please help me with this

Let Ramanujan pick a complex number $a+bi$, where $a$ and $b$ are real numbers. Then the product of the numbers chosen by Ramanujan and Hardy is:

(

�

+

�

�

)

(

5

+

3

�

)

=

5

�

+

3

�

�

+

5

�

�

+

3

�

�

2

=

(

5

�

+

3

�

)

+

(

5

�

+

3

�

)

�

−

3

�

(a+bi)(5+3i)=5a+3ai+5bi+3bi

2

=(5a+3b)+(5b+3a)i−3b

We want this product to be equal to $32-8i$. Equating the real and imaginary parts gives us two equations:

5

�

+

3

�

=

32

(1)

5a+3b=32(1)

5

�

+

3

�

=

−

8

(2)

5b+3a=−8(2)

We can solve this system of equations to find $a$ and $b$. Multiplying equation (1) by 3 and equation (2) by 5, and subtracting the resulting equations gives:

15

�

+

9

�

−

25

�

−

15

�

=

−

96

15a+9b−25b−15a=−96

Simplifying:

−

16

�

=

−

96

−16b=−96

Therefore, $b=6$. Substituting $b=6$ into equation (1) gives:

5

�

+

18

=

32

5a+18=32

Therefore, $a=2$. Thus, Ramanujan picked the complex number $2+6i$.

Based on the given conditions and informations provided, the complex number that was picked by thr Ramanujan is calculated to be  2+6i.

Let Ramanujan pick a complex number a+bi, where a and b are real numbers. Then the product of the numbers chosen by Ramanujan and Hardy is:

(a+bi)(5+3i) = 5a+3ai+5bi+3bi² = (5a+3b)+(5b+3a)i-3b

We want this product to be equal to 32-8i. Equating the real and imaginary parts gives us two equations:

5a+3b=32 (1)

5b+3a=-8 (2)

We can solve this system of equations to find a and b. Multiplying equation (1) by 3 and equation (2) by 5, and subtracting the resulting equations gives:

15a+9b-25b-15a=-96

Simplifying:

-16b=-96

Therefore, b=6. Substituting b=6 into equation (1) gives:

5a+18=32

Therefore, a=2. Thus, Ramanujan picked the complex number 2+6i.

Learn more about Complex Number :

https://brainly.com/question/28007020

#SPJ4

From given ratio of boys and girls, the total number of students at the school is 1240.

What exactly is a ratio?

A ratio is a comparison of two quantities or values by division. It is commonly used to express the relationship between two numbers, such as the ratio of the number of boys to the number of girls in a classroom.

The ratio of two numbers a and b can be expressed as a/b or as the fraction a:b. For example, if there are 20 boys and 30 girls in a classroom, the ratio of boys to girls is 20/30, which can be simplified to 2/3 or written as the fraction 2:3.

Now,

Let's represent the number of boys as 11x and the number of girls as 9x, where x is a common factor.

From the given information, we know that:

11x = 9x + 124

Simplifying

11x - 9x = 124

2x = 124

x = 62

Now,

Number of boys = 11x = 11(62) = 682

Number of girls = 9x = 9(62) = 558

Therefore, the total number of students at the school is:

Total number of students = Number of boys + Number of girls = 682 + 558 = 1240

To know more about ratios visit the link

brainly.com/question/13419413

#SPJ1

Step-by-step explanation:

We can use the simple interest formula:

R = P(1 + rt)

where R is the balance, P is the principal, r is the interest rate, and t is the time.

For Investment R, we have:

R = 1,500(1 + 0.0275*8)

R = 1,980

For Investment S, we can use the formula:

R = P(1 + r)^t

R = 900(1 + 0.0275)^8

R = 1,116.92

a. The difference between the balance of the two investments after 8 years is:

1,980 - 1,116.92 = 863.08

b. The interest earned for Investment R is:

I = Prt

I = 1,500*0.0275*8

I = 330

For Investment S, we can calculate the interest using:

I = R - P

I = 1,116.92 - 900

I = 216.92

The difference between the interest earned on the two investments is:

330 - 216.92 = 113.08

c. To compare the return on investment, we can use the concept of compound interest.

For Investment R, the total value after 8 years is:

R = 1,500(1 + 0.0275*8)

R = 1,980

For Investment S, we can calculate the effective annual rate:

EAR = (1 + r/n)^n - 1

EAR = (1 + 0.0275/1)^1 - 1

EAR = 0.0275

The total value after 8 years is:

R = 900(1 + 0.0275)^8

R = 1,116.92

The return on investment for Investment R is:

ROI = (1 + r/n)^nt - 1

ROI = (1 + 0.0275/1)^1*8 - 1

ROI = 0.2229

The return on investment for Investment S is:

ROI = (1 + EAR)^t - 1

ROI = (1 + 0.0275)^8 - 1

ROI = 0.2329

Therefore, Investment S had the better return on investment.

Answer:

It is 9 1/4

Step-by-step explanation:

This is because 9 stays the same as the whole number, and 25 divided by 100 is 4. To check our work we can multiply 25 by 4 to get 100. Now we know we have the right answer.

"The beautiful part of learning is that no one can take it away from you" (:

Potentially hazardous cold food that requires time and temperature control for safety (TCS food) should not be kept at a temperature above 41°F (5°C) for more than 6 hours total time, according to the FDA Food Code.

According to the U.S. Food and Drug Administration (FDA) Food Code, potentially hazardous cold food that requires time and temperature control for safety (TCS food) should not be kept at a temperature above 41°F (5°C) for more than 6 hours total time.

After that, the food must be thrown out to prevent the growth of harmful bacteria. These food are harmful to the health of consumer.

To know more about TCS:

https://brainly.com/question/14477469

#SPJ4

Answer:

Here is the correct answer:

15 m, 20 m, 25 m

Answer:

I think it will help u (:

Step-by-step explanation:

Assuming the given measurement is the diameter of the sphere, we can find the radius as follows:Radius (r) = Diameter / 2 = 7.2 / 2 = 3.6 cmThe formula for the volume of a sphere is:V = (4/3) * π * r³Substituting the values, we get:V = (4/3) * 3.14 * (3.6)³

V = 4.19 * 46.656

V = 195.4 cm³Rounding this to the nearest tenth, we get:V ≈ 195.4 cm³Therefore, the volume of the sphere is approximately 195.4 cm³.

After answering the presented question, we can conclude that As a expression result, the laptop costs $2500 without GST.

An expression in mathematics is a collection of representations, digits, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a combination of the two can be used as an expression. Mathematical operators include addition, subtraction, rapid spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the representation of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.

To get the laptop's price without GST, divide the inclusive price by (1 + GST rate as a decimal).

As a result, the laptop's price without GST can be estimated as follows:

Price excluding GST = Price including GST / (1 + GST rate as a decimal)

Price excluding GST = 2675 / (1 + 0.07)

Without GST, the price is $2500.

As a result, the laptop costs $2500 without GST.

To know more about expression visit :-

https://brainly.com/question/14083225

#SPJ9

The p-value for the test of the claims is roughly 0.004.

To test the claim of the bulb patron, we can use a thesis test with the following null and indispensable suppositions.

We can use the binomial distribution to calculate the probability of observing 1050 or further imperfect bulbs out of,000 if the imperfect rate is 0.01 or lower. This probability is the p-value of the test.

To calculate the p-value, we need to use the binomial distribution and find the probability of observing 1050 or further imperfect bulbs out of,000 if the imperfect rate is 0.01 or lower. Using a binomial calculator or statistical software, we can find that the probability of observing 1050 or further imperfect bulbs is roughly 0.004.

Since this p-value is lower than the significant position of 0.05, we can reject the null thesis and conclude that the imperfect rate of the bulbs is more advanced than 0.01.

Learn more about p-value;

https://brainly.com/question/13786078

#SPJ4

The area of triangle PQR is 25/2 * √(5).

In geometry, a triangle is a two-dimensional shape with three straight sides and three angles. It is one of the basic shapes in geometry and has many important properties and applications.

The sum of the interior angles of a triangle is always 180 degrees, and the exterior angles sum to 360 degrees. The three sides of a triangle can be classified according to their lengths and their angles.

To find the area of triangle PQR, we can use the formula:

Area = (1/2) * base * height

where the base and height are the lengths of two sides of the right triangle that meet at a 90-degree angle.

We can use the distance formula to find the lengths of the sides of triangle PQR:

PQ = √[(x_Q - x_P)² + (y_Q - y_P)²] = √[(-1 - (-6))² + ((-1) - 9)²] = √[25 + 100] = 5√(5)

PR = √[(x_R - x_P)² + (y_R - y_P)²] = √[(-9 - (-6))² + (5 - 9)²] = √[9 + 16] = 5

QR = √[(x_R - x_Q)² + (y_R - y_Q)²] = √[(-9 - (-1))² + (5 - (-1))²] = √[64 + 36] = 2√20) = 4√(5)

Since triangle PQR is a right triangle, the base and height are PQ and PR, respectively. Therefore, the area of triangle PQR is:

Area = (1/2) * PQ * PR = (1/2) * 5√(5) * 5 = 25/2 * √(5)

So the area of triangle PQR is 25/2 * √(5).

To know more about area visit:

https://brainly.com/question/29174852

#SPJ1

Answer: To have infinitely many solutions, the equation must simplify to a true statement, such as 0=0.

So, let's simplify the given equation:

4(x-4) = 4x + ?

We need to choose a value for the constant on the right-hand side so that the equation simplifies to a true statement.

We can start by distributing the 4 on the left-hand side:

4x - 16 = 4x + ?

Now we can see that the 4x terms cancel out, leaving:

-16 = ?

We can choose any value for the constant on the right-hand side, and the equation will be true. For example:

-16 = -16

So the complete equation with infinitely many solutions is:

4(x-4) = 4x - 16

Step-by-step explanation:

Answer:

right angles

Step-by-step explanation:

Answer:

The number is 6.

Step-by-step explanation:

[tex]3x + 4 = 22[/tex]

[tex]3x = 18[/tex]

[tex]x = 6[/tex]

The number which is thought is 6.

An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.

Given that,

a number multiply it by 3 add 4 and get 22.

Let the unknown number be x.

First multiply the given number by 3.

It becomes 3x.

Add 4 to it.

It becomes 3x + 4.

The equation we get is,

3x + 4 = 22

Subtracting both sides by 4,

3x = 18

Dividing both sides by 3,

x = 6

Hence the value of x is 6.

Learn more about Equations here :

https://brainly.com/question/29657983

#SPJ2

The total surface area of the triangular prism is 200 square inches.

A triangular prism is a three-dimensional geometric shape that consists of two parallel triangular bases connected by three rectangular faces.

To find the surface area of the triangular prism using the net provided, we need to find the area of each face and then add them together.

Using formulas,

Area of rectangle= [tex]length\times width[/tex] square unit.

Area of triangle = [tex]\frac{1}{2}\times Base \times Height[/tex] square unit.

The blue rectangle has a length of 5 inches and a width of 4 inches, so its area is

=> 5 x 4 = 20 square inches.

The orange rectangle has a length of 12 inches and a width of 4 inches, so its area is

=> 12 x 4 = 48 square inches.

The red rectangle has a length of 13 inches and a width of 4 inches, so its area is

=> 13 x 4 = 52 square inches.

The green triangles have a base of 12 inches and a height of 5 inches, so each triangle's area is

=> [tex]\frac{1}{2}\times12\times5[/tex]=  30 square inches.

To find the total surface area, we add the area of each face:

=> 20 + 20 + 48 + 52 + 30 + 30 = 200 square inches.

Therefore, the surface area of the triangular prism is 200 square inches.

To learn more about triangular prism refer the below link

brainly.com/question/4680982

#SPJ1

Answer:

In the images

Hope this helps!

Step-by-step explanation:

Clockwise is the direction the arrow is pointing, counter-clockwise is the other direction.

Answer:

x=2

Step-by-step explanation:

33=32x-31

64=32x

2=x

Answer:  [tex]x = 2[/tex]

The recursive and explicit formulae of the given arithmetic expression are:

Recursive formular = [tex]a_{n+1} = a_{n-13}[/tex]

Explicit formular is Tn = 75 - 13n

You can determine the value of any term in a series by using an explicit formula. The integers are divided into the natural numbers, which are 1, 2, 3, 4, and so on. If you know the value of the (n-1)th term in a sequence, you may use a recursive formula to determine the value of the nth term in the sequence. Recursive formulas give the value of a specific phrase depending on the previous term, whereas explicit formulas give the value of a specific term based on the position.

Given:

a1 = 52

a2 = 75 = 52 - 13 = a1 - 13

Recursive formular = [tex]a_{n+1} = a_{n-13}[/tex]

a = 52, d = -13

F(11) = a + (n - 1)d = 52 + (n - 1)-13 = 52 - 13n + 13 = 75 - 13n

Explicit formular is Tn = 75 - 13n

To learn more about explicit formulae, visit:

https://brainly.com/question/12620593

#SPJ1

The arithmetic sequence has a common difference of 23, recursive rule of this sequence is xₙ = [tex]x_{n-1}[/tex] + 23 and explicit rule is xₙ = 23n + 29. The 11th term is 282.

An arithmetic sequence is a mathematical progression in which each term is generated by adding a constant value to the preceding term, with the exception of the first term. This constant value is known as the common difference, and it governs the pattern of the sequence. There are two types of arithmetic sequences, one is an increasing sequence and the other is a decreasing sequence. Increasing sequence is obtained if the common difference is a positive number. Decreasing sequence is obtained if the common difference is a negative number.

The general format of the recursive rule is xₙ = [tex]x_{n-1}[/tex] + d, where xₙ is the nth term, [tex]x_{n-1}[/tex] is the (n-1)th term and d is the common difference of the arithmetic sequence.

In the given sequence x₁ = 52 and x₂ = 75

Common difference can be calculated by subtracting first term from second term = x₂ - x₁ = 75 - 52 = 23

Hence recursive rule for this sequence is xₙ = [tex]x_{n-1}[/tex] + 23

The general format of explicit rule is xₙ = x₁ + (n-1) × d.

For this sequence it is xₙ = x₁ + (n-1) × 23 = 52 + 23n - 23 = 23n + 29

To find f(11) we can use the explicit rule

f(11) = 23 × 11 + 29 = 282

To know more about arithmetic sequence visit:

brainly.com/question/25715593

#SPJ1

For the point (5, 4) to the directrix, the distance to the directrix is equal to the distance to the focus. For the other points on the list, the distance to the directrix is equal to the distance to the focus. The distance to the directrix is equal to the distance to the focus, we can conclude that (-1, 5) is on the parabola.

A perfect, U-shaped curve is referred to by the term parabola in mathematics. It is described as the collection of all plane points that are equidistant from a focus and a fixed broadband (the directrix). The point where the parabola's curve changes direction is known as its vertex, and the line that runs through it and is perpendicular toward the directrix is known as its axis of symmetry. The behavior of electromagnetic waves, projectile motion, reflector and antenna forms, and other real-world phenomena are frequently modelled using parabolas.

To determine whether a point is on a parabola with a given focus and directrix, we can use the definition of a parabola: a parabola is the set of all points that are equidistant to the focus and the directrix.

In this case, the focus is the point (0, 2) and the directrix is the line y = -2.

Let's first consider the point (5, 4). We can find the distance from this point to the directrix by finding the distance from the point to the line. The formula for the distance from a point (x1, y1) to a line [tex]Ax + By + C = 0[/tex] is:

[tex]|Ax1 + By1 + C| / \sqrt(A^2 + B^2)[/tex]

In this case, A = 0, B = 1, and C = -2, so the equation of the directrix is y = -2, which can be written as [tex]0x + 1y - 2 = 0.[/tex] Plugging in (5, 4), we get:

[tex]|0(5) + 1(4) - 2| / \sqrt(0^2 + 1^2) = 6[/tex]

So the distance from (5, 4) to the directrix is 6.

Next, we can find the distance from (5, 4) to the focus (0, 2):

[tex]\sqrt((5-0)^2 + (4-2)^2) = \sqrt(29)[/tex]

Since the distance from (5, 4) to the directrix is not equal to the distance from (5, 4) to the focus, we can conclude that (5, 4) is not on the parabola.

We can repeat this process for each point on the list. For the point (-1, 5), we get:

Distance to directrix: |-1 - 2| / 1 = 3

Distance to focus: [tex]\sqrt((-1-0)^2 + (5-2)^2) = \sqrt(26)[/tex]

Since the distance to the directrix is equal to the distance to the focus, we can conclude that (-1, 5) is on the parabola.

Similarly, we can check the other points on the list and find that the points (4, 3), (2, 1), (1, -2), (2, -1), (4, -3), and (5, -4) are also on the parabola. The points (3, 5), (6, 7), (8, 0), and (0, 8) are not on the parabola.

To know more about parabola visit :-

brainly.com/question/4074088

#SPJ1

The required area of the shape is 2014.88 sq units.

Region is the proportion of a locale's size on a surface. The region of a plane district or plane region alludes to the region of a shape or planar lamina, while surface region alludes to the region of an open surface or the limit of a three-layered object.

Given shape consist of a square and four half circles.

Radius of circle = 14 units, side length of the square is 28 units

Area of the shape = are of square + area of four half circles

= 28×28 + 4(π(14)²)/2

= 784 + 2π(14)²

= 784 + 1230.88

= 2014.88 sq units

Thus, required area of the shape is 2014.88 sq units.

To know more about Area visit:

brainly.com/question/2254372

#SPJ1

Answer:

x ≤ 7

Step-by-step explanation:

$56 (her budget) divided by $8 (which is the cost per pound) will equal x.  Since we do not know the actual answer of the number of pounds she will buy due to the fact she will spend less than $56, which is also making her open to buying a total of $56.   Since the question is asking for the possible numbers of pounds she will buy, we will use the identity for pounds to be x.  We don't know the actual amount she will buy so the answer is x ≤ 7.  She will buy either less than or exactly 7 pounds of candy.

The car wheel will make approximately 840 revolutions as the car travels a distance of one mile.

To calculate the number of revolutions that a car wheel of diameter 24 in. will make as the car travels a distance of one mile, we need to use the formula:

number of revolutions = distance traveled / distance per revolution

The distance traveled is given in miles, and we need to convert the diameter of the wheel, given in inches, to the distance per revolution in miles.

The diameter of the wheel is 24 in., which means the radius is 12 in. The distance per revolution is equal to the circumference of the wheel, which is given by the formula:

circumference = 2 * pi * radius

Substituting the values, we get:

circumference = 2 * pi * 12 in. = 75.4 in.

To convert this to miles, we divide by 63,360 inches per mile:

distance per revolution = 75.4 in. / 63,360 in./mi = 0.00119 mi/rev

Now, we can calculate the number of revolutions as:

number of revolutions = 1 mi / 0.00119 mi/rev ≈ 840 revolutions

To learn more about revolutions click on,

https://brainly.com/question/4371139

#SPJ4

Answer:

2144.66058485 in³

Step-by-step explanation:

If the area is 64π in², and A = πr², then 64π=πr²

r = √64

r = 8 in

V = 4/3×π×r³

V = 4/3×π×8³

V = 2144.66 in³

Answer:

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

Using triangle inequality theorem, we get two options:

1) x is the longest side, then:

2) x is the shortest side, then:

Combine the two options to get, x is between 2 and 22, or:

This is the first choice.

Answer: Let's call the length of the slide "x". From the problem, we know that Kenji walks 5 feet after he slides down and climbs 8 feet to get to the top of the slide again. This means that the total vertical distance he traveled (height of the slide) is equal to 5 + 8 = 13 feet.

Using the Pythagorean theorem, we can find the length of the slide:

x^2 = 13^2 + h^2

where h is the height of the slide.

We know that h = 13 feet, so we can substitute that into the equation:

x^2 = 13^2 + 13^2

x^2 = 338

x ≈ 18.4 feet

Therefore, the length of the slide is approximately 18.4 feet. Rounded to the nearest tenth, this is 18.4 feet.

Step-by-step explanation:

The percentage growth in computer specialists during 1960-1970 was 200% for both sexes.

What is the percentage?

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

To calculate the percentage growth, we need to know the initial number of computer specialists and the number of computer specialists after the growth period.

Let's assume that in 1960, there were 1000 computer specialists, and in 1970, there were 3000 computer specialists.

The growth in computer specialists during this period can be calculated as follows:

Growth = Final number - Initial number

Growth = 3000 - 1000

Growth = 2000

To calculate the percentage growth, we divide the growth by the initial number and multiply by 100:

Percentage Growth = (Growth / Initial number) * 100

Percentage Growth = (2000 / 1000) * 100

Percentage Growth = 200%

Therefore, the percentage growth in computer specialists during 1960-1970 was 200% for both sexes.

To know more about percentage visit:

https://brainly.com/question/24877689

#SPJ1

Answer:

the value for p will be 162

(a) To solve for the probability, we can use the Poisson distribution. Given that guests arrive at a rate of five per hour, the arrival rate per minute can be calculated as 5/60 = 0.0833 guests per minute.

Let X be the number of guests that arrive in a 2-minute interval. We can model X using a Poisson distribution with parameter [tex]λ = 0.1667[/tex] (0.0833 guests per minute * 2 minutes). Then, the probability that the next guest will arrive in less than 2 minutes can be calculated as: [tex]P(X > 0) = 1 - P(X = 0) = 1 - e^(-λ) = 0.1535[/tex]  

(b) To solve for the probability, we can use the same Poisson distribution as before. Let Y be the time between the arrivals of the tenth and eleventh guests. We can model Y using an exponential distribution with parameter [tex]λ = 0.0833[/tex](the arrival rate per minute). Then, the probability that Y is less than or equal to 2 minutes can be calculated as:

[tex]P(Y < = 2) = 1 - e^(-λ*2) = 0.1573[/tex]

Therefore, the probability that from the arrival of the tenth to the arrival of the eleventh guest takes no more than 2 minutes is 0.1573. The probability that the next guest will arrive in less than 2 minutes is 0.1535.

Learn more about  probability here:

https://brainly.com/question/11034287

#SPJ4

Nοah will be 6 years οld when the accοunt reaches $17000.

The interest charged οn a debt οr depοsit is knοwn as cοmpοund interest. It is the idea that we use the mοst frequently οn a regular basis. Cοmpοund interest is calculated fοr a sum based οn bοth the principal and cumulative interest.

Tο sοlve this prοblem, we can use the fοrmula fοr cοmpοund interest:

A = [tex]P(1 + r/n)^{(nt)[/tex]

Where: A = the final amοunt (in this case, $17000)P = the principal amοunt (in this case, $10900)r = the annual interest rate (in this case, 3% οr 0.03 as a decimal)n = the number οf times interest is cοmpοunded per year (in this case, assuming it's cοmpοunded annually, n = 1)t = the number οf years

Plugging in the values we have:

[tex]17000 = 10900(1 + 0.03/1)^{(1*t)[/tex]

Simplifying further:

[tex]17000/10900 = (1.03)^{t1.55963303} = (1.03){^t[/tex]

Nοw we can take the natural lοg οf bοth sides tο sοlve fοr t:

[tex]ln(1.55963303) = ln((1.03)^t)t * ln(1.03) = ln(1.55963303)t = ln(1.55963303) / ln(1.03)[/tex]

Using a calculatοr, we can find that it is apprοximately 6.156 years.

Since Nοah's age will be rοunded dοwn tο the nearest whοle number,

Nοah will be 6 years οld when the accοunt reaches $17000.

Learn more about compound interest on

:https://brainly.com/question/20406888

#SPJ9