*
The equation p(n) = 12/5n²-17 represents the overall profit
from production for a company where p is the total profit
in dollars and n is the number of hours per day that
the production line operates. Determine the profit when
n = 101/2 hours.

The following are solutions to the given problems given f(x)=x²- 5x+7 and g(x)=3x² + 4x - 2;

(f- g)(x) = -2x² - 9x + 9

(g-f)(x) = 2x² + 9x - 9

(f + g)(x) = 4 x² - x + 5

A function can be defined as a relation between a set of inputs having one output each. In other words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain (range). A function is generally denoted by f(x) where x is the input.

The general mathematical representation of a function is y = f(x).

f(x)=x²- 5x+7

g(x)=3x² + 4x - 2

(f- g)(x) = x²- 5x+7 - (3x² + 4x - 2)

(f- g)(x) = x²- 5x+7 - 3x² - 4x + 2

(f- g)(x) = -2x² - 9x + 9

(g-f)(x) = 3x² + 4x - 2 - (x²- 5x+7)

(g-f)(x) = 3x² + 4x - 2 - x² + 5x - 7

(g-f)(x) = 2x² + 9x - 9

(f + g)(x) = x²- 5x+7 + 3x² + 4x - 2

(f + g)(x) = 4 x² - x + 5

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The length of a  rectangular rug is 84 yards and the width is 36 yards

Let 'l' represents the length of the  rectangular rug and 'w' represents the width of a  rectangular rug.

The length of the rug is 12 yards more than two times the width.

So, we get an equation,

l = 12 + 2w

Also, a  perimeter of  a rectangular rug is 240 yards

We know that the formula for the perimeter of rectangle:

P = 2(w + l)

Using this formula the  perimeter of  a rectangular rug would be,

P = 2 × [w + (12 + 2w)]

240 = 2 (12 + 3w)

120 = 12 + 3w

108 = 3w

w = 118/3

w = 36

And the length would be,

l = 2 + 2( 36)

l = 84 yards

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The complete question is:

A rectangular rug has perimeter 240 yards. The length is 12 yards more than twice the width. Find the length and width of the rug.

The equation represented as f(x) = x + 1/x + 2 is a function

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

f(x) = x + 1/x + 2

The above equation is a rational equation whose numerator and denominator are linear functions

Equations with these properties are always functions

This is so because their x values do not have multiple y values

Hence, the equation f(x) = x + 1/x + 2 is a function

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the total FBT liability for this benefit is $329 ($700 x 47% = $329).

Fringe Benefits Tax (FBT) is imposed by the Australian Taxation Office (ATO) on certain non-cash benefits provided to employees. If a company provides a weekend stay in Sydney valued at $1,000 as a Christmas gift in December 2022, the FBT liability is calculated as follows:

The taxable value of the benefit is calculated by deducting the expenses that are not subject to FBT, i.e. $300 (flight) from the total value of the gift, i.e. $1,000. This leaves a taxable value of $700 ($1,000 - $300 = $700).

The FBT liability is then calculated by multiplying the taxable value, i.e. $700, by the FBT rate which is currently 47% (as of 2021). This results in an FBT liability of $329.

Therefore, the total FBT liability for this benefit is $329 ($700 x 47% = $329).

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About 40 meters the submarine’s depth be if it continues its dive for another 110 meters. So the option A is correct.

Since it is essentially a ratio problem, we can use the information given and apply the Rule of Three assuming that the submarine is moving at a constant speed.

In this rule, the depth after 110 meters is obtained by simply multiplying the ratio's diagonal values and dividing by the last value.

70min = 25 m

110min = x m

x = (110 × 25)/70

x = 2750/70

x = 39.28

x = 40(approx)

We can see that at 110 minutes, the depth would have been approximately 40 meters, but since the submarine travelled further during this time, we must add this measurement to the original 25 meters to determine the submarine's total depth.

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The volume of the solid is approximately 32π/3 cubic units.

The volume of the solid obtained by rotating the region bounded by the curves y = 4 - 1/2x, y = 0, x = 0, and x = 1 about the x-axis can be found using the method of disks/washers. The formula for the volume is given by:

V = π * ∫[a,b] (R^2 - r^2) dx

where R is the outer radius (4 - 1/2x), r is the inner radius (y = 0), a and b are the limits of x (x = 0 and x = 1), and dx represents a small slice of the solid.

Evaluating the definite integral:

V = π * ∫[0,1] (16 - x^2 - x^2) dx = π * ∫[0,1] (16 - 2x^2) dx = π * (16x - 4/3 x^3) evaluated at x = 1 - x = 0 = π * (16 - 4/3) = 32π/3 cubic units.

So the volume of the solid is approximately 32π/3 cubic units.

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To evaluate the expression 1+(-2/3)-(-m) where m = 9/2, we need to substitute the value of m into the expression and simplify it.

What is a mathematical expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)

1+(-2/3)-(-m)

1+(-2/3)-(-9/2)

Here, we have to simplify the inner expressions first,

-9/2 can be simplified to -4.5

1+(-2/3)-(-4.5)

Next, we add and subtract the terms:

1-2/3+4.5

Now, we simplify -2/3 by multiplying both numerator and denominator by -1:

1+4/3+4.5

Now, we add the fractions,

1+4/3+4.5

7/3+4.5

Finally, we get the value of the expression as

7/3+4.5 = 4.5 + 7/3 = 25/3.

So the final value of the expression 1+(-2/3)-(-m) where m=9/2 is 25/3.

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Answer:

Hi, I'm Za'Riah! I will gladly assist you with this problem. (see explanation)

Step-by-step explanation:

1. Repeating Decimal

2. Rational Number

This is all I can think of at the moment.

The variables are x = 1/2 and y = 5/3

From the figure, we have been told 21x = 5x + 8. So let's solve for x.

21x = 5x + 8

21x - 5x = 8

16x = 8

x = 8/16

x = 1/2 = 0.5

Using the Basic Proportionality Theorem or Thales’s Theorem, which states that "If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally". We can say:

(20y-2)/(17y + 3) = (5x+8)/(21x)

Substitute x = 0.5 and solve for y:

(20y-2)/(17y + 3) = (5(0.5) + 8)/(21(0.5))

(20y-2)/(17y + 3) = (10.5)/(10.5)

(20y-2)/(17y + 3) = 1

20y-2 = 17y + 3

20y - 17y = 3 + 2

3y = 5

y = 5/3

Thus, the variables are x = 1/2 and y = 5/3

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Paying off the lowest outstanding balance credit card first will not be beneficial over paying off the one with the highest. Instead, it's beneficial to pay the highest outstanding balance first.

Actually, paying the one with the highest interest rate is logically the right thing to do because you can shorten the amount that you spend on interest or charges by month, but paying the lowest outstanding balance credit card first makes more sense if you could transfer the highest interest balance into a low or maybe no interest rate that is being offered to another card, only if you are so certain that you will pay the remaining balance or on before its due. 

It should be remarked that paying off the lowest outstanding balance credit card first cannot be beneficial over paying off the one with the highest interest rate.

It's significant to pay the one that has the highest interest rate. This is because it'll assist in shortening the amount the person will pay as interest.

In conclusion, it's important to pay the highest Interest rate.

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The solution for the system of inequalities is (-0.43,4.15).

What is an inequality?

An inequality statement in algebra uses the inequality symbol to illustrate the relationship between two expressions. A symbol of inequality has non-equal expressions on both sides.

We are given a graph on which two inequalities are plotted

First, we need to frame the equations from the points.

For the blue line, we have points (0,2) and (-1,7)

So, using slope-intercept form, we get the equation as

y+5x>2

For the red line, we have points (0,4) and (3,3)

So, using slope-intercept form, we get the equation as

y+0.33x >4

Now, in order to find the solution, we will subtract both the equations

On subtracting, we get

⇒(y+5x) - (y+0.33x) = 2 - 4

⇒y+5x - y-0.33x = -2

⇒4.67x = -2

⇒x = -0.43

Substituting this value in the equation, we get

⇒y+5(-0.43) = 2

⇒y-2.15 = 2

⇒y = 4.15

Hence, the solution for the system of inequalities is (-0.43,4.15).

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For the given table of values the equation that could be the function rule is option 4: y = 3x - 2.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The table of input and output values are given for the function.

The slope-intercept form of an equation/function is - y = mx + b

To find the slope m use the formula -

(y2 - y1)/(x2 - x1)

Substituting the values in the equation -

[-2 - (-11)]/[0 - (-3)]

(-2 + 11)/(0 + 3)

9/3

3

So, the slope point is obtained as m = 3.

The equation becomes - y =3x + b

To find the value of b substitute the values of x and y in the equation -

-11 = 3(-3) + b

-11 = -9 + b

b = -11 + 9

b = -2

So, the value for b is -2.

Now, the equation becomes -

y = 3x - 2

The graph is plotted for the function.

Therefore, the equation is y = y = 3x - 2.

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After setting f (x) = 0, solve for to determine the intercept(s) of. Set y = f(0) and then find to determine the intercept of. Choose test points to establish the sign of each interval on the axis after using the intercept(s) to divide it into intervals. Determine the test points.

The Use of Zeros When Graphing Polynomials

You may graph a polynomial function using the steps below.

To assess how the graph will behave at its finish, use the leading-term test.

Choose the x− capture(s) of f(x) by putting f(x)=0 and then figuring up a solution for x .

Choose the y− a snare in f(x) by putting y=f(0) and discovering y.

Use the x− dividing intercept(s) x−

Divide the axis into intervals and then select test points to ascertain the sign of f(x) per interval.

the test points on a map.

To identify the general form of the graph, locate additional points as necessary.

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Therefore, the method (addition or subtraction) used to combine or separate the terms is the primary distinction between adding and subtracting linear expressions.

Another difference is that when we subtract linear expressions, we get a smaller expression, whereas when we add them, we get a larger one.

A mathematical expression that results from the linear combination of variables and constants is known as a linear expression.

Linear expressions can be combined or separated using two distinct operations: adding and subtracting.

By keeping the same variable while adding the coefficients (the numerical part) to linear expressions, like terms are combined. If we wanted to add 2x and 3x, for instance, we would get 5x.

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The details which describe the free African American who lived in the South is that they were counted as part of population statistics which is denoted as option D.

This is referred to an an entire set of items from which you draw data for a statistical study while on the other hand, population refers to the people who live in a particular area at a specific time.

From the passage we were told that there was about 225,964 free African americans of which only 6 percent were free black people which made them less than those found in the North. They were counted during census which means that they were part of the population statistics.

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If Steven made $4,800 this month, the value of his sales for the month is $51,000.

Steven's base monthly salary on sales of $17,000 = $2,700

First sales commission above $17,000 sales = 10%

Second bonus commission above $25,000 sales = 5%

Total earnings for the month = $4,800

First sales commission = $800 ($25,000 - $17,000 x 10%)

Additional bonus commission = $1,300 ($4,800 - $2,700 - $800)

$1,300 = 5%

Sales above $25,000 = $26,000 ($1,300/5%)

Sales value = $51,000 ($25,000 + $26,000)

Thus, we can assert that Steven generated Sales of $51,000 to earn a total of $4,800 for the month.

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=============================================

Work shown for problem 1

[tex](f - g)(\text{x}) = f(\text{x}) - g(\text{x})\\\\(f - g)(\text{x}) = (2\text{x}^2+\text{x}-3)-(\text{x}-1)\\\\(f - g)(\text{x}) = 2\text{x}^2+\text{x}-3-\text{x}+1 \\\\(f - g)(\text{x}) = 2\text{x}^2-2\\\\[/tex]

This leads to choice D as the final answer.

The domain is the set of all real numbers since the result is a polynomial. Any polynomial function has the domain "all real numbers". This is because we don't have anything like a division by zero error. We also don't have to worry about square roots of negative values.

------------------------

Work shown for problem 2

[tex](f * g)(\text{x}) = f(\text{x}) * g(\text{x})\\\\(f * g)(\text{x}) = (2\text{x}^2+\text{x}-3) * (\text{x}-1)\\\\(f * g)(\text{x}) = \text{x}(2\text{x}^2+\text{x}-3)-1(2\text{x}^2+\text{x}-3)\\\\(f * g)(\text{x}) = 2\text{x}^3+\text{x}^2-3\text{x}-2\text{x}^2-\text{x}+3\\\\(f * g)(\text{x}) = 2\text{x}^3-\text{x}^2-4\text{x}+3\\\\[/tex]

This leads to choice A as the final answer.

The domain is the set of all real numbers for the exact same reasoning as problem 1.

The Area of the circle, A is 530.66 inch.

An area of a circle is the amount of two-dimensional space within the boundaries of the circle. It is the region enclosed by a circle's circumference. The area of a circle can be found by using the formula A = πr2, where A denotes the area of the circle, r represents its radius, and π is equal to 3.1416. The radius of a circle is the distance from its center to its circumference. It can also be found by dividing the circumference by 2π.

Given,

Diameter, d = 26 inch

Radius, r = d/2 = 26/2 = 13 inch

π = 3.14 or 22/7

We apply all these above values in the formula, we get area of circle:

A = πr²

A = 3.14 x (13)²

A = 530.66 inch

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The probability of drawing a king then getting an 1 is P = 0.006

First we need to find the probability of randomly drawing a king from the deck, that is equal to the quotient between the number of kings on a deck and the total number of cards on the deck.

There are 4 kings and 52 cards, thus the probability is:

p = 4/52

Then we want to get a 1, the probability is computed in the same way, notice that now there are 51 cards on the deck because we already drawn one.

q = 4/51

The joint probability (first drawing a king and then a 1) is equal to the product between the individual ones.

P = (4/52)*(4/51) = 0.006

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The length of DH (diagonal) in the rectangular prism is approximately 3.5 inches.

A rectangular prism has three dimensions: length, width, and height. The diagonal of a rectangular prism is the distance between two opposite corners of the prism.

To find the diagonal of a rectangular prism, you can use the Pythagorean theorem, which states that the square of the diagonal is equal to the sum of the squares of the length, width, and height.

In this case, the length is 1.5 inches, the width is 1 inch, and the height is 3 inches. So, you can use the formula:

d² = 1.5² + 1² + 3²

d² = 2.25 + 1 + 9

d² = 12.25

Then take the square root of the d²

d = √12.25

d ≈ 3.5 inches.

Therefore, the measure of the diagonal of the rectangular prism is approximately 3.49 inches.

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Answer:

3.5

Step-by-step explanation:

I took the quiz

The value of the variable, h is 60 degrees

The properties of a trapezium are expressed as;

If the size of the angles of the trapezium are 2h, 2h, h and h

Then, it is represented as;

Add all the angles

2h + 2h + h + h

Equate the angles to the sum of the interior angles of a trapezium, we get

2h + 2h + h + h  = 360

Now, add the like terms

6h = 260

Divide both sides by the coefficient of h, we get;

6h/6 = 630/6

h = 60 degrees

Hence, the value is 60 degrees

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In the two highlighted rows show that for the same amount of yellow, Green #1 uses more blue than Green #2 and green #1 is bluer shade of green than Green # 2.

It should be noted that there are two tables which give the number of pints of blue and yellow that are used to make different amounts of two shades of green dye.

Green #1 is made by mixing blue and yellow in a ratio of 2 : 3.

Green #2 is made by mixing blue and yellow in a ratio of 1 : 2.

The first one has 2 parts of blue at 3 parts of yellow. In the two highlighted rows show that for the same amount of yellow, Green #1 uses more blue than Green #2.

In grade two the blue part is half of the yellow part. This means that green #1 is bluer shade of green than Green # 2.

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The range of h include the following: A. -4 ≤ h(x) ≤ 6.

In Mathematics, a range can be defined as the set of all real numbers that connects with the elements of a domain.

Generally speaking, the vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown, we can reasonably and logically deduce the following:

Range = [-4, 6]

In mathematical notation, the range of this graph can be rewritten as -4 ≤ h(x) ≤ 6.

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The population of interest and the sample size include the following:

Population: all potential customers of the parlor.

Sample size: 8.

In Statistics and science, a population can be defined as a set of similar items or events that comprises every member of a group and from which a researcher, scientist, or experimenter, obtains or generates data for a statistical study.

In Statistics and science, a sample can be defined as a set of data that is collected from a population based on a well-defined sampling procedure.

In this scenario, we can logically deduce that sample is the eight customers that were selected and questioned by an employee at the local ice cream parlor.

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Answer:

3 laps every 12 minutes

Step-by-step explanation:

If the size length of a square is 5x-7 and the perimeter is 125, then the value of x is 7.65 units.

What is perimeter?

The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges. Its dimensions are expressed in linear units like centimetres, metres, inches, and feet.

The length of the square is given as a = (5x - 7) units

The formula for the perimeter of square is -

P = 4a

The value for perimeter of the square is 125 units.

Substitute the values in the equation -

P = 4a

125 = 4(5x - 7)

125 = 20x - 28

20x = 125 + 28

20x = 153

x = 153/20

x = 7.65

Therefore, the value of x is obtained as 7.65 units.

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The amount of wood that they need to complete the supports is given as follows:

130.10 cm.

First we must obtain the side lengths of the right triangle.

The equal side lengths of the right triangle are obtained applying the Pythagorean Theorem, hence:

x² + x² = 92²

2x² = 92²

x² = 4232

x = square root of 4232

x = 65.05 cm.

They have two sides of 65.05 cm, hence the amount of wood needed is obtained as follows:

2 x 65.05 = 130.10 cm.

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Answer: 59

Step-by-step explanation:

(13 + 9) + 37

13 + 9 = 22

22 + 37

59

Aaron would make $73000 salary after 6 years. His salary after t years will be 52000 + 3500t. The solution has been obtained by using the linear expression.

What is a linear expression?

The expression with the highest degree of one is a linear expression. This demonstrates that there are no variables in a linear expression with an exponent bigger than one. Such an equation yields a straight line on the graph.

We are given that Aaron has just accepted a job at a new company where he will make an annual salary of $52000.

Also, for each year he stays with the company, he will be given a salary raise of $3500.

So, let's formulate a linear expression for the given situation.

The expression comes out to be

52000 + 3500t, where t denotes the number of years

Salary after 6 years will be

⇒52000 + 3500(6)

⇒52000 + 21000

⇒$73000

Hence, Aaron would make $73000 salary after 6 years and his salary after t years will be 52000 + 3500t.

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The total number of visitors to the zoo that day=3980.

Given:

cost of each ticket for an adult=$25

cost of each ticket for an adult=$10

Total ticket sales one day=$47,750.

⇒$25+$10=47,750

⇒25+10=47,750-----(1)

Also,

c=6a+270

6a-c=-270------(2)

multiplying eq(2) with 10 we get:

60a-10c=-2700----(3)

Adding eq(1) and eq(3) we get

25a+10c=47,750

60a-10c=-2700

--------------------------

85a=45050

--------------------

To calculate a value just divide 45050 by 85 we get

a=45050/85

a=530

Now to get the value of c substitutes a value.

c=60(530)+270

c=3450

Number of Adult visitors=530

Number of children visitors=3450

Total Number of visitors=3980.

Therefore, the total number of visitors to the zoo that day=3980.

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