Minh paints 12 rooms to paint 3 he needs 2 cans of paint . He wants to know how many cans he needs in all. Which tools would be appropriate for Minh to use to solve the problem? Choose all that apply

A rather lengthy solution using a neat method I just learned relying on complex analysis.

First observe that

[tex]e^{-x^2} \cos(2x^2) = \mathrm{Re}\left[e^{-x^2} e^{i\,2x^2}\right] = \mathrm{Re}\left[e^{a x^2}\right][/tex]

where [tex]a=-1+2i[/tex].

Normally we would consider the integrand as a function of complex numbers and swapping out [tex]x[/tex] for [tex]z\in\Bbb C[/tex], but since it's entire and has no poles, we cannot use the residue theorem right away. Instead, we introduce a new function [tex]g(z)[/tex] such that

[tex]f(z) = \dfrac{e^{a z^2}}{g(z)}[/tex]

has at least one pole we can work with, along with the property (1) that [tex]g(z)[/tex] has period [tex]w[/tex] so [tex]g(z)=g(z+w)[/tex].

Now in the complex plane, we integrate [tex]f(z)[/tex] along a rectangular contour [tex]\Gamma[/tex] with vertices at [tex]-R[/tex], [tex]R[/tex], [tex]R+ib[/tex], and [tex]-R+ib[/tex] with positive orientation, and where [tex]b=\mathrm{Im}(w)[/tex]. It's easy to show the integrals along the vertical sides will vanish as [tex]R\to\infty[/tex], which leaves us with

[tex]\displaystyle \int_\Gamma f(z) \, dz = \int_{-R}^R f(z) \, dz + \int_{R+ib}^{-R+ib} f(z) \, dz = \int_{-R}^R f(z) - f(z+w) \, dz[/tex]

Suppose further that our cooked up function has the property (2) that, in the limit, this integral converges to the one we want to evaluate, so

[tex]f(z) - f(z+w) = e^{a z^2}[/tex]

Use (2) to solve for [tex]g(z)[/tex].

[tex]\displaystyle f(z) - f(z+w) = \frac{e^{a z^2} - e^{a(z+w)^2}}{g(z)} = e^{a z^2} \\\\ ~~~~ \implies g(z) = 1 - e^{2azw} e^{aw^2}[/tex]

Use (1) to solve for the period [tex]w[/tex].

[tex]\displaystyle g(z) = g(z+w) \iff 1 - e^{2azw} e^{aw^2} = 1 - e^{2a(z+w)w} e^{aw^2} \\\\ ~~~~ \implies e^{2aw^2} = 1 \\\\ ~~~~ \implies 2aw^2 = i\,2\pi k \\\\ ~~~~ \implies w^2 = \frac{i\pi}a k[/tex]

Note that [tex]aw^2 = i\pi[/tex], so in fact

[tex]g(z) = 1 + e^{2azw}[/tex]

Take the simplest non-zero pole and let [tex]k=1[/tex], so [tex]w=\sqrt{\frac{i\pi}a}[/tex]. Of the two possible square roots, let's take the one with the positive imaginary part, which we can write as

[tex]w = \displaystyle -\sqrt{\frac\pi{\sqrt5}} e^{-i\,\frac12 \tan^{-1}\left(\frac12\right)}[/tex]

and note that the rectangle has height

[tex]b = \mathrm{Im}(w) = \sqrt{\dfrac\pi{\sqrt5}} \sin\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \sqrt{\dfrac{\sqrt5-2}{10}\,\pi}[/tex]

Find the poles of [tex]g(z)[/tex] that lie inside [tex]\Gamma[/tex].

[tex]g(z_p) = 1 + e^{2azw} = 0 \implies z_p = \dfrac{(2k+1)\pi}2 e^{i\,\frac14 \tan^{-1}\left(\frac43\right)}[/tex]

We only need the pole with [tex]k=0[/tex], since it's the only one with imaginary part between 0 and [tex]b[/tex]. You'll find the residue here is

[tex]\displaystyle r = \mathrm{Res}\left(\frac{e^{az^2}}{g(z)}, z=z_p\right) = \frac12 \sqrt{-\frac{5a}\pi}[/tex]

Then by the residue theorem,

[tex]\displaystyle \lim_{R\to\infty} \int_{-R}^R f(z) - f(z+w) \, dz = \int_{-\infty}^\infty e^{(-1+2i)z^2} \, dz  = 2\pi i r \\\\ ~~~~ \implies \int_{-\infty}^\infty e^{-x^2} \cos(2x^2) \, dx = \mathrm{Re}\left[2\pi i r\right] = \sqrt{\frac\pi{\sqrt5}} \cos\left(\frac12 \tan^{-1}\left(\frac12\right)\right)[/tex]

We can rewrite

[tex]\cos\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \sqrt{\dfrac{5+\sqrt5}{10}}[/tex]

so that the result is equivalent to

[tex]\sqrt{\dfrac\pi{\sqrt5}} \cos\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \boxed{\sqrt{\frac{\pi\phi}5}}[/tex]

Answer:

  8 < x

  see attached for the graph

Step-by-step explanation:

You want the compound inequality and the graph that represents the solution to the system ...

  2x-5>3 and -4x+7<-25

The first inequality has the solution ...

  2x -5 > 3 . . . . . given

  2x > 8 . . . . . . . add 5

  x > 4 . . . . . . . divide by 2

The second inequality has the solution ...

  -4x +7 < -25

  -4x < -32 . . . . . . subtract 7

  x > 8 . . . . . . . . divide by -4

The solution is the intersection of these solution spaces:

  4 < x    and   8 < x

The intersection is ...

  8 < x . . . . . solution to the compound inequality

The graph is attached.

We have the next operation

the result of this operation rounded to nearest hundredths place is 57.86

The dragon would take 2.16 seconds to eat a human weighing 90 kilograms.

Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.

Let x seconds it would take to eat a human weighing 90kilograms

500 kg →  12 seconds

90 kg →  x

500/90 = 12/x

x = (90×12)/500

x = 2.16 seconds

Thus, the dragon would take 2.16 seconds to eat a human weighing 90 kilograms.

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Answer: if you're asking how tall it will stand at 3 years of age it would be 6.

Step-by-step explanation: 2 times 3 equals 6.

Given the expressions:

You need to remember that the Distributive Property states that:

• In order to simplify the first expression, you only need to apply the Distributive Property, because there is a number multiplying a Sum. Therefore, get:

• In order to simplify the second expression, you need to follow these steps:

1. Apply the Distributive Property:

2. Solve the Addition:

Hence, the answers are:

• The first problem requires to use only the Distributive Property to simplify it. It is required because there is a number multiplying a Sum:

Notice that the second expression can be simplified using the Distributive Property as the first step.

• The second problem can be simplified by applying the Distributive Property and then adding the numbers.

Answer:

Part 1 is B, Part 2 is D

If you want an explanation I can give one :)

Answer:

The equation is W = 1600 + .03S

Answer: [tex]f(-2)=13, g(-6)=-8[/tex]

Step-by-step explanation:

[tex]f(-2)=-2(-2)^3 -3=13\\\\g(-6)=2(-6)+4=-8[/tex]

The values of functions f(-2) and g(-6) are 13 and -8 respectively.

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

It is given in the question  that:-

[tex]f (x) = -2x^3 - 3[/tex]

g(x) = 2x + 4

We have to find the values of f(-2) and g(-6).

Putting x = -2 in f(x), we get,

f(-2) = [tex]-2(-2)^3 - 3[/tex] = -2*(-8) - 3 = 16 - 3 = 13

Putting x = -6 in g(x), we get,

g(-6) = 2(-6) + 4 = -12 + 4 = -8

Hence, the values of functions f(-2) and g(-6) are 13 and -8 respectively.

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All the options occurred as a result of Roman expansion following the Punic Wars except; B: It allowed many Romans to buy large farming estates

The three Punic Wars between Carthage and Rome took place over about a century, starting in 264 B.C. and it ended with the event of the destruction of Carthage in the year 146 B.C.

Now, at the time the First Punic War broke out, Rome had become the dominant power throughout the Italian peninsula, while Carthage–a powerful city-state in northern Africa–had established itself as the leading maritime power in the world. The First Punic War commenced in the year 264 B.C. when Rome expressed interference in a dispute on the island of Sicily controlled by the Carthaginians. At the end of the war, Rome had full control of both Sicily and Corsica and this meant that the it emerged as a naval and a land power.

In the Second Punic War, the great Carthaginian general Hannibal invaded Italy and scored great victories at Lake Trasimene and Cannae before his eventual defeat at the hands of Rome’s Scipio Africanus in the year 202 B.C. had to leave Rome to be controlled by the western Mediterranean as well as large swats of Spain.

In the Third Punic War, we saw that Scipio the Younger led the Romans by capturing and destroying the city of Carthage in the year 146 B.C., thereby turning Africa into yet another province of the mighty Roman Empire.

Thus, we can see that  the cause of the Punic wars is that the Roman republic grew, so they needed to expand their territory by conquering other lands, including Carthage.

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Perfoming each operation, we have:

By interpreting the vertex form of the equation of parabola, the focus of the curve is equal to F(x, y) = (2, 0). (Correct choice: D)

In this problem we find the equation of a parabola in standard form, which has to be rearranged into its vertex form in order to determine the coordinates of its focus.

The focus of a parabola is a point outside the curve such that the least distance from any point of the parabola and the least distance between that the point on the parabola and directrix are the same.

First, rearrange the polynomial into its vertex form by algebraic handling:

y = (1 / 8) · (x² - 4 · x - 12)

y + (1 / 8) · 16 = (1 / 8) · (x² - 4 · x - 12) + (1 / 8) · 16

y + 2 = (1 / 8) · (x² - 4 · x + 4)

y + 2 = (1 / 8) · (x - 2)²

y + 2 = [1 / (4 · 2)] · (x - 2)²

Second, determine the vertex and the distance between the vertex and the focus:

The parabola has a vertex of (h, k) = (2, - 2) and vertex-to-focus distance of 2.

Third, determine the coordinates of the focus:

F(x, y) = (h, k) + (0, p)

F(x, y) = (2, - 2) + (0, 2)

F(x, y) = (2, 0)

The focus of the parabola is equal to F(x, y) = (2, 0).

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Answer:

please provide options

Step-by-step explanation:

The value of  k = -0.079

The Fahrenheit temperature of the turkey, to the nearest degree, after 5.5 hours =  T(5.5) = 159.865

What is Temperature?

Temperature is a numerical expression of how hot a substance or radiation is. There are three different types of temperature scales: those that are defined in terms of the average, like the SI scale;

Newton's law of cooling states that T=Ta + (To-Ta)e-kt

T = Ta + (To-Ta)e-kt

Ta is the object's ambient temperature.

To equal the starting temperature.

t = the number of hours.

after t hours, temperature equals T.

decay constant is k.

According to the formula,

To = 70 and

Ta = 325,

therefore

(To -Ta) = 70 - 325.

(To -Ta)           = -255

T = 325 + (-255 e^kt) or

325 -255e^kt.

After 2.5 hours, the turkey reaches a temperature of 116°F.

In other words,

T(2.5) = 116 F.

Consequently,

116 = 325 - 255e^k2.5

Following that,

116 - 325 = - 255e^2.5k

Now,

-209 = -255e^2.5k

e^2.5k = -209/-255

Using the function 'ln,'

now

ln(e2.5k) = ln(209 / 255)

Then, 2.5k = -0.198929

The value of k is now given by k = -0.198929 / 2.5, which is -0.079571.

The result is k = -0.079.

After 5.5 hours, the temperature is

T(5.5) = 325 - 255e^(-0.079 x 5.5)

Then, = 325 - 255e^(-0.4345)

Now,

         = 325 - (165.1350) (165.1350)

T(5.5) = 159.865 is the result.

Consequently, T(5.5) = 159.865 degrees Fahrenheit is the turkey's exact temperature after 5.5 hours.

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Answer:

Step-by-step explanation:

it is my first time doing dis so it is 12.

Answer:

x=20 and x=4

Step-by-step explanation:

The set of letters in the word 'woodpecker' using the most concise method is {c, d, e, k, o, p, r, w}

From the question, we are to write the set of letters of the given word.

The word is 'woodpecker'

We are to write the set using the listing (roster) method or the set builder notation.

The roster method or listing method is a way to show the elements of a set by listing the elements inside of brackets

Set builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy

The set builder notation is not a suitable method to list the elements of the given word.

The most concise method to list the elements of the given word, 'woodpecker', is the listing (roster) method.

Thus,

Using the listing (roster) method,

The set of letters of the given word is {c, d, e, k, o, p, r, w}

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Using the binomial distribution, there is a 0.8784 = 87.84% probability that at least 1 of them arrives late.

The probability mass function, giving the the probability of x successes is:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters of the distribution are given as follows:

In the context of this problem, of 20% shipments with 90% on time, the parameters are:

p = 0.9, n = 20.

The probability that all arrive on time is:

P(X = 20) = (0.9)^20 = 0.1216.

All on time and least one late are complementary events, hence the probability of at least one package arriving late is given as follows:

P(X < 20) = 1 - P(X = 10) = 1 - 0.1216 = 0.8784.

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On simplifying the given equation, the answer is [tex]\frac{5}{6}[/tex]

The fundamental guidelines and procedures to simplify any statement are as follows:

By multiplying factors, remove any grouping symbols like brackets and parenthesis.

If the words contain exponents, use the exponent rule to eliminate grouping.

Add or subtract like terms to combine them.

Add the constants together.

Given,

1 8/11 x 5/12 + 5/12 x 3/11

[tex]\frac{19}{11}[/tex]  ×  [tex]\frac{5}{12}[/tex]    +  [tex]\frac{5}{12}[/tex]   × [tex]\frac{3}{11}[/tex]

Taking [tex]\frac{5}{11 * 12}[/tex] as a common factor,

⇒ [tex]\frac{5}{11 * 12}[/tex] ( [tex]19 + 3[/tex])

⇒ [tex]\frac{5}{11 * 12}[/tex] × 22

On simplifying,

⇒ [tex]\frac{5}{6}[/tex]

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The maximum dimensions is a square of sides 36 ft by 36 ft;

The maximum area is 1,296 ft^2

Here, we want to know the diemensions of the largest area the farmer can enclose within the perimeter

Mathematically, the greatest dimension that can maximize the area of the plot is the shape being a square

In other words, we need the diemnsions of both sides to be equal so as to get the area

If the dimensions are equal, we can say the length and width are represented by x

The length is thus as follows;

The greatest possible or the maximum area is thus;

The coordinates of the point Y at the other endpoint is 41

From the question, the given parameters are

Because there is not y coordinates in the coordinates, we can make use of only the x coordinate

So, we have the following equation

Point = Proportion * (Y - X)

Substitute the known values in the above equation

So, we have the following equation

12 = 1/3 * (Y - 5)

Multiply through the equation by 3

So, we have the following equation

36 = Y - 5

Add 5 to both sides of the equation

So, we have the following equation

Y = 41

This means that the conclusion is the coordinate of the point Y is 41

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Answer:

I think it's D because it can't be factorized

We can find the x and y-intercept by substituting zero for x and y respectively.Part A

iven the eequation below;

When x=0

when y=0

Answer 1

Part B

When x=0

When y=0

Answer 2:

The savings account balance after six months with an 8% annual percentage rate and $200 monthly contributions is $202.6.

Savings Plan Formula

The future worth of a savings or investment account in which you make regular deposits can be calculated using the formula below.

EXAMPLE 1 Assume you put $100 into a savings account each month that pays 7.3% interest that is compounded monthly.

To determine the saving plan balance after 6 month

First of all we have to calculate the growth factor

Growth factor =(1+(0.08/6)) per month

Growth factor= 1.013

Now let calculate the savings plan balance after 6 month

Saving plan balance after 6

Months =$200.00x1.0133

             = $202.6

From conclusion the saving plan balance after 6 month is=$202.6

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Based on the percentage of fencers who said they used foil as their main weapon when surveyed, the number of fencers out of 28 fencers who would not use foil as their main weapon is 11 fencers

The random survey showed that in a given sample, there would be 60% of fencers who would prefer to use foil as their main weapon. This means that the percentage of fencers who would not use foil as their main weapon is:

= 1 - percentage who use foil as main weapon

= 1 - 60%
= 40%

If 28 fencers are randomly surveyed, the percentage of them who would not use foil as their main weapon would be expected to be:

= Number of fencers x Percentage who don't use foil as main weapon:

= 28 x 40%

= 11.2 fencers

= 11 fencers to the nearest whole number

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Answer:

c

Step-by-step explanation:

2-2/5=12/5

12/5=

2-2/5 is the mixed fraction for the number 12/5

well, the key and you are absolutely correct, so you're golden

[tex]\stackrel{reciprocals}{\cfrac{3}{2}\cdot \cfrac{8}{1}\cdot \cfrac{1}{5}}\implies \cfrac{3}{5}\cdot \cfrac{8}{2}\cdot \cfrac{1}{1}\implies \cfrac{3}{5}\cdot \cfrac{4}{1}\cdot \cfrac{1}{1}\implies \cfrac{12}{5}\implies \cfrac{10+2}{5} \\\\\\ \cfrac{10}{5}+\cfrac{2}{5}\implies 2+\cfrac{2}{5}\implies 2\frac{2}{5} ~~ \checkmark[/tex]

If the child uses the marbles at random what is the chance that the child gets less than six red marbles is given as   0.9561. See the calculation below.

Note that we are given the following:

p = 20% = 0.20

1 - p = 1 - 0.20 = 0.80

n = 14

Utilizing the binomial formula, we have:

P(X = x) = ((n! / x! (n - x)!) * px * (1 - p)n - x

P(X < 6) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)+  P(x = 4) + P(x = 5)  

Plugging in the value of n, we have:

=  ((14! / 0! (14)!) * 0.200 * (0.80)14 + ((14! / 1! (13)!) * 0.201 * (0.80)13 + ((14! / 2! (12)!) * 0.202 * (0.80)12 + ((14! / 3! (11)!) * 0.203 * (0.80)11 + ((14! / 4! (10)!) * 0.204 * (0.80)10 + ((14! / 5! (9)!) * 0.205 * (0.80)9

=  0.9561

As a result, the probability that the youngster would receive less than six red marbles is provided as 0.9561 if the marbles are used at random.

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Answer:

Maximum length = 30 cm

Step-by-step explanation:

Perimeter of a rectangle = 2 × (length + width)

According to the question,

2 × (length + width) < 71 cm (It can be 70 cm at maximum)

length + width < 71/2 cm

length + width < 36 cm

Since, width = 5 cm,

length + 5 cm < 36 cm

length < 36 - 5 cm

length < 31 cm

Therefore, the maximum length can be 30 cm

1/4(n-6)=1/4n-3/2. We can say left side is equal to right side.

Given that,

1/4(n-6)=1/4n-3/2

We have to find the n value.

We have to 1st prove left side is equal to right side

We can split an equation into two parts, the left hand side and the right hand side. We refer to the LHS and RHS of the equation in abbreviated form. If the LHS and RHS are equal, then the equation is true; otherwise, the equation does not hold for at least some real number values.

First take the left side which is

1/4(n-6)

We multiply 1/4 to n-6

1/4n-1/4(6)

Now we divide 6/4

1/4n-3/2

See the right side

1/4n-3/2

Therefore,

left side is equal to right side.

1/4(n-6)=1/4n-3/2

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The  three equivalent ratios are given below .

What are equivalent ratios?

Ratios that can be streamlined or decreased to the same value are said to be equivalent. In other words, if one ratio can be written as a multiple of the other, then they are said to be equivalent. The ratios 1:2 and 4:8, 3:5 and 12:20, 9:4 and 18:8, and others are some examples of analogous ratios. To put it another way, two ratios are said to be equivalent if one of them can be written as the multiple of the other. Therefore, we must multiply the two values (antecedent and consequent) by the same number in order to obtain the equivalent ratio of another ratio. This approach is comparable to the approach for determining equivalent fractions.

1:9 , 99:11 , 9:11 , 1:99

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