What is the length of AB?​

The expansion of  (x – 3)^5 is     [tex]x^5-15x^4+90x^3-270x^2+405x-243[/tex]

The Binomial Theorem is the method of expanding an expression that has been raised to any finite power. A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc. Binomial Expression: A binomial expression is an algebraic expression that contains two dissimilar terms.

Pascal's Triangle is a never-ending equilateral triangle in which the arrays of numbers arranged in a triangular manner. The triangle starts at 1 and continues placing the number below it in a triangular pattern

Using Binomial theorem,

=[tex]\sum _{i=0}^5\binom{5}{i}x^{\left(5-i\right)}\left(-3\right)^i[/tex]

=[tex]x^5-15x^4+90x^3-270x^2+405x-243[/tex]

Using Pascal Triangle  for (x-1)

                                               1

                                       1    1              

           1    2    1            

        1    3    3    1        

     1    4    6    4    1      

  1    5    10    10    5    1  

1    6    15    20    15    6    1

Accordingly replacing 1 with 3 we get

= [tex]x^5-15x^4+90x^3-270x^2+405x-243[/tex]

Thus the expansion of  (x – 3)^5 is      [tex]x^5-15x^4+90x^3-270x^2+405x-243[/tex]

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Answer:

B. -2

Step-by-step explanation:

To find a slope of a line, you have to identify x and y variables:

We use the formula;

m = ∆y/∆x

= (y2 - y1)/(x2 - x1)

Substitute the values into the formula above.

m = (7 - 3)/(2-4)

= 4/-2

= -2

Therefore, the slope of the line is passing through points (4,3) and (2,

7) is -2.

Answer:

x=4 and y=−9

Step-by-step explanation:

Problem:

Solve 7x+4y=−8;−8x−5y=13

Steps:

I will solve your system by substitution.

(You can also solve this system by elimination.)'

7x+4y=−8;−8x−5y=13

Step: Solve 7x+4y=−8 for x:

7x+4y+−4y=−8+−4y(Add -4y to both sides)

7x=−4y−8

7x/7 =  -4y-8/7 (Divide both sides by 7)

x=-4/7y + -8/7

Step: Substitute -4/7y + -8/7 for x in −8x−5y=13:

−8x−5y=13

-8(-4/7y + -8/7) −5y=13

-3/7y + 64/7 =13(Simplify both sides of the equation)

-3/7y + 64/7 + -64/7 = 13 + -64/7 (Add (-64)/7 to both sides)

-3/7y = 27/7

-3/7y/-3/7 = 27/7/-3/7 (Divide both sides by (-3)/7)

y=−9

Step: Substitute −9 for y in x= -4/7y + -8/7:

x = -4/7y + -8/7

x = -4/7 (-9) + -8/7

x=4(Simplify both sides of the equation)

Answer:

x=4 and y=−9

Answer:

  (a)  (x, y) = (3, 2)

Step-by-step explanation:

The correct answer to the given system of equations can be found by trying the offered choices.

The equations are ...

The first equation tells you the sum of the x and y values is 5. Here's what we get for a sum from the different answer choices:

  A: 3 +2 = 5 . . . . a viable choice

  B: 4 +4 = 8 . . . . doesn't work

  C: 6 +1 = 7 . . . . doesn't work

  D: 3 +5 = 8 . . . . doesn't work

__

Checking (x, y) = (3, 2) in the other equation, we have ...

  7(3) +3(2) = 21 +6 = 27 . . . . as required

The solution is x = 3; y = 2.

__

Additional comment

Often, the easiest way to find the answer to a multiple-choice question is to see which answer works to solve the problem. It can sometimes be easier to check the answer than to develop it in the first place.

Substitution can work for this problem to find the correct y-value. (The answer choices differ by y-values, so knowing y tells the answer.)

  7(5 -y) +3y = 27 . . . . substitute x=5-y

  35 -4y = 27 . . . . . . simplify

  8 = 4y . . . . . . . . . add 4y-27

  2 = y . . . . . . . . . divide by 2. Matches the first answer choice.

Answer:

12.22 cm

Step-by-step explanation:

We'll be using cosine laws to solve for QR side:

[tex]\displaystyle{QR^2 = QP^2+RP^2-2QP\cdot RP \cdot \cos P}[/tex]

We know that QP = 13 cm, RP = 4 cm and cosP = 70°. Hence:
[tex]\displaystyle{QR^2=13^2+4^2-2(13)(4)\cos 70^{\circ}}[/tex]

Then evaluate the expression:

[tex]\displaystyle{QR^2=169+16-104 \cos 70^{\circ}}\\\\\displaystyle{QR^2=185-104\cos 70^{\circ}}\\\\\displaystyle{QR^2=185-35.57}[/tex]

Square root both sides, since length can only be positive. The negative side will be cancelled:

[tex]\displaystyle{\sqrt{QR^2}=\sqrt{185-35.57}}\\\\\displaystyle{QR=\sqrt{185-35.57}}\\\\\displaystyle{QR=12.22}[/tex]

Therefore, the length of QR will be around 12.22 cm or 12 cm when rounded to nearest integer.

By applying the law of cosine, the length of side BM is equal to 7.05 cm..

A polygon can be defined as a two-dimensional geometric shape that comprises straight line segments and a finite number of sides. Also, some examples of a polygon include the following:

Note: The number of sides (n) of a pentagon is 5.

In Geometry, the interior angle of both a regular and irregular polygon is given by this formula:

Interior angle = 180 × (n - 2)/n

Interior angle = 180 × (5 - 2)/5

Interior angle = 180 × 3/2

Interior angle = 108°.

For the central angle, we have:

Central angle, O = 360/5

Central angle, O = 72°.

By applying the theorem of intersecting secants, angle OCB will be given by this formula:

<OCB = ½ × (180 - 72)

<OCB = ½ × 108

<OCB = 54°.

In order to determine the length of side BM, we would apply the law of cosine:

cos(θ) = Adj/Hyp

cos54 = BM/12

BM = 12 × cos54

BM = 12 × 0.5878

Side BM = 7.05 cm.

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Answer: (2, -1)

Step-by-step explanation:

[tex]y=-3x^2 +12x-13\\\\y=-3(x^2 - 4x)-13\\\\y=-3\left(\left(x-2 \right)^{2}-4 \right)-13\\\\y=-3(x-2)^{2}+12-13\\\\y=-3(x-2)^2 - 1[/tex]

So, the vertex is (2, -1).

Answer:

Arithmetic

Step-by-step explanation:

Arithmetic Sequence is a sequence that has common difference - a common difference is same difference in a sequence. If we take next term to subtract the previous term and you keep getting same difference or value then that's a common difference. It can be mathematically expressed as an equation of:

[tex]\displaystyle{d=a_{n+1}-a_n}[/tex]

Now, I'll show why the sequence is arithmetic - take the sequence to subtract for common difference:

Notice how we keep getting -16 after subtracting previous term with next term. Hence this sequence is arithmetic sequence for sure!

Geometric Sequence is a sequence with common ratio - a common ratio is similar to common difference except ratio is all about division. Since common difference is expressed as subtraction of next term and previous term then common ratio is division of next term and previous term which can be mathematically expressed as:

[tex]\displaystyle{r = \dfrac{a_{n+1}}{a_n}}[/tex]

However this sequence is not geometric sequence because if you divide next term with previous term then you'll keep getting difference result and common ratio means having same ratio or constant ratio for entire sequence.

Now see why this sequence cannot be geometric sequence:

Notice that none of values are equal and thus, this sequence doesn't have common ratio which leads to not being geometric sequence.

Answer:

(32÷(10-8)÷2)-3

Step-by-step explanation:

(32÷(10-8)÷2)-3

(32÷2÷2)-3

(16÷2)-3

8-3 = 5

✅

Answer: b

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The cross products of the proportion are equal.

According to the question, Sonny's equation can be written as follows.

16/x = 48/15

Now, the variable x is replaced by the constant 5. So the equation can now be written as follows.

16/5 = 48/15

When two fractions are equal to one another as shown above, the constants are cross multiplied with one another in order to form an equation.

16*15= 48*5

The values of both the products are equal which is 240.

Therefore, because the cross products of the proportion are equal.

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The p-value from the hypothesis test is 0.142 i.e., greater than the given significance level of 0.05. So, the null hypothesis is not rejected. The z-score for the given sample is 1.471.

The decision rule based on p-value states,

Since it is given that the valve would produce a mean pressure of 5.4 pounds/square inch. I.e., μ = 5.4 p/si

So, Defining the hypothesis:

Null hypothesis H0: μ = 5.4

Alternative hypothesis Ha: μ ≠ 5.4

It is given that,

The valve was tested on 24 engines. I.e., Sample size n = 24

The sample mean X = 5.7

Standard deviation σ = 1.0 and

The significance level = 0.05

Since the population distribution is approximately normal,

the test statistic is calculated as follows:

z = (X - μ)/(σ/[tex]\sqrt{n}[/tex])

On substituting the value,

z = (5.7 - 5.4)/(1.0/[tex]\sqrt{24}[/tex])

  = (0.3)/0.204

  = 1.471

Fron this z-score, the p-value is calculated as 0.142.

Since, the value of p > 0.05 (significance level), the null hypothesis is not rejected.

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Answer :

( 6 y^4 + 3 y^2 -7) - (12 y^4 - y^2 + 5)

Step-by-step explanation:

(6 y^4+ 3 y^2-7) - (12 y^4 - y^2 + 5 )

=6 y^4 +3 y ^2 - 7 - 12 y^4 + y^2 - 5

=6 y ^4 - 12 y ^4 +3 y ^2 + y ^2 - 7 -5

= -6 y ^4 + 4 y ^2- 12

Hope this helps!

Answer:

Exact form: 19 multiply 4200^28 + 1972^31 / 19

Decimal form: 3.55481422 multiply 10^101

Answer:

A 3 1 B 2-4 C4-3 then work x and y graph

Answer:

113.04 ft^2

Step-by-step explanation:

area of the circle= 3.14*(6)^2

Using the probability concept, the correct option of the probability of a day in May being cloudy in Sacramento is:

A. 0.43, because 13 of the 30 days were cloudy.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

For Sacramento, 30 days were considering, of which 13 days were cloudy, hence the probability of a day in May being cloudy in Sacramento is given as follows:

p = 13/30 = 0.43.

Hence the correct option is:

A. 0.43, because 13 of the 30 days were cloudy.

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Answer:

x ^ (1/3)

Step-by-step explanation:

solution:

the answer is in the picture I have sent.

I hope this will help you

Considering the least common factor of 15 and 18, it is found that they will depart from the central station at the same time at 11 AM.

To find the time that passes between the events happening at the same time, we need to find the least common multiple of the periods.

In this problem, the periods are of 15 and 18, hence their lcm is found as follows:

15 - 18|2

15 - 9|3

5 - 3|3

5 - 1|5

1 - 1

Hence:

lcm(15,18) = 2 x 3 x 3 x 5 = 90 minutes.

They will depart from the central station at the same time in 90 minutes from 9:30 AM, hence at 11 AM.

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The average rate of change in weight is 3/8.

According to the statement

Weight of baby during birth = 3 kg

Weight of baby after 8 months = 6 kg

Average rate of change of in weight is calculated by

Average rate = Increased weight - Real weight / Time period

Average rate = 6-3 / 8

Average rate = 3/8

So, the average rate of change in weight is 3/8.

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The pound was stronger in 2008 against the dollar than in 2017.

The Pound to greenback rate reached a high of $2.649 on the sixth Mar 1972. That remains the most powerful the Pound has been towards USD because it freely floated in 1971.

One might count on that the strongest economies might have the most powerful international currencies; but, that is not always the case. It turns out that lengthy-term movements in forex costs are extra essential than alternate quotes, that's why the British pound is worth extra than the U.S. dollar.

What topics is how the price of that forex adjustments over the years relative to different currencies. Historically, for over 20 years one U.S. dollar has been really worth less than one British pound. As of July 31, 2020, the greenback is sitting around 1.32 to at least one pound. 2  this is down from 1.68 in may additionally 2014 and 1. forty in March 2018.

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The Volume of cylinder represented by the option B.

According to the statement

we have given that the volume formula of cylinder is V= (pi)r^2h, and we have to find that the which expression verify the volume formula for the cylinder given in diagram.

So,

We know that height of the cylinder is given by h = 2x + 7 and

radius r = x - 3.

We know that the formula of volume of cylinder is:

Volume of a cylinder = (pi)r^2h

and Substituting the given values in the above formula

And the volume becomes

Volume = (pi)r^2h

Volume = (pi)( x-3 )^2 (2x+7)

Volume = (pi) ( x^2 + 9 - 6x ) (2x+7)

Volume = (pi) ( 2x^3 + 7x^2 +18x +63 - 42x)

So, The Volume of cylinder represented by the option B.

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Disclaimer: This question is incomplete. Please find the full content below.

Question:

The volume of a cylinder is given by the formula V= pi^2h, where r is the radius of the cylinder and h is the height: which expression represents the volume of this cylinder?

a) Volume = (pi) ( 3x^3 + 7x^2 +14x +63 - 42x)

b) Volume = (pi) ( 2x^3 + 7x^2 +18x +63 - 42x)

c) Volume = (pi) ( 7x^3 + 7x^2 +11x +63 - 42x)

d) Volume = (pi) ( 11x^3 + 7x^2 +13x +63 - 42x)

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The vendor has to sell 88 gingerbread houses to earn a profit of $665.60 and there is no chance that the vendor will earn $1500.

Given an equation showing profits of A Christmas vendor as

P=-0.1[tex]g^{2}[/tex]+30g-1200.

We have to find the number of gingerbread houses that the vendor needs to sell in order to earn profit of $665.60 and $1500.

To find the number of gingerbread houses we have to put P=665.60 in the equation given which shows the profit earned by vendor.

665.60=-0.1[tex]g^{2}[/tex]+30g-1200

0.1[tex]g^{2}[/tex]-30g+1200+665.60=0

0.1[tex]g^{2}[/tex]-30g+1865.60=0

Divide the above equation by 0.1.

[tex]g^{2}[/tex]-300g+18656=0

Solving for g we get,

g=[300±[tex]\sqrt{(300)^{2}-4*1*18656 }[/tex]]/2*1

g=[300±[tex]\sqrt{90000-74624}]/2[/tex]

g=[300±[tex]\sqrt{15376}[/tex]]/2

g=(300±124)/2

g=(300+124)/2       , g=(300-124)/2

g=424/2,  g=176/2

g=212,88

Because 212 is much greater than 88 so vendor prefers to choose selling of 88 gingerbread houses.

Put the value of P=1500 in equation P=-0.1[tex]g^{2}[/tex]+30g-1200.

-0.1[tex]g^{2}[/tex]+30g-1200=1500

0.1[tex]g^{2}[/tex]-30g+1500+1200=0

0.1[tex]g^{2}[/tex]-30g+2700=0

Dividing equation by 0.1.

[tex]g^{2}[/tex]-300g+27000=0

Solving the equation for finding value of g.

g=[300±[tex]\sqrt{300^{2} -4*1*27000}[/tex]]/2*1

=[300±[tex]\sqrt{90000-108000}] /2[/tex]

=[300±[tex]\sqrt{-18000}[/tex]]/2

Because [tex]\sqrt{-18000}[/tex] comes out with an imaginary number so it cannot be solved for the number of gingerbread houses.

Hence the vendor has to sell 88 gingerbread houses to earn a profit of $665.60 and there is no chance that the vendor will earn $1500.

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Answer:

C

Step-by-step explanation:

The temperature rises by 30 degrees, so lets say that it goes from 0 to 30.

The temperature is now 30. But then, it falls by 30 degrees. It is now 0 again. The quantities even out, or make 0.

Answer:

Step-by-step explanation:

A parallel line will have the same slope as the line on the graph. A perpendicular line will have a slope that is the opposite reciprocal of the slope of the graphed line.

The line on the graph rises 1 grid square for each run of 2 grid squares to the right. Its slope is ...

  m = rise/run = 1/2

The opposite reciprocal of this slope is ...

  -1/(1/2) = -2

A perpendicular line will have a slope of -2.

The slope-intercept form of the equation for a line is ...

  y = mx +b

where the slope is m and b is the y-intercept. For the purpose here, we don't care about the y-intercepts of any of the lines. We only care about the slope: the coefficient of x.

This means the equations we're looking for are of the form ...

  parallel line: y = 1/2x + b

  perpendicular line: y = 2x +b . . . . . for some constant b

Of the equations offered, the only one with an x-coefficient of 1/2 is ...

  y = 1/2x +3

Of the equations offered, the only one with an x-coefficient of -2 is ...

  y = -2x +1

Answer:

Step-by-step explanation:

you are welcome

The calculation of the dividends per share on each class of stock for each of the four years for Lightfoot Inc. are:

                                 1st Year       2nd Year       3rd Year      4th Year

Preferred stock          $0.19            $0.32            $0.99          $0.50

Common stock           $0                 $0                $0.79          $2.69

How is dividend per share calculated?

The dividend per share for each class of stock can be computed by dividing the total dividend distributed by the number of shares outstanding.

For cumulative preferred stock, the unpaid dividends in a year are carried forward for distribution when profits are made.  Note also that preferred stock dividends enjoy preferential payment before common stock dividends.

Cumulative preferred stock = 30,000 shares

Par value of preferred stock = $25

Dividend rate = 2%

The total value of preferred stock = $750,000 ($25 x 30,000)

Expected annual dividend for preferred stockholders = $15,000 ($750,000 x 2%)

Common stock = 38,000 shares

Par value of common stock = $75

The total value of ordinary stock = $2,850,000 ($75 x 38,000)

                                        1st Year       2nd Year       3rd Year      4th Year

Dividends distributed    $5,700           $9,600       $59,720       $117,220

Preferred stock             $5,700            $9,600      $29,700        $15,000

Preferred stock per share $0.19           $0.32        $0.99             $0.50 ($15,000/30,000)

Common stock                  $0                   $0         $30,020      $102,220

Common stock per share $0                  $0            $0.79           $2.69 ($102,220/38,000)

Thus, the dividends distribution of Lightfoot Inc. shows that the common stockholders started earning dividends after the second year.

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Calculate the dividends per share on each class of stock for each of the four years. Round all answers to two decimal places. If no dividends are paid in a given year, enter "O."

Check the picture below.

part A

since the base of the triangular base is 16, and the altitude "h" splits the base in two equal halves, half that is just 8, so we're looking at a right triangle with a hypotenuse of 17 and a side of 8, thus

[tex]\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2 - a^2}=b \qquad \begin{cases} c=\stackrel{hypotenuse}{17}\\ a=\stackrel{adjacent}{8}\\ b=\stackrel{opposite}{h}\\ \end{cases} \\\\\\ \sqrt{17^2-8^2}=h\implies \sqrt{225}=h\implies \boxed{15=h}[/tex]

part B

well, the prism is simply two triangles and 3 rectangles, le's simply add their areas.

[tex]\stackrel{two~triangles}{2\left[ \cfrac{1}{2}(\stackrel{base}{16})(\stackrel{height}{15}) \right]}~~ + ~~\stackrel{two~rectangles}{2(20)(17)}~~ + ~~\stackrel{one~rectangle}{(20)(16)} \\\\\\ 240~~ + ~~680~~ + ~~320\implies \text{\LARGE 1240}[/tex]

Answer:

step 2

Step-by-step explanation:

Erin's mistake is in step 2 it should be 8*2 first and 16* 5 which equals to 86

The dimensions of the given cylinder are that its height is 12cm, while its base radius is 4cm, which implies its diameter is 8 cm.

Thus, the first option, The height is 12 cm, and the diameter of the base is 8 cm, is the right choice.

The volume of a cylinder is calculated using the formula, V = πr²h, where V is its volume, r is its radius, and h is its height.

The expression used by Aiden to calculate the volume of a cylinder is π(16)(12), or, π(4²)(12).

Comparing this expression with the general formula of the volume of a cylinder, V = πr²h, where V is its volume, r is its radius, and h is its height, we can say that the radius can be 4 units and the height can be 12 units.

Assuming the units to be cm, we can say that:

The dimensions of the given cylinder are that its height is 12cm, while its base radius is 4cm, which implies its diameter is 8 cm.

Thus, the first option, The height is 12 cm, and the diameter of the base is 8 cm, is the right choice.

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The complete question is:

Aiden evaluated the expression π(16)(12) to find the volume of a cylinder. What could be the dimensions of the cylinder?

The height is 12 cm, and the diameter of the base is 8 cm.

The height is 12 cm, and the area of the base is 16 cm².

The height is 16 cm, and the diameter of the base is 6 cm.

The height is 16 cm, and the area of the base is 12 cm².