0 + 2.17 * 0.002 = 1.00434

LCL = 1.0 - 2.17 * 0.002 = 0.99566

Lastly, the process is not in control because the sample means 1.005 and 0.995 are outside the control range.

The complete question that you must be looking for is given below.

An automatic filling machine is used to fill 1-liter bottles of cola. The machine output is approximately normal with a mean of 1.0 liter and a standard deviation of .01 liter. Output is monitored using means of samples of 25 observations.

a. Determine upper and lower control limits that will include roughly 97 percent of the sample means when the process is in control.

b. Given these sample means: 1.005, 1.001, .998, 1.002, .995, and .999, is the process in control.

c. What is the z-value we should use if the control limits will include roughly 97% of the sample means?

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The equation of the straight line perpendicular to the line 3x - 6y = - 8 is y-2x+6=0 and of line parallel to the line 3x - 6y = - 8 is y=0.

The general equation of a straight line is -

y=mx+c

where -

m is slope of line which tells the unit rate of change of y with respect to x.

c is the y - intercept i.e., the point where the graph cuts the y axis.

The equation of a straight line can be also written as -

Ax + By + C = 0

By = - Ax - C

[tex]y=\frac{-A}{B} x-\frac{C}{B}[/tex]

Given that: -

[tex]3x-6y=8\\\\6y=3x-8\\\\y=\frac{1}{2}x-\frac{4}{3}[/tex]

then,

[tex]m=\frac{1}{2}\\\\c=\frac{4}{3}[/tex]

For a parallel line,

[tex]m=\frac{1}{2}\\and,(4,2)[/tex]

[tex]y-y_{1}=m(x-x_{1})\\\\y-2=\frac{1}{2}(x-4)\\\\2y-4=x-4\\\\y=\frac{x}{2}[/tex]

The equation of parallel line is y=0.5x

For a perpendicular line

[tex]m=2\\\\y-2=2(x-4)\\\\y-2=2x-8\\y-2x=-6\\y-2x+6=0\\[/tex]

Therefore, the equation of the line perpendicular to the line 3x - 6y = - 8 is y -2x+6=0 and of line parallel to the line 3x - 6y = - 8 is y = 0.5x.

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[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$100000\\ r=rate\to 4.2\%\to \frac{4.2}{100}\dotfill &0.042\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 100000e^{0.042\cdot 6} \implies A = 100000e^{0.252}\implies A \approx 128659.60[/tex]

If the image of point (x,5) is (-6, y) when it is translated by the vector t(6 -2), then the values of x = -12 and y = 3.

Vector translation is a mathematical concept that describes how an object is moved in space. It involves the use of a vector, which is a quantity that has both magnitude and direction. A vector can represent a displacement, a velocity, an acceleration, or any other physical quantity that has both magnitude and direction.

When an object is translated by a vector, it moves a certain distance in a specific direction. This is usually done by adding the vector to the current position of the object. The magnitude of the vector determines the distance of the translation, while the direction of the vector determines the direction of the translation.

Additionally, the vector can be used to rotate the object. This is done by multiplying the vector by the object’s rotation matrix. The result is the object's new orientation in space, which can be used to orient the object in a different direction.

Vector translation is used in a variety of applications, including computer graphics, robotics, and navigation systems. It i also used in mathematics and physics to represent physical phenomena, such as forces, velocities, or accelerations.

Let the point P(x,5) be translated by the vector t(6,-2).

The image of point P is (x',y') = (x+6,5-2) = (-6,y)

Therefore, x+6 = -6

=> x = -6-6 = -12

Similarly, 5-2 = y

=> y = 5-2 = 3

So the value of x is -12 and y is 3.

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Answer:

7/18.

Step-by-step explanation:

We know that the result is the same because multiplication is Commutative.

The order of the operation does not matter.

Answer: About 31%

Step-by-step explanation:

1. Divide 114 to 361

The value of x that best represent this expression √x = 11 is 121.

The square root of a number is the part of the number that, when multiplied by itself, produces the original number. Squares and square roots are examples of special exponents. The portion of a number that yields the original number when multiplied by itself is known as the square root of the number.

Examples of special exponents include squares and square roots. Think about the number 9. Nine is the result of multiplying 3 by itself. This can be expressed as either or 3*3. We refer to this as a square since it has a 2 exponent. When the exponent is now 1/2, the integer's square root is being discussed.

An integer or number (not a fraction) multiplied by itself produces a "Square Number," which is the result. For instance, 3 x 3 = or 3 squared is the result of 3 times 3.

Given:

√x+11 = 0

√x = -11

can be written as

x = -11²

x = 121

The value of x that best represent the expression √x+11 = 0 is 121.

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Correct question:

For which values of x does the expression √x + 11 = 0 make sense? ​

Answer: To find r'(1), we need to use the chain rule of differentiation, which states that:

r'(x) = f'(g(h(x))) * g'(h(x)) * h'(x)

In this case, we know that:

h(1) = 2, g(2) = 4, h'(1) = 4, g'(2) = 4, and f '(4) = 7

Therefore:

r'(1) = f'(g(h(1))) * g'(h(1)) * h'(1) = f'(g(2)) * g'(2) * h'(1) = 7 * 4 * 4 = 112

So, the derivative of r(x) with respect to x at x = 1 is 112

Step-by-step explanation:

The frequency Asked in the problem are listed below in order:

To make the information easier to understand, frequency distributions are visual displays that organise and present frequency counts. Absolute frequencies as well as relative frequencies, such as percentages or ratios, can be displayed in frequency distributions.

Given, a partial relative frequency distribution is given. class relative frequency a 0.22 b 0.18 c 0.40 d

Since all relative frequencies add up to be 1.

So,

0.22 + 0.18 + 0.40 + x = 1

x = 0.20

For b)

total sample size * relative frequency = frequency of a class

200*0.2 = 40

For c)

Frequency distribution is calculated as the frequency of a class = total sample size * relative freq.

A = 44

B = 36

C = 80

D = 40

TOTAL = 200

For d)

Percent frequency is calculated as Percent frequency of a class = 100 * relative freq.

A = 22

B = 18

C = 40

D = 20

TOTAL = 100

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Complete question:

The area of the trapezoid is 42.5 m².

A trapezium is a quadrilateral with four sides where two sides are parallel to each other.

We have,

Trapezoid:

Short base = 3 m

Height = 5 m

Large base.

= 4 + 3 + 7

= 14 m

Now,

Area of a trapezoid.

= 1/2 x (Short base + Large base) x h

= 1/2 x (3 + 14) x 5

= 1/2 x 17 x 5

= 42.5 m²

Thus,

Trapezoid area is 42.5 m².

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On Monday, for completing the daily target Philip needs to travel 9/8 more miles.

As per the question,

As Philip wants to run a total of 3 miles each day.

As on Monday morning, he ran only 1(7/8) miles.

First, we will convert the given mixed fraction in form of improper fractions.

So, converting 1(7/8),

It will become equal to (8*1+7)/8 = 15/8

From above,

We can say, out of the 3 miles he only travel 15/8 miles on Monday.

So, for calculating the required distance we need to subtract the given value 15/8 from 3

Thus we will get it as:

3-15/8 = (24-15)/8 = 9/8 miles.

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The length of the line segment AC and AG will be 6√2 units and 6√3 units, respectively.

The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.

The Pythagoras theorem formula is given as

H² = P² + B²

AB = 6

BC = 6

CG = 6

Then the length of diagonal AC is given as,

AC² = AB² + BC²

AC² = 6² + 6²

AC = 6√2

Then the length of diagonal AG is given as,

AG² = AC² + CG²

AG² = (6√2)² + 6²

AG = 6√3

The length of the line segment AC and AG will be 6√2 units and 6√3 units, respectively.

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Click It or Ticket is a nationwide campaign to enforce seat belt laws and increase seat belt use, supported by major organizations.

Click It or Ticket is a nationwide mobilization campaign to enforce seat belt laws and increase seat belt use. It is a joint effort of state and local law enforcement, highway safety offices, and the National Highway Traffic Safety Administration and is supported by national and state-level organizations, including:

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As per the unitary method, Gordon is correct because the Gordon would charge more than Ironing Palace

In math the term unitary method is defined as the value of a single unit and then find the value of more or fewer units by multiplying their quantity with the value of a single unit.

Here we have given that the total weight of these shirts is 9. 5 kg.

And here we have also given that each shirt weighs 190 g.

And Gordon charges £38 to iron these shirts.

And the charges in the Ironing Palace is 90p for each.

Here we have to use the unitary method to calculate the values for each of them,

The Total cost for Gordon is calculated as,

=> 38 x (9.5 / 0.19)

=> 1900

And the total cost for Ironing Palace is calculated as,

=> 90 x 9.5

=> 855

Therefore, while we looking into the values we have identified that Gordon is correct.

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Answer:

8 square units

Step-by-step explanation:

For a triangle,

area = base × height / 2

We need to identify a base and a height for this triangle, and then use them in the area formula above.

Use BC as the base.

A height of a triangle is usually inside the triangle, but not always.

The height of a triangle is a segment that is perpendicular to the a side, or the line containing a side, and intersects the opposite vertex.

In this case, for base BC, the height is a segment from point (-2, 2) to point A.  The length of the height is 4.

area = 4 × 4 / 2

area = 8 square units

Answer:

Step-by-step explanation:

The next time that they have an appointment start at the same time is at  10:50 a.m.

Now, According to the question:

Ist Step

For the same time, take LCM of 20 and 35.

20 = 2 × 2 × 5

35 = 5 × 7

IInd Step

HCF (20, 35) = 2 × 2 × 5 × 7

HCF (20, 35) = 140

Hence, it will take the appointment after 140 minutes which equals to 2 hours 20 minute.

Now, from 8.30 am, the addition of the 2 hours 20 minute gives us 10:50 a.m. which is the next time that they have an appointment start at the same time.

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After simplification, the value of x is 6.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

The expression is,

⇒ 3x + 30 + x = 10 + 2x + 5x + 2

Now, We can solve as;

⇒ 3x + 30 + x = 10 + 2x + 5x + 2

Combine like terms;

⇒ 4x + 30 = 12 + 7x

Subtract 4x both side,

⇒ 4x + 30 - 4x = 12 + 7x - 4x

⇒ 30 = 12 + 3x

Subtract 12 both side,

⇒ 30 - 12 = 3x

⇒ 18 = 3x

⇒ 3x = 18

Divide by 3;

⇒ x = 6

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The two-way frequency table to display the data will be:

                         Fishing         Not Fishing Total

Swimming        15                     30           45

Not Swimming 5                 45           50

Total                  20                 75         95

20% of 120 people is 24 people, but since 20% of the people went fishing, we know that 24 people went fishing.

75% of 24 people who went fishing is 18 people.

50% of 96 people who did not go fishing is 48 people.

Total number of people who went swimming is 45 (15+30)

Total number of people who did not go swimming is 50 (5+45)

Total number of people in the survey is 95 (24+71)

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In a survey of 120 people who visited the a lake, 20% went fishing.of those who went fishing,75% also went swimming. Of those who did not go fishing ,50% went swimming. Construct a two-way frequency table to display the data.

Valerie's table is the table for which the y-value is given by the multiplication of the x-value by -6.

A proportional relationship is modeled as follows:

y = kx.

In which k is the constant of proportionality.

The constant for the relationship in this problem is given as follows:

k = -18/3

k = -6.

Hence the relationship is given as follows:

y = -6x.

The problem is incomplete, hence the answer was given in general terms.

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According to Bernoulli's theorem, the success rate for the situation is 0.7173.

In math Bernoulli's theorem is states that that random experiment with exactly two possible outcomes, "success" and "failure" in which the probability of success is the same every time the experiment is conducted.

In order to calculate p we must sum the possible permutations of getting 0 success of 10 trials, 1 success of 10 trials, and 2 successes of 10 trials, and divide this sum by the total possible subsets from 10 elements, then it can be written as,

=> P = (10 1) (10 2) (10 1) / 2¹⁰

When we simplify this one, then we get,

=> P = 56/1024

=> P = 0.0547

When we apply the value on Bernoulli's theorem, then we get,

=> B(340, 0.0547, 0) = ( 340 0 ) x (1/1024) x (1023/1024)³⁴⁰

When we simplify this one then we get,

=> B = 0.7173

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The number of bicycles there is given as follows:

63.

The number of bicycles and tricycles is obtained using a system of equations, for which the variables are given as follows:

The number of bicycles is 1 less than 4 times the number of tricycles, hence:

x = 4y - 1.

There is a total of 174 wheels, hence:

2x + 3y = 174.

The number of tricycles is obtained as follows:

2(4y - 1) + 3y = 174

11y = 176

y = 176/11

y = 16.

Hence the number of bicycles is of:

x = 4(16) - 1

x = 63.

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The inflation rate is 16.71 %

As we know the inflation rate is given by the percentage error.

We know that the formula for the percentage error:

Percentage Error = [ ( | Measured Value - True Value | ) / True Value ] x 100

Let us assume that A represents the inflation rate of percentage error.

Here, the cost of the Jolly Peanut butter is = $ 3.59

So , the original cost = $ 3.59

The increased cost of Jolly Peanut butter is = $ 4.19

So , the new rate = $ 4.19

Now , the inflation rate is given by the formula of percentage error,

Percentage Error = [ ( | Measured Value - True Value | ) / True Value ] x 100

A = [ ( 4.19 - 3.59 ) / 3.59 ] x 100

A =  ( 0.6 / 3.59 ) x 100

A = 0.16713 x 100

A = 16.71 %

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The dimension of soccer field is 4252.176 in long and 2204.832 in wide.

The power to which the fundamental quantities are elevated in order to express a physical quantity is known as its dimension.

The dimension of soccer field is 108 meters long and 56 meters wide.

The dimension of soccer field in ft is 354.348 ft long and 183.736 ft in width.

As it is given that 1 m =3.28 ft

Therefore the length of the soccer field in ft is

108*3.281 = 354.348 ft

And width of the field is

56*3.281 = 183.736 ft

We know that 1 ft =12 in

Therefore, the length of the soccer field in inches is

354.348 × 12 = 4252.176 in

And width of the field is

183.736 *12 = 2204.832 in

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The percent discount is 18%.

Now, According to the question:

Based on the given condition:

Formulate:

(34 - 27.88) ÷ 34

Calculate the sum or difference:

= 6.12/34

Convert decimal into fraction:

[tex]=\frac{612}{\frac{100}{34} }[/tex]

Divide a fraction by multiplying its reciprocal:

[tex]\frac{612}{100}[/tex] × [tex]\frac{1}{34}[/tex]

Write as a single fraction:

= 612/ (100 × 34)

Cross out the common factor:

= 18/100

= 9/50

Multiply a number to both numerator and denominator:

[tex]\frac{9}{50}[/tex] × [tex]\frac{2}{2}[/tex]

= 18/100

Rewrite a fraction with denominator equals 100 to a percentage: 18%

Hence,  the percent discount is 18%.

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The amount of type A used is A=31 lbs.

Assign variables for unknowns.

Let A be the number of pounds of Type A coffee.

Let B be the number of pounds of Type B coffee.

If one pound of Type A costs $4.10 then A pound costs 4.10A

If one pound of Type B costs $5.45 then B pounds cost 5.45B

At this point, you have two equations with A and B as variables.  You can use the substitution or elimination method to solve for A.

Equations:

B = 2A----(1)

4.10A + 5.45B = 465.00

4.10A + 5.45(2A) = 465.0

4.10A + 10.9A = 465.00

15A = 465.00

A = 31 lbs (amt. of type A coffee used)

To know the amount of type B coffee used just plug the A value in equation(1) we get:

B=2(31)

B=62.

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Here we would like to find out the value of given trigonometric ratio . A right angled triangle is given to us with perpendicular= 8 and hypotenuse= 11 . Firstly let's calculate the base using Pythagoras theorem as ,

[tex]\longrightarrow h^2=p^2+b^2\\[/tex]

[tex]\longrightarrow 11^2 = 8^2+b^2\\ [/tex]

[tex]\longrightarrow 121 = 64 + b^2\\ [/tex]

[tex]\longrightarrow b^2=121-64=57\\ [/tex]

[tex]\longrightarrow b =\sqrt{57}=7.54 \\ [/tex]

Now we know that,

[tex]\longrightarrow \tan\theta =\dfrac{p}{b}\\ [/tex]

so substitute the respective values,

[tex]\longrightarrow \tan\theta =\dfrac{8}{7.54}\\ [/tex]

[tex]\longrightarrow\underline{\underline{ \tan\theta = 1.06 \approx \boxed{1}}} [/tex]

and we are done!

Answer: The distance between the points (-4,7) and (-4,-5) is 12.

Step-by-step explanation:

To calculate the distance between two points, we apply the following formula:

[tex]\huge \boxed{\sf{d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }}[/tex]

The points are:

To solve the process is:

We substitute the points in the previous formula given, then

[tex]\huge \boxed{\sf{d=\sqrt{(-4-(-4))^2+(-5-7)^2 } }}[/tex]

[tex]\huge \boxed{\sf{d=\sqrt{0^2+(-12)^2 }=\sqrt{0+144} }}[/tex]

[tex]\huge \boxed{\sf{d=\sqrt{144}=12 }}[/tex]

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a. The party that won the highest percentage of votes and seats in 1982 is democrat.

b. A change between the percentage of seats won by republicans in 1982 and 2012 is 4%

c. The difference between votes percentage and seats percentage for democrat is 4% and 26% respectively. The difference between votes percentage and seats percentage for republican is 3% and 26% respectively.

The pie chart is given below:

a. We have to identify the party that won the highest percentage of votes and seats in 1982.

We can see that in diagram that gray part of circle shows the vote percentage of democrat and white part of circle shows the vote percentage of republican.

Republican gain votes in 1982 = 52%

Democrat gain votes in 1982 = 49%

So the democrats get the maximum votes in 1982.

b. We have to describe a change between the percentage of seats won by republicans in 1982 and 2012.

In 2012 republican get votes percentage = 48%

In 1982 republican get votes percentage = 52%

So the change between 1982 and 2012 = (52 - 48)% = 4%

c. We have to draw a conclusion about the difference in percentage of votes won and seats.

In 2012 republican get votes percentage = 48%

In 1982 republican get votes percentage = 52%

So, the difference between vote percentage is 4%.

In 2012 republican get seats percentage = 25%

In 1982 republican get seats percentage = 51%

So, the difference between seats percentage is 26%.

In 1982 democrat get votes percentage = 49%

In 2012 democrat get votes percentage = 52%

So, the difference between vote percentage is 3%.

In 2012 democrat get seats percentage = 75%

In 1982 democrat get seats percentage = 49%

So, the difference between seats percentage is 26%.

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