What is the sum of the rational expressions below?
2x-1
7x+x=2
X-2
OA. 9x²-5x+2
7x²-2
OB. 9x²-5x+2
7x²-14x
O C.
O D.
2x²-5x+2
7x²-14x
3x-1
8x-2

The percentage of saline solution in brand B is 3%.

Let x be Percentage of saline solution in brand B

We know that the overall volume of saline solution in the combination is 18% of 6 gallons plus x% of (18 - 6) gallons since Danielle blended brands A and B to make a total of 18 gallons of 8% saline solution.

Then the equation is given as,

0.18(6) + 0.01x(18 - 6) = 0.08(18)

Simplifying and solving for x, we get:

1.08 + 0.12x = 1.44

0.12x = 0.36

x = 3

Therefore, the percentage of saline solution in brand B is 3%.

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The transformation sequence of the function is:

The function is first shifted to the left, then vertically stretched, and finally vertically shifted up and down to get the final graph.

We have,

The transformations of f(x) = 2^x to get to g(x) = -5 + 2^{x + 6} + 3 are as follows:

Translation:

f(x) is translated 6 units to the left, which gives 2^{x+6}.

Vertical stretch/compression:

The entire function is multiplied by 2, which makes it taller.

Vertical shift:

The entire function is shifted up 3 units and down 5 units, which raises the entire graph 3 units and then lowers it 5 units from its original position.

Thus,

The function is first shifted to the left, then vertically stretched, and finally vertically shifted up and down to get the final graph.

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Answer:

[tex] - 8(45) + 482 = - 360 + 482 = 122[/tex]

[tex]7^n - 1[/tex] is divisible by 6 for every positive integer n.

To prove that [tex]7^n - 1[/tex] is divisible by 6 for every positive integer n, we will use mathematical induction.

First, we will check the base case, n = 1:

[tex]7^1 - 1 = 6[/tex], which is divisible by 6.

Now, we assume that [tex]7^k - 1[/tex]is divisible by 6 for some positive integer k. This means that there exists an integer m such that:

[tex]7^k - 1 = 6m[/tex]

Next, we need to show that this implies that 7^(k+1) - 1 is also divisible by 6. We can use the following algebraic manipulation:

[tex]7^{k+1} - 1 = 7*7^k - 1 = 6\\7^k + (7^k - 1)[/tex]

Since we know that [tex]7^k - 1[/tex] is divisible by 6 (by the induction hypothesis), we can substitute 6m for [tex]7^k - 1[/tex] in the above equation:

[tex]7^{k+1} - 1 = 67^k + (7^k - 1) = 67^k + 6m = 6(7^k + m)[/tex]

This shows that [tex]7^{k+1} - 1[/tex] is also divisible by 6, since it is a multiple of 6. Therefore, by mathematical induction, we can conclude that [tex]7^n - 1[/tex] is divisible by 6 for every positive integer n.

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The equation that shows the total number of miles Jill walked, w, on Tuesday and Wednesday is w = 3 + 2

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

Tuesday = 3 miles

Wednesday = 1 less than Tuesday

This means that

Wednesday = 3 miles - 1 mile

Evaluating the like terms, we have

Wednesday = 2 miles

The equation shows the total number of miles Jill walked, w, on Tuesday and Wednesday is represented as

w = Tuesday + Wednesday

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

w = 3 + 2

Evaluate

w = 5

Hence, the equation is w = 3 + 2 and the number of miles s 5

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Answer:

Step-by-step explanation:

You want the values of 'a', 'h', and 'k' for the transformed function f(x) = 2·log₄(x +3) -2.

When a function is scaled by the factor 'a' and translated by (h, k), it becomes ...

  f(x) ⇒ a·f(x -h) +k

Comparing the function f(x) = log₄(x) to f(x) = 2·log₄(x +3) -2, we see that  has been scaled by the factor a=2, and translated by (h, k) = (-3, -2).

The values of the transformation constants are ...

__

Additional comment

In the above transformation, the value 'h' represents a translation to the right. If the form f(x+h)+k is used for the transformed function, then h represents a translation to the left. We don't have information regarding what your values of 'a', 'h', and 'k' are supposed to represent, so we have used the usual meaning.

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The answer is  960in² (Option C) This is a geometry prompt involving hexagonal shape.

Since there are 6 congruent triangles, you find the area of 1 of the triangles and multiply by 6

Area of 1 triangle:

1/2bh = 1/2 (9 in) (10 in)

=1/2 (90 sq in.)

= 45 sq in.

Area of 1 hexagon is 6 (45 sq. in) = 270 sq. in.

There are 2 hexagons in the net, so multiply by 2

Area of 2 hexagons is...

2 (270 sq. in) = 540 sq. in.

Then find the area of the rectangular sides...which will be the 6 small squares shown on the first picture

**the rectangles each have 1 side that is equal to 1 side of the hexagon and 1 side is equal to the height of the prism.

Area of 1 rectangle side

lw = (10 in) (7in.)

= 70 sq. in.

multiply all of the 6 sides of the area

=6 (70 sq.in)

=420 sq. in

Then add the area of the 2 hexagons to the area of the 6 rectangular sides

Total area = 540sq. in + 420sq. in

= 960sq. in.

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A linear equation in slope intercept form for each of this line are as follows;

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

At data point (2, 7) and a slope of -7, a linear equation in slope intercept form for this line can be calculated by using the point-slope form as follows:

y - 7 = -7(x - 2)  

y - 7 = -7x + 14

y = -7x + 21

y = 5x - 6

At data point (-1, -7) and a slope of 1, a linear equation in slope intercept form for this line can be calculated by using the point-slope form as follows:

y - (-7) = 1(x - (-1))  

y + 7 = x + 1

y = x - 8

At data point (6, 7) and a slope of 1/2, a linear equation in slope intercept form for this line can be calculated by using the point-slope form as follows:

y - 7 = 1/2(x - 6)  

y - 7 = x/2 - 3

y = x/2 + 4

At data point (9, 0) and a slope of -2/3, a linear equation in slope intercept form for this line can be calculated by using the point-slope form as follows:

y - 0 = -2/3(x - 9)  

y - 0 = -2x/3 + 6

y = -2x/3 + 6

At data point (5, 19) and a slope of 5, a linear equation in slope intercept form for this line can be calculated by using the point-slope form as follows:

y - 19 = 5(x - 5)  

y - 19 = 5x - 25

y = 5x - 6

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The triangle with the side length of 4.8, 28.6 and 29 is a right triangle.

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The sum of angles in a triangle is 180 degrees.

The sides of a right angle triangle can be found using Pythagoras's theorem as follows:

c² = a² + b²

where

Therefore, lets check if the triangle is a right triangle using Pythagoras's theorem as follows:

4.8² + 28.6² = 29²

23.04 + 817.96 = 841

841 = 841

Therefore, the triangle is a right triangle.

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Answer:x=10 y=14

Step-by-step explanation:

The amount that Peyton would need to invest for the value of the account to reach $39,000 in 17 years is given as follows:

A(0) = $19,097.58.

The balance of an account after t years, using continuous compounding, is  modeled by the equation presented as follows:

[tex]A(t) = A(0)e^{kt}[/tex]

In which:

The parameters for this problem are given as follows:

A(t) = 39000, t = 17, k = 0.042.

Hence the investment is obtained as follows:

[tex]A(0)e^(0.042 \times 17) = 39000[/tex]

[tex]A(0) = \frac{39000}{e^(0.042 \times 17)}[/tex]

A(0) = $19,097.58.

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The number of article sold for Rupees 20 to gain a profit of 20% is equal to 4.

Number of article sold for Rupees 20 = 6

Percent of loss in selling 6 article for Rupees 20 = 20%

Let us start by finding the cost price of one article.

We know that if 6 articles are sold for 20 rupees,

then the selling price of one article is equal to,

Selling price of 1 article = 20/6

⇒ Selling price of 1 article = 3.33 rupees

There was a loss of 20% when 6 articles were sold for 20 rupees.

This means that the cost price of 6 articles is equal to,

Cost price of 6 articles

= Selling price of 6 articles / (1 - Loss%)

= 20 / ( 1 - 20% )

= 20 / (1 - 0.2)    

= 25 rupees

To gain 20% profit, the selling price of the articles should be equal to,

Profit% is 20% = 0.2

Selling price of 6 articles

= Cost price of 6 articles × (1 + Profit%)

= 25 × (1 + 0.2)  

= 30 rupees

Now , the selling price of one article to calculate how many articles can be sold for 20 rupees,

Selling price of 1 article

= Selling price of 6 articles / 6

= 30 / 6

= 5 rupees

This implies,

The number of articles that can be sold for 20 rupees to gain a 20% profit is equal to,

Number of articles

= 20 / 5

= 4 articles

Therefore,  4 articles must be sold for 20 rupees to gain a 20% profit.

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The amount more in income tax that Cho pays than Helen each month would be £56.

First, find Helen's yearly salary :

= 1, 720 x  12

= £ 20, 640

Both Cho's and Helen's annual salaries fall within the Basic rate tax bracket.

Cho 's monthly tax would be:

= (( 24, 000 - 12, 570 ) x 20 % ) / 12

= £ 190. 50

Helen's monthly tax :

= (( 20, 640 - 12, 570 ) x 20 % ) / 12

= £ 134.50

The difference is therefore :

= 190. 50 - 134.50

= £56

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A line segment that measures 92 millimeters in length can be drawn using scale.

Given that,

A line segment has to be drawn which has a measure of length 92 millimeters.

We know that,

10 millimeters = 1 centimeter

1 millimeter = 1/10 centimeters

92 millimeters = 92/10 = 9.2 centimeters

So it is enough to draw a line segment of length 9.2 centimeters.

In the scale, between a centimeter, there are 9 small lines which indicates the millimeters.

In between 9 and 10, there are 9 lines which indicates, 9.1, 9.2, 9.3, ....., 9.9 and after that is 10 cm.

So draw a line segment starting from 0 to 9.2.

Hence the line segment is drawn with scale.

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B

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular.

An equilateral triangle can never be obtuse. 

Answer: The second candle can burn for 20 hours before its length is zero inches.

Step-by-step explanation:

If the second candle is 8 inches in length and burns down at a rate of 2/5 inches per hour, we can use a simple formula to find how long it can burn before its length is zero inches:

Number of hours = Length ÷ Rate of burn

Number of hours = 8 inches ÷ (2/5 inches per hour)

Number of hours = 8 inches × (5/2) hours per inch

Number of hours = 20 hours

Therefore, the second candle can burn for 20 hours before its length is zero inches.

The rules for the two graphs above are: A. first graph: y = 3 + x; second graph : y = 6 + x.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of graph 1;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (4 - 3)/(1 - 0)

Slope (m) = 1/1

Slope (m) = 1

At data point (0, 3) and a slope of 1, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 3 = 1(x - 0)  

y = x + 3

Next, we would determine the slope of graph 2;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (7 - 6)/(1 - 0)

Slope (m) = 1/1

Slope (m) = 1

At data point (0, 6) and a slope of 1, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 6 = 1(x - 0)  

y = x + 6

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Answer:

Step-by-step explanation:

To find the median of a set of data, we first need to arrange the data numbers from smallest to largest. The median is the middle data number in the list.

Given:

5, 0, 1, 3, 9, 0

Arrange the numbers:-

0, 0, 1, 3, 5, 9

Median = 1, 3

Since there are even amount of numbers, the median will be the average of the two middle data numbers.

To find the median, we'll add the numbers 1 + 3 and then divide them by two. (since they are 2 set numbers):-

Therefore, the median of this set of data is 2.

__________________________

Hope this helps! :)

The probability that both utensils are spoons is 0.34.

The total number of utensils in the drawer is 18 + 12 = 30.

The probability of choosing a spoon on the first draw is 18/30, since there are 18 spoons in the drawer out of 30 utensils total.

After the first spoon is drawn, there are 17 spoons and 30 - 1 = 29 utensils left in the drawer.

The probability of choosing another spoon on the second draw is 17/29, since there are 17 spoons left in the drawer out of 29 utensils total.

To find the probability of both events happening (drawing two spoons), we need to multiply the probabilities of each event:

(18/30) * (17/29) = 0.34

Therefore, the probability of randomly choosing two spoons from the drawer is approximately 0.34 or 34%.

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1. The Pythagoras theorem is (6+r)² = r²+9²

2. 36+12r+r² = r²+9²

3. The value of r is 3.75

Pythagoras theorem states that; the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.

This means that, c² = a² + b²

In the triangle, 6+r is the hypotenuse, 9 and r are the other two legs.

therefore;

(6+r)² = r²+9²

= (6+r)(6+r) =r²+9²

36+6r+6r+r² = r²+9

36+12r+r² = r²+9

Collecting like terms

12r+r²-r²= 81-36

12r = 45

r = 45/12

r = 3.75

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Answer:r=3sinθ/4cos^2 θ.

Step-by-step explanation:

To convert to polar coordinates, we use the relationships

xy=rcosθ=rsinθ.

Substituting these into the given equation, we find

3y3rsinθ3rsinθ=4x2=4(rcosθ)2=4r2cos2θ

Divide each side of the equation by the common factor r to get

The Dlamini household bought 367.13 units of electricity, and the cost of purchasing between 0-600 kWh of units is given by the equation cost = (units * 229) / 100. Using this equation, the cost of purchasing 367.13 units of electricity is R842.36.

a) The Dlamini household bought 532.56 - 165.43 = 367.13 kWh of electricity.

b) The cost of purchasing between 0-  600 k wh of units can be calculated using the following equation

Cost = Units x Rate

where the Rate depends on the block the units fall into.    

 For Block 1, the rate is 229.00 c  /kw h and for Block 2, the rate is 278.46 c/kWh. The equation for Block 1 can be written as.....

Cost = Units x 229.00 c/kWh

However, if the units purchased are more than 600 kWh, they will fall into Block 2, and the equation for Block 2 can be written as:

Cost = (600 x 229.00 c/kWh) + (Units - 600) x 278.46 c/kWh

c) From part (a), we know that the Dlamini household bought 367.13 kWh of electricity, which falls entirely in Block 2. Therefore, we can use the equation for Block 2 to calculate the cost

Cost = (600 x 229.00 c/kWh) + (367.13 - 600) x 278.46 c/kWh

= (137400 c) + (-7388.38 c)

   = 130011.62 c

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The number 889/100 as a decimal expression is 8.89

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

8

89

100

Express the number properly

So, we have the following representation

889/100

When 889 is divided by 100, we have

889/100 = 8.89

This means that the number as a decimal is 8.89

Hence. the number 889/100 as a decimal expression is 8.89

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Answer: 1

Step-by-step explanation:

you can create a ratio because they are similar you can see that if you multiply 8 by 2 you get 16 so you divide 2 by 2 and you get x

The expression which is used to represents the statement "one-more than the quotient of a number x" and 4 is "x/4 + 1".

A "Quotient" is the result of a division operation between two numbers, where one number is divided by another. It represents the number of times one number is contained within another number.

An "Expression" is defined as "mathematical-phrase" which contain numbers, variables, and operators such as addition, subtraction, multiplication, and division. It can also contain functions and other mathematical symbols.

The quotient of a number"x" and 4 can be written as x/4.

So, one more than the quotient of a number x and 4 can be represented by the expression:

⇒ x/4 + 1,

Therefore, required mathematical expression is "x/4 + 1".

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The given question is incomplete, the complete question is

Write an expression to represent : One more than the quotient of a number x and 4.

The indicated value (g o f)(-7)  has a value of -14 when evaluated

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

f(x) = -2x - 13

g(x) = 2x² - 6x - 10

The indicated value of the function is

(g o f)(-7)

This means that

(g o f)(-7) = g(f(-7))

Calculate g(-7)

So, we have

f(-7) = -2(-7) - 13

f(-7) = 1

This means that

(g o f)(-7) = g(1)

Recall that

g(x) = 2x² - 6x - 10

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

(g o f)(-7) = 2(1)² - 6(1) - 10

Evaluate

(g o f)(-7) = -14

Hence, the value is -14

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Answer:

Part A:

1. Computation of Basic EPS

Basic EPS = (Net Income - Preferred Dividends) / Weighted Average Shares Outstanding²

Net Income = $120,000

Preferred Dividends = 4% x $50,000 = $2,000

Weighted Average Shares Outstanding = 100,000 (assuming no change in common stock during the year)

Basic EPS = ($120,000 - $2,000) / 100,000

Basic EPS = $1.18

2. Individual dilution/anti-dilution with decision making

We need to check if any of the convertible securities or stock options are dilutive or anti-dilutive by comparing their impact on EPS with the basic EPS.

Convertible Preferred: Each preferred share can be converted into 1.9 common shares (1,900 / 1,000). The conversion would increase the common shares outstanding by 1,900 and decrease the preferred dividends by $2,000.

EPS after conversion = ($120,000 - $0) / (100,000 + 1,900)

EPS after conversion = $1.16

Since this is lower than the basic EPS of $1.18, the convertible preferred is dilutive and should be included in the diluted EPS calculation.

Stock Options - A: Each option can be exercised at $12 to buy one common share when the average market price is $11. This means that the options are out of the money and would not be exercised by the holders. Therefore, they have no impact on EPS and are anti-dilutive.

Stock Options - B: Each option can be exercised at $7 to buy one common share when the average market price is $8. This means that the options are in the money and would be exercised by the holders. The exercise would increase the common shares outstanding by 4,000 and generate cash proceeds of $28,000 ($7 x 4,000). The cash proceeds can be used to buy back some common shares at the market price of $8. The number of shares bought back is 3,500 ($28,000 / $8). The net increase in common shares outstanding is 500 (4,000 - 3,500).

EPS after exercise = ($120,000 - $2,000) / (100,000 + 500)

EPS after exercise = $1.17

Since this is lower than the basic EPS of $1.18, the stock options - B are dilutive and should be included in the diluted EPS calculation.

Convertible Bonds: Each bond can be converted into 40 common shares (40,000 / 1,000). The conversion would increase the common shares outstanding by 40,000 and decrease the interest expense by $50,000 ($1,000,000 x 5%). The interest expense is net of tax, so we need to multiply it by (1 - tax rate) to get the after-tax amount. The tax rate is 30%.

EPS after conversion = ($120,000 - $2,000 + $50,000 x (1 - 0.3)) / (100,000 + 40,000)

EPS after conversion = $1.19

Since this is higher than the basic EPS of $1.18, the convertible bonds are anti-dilutive and should be excluded from the diluted EPS calculation.

3. Ranking of potential dilutors

We need to rank the potential dilutors from the most dilutive to the least dilutive based on their impact on EPS.

The most dilutive security is the convertible preferred with an EPS of $1.16.

The second most dilutive security is the stock options - B with an EPS of $1.17.

The least dilutive security is the convertible bonds with an EPS of $1.19.

4. Step-by-step inclusion of individual dilutors to arrive at reportable EPS with decision making.

We need to include the potential dilutors in order of their dilution until we reach a point where the next security would be anti-dilutive.

The first security to include is the convertible preferred.

EPS with convertible preferred = ($120,000 - $0) / (100,000 + 1,900)

EPS with convertible preferred = $1.16

The next security to include is the stock options - B.

EPS with stock options - B = ($120,000 - $0) / (100,000 + 1,900 + 500)

EPS with stock options