Please answer both parts

We can expand the given expression:

(x + 4)²*(x - 2)

To get the equivalent one:

x³ + 6x² - 32

Here we want to expand the given expression, which is:

(x + 4)²*(x - 2)

First remember the rule:

(a + b)² = a² + 2ab + b²

So we can rewrite the first factor as:

(x + 4)² = x² + 2*4*x + 4²

= x² + 8x + 16

Replacing that in our expression we get:

(x² + 8x + 16)*(x - 2)

Now we can expand that to get:

x²*x + x²*(-2) + (8x)*x + 8x*(-2) + 16*x + 16*(-2)

= x³ - 2x² + 8x² - 16x + 16x - 32

= x³ + 6x² - 32

That is the expression expanded.

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The triangles ABC and CDA are congruent by SAS criteria.

Two triangles are said to be congruent when all of their corresponding sides and angles are equal. For this relation between two triangles, there are many criteria such as SSS, SAS, ASA and RHS.

Given that,

In quadrilateral ABCD, AD = BC and AD ║BC.

Now, the given relation ΔABC ≅ ΔCDA can be proved as follows,

In ΔABC and ΔCDA, it is found as,

AC = AC             (Common sides)

AD = BC              (Given data)

And, ∠ACB = ∠DAC (Interior angles between parallel lines AD and BC)

Thus, ΔABC ≅ ΔCDA by SAS congruence criteria.

Hence, the given relation ΔABC ≅ ΔCDA is proved by SAS criteria.

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Answer:

since , 9^x / 9^3 = 9^5

and since there is a law for exponents that says if the bases of the exponent are equal and divided by each other we subtract the exponents

, so 9^x-3 = 9^5

then x-3 =5 , therefore the final answer will be X= 8 .

Therefore the value of x from the given equation comes out to be 13.5

A statement proving the equality of two mathematical expressions is known as an equation. The equal sign (=) is used to represent an equation between two mathematical expressions. Anything that contains the equal sign is an equation, to put it simply. Examples of equations include x=7, 2x+5=7, etc.

Here,

The equation given is 3(x+2)=7x-48

thus we find the value of x from the given equation

=> 3(x+2)=7x-48

=> 3x+6=7x-48

=> 7x-48-3x-6=0

=>  4x-54=0

=> 4x=54

=> x=54/4

=> x=13.5

Therefore the value of x from the given equation comes out to be 13.5

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Based on the slope formula applied, we can conclude that, the line that passes through P(9, 7) and Q(16, 2) is not perpendicular to the line that passes through  R(3, 8) and S(11, 0).

Using the slope formula, the slope of a line is calculated as:

m = change in y / change in x.

If two lines are such that one is perpendicular to the other, both lines will have slopes that are negative reciprocal to each other and will therefore have a product of -1.

For example, if the slope of a line is -1/2, the negative reciprocal of -1/2 is 2, which will be the slope of the line that it is perpendicular to.

Find the slope of PQ and RS using the slope formula:

Slope of PQ = change in y / change in x = (7 - 2) / (9 - 16) = 5/-7

Slope of PQ = -5/7

Slope of RS = change in y / change in x = (8 - 0) / (3 - 11) = 8/-8

Slope of RS = -1.

-1 is not the negative reciprocal of -5/7, this implies that both lines are not perpendicular.

Therefore, we can state that lines PQ and RS are not perpendicular to each other.

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Answer:  

 a.  (-2,-1)

  b.   y = -(1/2)x-2

Step-by-step explanation:

We will look for an equation of the form y=mx+c, where m is the slope and b is the y-intercept (the value of y when x = 0).

Put the two given points into a table to make it easier to calculate the slopes for Lines A and B.  See the attachment for the table.  Slope is the change in y for a change in x, often termed the "Rise/Run."  Using Line A as the example, the change in y (the Rise) is obtained by taking the Point 2 value for y (7) and subtracting the Point 1 value for y (3):  Rise = (7 - 3) = 4.  The Run is obtained by taking the Point 2 value for x (2) and subracting the Point 1 value of x (0).  The ratio of Rise/Run is 4/2, so the slope is 2.  The same process is used for Line B to obtain a slope of 3 for that line.

The two equations can be written with these slopes:

Line A:  y = 2x + c

Line B:  y = 3x + c

We need to find a value of c for each equation that will force the line through either of their two given points.  c can be determined by entering either of that line's points into the equation(s) above and solving for c:

Line A:  

y = 2x + c

3 = 2*(0) + c    for point (0,3)

c = 3

The equation for Line A is y = 2x + 3

Line B:

y = 3x + c

2 = 3*(-1) + c  for point (-1,2)

c = 5

The equation for Line B is y = 3x + 5

The graph of these lines is attached.  They intersect at point (-2,-1).

[As a note, the solution may also be found by using substitution and solving for both x and y:

y = 2x+3

y = 3x+5

The solution exists when both y's and x's are equal:

Assume y is equal:

 2x+3 = 3x+5

 -x = 2;  x = -2

----

Assume x is equal:

Rearrange:

 y = 2x+3

  2x = y-3

   x = (y-3)/2

Use this definition of x in the second line equation:

 y = 3x+5

 y = 3((y-3)/2)+5

 y = (3y - 9)/2 + 5

2y = 3y - 9 + 10

-y = 1

 y = -1

The solution is (-2,-1), the same as was found by graphing.]

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

A perpendicular line has a slope that is the negative inverse of the referenced line.  For Line A, the perpendicual line would be -(1/2).  The equation for this line would take the form y = -(1/2)x + c.  We want this perpendicular line to intersect the same point that is the solution to the first problem:  (-2,-1).  Enter this point to find c:

y = -(1/2)x + c

-1 = -(1/2)*(-2) + c  for (-2,-1)

-1 = 1+ c

c = -2

The perpendicular line is y = -(1/2)x - 2

See the attached graph.

The category of the polynomial  2x³ - 5 + 9x² is a trinomial

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

2x^3-5+9x^2

Rewrite this expression properly

So, we have the following representation

2x³ - 5 + 9x²

This expression can be further rewritten as

2x³ + 9x² - 5

The degree of a polynomial is the highest power of the polynomial

In this case, the degree of the above polynomial is 3

As a general rule, a  trinomial is any polynomial that has a degree of 3

Hence, the expression is a trinomial

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The simplification of the ratio 1320 feet : 1 mile will be 1:4

What is Ratio?

A ratio in mathematics illustrates how many times one number contains another. The term ratio is a straight borrowing from the Latin word ratio. In Latin, ratio could imply everything from "reason" to "calculation" to "proportion," and it was this last meaning that was used to create the mathematical and English ratios.

Given taht we need to simplify the ratio 1320 feet : 1 mile

For simplifying ratios we need to convert the terms in same units

and we know that 1 mile = 5280 feet

So,

1320 feet : 1 mile can be written as 1320 feet : 5280 feet

The simplification wil be 1:4

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The system of equations to model is s=220 and a=241 when 461 tickets for a school performance were sold. Adult tickets are $4 and cost $3 for students.

Given that,

461 tickets for a school performance were sold. Adult tickets are $4 and cost $3 for students. Sales of tickets totaled $1624.

We have to find the system of equations to model.

We know that,

We get,

a + s = 461

s = 461 -a

The student tickets sold = s × 3   = 3 × (461-a)

The adult tix sold =  4 a              

Summed they total 1624

We get system of equations as

s = 461-a

3 (461-a) + 4a = 1624

1383 -3a +4a = 1624

a = 241      

Then  

s = 461 -a = 220

Therefore, The system of equations to model is s=220 and a=241 when 461 tickets for a school performance were sold. Adult tickets are $4 and cost $3 for students.

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The Scooter Company's total profit function is represented by -2x²+250x-2000.

Total profit of a company is the difference between total revenue and total cost.

The profit or gain is equal to the selling price minus the cost price. Loss is equal to the cost price minus the selling price.

Company earns a total revenue that can be determined by the function R(x) = 400x - 2x²

Each scooter cost $150 to produce, and the company has a fixed cost of $2,000.

Variable cost = Cost of each scooter × Number of scooter sold

Total cost = fixed cost + variable cost

= 2000 + (150×x)

Cost, C(x) = 2000 + 150x

Revenue, R(x) = 400x - 2x²

Total profit = Revenue - Cost

= 400x-2x²-(2000 + 150x)

= 400x-2x²-2000-150x

= -2x²+250x-2000

Therefore, the Scooter Company's total profit function is represented by -2x²+250x-2000.

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Answer:

D) The economy has done well for a while and is at its highest point of the cycle.

Step-by-step explanation:

I believe that it is "D" considering this is the highest point on the graph. Also the fact that, that's the only statement that makes sense for point "A".

I hope this helps!!!

Answer: C

Step-by-step explanation:

A just means that the growth of the cycle will keep increasing and will not decrease, which is wrong.

B means that point A is at its lowest point in the cycle, which is wrong too.

C is correct because before the dot that’s labeled A the growth was going down. But then point A showed the cycle growing slowly.

D is incorrect because point a isn’t the highest point, point A and point C are equal.

An apple pie requires 7 apples and an apple tart requires 4 apples

To set up this problem, we can let p represent the number of apples needed for one pie and t represents the number of apples needed for one tart.

We can then set up the following system of equations to represent the information given in the problem:

2p + 6t = 38

2p + 8t = 46

To solve this system, we can subtract the first equation from the second to eliminate the p terms:

2p + 8t - 2p - 6t = 46 - 38

2t = 8

t = 4

Substituting this value back into the first equation, we get:

2p + 6(4) = 38

2p = 14

p = 7

Thus, an apple pie requires 7 apples and an apple tart requires 4 apples.

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Answer:

y = 3x - 2

Step-by-step explanation:

Slope intercept form is y= mx+b

m being the slope

and b being the y intercept

we have the slope, but we need to find the y intercept. first, plot (1,1) you can use a sheet of graph paper, demos or just a little grid on scrap paper. then you want to create another point on your graph using the slope, in this case 3 would be 3/1 so you go from your plotted point 3 units down, then 1 unit to the left. keep on plotting points until you identify your y intercept, where a plotted point goes through y axis. this is -2, which will be B in the equation.

Answer:

.05 yd

Step-by-step explanation:

5/91.44=.05468066

When a freeway is 600 miles long, its length on a map will be 6 inches, and 50 miles will be represented by 0.5 inches.

A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls) There are 1 in 4 boys and 3 in 4 girls. 0.25 are male (by dividing 1 by 4). The relationship between two quantities is constant for all values when they are proportional, which indicates that as one quantity rises, the other rises as well. An illustration would be the ratio of a circle's circumference to its diameter, which is equal to pi.

Here,

Length of freeway=600 miles

if 0.5 inch make 50 miles in map,

600:x=50:0.5

=0.5*600/50

=6 inch

The length of the freeway on the map will be 6 inch when it is 600 miles long and 0.5 inch make 50 miles in map.

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Answer:

Power functions are functions where y = x^n where "n" is any real constant number.

Answer: 50.5

Step-by-step explanation: To calculate the median, you add up all the #'s to get 801. Then you divide 801 by how many #'s there are (in this case 16) Then, you would get an answer close to 50.5, but since this is a test score, it would be around 51%. I hoped this helped you!

Molly would need 43 feet of fencing to fence the triangular area.

What is Pythagorean theorem?

Pythagorean theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle).

According to the given question:

We can see that the area is in the form of a right-angled triangle. Since only two sides of the triangle is given, we could utilize Pythagoras theorem to our advantage to find out the third side.

In Δ ABC

AB² (base) + AC² (perpendicular) = BC² (hypotenuse)

Substituting the values of AB as 10 feet and BC as 18 feet

AC²= BC²- AB ²

∴AC²= 224  

∴AC= 14.96 cm

Amount of fence required by molly would equal perimeter of the area.

⇒The perimeter of the area would equal AB+BC+AC

⇒Summing up the values of AB, AC and BC

Perimeter= 10+14.96+18= 42.96 feet which is about 43 feet

Hence, Molly would require 43 feet of fencing.

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By solving a linear equation, it can be calculated that

Time for which new machine prints = 64 minutes

What is linear equation?

Equation shows the equality between two algebraic expressions by connecting the two algebraic expressions by an equal to sign.

A one degree equation is known as linear equation.

For the new machine

The line passes through (0, 0) and (6, 75)

Slope = [tex]\frac{75 - 0}{6-0}[/tex] = 12.5

Equation of line for new machine

y - 0 = 12.5(x - 0)

y = 12.5x

Equation of line for old machine

y = 9x

Now,

During the 14 minutes, the number of pages printed = 12.50 [tex]\times[/tex] 14 = 175

Number of pages printed earlier = 1250 - 175 = 1050

By the problem,

12x + 9x = 1050

21x = 1050

x = [tex]\frac{1050}{21}[/tex]

x = 50

Time for which new machine prints = 50 + 14 = 64 minutes

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Answer:

1/16IB

Step-by-step explanation:

Differnce means to subtract which means we need to subract the two fractions. First find the the lowest of common factor for both fractions. In this case it is 16 because 4 can go into 16 four times as 4x4=16.

Since we know that 4 goes into 16 four times we get the 4 and then multiply it by the numerator (the top number of the fraction) for the fracion 1/4. Now the fraction 1/4 is 4/16. Keep the 3/16 the same as the denominator  (bottom number of fraction) is already our common factors between the two fractions.

Work Shown:

Orginal Equation: 1/4-3/16

Get common Denomiantors: Convert 4 to 16. 4 goes into 16 four times, multiply by 4 by numerator. 4X1=4.

Final Step: Since both are now common fractions just subtract the numerators keeping the denominators the same of the fractions.

4/16-3/16= 1/16

Final Answer: 1/16IB

Yes. The duration of the ride can be represented as a function of the time Jayden got on the bus.

We have given that,

Time (minutes)           18  20  20  22  22  23  25  30

Duration (minutes)    38  40  42   41   42  44  47   52

As we can see the duration of the ride seems to increase as the amount of time after 8 am that Jayden got on the bus.

This means that the duration of the ride and the time Jayden got on the bus have the linear relationship.

Therefore the duration of a ride can be represented as a function of the time Jayden got on the bus.

Following are the function of a time:

s(t) = distance a particle travels from tome 0 to t.

v(t) = velocity of a particle at time t.

a(t) = acceleration of a particle at time t.

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Using trigonometric ratios, we have been able to deduce that;

BE = 10 cm; sin B = 3/5 and cos B = 4/5

The basic trigonometric ratios for an angle θ are expressed as;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

Now, looking at the given triangle NEB, we are given that;

NE = 7.5 cm

tan B = 3/4

angle E = 90°

If tan B = 3/4, then it means that;

3/4 = NE/BE

since NE = 7.5, then we can say that;

3/4 = 7.5/BE

BE = (7.5 * 4)/3

BE = 10 cm

By Pythagoras theorem, we can say that BN = 12.5 cm.

Thus;

sin B = opposite/hypotenuse = 7.5/12.5 = 3/5

cos B = adjacent /hypotenuse = 10/12.5 = 4/5

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Using trigonometric ratio, the side BN is 12.5 units, sin and cos angle of B are 36.9 degrees respectively

The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). In geometry, trigonometry is a branch of mathematics that deals with the sides and angles of a right-angled triangle. Therefore, trig ratios are evaluated with respect to sides and angles.

In this problem;

A)

Tan B = 3/4

BN = ?

Tan B = NE / BE

3/4 = NE / BE

BE = NE / 3/4

BE = 7.5 / 0.75

BE = 10

The side BN can found using Pythagoras theorem

BN² = BE² + NE²

BN² = 10² + 7.5²

BN = 12.5

B)

Sin B = NE / NB

sin B = 7.5 / 12.5

sin B = 0.6

B = sin⁻¹(0.6)

B = 36.9°

C)

Cos B = BE / NB

cos B = 10 / 12.5

B = cos⁻¹(10/12.5)

B = 36.9°

The length of Bn is 12.5cm, and sin and cos angle of B is 36.9 degrees respectivelt

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Answer:

12 meters

Step-by-step explanation:

The length of the volleyball net is "L"

   6  + 1/2 L   = L    <======this is given , solve for L

    6 = 1/2 L

     L = 12 Meters

Reaction rates decline with time because the reactants are consumed over time, reducing the number of reactant particles available for reaction.

The components that are involved in a chemical reaction are known as reactants. They are the beginning ingredients that are changed into new compounds, known as products, through a chemical process. The left side of a chemical equation is often used to represent reactants, whereas the right side is used to indicate products. The quantity and kind of reactants and products affect the kind of chemical reaction that takes place. Pure substances, such as elements or compounds, or combinations of substances, can function as reactants with time. In a chemical reaction, reactants are changed into products by chemical bonds being broken and formed, releasing or absorbing energy in the process.

How to solve?

As the number of reactant particles decreases, the rate of reaction also decreases. This phenomenon can be observed in many chemical reactions, including reactions that involve the decay of radioactive elements.To correctly process data on the decline of reaction rates with time, it is important to choose the appropriate data points for linear regression. This typically involves selecting data points at regular intervals, such as every minute or every hour, to accurately represent the decline in reaction rates over time. By choosing the right data points and performing linear regression, it is possible to accurately model the decline in reaction rates and make predictions about future reactions.

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On solving the question, we got that - The dimensions of the rug should be 18 feet and 24 feet

By multiplying the rectangle's length by its breadth, we may get its area.

Equiangular quadrilaterals, like rectangles, are known in this way. This is so because a rectangle is a four-sided quadrilateral shape with parallel sides that are equal to one another and four corners with 90o angles. Rectangles are sometimes known as equiangular quadrilaterals since all of its angles are 90 degrees.

the parameters provided;

The room is 20 feet by 26 feet in size.

432 feet2 is the largest area rug she can afford.

If the floor will be a uniform stripe around the rug, then each dimension should have its uniform extra length of floor =  (y).

[tex]Area = Length X Width\\432 = ((20=y) X (26-y)\\432 = 520 - 20y - 26y + y^2\\y^2 - 46y+88=0\\[/tex]

here, a= 1, b= -46, c =88

[tex]y = \frac{-b +/- \sqrt{b^2 - 4ac} }{2a} \\\\[/tex]

[tex]y = \frac{-(-46) +/- \sqrt{(-46)^2 - 4X1X88} } {2} \\\\y = 44 \\or\\ y = 2[/tex]

Therefore, Length = 20-2 = 18 ft

and width = 26-2 = 24ft

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13>10 is the answer on the number line.

Explanation:

x.6 + 4x -1 > 10

6.6 = 4.6 - 1

36 = 24 - 1

36 - 23 = 13>10.

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13>10 is the answer on the number line.

A system of linear equations has a solution, which is the quantity that makes all of the equations true. This is the intersection of the two graphs for two variables and two equations. The two variables in the two equations will have a solution at these coordinates.

When x=6, x will remain 6 regardless of what y is. Therefore, simply draw a vertical straight line through (6, 0), parallel to the y axis, and extend it on both sides. The finished graph is there.

x.6 + 4x -1 > 10

6.6 = 4.6 - 1

36 = 24 - 1

36 - 23 = 13>10.

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The equations that represent the hanger are -

x + x + x = 1 + 1 + 1 + 1 + 1 + 1 and 3x = 6.

Given are the expressions as given in the question.

The equations that represent the hanger are -

x + x + x = 1 + 1 + 1 + 1 + 1 + 1

and

3x = 6

Therefore, the equations that represent the hanger are -

x + x + x = 1 + 1 + 1 + 1 + 1 + 1 and 3x = 6.

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