A retailer sells model solar systems for $72 that were acquired at a cost of $24. What is the mark-up percentage?

The inequalities shown by the graph are y < 3, x > -1, x < 3, and y >-2.

The given graph has four lines showing different inequalities.

One undotted line is parallel to the x-axis and is at y = 3. The shaded region is below y = 3

Hence, the inequality will be y < 3

Another dotted line is parallel to the x-axis and is at y = -3. The shaded region is above y = -2

Hence, the inequality will be y > -2

Another dotted line is parallel to the y-axis and is at x = -1. The shaded region is to the right of x = -1

Hence, the inequality will be x > -1

Another dotted line is parallel to the y-axis and is at x = 3 The shaded region is to the left of x = 3

Hence, the inequality will be x < 3

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The value of expression (9.5×10-6)÷(5×104) is 0.1712

Consider a mathematical expression (9.5×10-6)÷(5×104)

We use PEDMAS rule to solve this expression.

We know that the PEMDAS rule gives the order of mathematical operations.

PEMDAS means we solve an expression in following order : parentheses, exponents, multiplication, division, addition, and subtraction.

Here (9.5×10-6)÷(5×104) we have two parentheses

Consider the first parentheses

(9.5×10 - 6)

= 95 - 6

= 89

Consider the second parentheses

(5×104) = 520

Now we solve (9.5×10 - 6)÷(5×104) = 89 ÷ 520

                                                       = 0.1712

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Solving a quadratic equation we can see that the 3 consecutive numbers are:

4, 5, and 6.

First, we can write 3 consecutive integers as:

x, (x + 1), and (x +2 )

The square of the first increased by the last is 22, then we can write:

x^2 + (x + 2) = 22

x^2 + x + 2 - 22 = 0

x^2 +x  - 20 = 0

Using the quadratic formula we will get the solutions:

[tex]x = \frac{-1 \pm \sqrt{1^2 - 4*1*-20} }{2} \\\\x = \frac{-1 \pm 9}{2}[/tex]

We know that x is positive, then the solution is:

x = (-1 + 9)/2 = 4

The 3 consecutive numbers are:

4, 5, and 6.

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b.4+3cos²x is the simplification of the given trigonometric identity

Explanation:
Using the identity sin²x + cos²x = 1, we can simplify the expression:

4-4sin²x/(cot²x) + 7cos²x
= 4 (1-sin²x)/(cot²x) + 7cos²x
= 4 cos²x/cot²x + 7cos²x
= 4sin²x + 7cos²x

= 4sin²x + 4cos²x + 3cos²x

=4+3cos²x

Hence, the answer is b. 4 + 3cos²x

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A) The probability of getting 3 or fewer students who play a sport in your sample is 0.014, or 1.4%.

B) The probability of getting 3 or fewer students who play a sport in your sample is much lower than the expected probability of 0.4 (40%). This suggests that your estimate of 40% of students in the whole school playing a sport may be too high.

C) There are 1140 different combinations of 3 students who play sports and 17 students who don't in your sample.

A) To find the probability of getting 3 or fewer students who play a sport in your sample, you can use the binomial probability formula:

P(X = x) = nCx * p^x * (1-p)^(n-x)

where n is the sample size, x is the number of successes, p is the probability of success, and nCx is the number of combinations of x successes in n trials.

For x = 0, 1, 2, and 3, the probabilities are:

P(X = 0) = 20C0 * 0.4^0 * 0.6^20 = 0.000006
P(X = 1) = 20C1 * 0.4^1 * 0.6^19 = 0.00016
P(X = 2) = 20C2 * 0.4^2 * 0.6^18 = 0.0019
P(X = 3) = 20C3 * 0.4^3 * 0.6^17 = 0.012

Adding these probabilities gives:

P(X <= 3) = 0.000006 + 0.00016 + 0.0019 + 0.012 = 0.014



C) The number of different combinations of 3 successes and 17 failures is given by:

20C3 = 20! / (3! * 17!) = 1140


D) The formula for the number of combinations of x successes in n trials is:

nCx = n! / (x! * (n-x)!)

For 20C2 and 20C17, the formulas are:

20C2 = 20! / (2! * 18!)
20C17 = 20! / (17! * 3!)

Since 2! * 18! = 17! * 3!, these two formulas are equivalent and give the same result. This is why 20C2 = 20C17.

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We have the solve using the quadratic equation as;

Step 2:

a = 1

b = -5

c = -14

Step 3:

What 1: (-5)²

What 2: 1

What 3: -14

What 4: 1

The quadratic equation is expressed as;

ax² + bx + c = 0

x = -b ± [tex]\sqrt{b^2 - 4ac}[/tex]/2a

From the information given, we have that;

x² -5x - 14 = 0

Now, substitute the values, we get;

x = - (-5) ± [tex]\sqrt{(-5)^2 - 4 * 1 * -14}[/tex]/ 2(1)

find the square and substitute the values, we have;

x = 5 ± [tex]\sqrt{25 + 56}[/tex]/2

Add the values, we have;

x = 5 ± √81/2

Find the square root

x = 5 ±9/2

x = 5 ± 4. 5

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the length of the actual kitchen wall is 144 inches.

The ratio used to depict the relationship between the dimensions of a model or scaled figure and the corresponding dimensions of the real figure or object is called the scale. On the other hand, a scale factor is a value that is used to multiply all of an object's parts in order to produce an expanded or decreased figure.

Given, On a scale drawing, a kitchen wall is 6 inches long. The scale factor is 1/24

If the scale factor is 1/24, it means that every 1 inch on the drawing represents 24 inches in real life.

Let's set up a proportion:

1 inch on the drawing : 24 inches in real life = 6 inches on the drawing : x inches in real life

where x is the length of the actual wall.

To solve for x, we can cross-multiply:

1 inch on the drawing * x inches in real life = 6 inches on the drawing * 24 inches in real life

x = 6 inches on the drawing * 24 inches in real life / 1 inch on the drawing

x = 144 inches in real life

Therefore, the length of the actual kitchen wall is 144 inches.

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Answer:

6x-4y=18

-x-6y=7

-x-6y=7

-x=7+6y

x=-7-6y

6x-4y=18

6(x-7-6y)-4y=18

-42-40y=18

-40y=18+42

-40y=60

y=-60/40

y=-3/2

x=-7-6y

x=-7-6(-3/2)

x=-7+9

x=2

(x,y) = (2,-3/2)

a. The point estimate for the birth weights is 2714.58, b. the value of tq is 1.70814, and c. the margin of error for the confidence interval is 124.3. d. The confidence interval for birth weights is (2503, 2927). e. No conclusions can be drawn since the confidence interval contains 2500.

a. The point estimate for the birth weights is 2714.58. This is the average of the sample data and is used as an estimate for the population mean.

b. The value of tq is 1.70814. This is the t-value associated with the given confidence level and degrees of freedom.

c. The margin of error for the confidence interval is 124.3. This is calculated using the formula ME = tq * (s/√n), where tq is the t-value, s is the sample standard deviation, and n is the sample size.

d. The confidence interval for birth weights is (2503, 2927). This is calculated by taking the point estimate and subtracting and adding the margin of error to get the lower and upper bounds of the interval.

e. No conclusions can be drawn since the confidence interval contains 2500. This means that the true population mean could be less than 2500, greater than 2500, or equal to 2500.

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a)0.1359, or 13.59%

b)85,100,115

c)0.0062, or 0.62%.

d)0.9705, or 97.05%

a. The probability of a randomly selected person's IQ being over 120 is 0.1359, or 13.59%.

b. Q1 for IQ is 85, Q2 is 100, and Q3 is 115.

c. The probability of an outlier for IQ for a single person is 0.0062, or 0.62%.

d. The probability of the average IQ of 10 randomly selected people being over 105 is 0.9705, or 97.05%.

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The probability is 4/8 = 1/2. The probability that the student answered True at least two times is 1/2.

The number of residents in a city is a discrete variable because it is a countable number of people. The level of measurement is a ratio level because there is a true zero point (no residents in a city) and the difference between values is meaningful. Therefore, the correct answer is discrete, ratio level.

The sample space for the three True (T) or False (F) questions is: {TTT, TTF, TFT, FTT, FTF, FFT, TFF, FFF}

The outcomes corresponding to the event "The student answered True at least two times" are: {TTT, TTF, TFT, FTT}

To evaluate the probability of the event "The student answered True at least two times", we need to divide the number of favorable outcomes by the total number of possible outcomes. The number of favorable outcomes is 4 (TTT, TTF, TFT, FTT) and the total number of possible outcomes is 8 (TTT, TTF, TFT, FTT, FTF, FFT, TFF, FFF). Therefore, the probability is 4/8 = 1/2. The probability that the student answered True at least two times is 1/2.

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Answer: To determine the effective rate of Zoe's loan, we can use the formula:

Effective rate = (1 + (nominal rate/number of compounding periods))^number of compounding periods - 1

Since the nominal rate is compounded monthly, the number of compounding periods is 12.

Plugging in the values, we get:

Effective rate = (1 + (0.039/12))^12 - 1

Effective rate = 0.0407 or 4.07%

Therefore, the effective rate of Zoe's loan is 4.07%, rounded to the nearest hundredth. The correct response is 3.97%.

Step-by-step explanation:

The time needed for the number of people infected with the flu to reach 3072 is given as follows:

4 days.

The exponential function that models the situation has the definition given as follows:

y = ab^x.

In which the parameters are given as follows:

Assume a total of 12 people have the flu as of today, hence the parameter a is given as follows:

a = 12.

Each day the total number people who have the flu quadruples, hence the parameter b is given as follows:

b = 4.

Hence the function giving the number of people with the flu after x days is given as follows:

y = 12(4)^x.

The number of days needed for the number to reach 3072 is obtained as follows:

12(4)^x = 3072

4^x = 3072/12

4^x = 256.

2^(2x) = 2^(8)

Hence the value of x is obtained as follows:

2x = 8

x = 4.

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The value of the machine after 4 years can be calculated using the formula:
V = P * (1 - r) ^ n
Where:
V = value after n years
P = initial purchase price
r = annual depreciation rate
n = number of years

In this case, P = 3,000.00, r = 0.10 (10%), and n = 4.
Plugging these values into the formula, we get:
V = 3,000.00 * (1 - 0.10) ^ 4
V = 3,000.00 * 0.90 ^ 4
V = 3,000.00 * 0.6561
V = 1,968.30

Therefore, the value of the machine after 4 years is $1,968.30.

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In order to fill the entire shape, one must use 1 cm x 1 cm x 1 cm wooden blocks, with the total number of blocks equaling the volume of the shape in cm³. In this case, the number of blocks needed is 1 cm³, or one block.

Length is a measurement of distance or amount. It is commonly used to describe the size of an object or the distance between two points. Length can be measured in a variety of ways including inches, feet, yards, centimeters, miles, and kilometers. Length is also used to measure time, the extent of something, and the amount of material or substance.

To figure out how many 1 cm x 1 cm x 1 cm wooden blocks are required to fill the entire shape, one must first calculate the volume of the shape. The volume is calculated by multiplying the length, width, and height of the shape, which in this case is all 1 cm. Therefore, the volume of the shape is 1 cm x 1 cm x 1 cm = 1 cm³.

Next, one must calculate the total volume of all the blocks needed to fill the shape. Since each block is 1 cm x 1 cm x 1 cm, the total volume of the blocks is equal to the volume of the shape, which is 1 cm³.

Therefore, in order to fill the entire shape, one must use 1 cm x 1 cm x 1 cm wooden blocks, with the total number of blocks equaling the volume of the shape in cm³. In this case, the number of blocks needed is 1 cm³, or one block.

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In order to fill the entire shape, one must use 1 cm x 1 cm x 1 cm wooden blocks, with the total number of blocks equaling the volume of the shape in cm³. In this case, the number of blocks needed is 1 cm³, or one block.

Length is a measurement of distance or amount. It is commonly used to describe the size of an object or the distance between two points. Length can be measured in a variety of ways including inches, feet, yards, centimeters, miles, and kilometers. Length is also used to measure time, the extent of something, and the amount of material or substance.

To figure out how many 1 cm x 1 cm x 1 cm wooden blocks are required to fill the entire shape, one must first calculate the volume of the shape. The volume is calculated by multiplying the length, width, and height of the shape, which in this case is all 1 cm. Therefore, the volume of the shape is 1 cm x 1 cm x 1 cm = 1 cm³.

Next, one must calculate the total volume of all the blocks needed to fill the shape. Since each block is 1 cm x 1 cm x 1 cm, the total volume of the blocks is equal to the volume of the shape, which is 1 cm³.

Therefore, in order to fill the entire shape, one must use 1 cm x 1 cm x 1 cm wooden blocks, with the total number of blocks equaling the volume of the shape in cm³. In this case, the number of blocks needed is 1 cm³, or one block.

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Complete questions as follows-

A solid wooden block in the shape of a rectangular prism has a length, width, and height of centimeter, centimeter, and centimeter, respectively. The volume of the block is cubic centimeter. The number of cubic wooden blocks with a side length of centimeter that can be cut from the rectangular block is .

C) (i) How many 1 cm x 1 cm x 1 cm wooden blocks will he need to fill the entire shape outlined?​

let's firstly convert the mixed fractions to improper fractions and them sum them up.

[tex]\stackrel{mixed}{7\frac{5}{12}}\implies \cfrac{7\cdot 12+5}{12}\implies \stackrel{improper}{\cfrac{89}{12}}~\hfill \stackrel{mixed}{11\frac{2}{3}} \implies \cfrac{11\cdot 3+2}{3} \implies \stackrel{improper}{\cfrac{35}{3}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{89}{12}+\cfrac{35}{3}\implies \cfrac{(1)89~~ + ~~(4)35}{\underset{\textit{using this LCD}}{12}}\implies \cfrac{89+140}{12}\implies \cfrac{229}{12}\implies {\Large \begin{array}{llll} 19\frac{1}{12} \end{array}}[/tex]

To obtain 400 liters of 11% acid solution, you need to mix  390 liters of the 10% acid solution and 10 liters of the 50% acid solution to obtain 400 liters of 11% acid solution.

Let x be the number of liters of 10% acid solution and y be the number of liters of 50% acid solution.

We can set up a system of two equations to represent the given information:

x + y = 400 (total volume of the mixture is 400 liters)

0.10x + 0.50y = 0.11(400) (the amount of acid in the mixture is 11% of the total volume)

Simplifying the second equation:

0.10x + 0.50y = 44

We can now use either substitution or elimination method to solve for x and y.

Using substitution, we can solve for y in terms of x from the first equation:

y = 400 - x

Substituting this into the second equation:

0.10x + 0.50(400-x) = 44

Simplifying and solving for x:

0.10x + 200 - 0.50x = 44

-0.40x = -156

x = 390

So the chemist needs 390 liters of the 10% acid solution and 10 liters of the 50% acid solution to obtain 400 liters of 11% acid solution.

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The height of the container if it's a cylinder with a radius of 3 cm is 5 cm.

The cost of coffee is $2.83.

The height of the container if it's a cylinder with a radius of 5 cm is 1.8 cm.

The cost of hot chocolate powder is $9.90.

Mathematically, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

Picking a volume of 45π cm³, the height of this container is given by:

h = V/πr²

Height, h = 45π/π3²

Height, h = 5 cm.

For the cost of coffee, we have:

Cost of coffee = 45π cm³ × $0.02

Cost of coffee = $2.83.

When the radius is 5 cm, the height of this container is given by:

h = V/πr²

Height, h = 45π/π5²

Height, h = 1.8 cm.

For the cost of hot chocolate powder, we have:

Cost of hot chocolate powder = 45π cm³ × $0.07

Cost of hot chocolate powder = $9.90.

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Answer:

Last answer choice: 4

Step-by-step explanation:

The set of inputs to the graphed function is he set of all x values for which the function has a defined value

Looking at the answer choices we see that the graph does not go the negative x region nor does it have a value for 0

So x = 4 is an input, and the resultant output y from the graph = 0

Answer

Last answer choice: 4

Answer:

376, and 122

Step-by-step explanation:

To solve for the surface area, we add up the areas for each side together.

([8×6] × 2) + ([8×10] × 2) + ([10×6] × 2) = 376

376 is the surface area for the 1st shape.

(10 × 5) + (4×3) + ([3×10] ×2)= 122

122 is the surface area for the 2nd shape

Answer:

9)376ft^2 10)132ft^2

Step-by-step explanation:

the first one is pretty easy use the formula 2(l*w+l*b+w*b) where l is length b is breadth and w is the width, l=10 b=8 w=6

2(10*6+10*8+6*8)

=376ft^2

the second one is also not that hard, first find out the area of the triangles in which the base is 3 and height is 4 so 4*3/2

=6

since there are two triangles, the surface area of the triangles combined is 12

now lets move the base where the width is 3 and the length is 10 as it also corresponds, now l*w is the formula to find the surface area of a rectangle so 10*3 is 30 now lets find the surface area of the square on the front which is just 10*5 which equals to 50 and lastly the rectangle at the back, for which we know that the width is 4 and length is 10 so 10*4 is 40 now simply just add all ofn these areas, 12+50+40+30

=132ft^2

By answering the above question, we may infer that The answer is yes, equation and Kaylee now has 26 nickels. Option B, or 50, is the correct response, therefore.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

Hence, she has twice as many nickels and quarters as normal.

The combined worth of Kaylee's coins is $30. This may be expressed as an equation:

[tex]0.05(2x) + 0.10x + 0.25(4x) = 30\\0.10x + 0.10x + 1.00x = 30\\2.20x = 30\sx = 13.64[/tex]

Kaylee has 2x = 28 nickels and 4x = 56 quarters if x = 14. These coins are worth 0.05(28), 0.10(14), and 0.25(56), for a total of $7.70.

This does not equal $30, hence the suggested solution is invalid.

Kaylee has 2x = 26 nickels and 4x = 52 quarters if x = 13. These coins are worth [tex]0.05(26) + 0.10(13) + 0.25(52) = $30.[/tex]

The answer is yes, and Kaylee now has 26 nickels.

Option B, or 50, is the correct response, therefore.

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To find an equation that goes through the point (8.1) and is perpendicular to 2y + 4x = 12, we can first rearrange the given equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept:

2y + 4x = 12

2y = -4x + 12

y = -2x + 6

So the slope of the given equation is -2.

Since we want a line that is perpendicular to this line and passes through the point (8,1), we know that the slope of our new line will be the negative reciprocal of -2, which is 1/2.

Now we can use the point-slope form of the equation of a line to write the equation:

y - 1 = (1/2)(x - 8)

Simplifying this equation, we get:

y - 1 = (1/2)x - 4

y = (1/2)x - 3

Therefore, the equation that goes through the point (8,1) and is perpendicular to 2y + 4x = 12 is y = (1/2)x - 3.

Answer:

To find the equation of a line that goes through a given point and is perpendicular to a given line, we can use the following steps:

Rewrite the given line in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.

Determine the slope of the line that is perpendicular to the given line. The slope of a line perpendicular to a line with slope m is -1/m.

Use the point-slope form of the equation of a line to write the equation of the line that goes through the given point with the slope found in step 2.

Given the point (8, 1) and the line 2y + 4x = 12, we can rewrite the line in slope-intercept form by solving for y:

2y + 4x = 12

2y = -4x + 12

y = -2x + 6

The slope of the given line is -2.

The slope of the line perpendicular to the given line is -1/-2 = 1/2.

Using the point-slope form of the equation of a line, we can write the equation of the line that goes through the point (8, 1) with slope 1/2:

y - 1 = (1/2)(x - 8)

Simplifying this equation, we get:

y - 1 = (1/2)x - 4

y = (1/2)x - 3

Therefore, the equation of the line that goes through the point (8, 1) and is perpendicular to the line 2y + 4x = 12 is y = (1/2)x - 3.

Step-by-step explanation:

\[ \left[\begin{array}{rr}-35 & 7 \\ 5 & -1\end{array}\right] \]

In Exercises 1-12, use the Gauss-Jordan method to compute the inverse, if it exists, of the matrix.

1. \( \left[\begin{array}{ll}7 & 3 \\ 5 & 2\end{array}\right] \)

To find the inverse of this matrix, we need to use the Gauss-Jordan method. We begin by writing the augmented matrix:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
5 & 2 & 0 & 1
\end{array}\right]\]

Next, we subtract 5 times the first row from the second row to obtain:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
0 & -11 & -5 & 1
\end{array}\right]\]

Now, we divide the second row by -11 to obtain:

\[ \left[\begin{array}{cc|cc}
7 & 3 & 1 & 0 \\
0 & 1 & 5 & -1
\end{array}\right]\]

Finally, we subtract 7 times the second row from the first row to obtain:

\[ \left[\begin{array}{cc|cc}
1 & -20 & -35 & 7 \\
0 & 1 & 5 & -1
\end{array}\right]\]

Therefore, the inverse of the given matrix is:

\[ \left[\begin{array}{rr}-35 & 7 \\ 5 & -1\end{array}\right] \]

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(i)The value of x is either 29.326 or -28.226, such that such that |a + b| = |a| + |b|

To find the value of x such that |a + b| = |a| + |b|, we need to first find the magnitudes of a, b, and a+b.

The magnitude of a is |a| = √(1^2 + 2^2 + 2^2) = √9 = 3.

The magnitude of b is |b| = √(2^2 + (6x-3)^2 + 4^2) = √(4 + 36x^2 - 36x + 9 + 16) = √(36x^2 - 36x + 29).

The magnitude of a+b is |a+b| = √((1+2)^2 + (2+6x-3)^2 + (2+4)^2) = √(9 + (6x-1)^2 + 36) = √(6x^2 - 12x + 46).

Now, we can set |a+b| = |a| + |b| and solve for x:
√(6x^2 - 12x + 46) = 3 + √(36x^2 - 36x + 29)

Squaring both sides gives:
6x^2 - 12x + 46 = 9 + 36x^2 - 36x + 29 + 6√(36x^2 - 36x + 29)

Simplifying and rearranging terms gives:

-30x^2 + 24x - 8 = 3√(36x^2 - 36x + 29)
Squaring both sides again gives:
900x^4 - 1440x^3 + 672x^2 - 128x + 64 = 324x^2 - 324x + 87
Simplifying and rearranging terms gives:
900x^4 - 1440x^3 + 348x^2 + 196x - 23 = 0

Using the quadratic formula, we can find the value of x:

x = (-(-1440) ± √((-1440)^2 - 4(900)(348)(196)(-23)))/(2(900))
x = (1440 ± √(2073600 + 2731680000))/(1800)
x = (1440 ± √(2733753600))/(1800)
x = (1440 ± 52344)/(1800)
x = (1440 + 52344)/(1800) or x = (1440 - 52344)/(1800)
x = 29.326 or x = -28.226

Therefore, the value of x is either 29.326 or -28.226.

(ii) The value of y such that there exists a vector d satisfying a × d = c is -1.75, we need to first find the cross product of a and d:

a × d = [(2d3 - 2d2), (2d1 - d3), (d2 - 2d1)]
Now, we can set a × d = c and solve for y:
[(2d3 - 2d2), (2d1 - d3), (d2 - 2d1)] = [-2, 8y + 4, 1]

This gives us the system of equations:

2d3 - 2d2 = -2
2d1 - d3 = 8y + 4
d2 - 2d1 = 1

Solving for d1, d2, and d3 in terms of y gives:

d1 = (8y + 10)/3
d2 = (16y + 22)/3
d3 = (24y + 34)/3

Substituting these values back into the first equation gives:

2(24y + 34)/3 - 2(16y + 22)/3 = -2

Simplifying and rearranging terms gives:

8y + 12 = -2
8y = -14
y = -14/8
y = -1.75
Therefore, the value of y is -1.75.

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Answer:

[tex]\dfrac{x}{x + 1}\\\\\text{which is the first answer choice }[/tex]

Step-by-step explanation:

We are given
[tex]f(x) = x^2 - x\\g(x) = x^2 - 1\\\\\text{and we are asked to find $ \left(\dfrac{f}{g}\right)\left(x\right)$}[/tex]

[tex]\left(\dfrac{f}{g}\right)\left(x\right) = \dfrac{f(x)}{g(x)}\\\\\\= \dfrac{x^2-x}{x^2 - 1}[/tex]

[tex]x^2 - x = x(x - 1)\text{ by factoring out x}\\\\x&2 - 1 = (x + 1)(x - 1) \text{ using the relation $a^2 - b^2 = (a + 1)(a - 1)$}[/tex]

Therefore,

[tex]\dfrac{x^2-x}{x^2 - 1} = \dfrac{x(x-1)}{(x + 1)(x - 1)}[/tex]

x - 1 cancels out from numerator and denominator with the result
[tex]\dfrac{x}{x+1}[/tex]

So

[tex]\left(\dfrac{f}{g}\right)\left(x\right)$} = \dfrac{x}{x + 1}[/tex]

The value of y when x is 12 is 2 (option C)

Inverse variation is the relationship between two variables, such that if the value of one variable increases then the value of the other variable decreases. Example is the price of a commodity and the quantity acquired, the higher the price, the lower of commodity bought and vice- versa.

Inverse relationship between two quantities x and y is expressed as:

x= ky

where k is the constant

k = 6

when x = 12

12 = 6y

divide both sides by 6

y = 12/6

y = 2

therefore the value of y when x is 12 is 2

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The converse of Theorem 6.6 states that if a group G is such that every proper subgroup is cyclic, then G is cyclic. This statement is false, as shown by the following counterexample:

Let G = {e, a, b, ab}, where e is the identity element, and a and b are two distinct elements such that ab = ba.

This group is not cyclic because it does not contain an element of order 4, and thus cannot be generated by a single element. However, every proper subgroup is cyclic. For example, the subgroup {e, a} is cyclic, and the subgroup {e, b} is cyclic.

Therefore, this example provides a counterexample to the converse of Theorem 6.6.

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48500 has been invested in total. Total invested at the end: $289,279.40. This exemplifies the need to allow your savings to increase over time.

An investments is a procurement made with both the intention of making money or increasing capital. Appreciate seems to be the term for just an asset's value growing over time. When someone invests in something, they do it with the intention of using it to generate cash later on instead of for current consumption.

Investing is a wise method to use your money and may possibly increase your wealth. When your make intelligent investment choices, your cash can increase in value and outpace inflation.

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