6.4: Spot the Differences
Let /(x) = (x-2)(x+3)(x-7) and g(x)=(x-2)(x+3)(x-7).
Mathematics
1. Use graphing technology to graph both functions in the same window of
-10 ≤x≤ 10 and -100 ≤ y ≤ 100. Describe how the two graphs are the same and
how they are different.

The equation of this circle in standard form is x² + y² - 8x  - 16y + 73.75 = 0

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

Endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5)

These endpoints represent endpoints of the diameter

So, we have the diameter to be

diameter = √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute the known values in the above equation, so, we have the following representation

diameter = √[(4 - 4)² + (5.5 - 10.5)²]

Evaluate

diameter = 5

So, we have

radius, r = 5/2

The midpoint is

(a, b)= 1/2[(4 + 4, 5.5 + 10.5)]

So, we have

(a, b)= (4, 8)

The circle equation is calculated as

(x - a)² +(y - b)² = r²

Substitute the known values in the above equation, so, we have the following representation

(x - 4)² +(y - 8)² = 2.5²

This gives

x² - 8x + 16 + y² - 16y + 64 = 6.25

Evaluate the like terms

x² + y² - 8x  - 16y + 73.75 = 0

Hence, the equation is x² + y² - 8x  - 16y + 73.75 = 0

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The measure of side a is  4.1 unit.

Let there is triangle ABC such that |AB| = a units, |AC| = b units, and |BC| = c units and the internal angle A is of θ degrees, then we have:

c² = a² + b² - 2ab cos C

(c is opposite side to angle A)

In the given triangle ABC, we have sides BC = a, AB = b and AC = c

Now we know as per cosine rule for all three sides of the triangle.

c² = a² + b² - 2ab cosC

Where;  b = 9 and c = 7 degrees

Angle A = 35 degrees

Plugging these values into the equation, we have;

a² = c² + b² - 2cb cos A

a² = 9² + 7² - 2(9)(7) cos (35)

a² = 81 + 49 - 126 x (-0.9036)

a = 4.1

Therefore, the side a is 4.1 unit.

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The range of the given function in the figure is R - {0, 1, 2, 3}.

The given mapping shows a function f(x).

There are 4 inputs to the function and 5 outputs.

So, the domain set for the given function is written as D = {0, 5, 10, 15}

And the range set for the given function is written as R = {0, 1, 2, 3}

(Since the 5th output is repeating, in the range it is considered only once).

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When selecting the ply of the mat which we will use for an image, what we choose is the thickness of the mat. The correct answer is option A.

The width of our mat really influences how our art is showed. A perfect rule of thumb is to select a mat which is at least 1 inch wider than our moulding. Many mats are 2-4 inches wide; usually, larger artwork can use a wider mat, and smaller artwork should be paired with a narrower mat.

Mats available in 4 ply (0.040 to 0.052″) and 8 ply (0.125″); but it does not mean all ply are the same. For instance, there are 0.052″ 4 ply and 0.040″ 4 ply. Double or triple 4 ply mats are possible also to be used in area of 8 ply mats to produce a similar effect.

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Answer:

29=n/3

Step-by-step explanation:

Answer:

29 = n/3

Step-by-step explanation:

[tex]29 = \frac{1}{3} of \: n \\ 29 = \frac{1}{3} \times n \\ 29 = \frac{n}{3} [/tex]

The last remaining non - congruent side measure = 48°

Non congruent means "not congruent," i.e., not the same shape, and the sides. Congruent shapes are those that are copies of one another that have been rotated, reflected, and translated.

given

the sum of 6 exterior angles = 310°

remaining angle = ([tex]6x+2[/tex])

All sides of exterior angles sum= 360°

so , remaining angle + sum of all exterior angles=360°

[tex](6x+2)[/tex]  +  310°  =  360°

[tex](6x+2)[/tex]  =  360° - 310C

 [tex](6x+2)[/tex]  =  50°

[tex]x[/tex]  =  48°

The last remaining non - congruent side measure = 48°

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Answer:

y= 2x +4 is the equation

The fit statistic that can be used across different series that are in fact measured in different units is; MAPE

There are different statistics models in this regards such as;

Mean Absolute Error (MAE)

Mean Absolute Scaled Error (MASE)

Accuracy Percent (Accuracy %)

Mean Squared Error (MSE)

Root Mean Squared Error (RMSE)

Mean Absolute Percent Error (MAPE)

Now, we are told that we need the fit statistic that can be used across different series that are in fact measured in different units.

The correct one will be Mean Absolute Percent Error (MAPE) because the average absolute percent difference between the values that are fitted by the model and the observed data values.

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Answer:

3ab-12a/5-31b/6+4x

Step-by-step explanation:

The interest earned is $50

Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments. ​Simple Interest=P×r×t

               where:

                     P=Principal

                    r= interest rate per annum

                    t=Number of years

​

Given here, P = $1000 , r = 5%  t=1

thus using the formula we get    SI = 1000×5/100×1

                                                          = $50

Hence, the interest earned is equal to $50

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Option 2  and option 4 are the respective answers.

The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

Given here -3x²+6x+2 = 0

therefore the discriminant is (6)² - ( -3×4×2) = 60

Thus the parabola formed by the curve would intersect the x-axis twice

Now for the second part of the answer, the equation is given as

3(2x-1) (x-4) = 3( 2x² - 9x + 4)

=6x² - 27x + 12

Applying the direct formula for root we get                                                 x =  27 + √(27² − 4×6×12)/12

Hence, option 2 is the correct answer for part 1, and option 4 is the correct answer for the latter question.

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To determine Mary's required speed, we need to first understand the relationship between speed and time.

If Mary's speed is inversely related to her drive home, then this means that as her speed increases, her drive time decreases, and vice versa. In other words, the drive time is inversely proportional to the speed. We can represent this relationship using the equation t = k/s, where t is the drive time, s is the speed, and k is a constant. If we plug in the given values, we get t = 40 minutes = k/55 mph.

Now, if Mary wants to arrive home 15 minutes earlier, we can set the desired drive time equal to 25 minutes (40 minutes - 15 minutes) and solve for the required speed. Substituting this value into the equation above, we get s = k/(25 minutes) = (55 mph)/(25 minutes) = 2.2 mph.

Therefore, Mary needs to drive at a speed of 2.2 mph in order to arrive home 15 minutes earlier. Note that this answer is not realistic, as it is much lower than the speed limit on most roads. This suggests that it may not be possible for Mary to arrive home 15 minutes earlier without breaking the law or increasing her risk of an accident.

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Complete Question:

Mary's speed is inversely related to her drive home. Mary drives home from work in 40 minutes driving 55mph. If Mary needs to get home 15 minutes earlier, how fast dose she need to drive?

Answer:

[tex]1.25x + .50y \geqslant 150[/tex]

(a) The median of variable x is 70.

(b) The third quarterly of variable y is 94.

(c) Variable y has more dispersion.

(d) Shape of variable x is Symmetric.

(e) The shape of y variable is Skewed left.

a) The median of the variable x, which sits in the middle of the provided box plot, is 70 (roughly).

b) Variable Y's third quarterly value is 94. (approximately).

c) Because Q1 is farther away from the median than Q2, variable y has a higher inter quarterly range than variable x.

d) The shape of the variable x is symmetrical because both the left and right whiskers are roughly the same length and the median is located in the center of the box.

e) The left whisker of the y variable is longer than the right whisker, and the median is located outside the box's center.

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The complete question is given below:

The correct answer is [B] You are increasing your liability.

When you buy a pair of shoes for $85, your cash balance decreases by $85, which means that your wealth also decreases by $85. This is represented by [A] and [C].

[D] You are incurring an expense is also true, because you are spending money on the shoes.

[E] You are acquiring (non-cash) asset is also true, because you are obtaining an asset (the shoes) that you can use or sell in the future.

[B] You are increasing your liability is not true, because a liability is a debt or an obligation that you owe to someone else. In this case, you are not taking on any debt or obligation by buying the shoes, so your liability does not increase.

Answer: 0.05

Step-by-step explanation:

The equation is k = y/x, which is a constant change rate

Take 5 divided by 100 = 0.05

5% = 0.05

So the constant of proportionality is 0.05

Answer:

A and D

Step-by-step explanation:

> means larger than. 11.436 is larger than 11.422 and 11.286, but not 11.484 or 11.621.

When the spring is stretched and the distance from point a to point b is 5.3 feet, the value of θ is 53.13  degrees

The distance between point a to point b = 5.3 feet

The length of the top side = 3 feet

Therefore, it will form a right triangle

Here we have to use trigonometric function

Here adjacent side and the hypotenuse of the triangle is given

The trigonometric function that suitable for the given conditions is

cos θ = Adjacent side / Hypotenuse

Substitute the values in the equation

cos θ = 3 / 5

θ = cos^-1(3 / 5)

θ = cos^-1(0.6)

θ = 53.13 degrees

Therefore, the value of θ is 53.13  degrees

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Probability of the person has the diseases and the test result is positive = 1/90 = 0.011

Probability  refers to the possibility of occurring a certain event. It is expressed by the ratio of the number of favorable outcome to the total possible outcome.

Given, percentage of people who have disease and test result positive =90

percentage of people who has not disease but test result positive =3.1

Suppose 6.0 percent of population are infected

Now, probability of a  randomly selected person  who has diseases and test positive

                          = 1/90

                       =   0.011

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Term matrix is not a type of matrix among the given options.

A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined.

There are several types of matrices, including:

1. Square Matrix: A square matrix is a matrix in which the number of rows is equal to the number of columns.

2. Scalar Matrix: A scalar matrix is a type of matrix in which all the elements are equal to a single number (the scalar).

3. Trace Matrix: A trace matrix is a type of matrix in which the trace (sum of the diagonal elements) is a scalar multiple of the identity matrix.

4. Identity Matrix: An identity matrix is a type of matrix in which all the elements on the main diagonal are 1s and all other elements are 0s.

5. Symmetric Matrix: A symmetric matrix is a type of matrix in which the elements above and below the main diagonal are equal.

6. Diagonal Matrix: A diagonal matrix is a type of matrix in which all the elements outside the main diagonal are 0s.

7. Triangular Matrix: A triangular matrix is a type of matrix in which all the elements above or below a specific diagonal are 0s.

8. Orthogonal Matrix: An orthogonal matrix is a type of matrix in which the product of two matrices is the identity matrix.

9. Singular Matrix: A singular matrix is a type of matrix in which the determinant is 0.

10. Inverse Matrix: An inverse matrix is a type of matrix in which the product of the matrix and its inverse is the identity matrix.

Therefore, the answer is D. Term Matrix, as it is not a type of matrix.

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Option D is Answer. Compared with the given matrix, The Term matrix is not a type of matrix.

Term Matrix: It defines a document-term matrix as simply a matrix describing the frequencies of all terms occurring in the collection of text documents.

square matrix: A matrix which is having the same number of rows and columns. It is also called an n*n matrix.

Example:

                [tex]\left[\begin{array}{ccc}1&2\\4&5\\\end{array}\right][/tex]

Scalar matrix: It is a type of square matrix, whose diagonal elements are equal and off-diagonal elements are zero. It is basically a multiple of the square identity matrix.

Example:

                [tex]3I=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] =\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right][/tex]

Where I define the Identity matrix.

In a square matrix, the sum of elements of the principal diagonal is called the ‘trace of a matrix’. We denote the trace of the matrix A by ‘tr A’.

Example: If A =

                    [tex]\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]

, then tr A = 1 + 5 + 9 = 15.

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On solving the provided question by slope intercept form, equation formed id=s 3x + 4y - 14= 0 .

One of the most popular ways to express a line's equation is in the slope intercept form of a straight line. Given the slope of the straight line and the y-intercept, the slope intercept formula may be used to get the equation of a line ( the y-coordinate of the point where the line intersects the y-axis). A line's equation is the equation that each point on the line fulfills. There are several ways to solve the equation for a straight line given as, - Slope-intercept form, Point slope form Two-point form and Intercept the form

slope = -3/4

points ( -2,5 )

equation,

(y - y1) = m(x - x1)

( y - 5) = -3/4(x+2)

4y - 20 = -3x - -6

3x + 4y - 14= 0

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Answer:

The slope of a line is a measure of its steepness or incline. It is calculated by dividing the difference in the y-coordinates of two points on the line by the difference in their x-coordinates. In the case of the line (5,-9) and (5,2), the difference in the y-coordinates is -9 - 2 = -11 and the difference in the x-coordinates is 5 - 5 = 0. Since division by 0 is undefined, the slope of the line (5,-9) and (5,2) is undefined.

In general, when the x-coordinates of two points on a line are the same, the line is vertical and has no slope. This means that the line (5,-9) and (5,2) is a vertical line and has no slope.

The slope of a line is calculated as the rise over the run, which is the difference in the y-coordinates divided by the difference in the x-coordinates. In this case, the line has the same x-coordinate for both points, so the rise and run are both zero, and the slope is undefined.

It is approximately equal to 83.3%

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”

Given here, 5/6 thus in percentage terms it will be = 5/6 × 100

                                                                                    = 83.3%

Hence the answer is  83.3%

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The probability of drawing a red marble from the jar is 5/11 and the probability of drawing a blue marble from the jar is 6/11.

There are 22 marbles in the jar in total, 10 of which are red and 12 of which are blue. To find the probability of drawing a red marble, we can use the formula for probability:

Probability = Number of favorable outcomes / Total number of outcomes

In this case, the number of favorable outcomes is 10 (the number of red marbles), and the total number of outcomes is 22 (the total number of marbles in the jar). Plugging these values into the formula, we get:

Probability of drawing a red marble = 10/22

This fraction can be reduced to 5/11 by dividing both the numerator and denominator by 2. So, the probability of drawing a red marble from the jar is 5/11.

To find the probability of drawing a blue marble, we can use the same formula:

Probability = Number of favorable outcomes / Total number of outcomes

In this case, the number of favorable outcomes is 12 (the number of blue marbles), and the total number of outcomes is 22 (the total number of marbles in the jar). Plugging these values into the formula, we get:

Probability of drawing a blue marble = 12/22

This fraction can also be reduced to 6/11 by dividing both the numerator and denominator by 2. So, the probability of drawing a blue marble from the jar is 6/11.

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For the sample to be 99% confident and sample proportion to not differ more than 5% the sample size is 664  .

The term Sample Size is defined as number of observations that is used for determining estimations of a given population .

it is given that , sample proportion should not differ by 5% ,

that means , the Margin of Error (ME) = 0.05 .

Since , the prior information is unknown

we assume the sample proportion(p) = 0.05  .

and level of significance (α) ⇒ 0.01 ;

By using the z table ,critical value for 99% confidence interval is = 2.576

The sample size(n) can be calculated by the formula ,

n = p(1 - p)[(critical value)/ME]²  ;

substituting the values of p , critical value and ME in the above equation ,

we get ;

n = 0.05×(1 - 0.05)×[2.576/0.05]²

n = 0.25 × 2654.3104

n = 663.5776

n ≈ 664

Therefore , the required sample size is 664 .

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Answer:

Step-by-step explanation:

Answer: 25 units

Step-by-step explanation:

Answer:

t1/2=1m => 30days

A0=500g =0.5kg

t=8m = 240days

A=?

λ=ln2÷t1/2

λ=0.6931÷30

λ=0.0231033333

A=A0×e^-λ×t

A=0.5×e^-0.0231033333×240

A=1.953125grams

The correct option D. f(x) = 5(5)ˣ and E. f(x) = 5(1/5)⁻ˣ, for the reflected function over the y-axis.

We are aware of:

f(x) = 5(1/5)ˣ

Positive x-values and negative x-values are switched by a reflection over the y-axis, which also swaps the values of x and x.

Therefore:

f(ax) = f(x)

The following will be the value of the supplied function's y-axis reflection:

(Remember to swap out positive and negative x-values.)

f(x) = 5(1/5)⁻ˣ

A comparable form will be:

f(x) = 5(1/5)⁻ˣ

f(x) = 5(1/5)⁽⁻¹⁾⁽ˣ⁾

f(x) = 5[(1/5)⁻¹]ˣ

f(x) = 5(5)ˣ

Therefore

The following equations depict the reflected function:

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The complete question is-

The function f(x) = 5(1/5)ˣ h is reflected over the y-axis. Which equations represent the reflected function? Select two options.

A. f(x) = One-fifth

B. f(x) = One-fifth.One-fifth.One-fifth

C. f(x) = 5One-fifth

D. f(x) = 5(5)ˣ

E. f(x) = 5(1/5)⁻ˣ