Henry is trapped at the roof top of a burning building, which is 24 feet high. In order to rescue him, his father, Diego, must find a ladder which will be used to cross the river, which is 7 feet wide, and go up directly to the top of the building. Help Diego to find the appropriate length of ladder from the other side of the river to the top of the building

Answer:

14

Step-by-step explanation:

all you have to do is multiply 15x3 which is 45 and subtract 3 or 1 which is 14 she has 14 friends.

Answer:

Area of triangle = 1/2 × base × height

Step-by-step explanation:

The two big traingles have same dimensions..

its height = 24ft

base=32 ft

so area of one big traingle = 1/2 × 32 × 24

= 384

similarly area of another big traingle = 384 as it has same dimensions.

now one small traingle... both have same dimension

its base =7

height=24

its area = 1/2 × 7 ×24

= 84

another small traingle area similarly = 84

Total area = 384+ 384 +84+ 84

=936 units ...

Thank you

Hope it helps you

This calculation shows that Mary Ward's lease term is 36 months.

The question is asking us to find out for how many months Mary Ward's car lease lasts. To do so, we need to calculate the total monthly payments and then subtract the initial $1600 fee to find the total cost of the monthly payments alone. Then, we divide this amount by the monthly lease payment to determine the total number of months for the lease duration.

First, we deduct the initial fee from the total lease cost:

Total lease cost - Initial fee = Total of monthly payments

$26,800 - $1,600 = $25,200

Next, we divide this result by the monthly lease payment to find out the number of months:

$25,200 / $700 = 36 months

Therefore, the lease agreement lasts for 36 months, which is typically the term for most vehicle leases.

Answer:

  y = -9

Step-by-step explanation:

Standard form of the equation of a line is ...

  ax + by = c

When that line is a horizontal line, this can be reduced to ...

  y = c

The value of c must be the same as the y-coordinate of the point you want this line to go through.

  y = -9 . . . . . . horizontal line through (3, -9)

Answer: the solutions of the equation are

x = - 1

x = 8

x = 9

Step-by-step explanation:

The given cubic equation is expressed as

x³ - 16x² + 55x + 72 = 0

The first step is to test for any value of x that satisfies the equation when

x³ - 16x² + 55x + 72 = 0

Assuming x = - 1, then

- 1³ - 16(-1)³ + 55(-1) + 72 = 0

- 1 - 16 - 55 + 72 = 0

0 = 0

It means that x + 1 is a factor.

To determine the other factors, we would apply the long division method. The steps are shown in the attached photo. Looking at the photo, we would factorize the quadratic equation which is expressed as

x² - 17x + 72 = 0

x² - 9x - 8x + 72 = 0

x(x - 9) - 8(x - 9) = 0

(x - 9)(x - 8) = 0

Answer:

Step-by-step explanation:

According to the flaring, the triangle expressed in the figure is formed.

Now we have a vector A and vector B, whose components are Ax, Ay and Bx and By respectively.

To calculate these components you must use the following formulas:

Ax = A * cos [angle a] = 32 * cos 37 ° = 25.6

Ay = A * sin [angle a] = 32 * sin 37 ° = 19.3

Bx = B * cos [angle a] = 21 * cos 23 ° = 19.3

By = B * sin [angle b] = 21 * sin 23 ° = - 8.2

In the figure it can be seen that vector B is the result of vector A and vector C

Thus:

B = A + C

reorganizing is:

C = B - A

Now to calculate Cx and Cy, we will do it with the previously calculated components:

Cx = Bx - Ax = 19.3 - 25.6 = -6.3

Cy = By - Ay = -8.2 - 19.3 = -27.5

Now to calculate the value of vector C, we apply the following formula:

C ^ 2 = Cx ^ 2 + Cy ^ 2

Rearranging:

C = (Cx ^ 2 + Cy ^ 2) ^ (1/2)

C = [(-6.3 ^ 2) + (27.5 ^ 2)] ^ (1/2) = 28.2

Then the distance would be 28.2 meters.

Answer:

u => 4,028

Step-by-step explanation:

To find the answer, we have the following formula:

u => m - t (alpha, n-1) * [sd / (n) ^ (1/2)]

where m is the mean.

where sd is the standard deviation.

where n is the sample size.

t is a parameter that depends on the confidence interval and the sample size.

alpha = 1 - ci

ci = 90% = 0.9

Therefore, alpha = 1 - 0.9 = 0.1.

n - 1 = 25 - 1 = 24

So it would come being t (0.1, 24), if we look in the table, which I will attach the value of t is equal to 1.318.

We know the rest of the values, m = 4.05; sd = 0.08; n = 25

u => 4.05 - 1,318 * [0.08 / (25) ^ (1/2)]

u => 4.028

Which means that the interval with a 90% confidence of the wall thickness measurement is:

u => 4.028

Answer:

D. (-7, 7)

Step-by-step explanation:

Count left 7 on X-axis = -7

Count up 7 on Y-axis = +7

The coordinate pair is (-7, 7)

Answer:

The tank would hold 40 gallons of water when full.

Step-by-step explanation:

The tank is already 3/4 full of water

If Henry added 5 gallons, it will be 7/8 full.

Therefore the fraction added by Henry =7/8-3/4=1/8

It means that 1/8 of the total=5 gallons of water

Let the total capacity of the tank=x

[tex]\frac{1}{8}Xx=5 gallons\\ x=8X5=40 gallons[/tex]

The tank would hold 40 gallons of water when full.

Answer:

P [  1689  ≤   X  ≤  2267 ]  = 54,88 %

Step-by-step explanation:

Normal Distribution

Mean        μ₀  =  1730

Standard Deviation      σ  = 257

We need to calculate  z scores for the values   1689     and      2267

We apply formula for z scores

z =  ( X -  μ₀ ) /σ

X = 1689     then

z = (1689 - 1730)/ 257      ⇒ z = - 41 / 257

z  = -  0.1595

And from z table we get  for  z =  - 0,1595

We have to interpolate

        - 0,15          0,4364

        - 0,16          0,4325

Δ  =   0.01           0.0039

0,1595  -  0,15  =  0.0095

By rule of three

0,01                  0,0039

0,0095                 x ??      x  =  0.0037

And    0,4364  -  0.0037  = 0,4327

Then    P [ X ≤ 1689 ]  =  0.4327     or    P [ X ≤ 1689 ]  = 43,27 %

And for the upper limit  2267  z  score will be

z  =  ( X - 1730 ) / 257       ⇒  z =  537 / 257

z  =  2.0894

Now from z table   we find  for score   2.0894

We interpolate and assume  0.9815

P [ X ≤ 2267 ]  =  0,9815

Ths vale already contains th value of   P [ X ≤ 1689 ]  =  0.4327

Then we subtract  to get    0,9815  -  0,4327   = 0,5488

Finally

P [ 1689  ≤   X  ≤  2267 ]  =  0,5488  or  P [  1689  ≤   X  ≤  2267 ]  = 54,88 %

Answer:

The answer to your question is Positive

Step-by-step explanation:

Function

                  g(x) = x⁴ - 8x³ + x² + 42x

To know if the function is positive or negative in the interval (0, 3), look for two numbers between this interval and evaluate the function.

The numbers I chose were 1 and 2

-  g(1) = (1)⁴ - 8(1)³ + (1)² + 42(1)

         = 1 - 8 - 1 + 42

        = + 36    positive

-  g(2) = (2)⁴ - 8(2)³ + (2)² + 42(2)

          = 16 - 64 + 4 + 84

          = + 40

Conclusion

The function is positive in the interval (0, 3)  

Answer:

45,000

Step-by-step explanation:

If P=45,000−1,000m,

where P=The number of people left in the stadium m minutes after the end of the game

We want to determine how many people were present when the game ended but before people started to leave.

Note that immediately the game ended,

m=0

Therefore, the number of people left in the stadium

P=45000−(1000 X 0)

P=45000

There were 45,000 people.

Answer:

9 boxes of chopsticks.

Step-by-step explanation:

The first thing is to calculate the number of chopsticks spent in a day, the problem tells us that they are 15 ^ 2 = 15 * 15 = 225 chopsticks daily.

To calculate the number of chopsticks in 30 days, it is to calculate the previous amount by 30:

225 * 30 = 6750 chopsticks spent in one month.

To know how many boxes you should order is to divide the total number of chopsticks in a month and the number of chopsticks that a box brings that are 750:

6750/750 = 9 boxes of chopsticks.

Therefore you should order exactly 9 boxes of chopsticks for the restaurant's need.

Answer:

The correct option is

d) Five-sixths = StartFraction 5 times 3 over 6 times 3 EndFraction = Fifteen-eighteenths = 15/18 ÷ 1 = 15/18 ÷ 3/3 = 5/6

Step-by-step explanation:

Which method can be used to find a fraction equivalent to Five-sixths?

a) Five-sixths = StartFraction 5 times 2 over 6 times 3 EndFraction = Ten-eighteenths

Here 10/16 = 5/8 ≠ 5/6

b) Five-sixths = StartFraction 5 times 3 over 6 times 4 EndFraction = StartFraction 15 over 24 EndFraction

15/24 = 5/8 ≠ 5/6

c) Five-sixths = StartFraction 5 times 5 over 6 times 6 EndFraction = StartFraction 25 over 36 EndFraction

25/36 ≠ 5/6

d) Five-sixths = StartFraction 5 times 3 over 6 times 3 EndFraction = Fifteen-eighteenths

15/18 ⇒ 15/18÷3/3 = 5/6

note that 3/3 = 1

Answer:

Its D buddy!

Step-by-step explanation:

Cuz 3 People said it was D.

Step-by-step explanation:

To find the inverse of a function y = f(x), switch x and y, then solve for y.

y = ln(x + 5)

x = ln(y + 5)

eˣ = y + 5

y = eˣ − 5

y = eˣ + 4

x = eʸ + 4

x − 4 = eʸ

y = ln(x − 4)

Step-by-step explanation:

If the segments bisect each other, they will share the same midpoint.

Midpoint formula is:

(x, y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Midpoint of MN is:

(x, y) = ((-5 + 5)/2, (6 + -2)/2)

(x, y) = (0, 2)

Midpoint of PQ is:

(x, y) = ((5 + -5)/2, (3 + 1)/2)

(x, y) = (0, 2)

So the segments do bisect each other.

Answer:

The answer to your question is Yes.

Step-by-step explanation:

The math in this problem is that we need to use the Pythagorean theorem to solve it. Pythagorean theorem is part of trigonometry a branch of Maths.

Data

base = 9 ft                      

length = 37 ft

height = ?

Pythagorean theorem

                                     c² = a² + b²

length = c

base = a

height = b

                                   37² = 9² + b²

-Solve for b

                                    b² = 37² - 9²

-Simplify

                                    b² = 1369 - 81

                                    b² = 1288

-Result

                                   b = 35.88

-Conclusion

The ladder will reach a height higher than 35 ft.  

Answer:

  $1550.69

Step-by-step explanation:

Deposits get added and withdrawals and bill payments get subtracted from the balance. The new balance is ...

  $1349.85 -80 +699.65 -215.70 -53 -49.76 -100.35 = $1550.69

Answer:

4

Step-by-step explanation:

{ }

{Apple}

{Banana}

{Apple, Banana}

Answer:

C

Step-by-step explanation:

2.Why did people initially believe that a horse named "Clever Hans" could do math and that a procedure called "facilitated communication" could enable autistic children to type out complex messages?

A.In both cases, the unusual behavior was initially demonstrated under carefully controlled conditions

B.In both cases, scientists falsified their data

C.In both cases, people initially failed to recognize alternative explanations for the observed behavior

D.Both cases were very elaborate hoaxes designed by con artists who were intentionally trying to fool people

People believed in Clever Hans' mathematical abilities and in facilitated communication for autistic children due to misinterpretation of cues and expectations, known as the Clever Hans Effect and the influence of facilitators, respectively.

People initially believed that a horse named Clever Hans could do math and that a procedure called facilitated communication could enable autistic children to type out complex messages due to a misunderstanding of the animals' and participants' capabilities.

Regarding Clever Hans, the horse's trainer, Wilhelm von Osten, would ask Hans a math problem, to which the horse would tap out the answer with his hoof.

It was discovered that Hans was not performing mathematical calculations; rather, he was responding to physical and perhaps subconsciously given cues from his human observers, a phenomenon now known as the Clever Hans Effect.

Similarly, efforts to interpret autistic children's communications were influenced by the expectations and subtle suggestions from the facilitators in facilitated communication.

Answer: the rate of the cabin cruiser in calm water is 12 mph.

the rate of the current is 3 mph.

Step-by-step explanation:

Let x represent the rate of the cabin cruiser in calm water.

Let y represent the rate of the current.

A cabin cruiser traveling with the current went 45 miles in 3 h. This means that the total speed is (x + y) miles per hour.

Distance = speed × time

Distance travelled with the current is

45 = 3(x + y)

Dividing through by 3, it becomes

15 = x + y - - - - - - - - - - -1

Traveling against the current it took 5 hours to go the same distance. This means that the total speed is

(x - y) miles per hour.

Distance travelled against the current is

45 = 5(x + y)

Dividing through by 5, it becomes

9 = x - y - - - - - - - - - - -2

Adding equation 1 to equation 2, it becomes

24 = 2x

x = 24/2

x = 12 mph

Substituting x = 12 into equation 1, it becomes

15 = 12 + y

y = 15 - 12

y = 3 mph

      Rate of the cabin cruiser will be 12 miles per hour and the rate of the current is 3 miles per hour.

  Let the speed of the current = r miles per hour

And the speed of the cabin cruiser in the calm water = c miles per hour

Speed of the cabin cruiser against the current = (c - r) miles per hour

Speed of the cabin cruiser with the current = (c + r) miles per hour

Since, expression for the speed is given by,

Speed = [tex]\frac{\text{Distance}}{\text{Time}}[/tex]

Therefore, speed of the cabin cruiser with the current = [tex]\frac{45}{3}[/tex] = 15 miles per hour

So the equation for the speed will be,

c + r = 15 ------ (1)

Time taken to cover 45 miles against the current = 5 hours

Therefore, equation for this situation will be,

c - r = [tex]\frac{45}{5}[/tex]

c - r = 9 ------- (2)

Now solve this system of equations,

Add equation (1) and (2),

(c + r) + (c - r) = 15 + 9

2c = 24

c = 12 miles per hour

Substitute the value of 'c' in equation (1),

12 - r = 9

r = 12 - 9

r = 3 miles per hour

     Therefore, speed of the cabin cruiser will be 12 miles per hour and the rate of the current is 3 miles per hour.

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Step-by-step explanation:

Given : An equation for the depreciation of a car is given by [tex]y = A(1-r)^t[/tex], where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%.

To find : Approximately how old is the car?

Solution :

The value of a car is half what it originally cost i.e. [tex]y=\frac{1}{2} A[/tex]

The rate of depreciation is 10% i.e. r=10%=0.1

Substitute in the equation, [tex]y = A(1-r)^t[/tex]

[tex]\frac{1}{2} A= A(1-0.1)^t[/tex]

[tex]\frac{1}{2}= (0.9)^t[/tex]

Taking log both side,

[tex]\log(\frac{1}{2})=t\log (0.9)[/tex]

[tex]t=\frac{\log(\frac{1}{2})}{\log (0.9)}[/tex]

[tex]t=6.57[/tex]

[tex]t\approx 6.6[/tex]

Therefore, the car is about 6.6 years old.

Answer:

The car is 6.5 years old

Step-by-step explanation:

An equation for the depreciation of a car is given by [tex]y = A(1 - r)^t[/tex]

y = current value of the car

A = original cost

r = rate of depreciation

t = time in years

The value of a car is half what it originally cost

So, [tex]y = \frac{A}{2}[/tex]

The rate of depreciation is 10% = 0.1 =r

Substitute the values in equation

[tex]\frac{A}{2} = A(1 - 0.1)^t[/tex]

[tex]\frac{1}{2} =(1 - 0.1)^t[/tex]

[tex]\frac{1}{2} =(0.9)^t[/tex]

[tex]0.5=0.9^t[/tex]

t=6.57

Hence The car is 6.5 years old

In a state where the license plates have two letters followed by four numerals, the probability of getting a license plate that has a repeated letter or digit is approximately 51.5%.

To solve this question, we need to understand the fundamental counting principle and the concept of probability. First, let's explore the total number of possibilities for a license plate of this format. The choice for each spot on the license plate is independent of each other. Therefore, there are 26 possibilities (26 letters in the English alphabet) for each of the first two spots, and 10 possibilities for each of the next four spots (0-9 digits). The total number of possibilities is thus 26 * 26 * 10 * 10 * 10 * 10, or 67,600,000 combinations.

Next, we will calculate all the combinations in which no letters or digits are repeated. There are 26 options for the first letter, 25 remaining options for the second letter, 10 options for the first digit, and then 9, 8, and 7 options for the next three positions, giving us 26 * 25 * 10 * 9 * 8 * 7 = 327,600,000 combinations.

To get the probability of a repeated letter or digit, subtract the non-repeating combinations from the total combinations, divide by the total, and then multiply by 100 to get a percentage. So the probability is (676,000,000 - 327,600,000) / 676,000,000 * 100, which gives approximately 51.5%

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Answer:

One

Step-by-step explanation:

9/sin(34) = 6/sinC

sinC = 0.372795269

C = 21.9, 158.1

Since 158.1 + 34 = 192.1 > 180

Only one triangle can be formed with angle 21.9° at C

Sin(146° + ∠B) cannot exceed 1, there is no solution for b that satisfies the triangle inequality. Therefore, no distinct triangle can be drawn with the given measurements.

To determine the number of distinct triangles that can be drawn with the given measurements using the Law of Sines, we need to consider the Ambiguous Case (also known as the SSA case or the "side-side-angle" case).

The Law of Sines states:

sin(A) / a = sin(B) / b = sin(C) / c

Given:

m∠A = 34°

a = 9

c = 6

We want to find ∠B and side b.

First, find ∠B using the Law of Sines:

sin(B) / b = sin(A) / a

sin(B) / b = sin(34°) / 9

Now, find sin(34°):

sin(34°) ≈ 0.5592

Now, solve for sin(B) by multiplying both sides by b:

sin(B) = (0.5592 * b)

Next, find ∠C:

Since the sum of angles in a triangle is 180°:

∠C = 180° - ∠A - ∠B

∠C = 180° - 34° - ∠B

Determine the value of side b:

Using the Law of Sines again, we have:

sin(C) / b = sin(A) / a

sin(C) / b = sin(34°) / 9

Find sin(C):

sin(C) = sin(180° - 34° - ∠B)

We can now use the fact that sin(C) = sin(180° - angle) to find sin(C):

sin(C) = sin(180° - 34° - ∠B) = sin(146° + ∠B)

Now, solve for b by multiplying both sides by b:

sin(146° + ∠B) = (sin(34°) / 9) * b

Since sin(146° + ∠B) cannot be greater than 1 (the maximum value for sine), the number of distinct triangles will depend on the possible values of b.

In this case, we have sin(146° + ∠B) = (sin(34°) / 9) * b, and since sin(34°) ≈ 0.5592, the maximum value for (sin(34°) / 9) * b is approximately 0.5592 * 6 ≈ 3.3552.

Since sin(146° + ∠B) cannot exceed 1, there is no solution for b that satisfies the triangle inequality. Therefore, no distinct triangle can be drawn with the given measurements.

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Answer:

If there were 2 people, they would each pay $34. If there were 4 people, they would each pay $18

Answer:

He have to eat each piece in  [tex]\frac{1}{3} \ minute.[/tex]

Step-by-step explanation:

Given:

Brian claims that he can eat a pie that is divided into six equal pieces In two minutes.

If he can eat the whole pie in two minutes.

Now, to find the time he have to eat for each piece.

So, to solve by using unitary method:

If Brian can eat 6 pieces in = 2 minutes.

Then, he can eat 1 piece in = [tex]\frac{2}{6}[/tex] [tex]=\frac{1}{3} \ minute.[/tex]

Therefore, he have to eat each piece in [tex]\frac{1}{3} \ minute.[/tex]

Final answer:

Brian has 20 seconds to eat each of the six equal pieces of the pie if he eats the whole pie in two minutes.

Explanation:

If Brian can eat an entire pie divided into six equal pieces in two minutes, we calculate the time taken to eat each piece by dividing the total time by the number of pieces. So, it is a simple division problem: we divide 2 minutes by 6 pieces to find out how long Brian has to eat each piece.

To find the time for each piece, we perform the calculation: 2 minutes ÷ 6 pieces = 0.333 minutes per piece. Since there are 60 seconds in a minute, to convert this to seconds, we multiply by 60.

Therefore: 0.333 minutes × 60 seconds = 20 seconds per piece. So, Brian has 20 seconds to eat each piece of pie.

Answer:

  A, C

Step-by-step explanation:

You want the volume of a cylinder with diameter 5 m and height 10 m.

The volume is given by the formula ...

  V = πr²h . . . . . . . where r = radius = half the diameter

The diameter is 5 m, so the radius is 5/2 = 2.5 m. Using the given values in the formula, we find the volume of the cylinder to be ...

  V = π(2.5)²(10) m³ . . . . . . matches choice A

  = 62.5π m³ . . . . . . . . . . . matches choice C

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[tex]\frac{1}{3}[/tex]Answer:

2/9

Step-by-step explanation:

given that Tyler selects one card from the three(4,5, and a King), and rolls a number cube.

We find that A the event of selecting one card and B getting a number on rolling a number cube are independent events.

No of cards = 3

Prob of selecting 5 from 3 cards = [tex]P(1,2,3, or 4) = \frac{2}{3}[/tex]

When rolling a number cube (assuming fair) there is equally likely for all numbers to appear from 1 to 6

Prob of getting 5 =[tex]\frac{1}{6}[/tex]

Prob of getting less than 5 =

Since these two events are independent,

the probability that she selects the 5, and rolls a number less than 5

= Product of probabilities

= [tex]\frac{1}{3}[/tex]*[tex]\frac{2}{3}[/tex]

=[tex]\frac{2}{9}[/tex]

The simplified expressions for the area and perimeter of the rectangle are 16x and 4x, respectively.

To simplify the expressions for the area and perimeter of the rectangle, we can extract common factors:

Area: [tex]\(8(x + x) = 8 \times 2x = 16x\)[/tex]

Perimeter: [tex]\(2(x + x) = 2 \times 2x = 4x\)[/tex]

Therefore, the simplified expressions are:

Area: 16x

Perimeter: 4x

Answer:

[tex]\left(3+x\right)^4:\quad x^4+12x^3+54x^2+108x+81[/tex]

Step-by-step explanation:

Considering the expression

[tex]\left(3+x\right)^4[/tex]

Lets determine the expansion of the expression

[tex]\left(3+x\right)^4[/tex]

[tex]\mathrm{Apply\:binomial\:theorem}:\quad \left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i[/tex]

[tex]a=3,\:\:b=x[/tex]

[tex]=\sum _{i=0}^4\binom{4}{i}\cdot \:3^{\left(4-i\right)}x^i[/tex]

Expanding summation

[tex]\binom{n}{i}=\frac{n!}{i!\left(n-i\right)!}[/tex]

[tex]i=0\quad :\quad \frac{4!}{0!\left(4-0\right)!}3^4x^0[/tex]

[tex]i=1\quad :\quad \frac{4!}{1!\left(4-1\right)!}3^3x^1[/tex]

[tex]i=2\quad :\quad \frac{4!}{2!\left(4-2\right)!}3^2x^2[/tex]

[tex]i=3\quad :\quad \frac{4!}{3!\left(4-3\right)!}3^1x^3[/tex]

[tex]i=4\quad :\quad \frac{4!}{4!\left(4-4\right)!}3^0x^4[/tex]

[tex]=\frac{4!}{0!\left(4-0\right)!}\cdot \:3^4x^0+\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1+\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2+\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3+\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4[/tex]

[tex]=\frac{4!}{0!\left(4-0\right)!}\cdot \:3^4x^0+\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1+\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2+\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3+\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4[/tex]

as

[tex]\frac{4!}{0!\left(4-0\right)!}\cdot \:\:3^4x^0:\:\:\:\:\:\:81[/tex]

[tex]\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1:\quad 108x[/tex]

[tex]\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2:\quad 54x^2[/tex]

[tex]\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3:\quad 12x^3[/tex]

[tex]\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4:\quad x^4[/tex]

so equation becomes

[tex]=81+108x+54x^2+12x^3+x^4[/tex]

[tex]=x^4+12x^3+54x^2+108x+81[/tex]

Therefore,