Area of a circle 12 mm diameter

The sum of the given convergent series is 15.

What is convergent series with example?

An easy example of a convergent series is ∞∑n=112n=12+14+18+116+⋯ The partial sums look like 12,34,78,1516,⋯ and we can see that they get closer and closer to 1. The first partial sum is 12 away, the second 14 away, and so on and so forth until it is infinitely close to 1.

as given, ∈ from n=1 to n=∞ for ( 20 / n (n+2) )

20 / n(n+2) = (A / n) + (B / n+2)

∴ A = 10

∴ B= -10

∈ from n=1 to n=∞ for ( 20 / n (n+2) )

= 10 [1+0.5]

= 10 x 1.5

= 15

for complete solving process, please refer below attached image.

Thus, the sum of the given convergent series is 15.

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The sum of the given convergent series is 15.

What is convergent series with example?

An easy example of a convergent series is ∞∑n=112n=12+14+18+116+⋯ The partial sums look like 12,34,78,1516,⋯ and we can see that they get closer and closer to 1. The first partial sum is 12 away, the second 14 away, and so on and so forth until it is infinitely close to 1.

as given, ∈ from n=1 to n=∞ for [tex]( 20 / n (n+2) )[/tex]

[tex]20 / n(n+2) = (A / n) + (B / n+2)[/tex]

∴ A = 10

∴ B= -10

∈ from n=1 to n=∞ for [tex]( 20 / n (n+2) )[/tex]

= 10 [1+0.5]

= 10 x 1.5

= 15

Thus, the sum of the given convergent series is 15.

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By exploring the behavior of the tangent line at different intervals, the shape and behavior of the curve 8x², the slope is 16.

The tangent line to a curve is a straight line that just touches the curve at a single point, without crossing over it. The slope of the tangent line at that point is equal to the rate of change of the curve's slope at that point.

In the case of the curve 8x², we can find the tangent line at different points by using calculus. Calculus provides us with tools to find the slope of the curve, and from that, the equation of the tangent line.

So, to find the tangent line at a point on the curve

=> 8x²,

we need to find the derivative of the curve and evaluate it at the point of interest.

The derivative of

=> 8x² = 16x,

so the slope of the tangent line at any given point x is 16x.

To find the equation of the tangent line, we also need to know the y-coordinate of the point where the line touches the curve.

We can use this information to write the equation of the tangent line in the form y = mx + b, where m is the slope and b is the y-intercept.

Then the value of slope is 16.

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Answer:

$200

Step-by-step explanation:

If tickets cost $1/ticket then it'll be $400 for all the tickets combined. If you add all the winners up then you will get $200 that is given back to them.

For example:

1 $100 winner =$100

2 $25 winner =$50

5 $10 winners =$50

All together it will add up to $200

So then we do the prize money minus our starting cost which is $400-$200. Which equals $200. You will give back $200 as prize money, and then $200 is a profit you will keep.

Answer:

16.2

Step-by-step explanation:

785.7 pounds of food divided by 48.5 pounds of food equals 16.2 days that the adult seal will have food for.

Therefore , the solution of the given problem of complementary angles comes out to be 29° and 61° QED.

Two angles are considered complementary when their sum equals 90 degrees. Two angles are considered submissive when their sum equals 180 degrees. For example, the angles of 40 - 60 degrees complement one another. The difference between a supplementary angle and a supplementing angle is that a Triangle angle has two angles joined to produce a 90 degree angle. When two angles add up to 90 degrees, they are said to be complementary. Alternatively, the complementary angles result in a right angle (90 degrees).

Here,

Given :

Allow x to equal the complementary angle.

Given: x plus 32° the opposite angle

Recognized: Complementary angles sum to 90°

=>x + x + 32° = 90° (Adding the two angles) (Adding the two angles)

2x + 32° = 90° (adding the unknown factors) (adding the unknown variables)

=>2x = 90° - 32° (subtract 32 from both sides) (subtract 32 from both sides)

=> 2x = 58° (assess the right side) (evaluate the right side)

=>x = 29° (split both sides by 2) (divide both sides by 2) (One reply)

=> x + 32° = 61° (Add 32 to x) (Add 32 to x) (The other response)

There are two angles: 29° and 61° QED.

Therefore , the solution of the given problem of complementary angles comes out to be 29° and 61° QED.

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The complete question is "The measure of one angle is 32° greater than the measure of its complement. What are the measures of the angles?"

Part I.

g(x) = 3x-4

Part II.

f(x) = √(x)

Part III.

f(g(x)) = f(3x-4) = √(3x-4)

In Part I, the function F(x) is the square root of 3x-4. Therefore, g(x) is the expression inside the square root: g(x) = 3x-4.

In Part II, the function f(x) is the square root operation applied to its argument, so f(x) = √(x).

Finally, in Part III, we substitute g(x) into f(x) to obtain f(g(x)) = f(3x-4) = √3x-4), which is equal to F(x)

So,

f(g(x))= √(3x-4) = F(x)

Therefore, F(x) = f(g(x)) where f(x) = √(x) and g(x) = 3x-4

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The given statement, "in 2014 the number of mobile phones was greater than the planet's population" is true (T) because there were 5.8 billion mobile users globally, about 80% of the world's population, and they each had 1.5 devices.

In 2014, there were more mobile phones than there were people on the planet. This is because many people own more than one phone, either as a personal and work phone or as a backup device. Additionally, the rise of low-cost mobile devices in emerging markets, where many people were purchasing their first phones, contributed to the high number of mobile phones relative to the population.

This trend has continued, with the number of mobile phones in the world continuing to grow, outpacing population growth and driving down the cost of devices so that even more people have access to them.

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For the given tossing of number cube and its frequency outcome, the probability on landing a number greater than 4 is 18/60.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

here, we have,

As given in the question

Total number of tossing a number cube = 60

Number of times greater than 4 comes up = 10 + 8 (frequency of is 10 and frequency of 6 is 8)

⇒Number of times greater than 4 comes up = 18

Probability of getting number greater than 4

=  (frequency of getting 5 and 6) / (total number of trials)

⇒ probability of getting number greater than 4  = 18 / 60                                                                      

Therefore, for the given tossing of number cube and its frequency outcome, the probability on landing a number greater than 4 is 18/60.

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Regular expressions are patterns that can be used to match character combinations in strings.

Given data :

Given some Weekdays, the task is to check if they are valid or not using regular expressions.

Rules for the valid Weekdays :

It should contain specific only words as a string. They are mentioned below:

Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.

Mon, Tues, Wed, Thurs, Fri, Sat, Sun.

Mon., Tues., Wed., Thurs., Fri., Sat., Sun.

mon, tues, wed, thurs, fri, sat, sun.

Examples:

Input: “Monday”

Output: True

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Answer:

Step-by-step explanation:

Let's call the common difference of the arithmetic sequence "d", and the first term of the sequence "a". Then we can write two equations based on the information given:

a + 3d = 57 (the 11th term is 57)

a + a + 3d = 29 (the sum of the first and fourth terms is 29)

Solving for the first equation for "a", we get:

a = 57 - 3d

Substituting this expression for "a" into the second equation, we get:

57 - 3d + 3d = 29

57 = 29

So the first three terms of the arithmetic sequence are:

a = 57 - 3d

a + d = 57 - 2d

a + 2d = 57 - d

Solving for "d", we can find the value of "a", and then use the equations above to find the first three terms of the sequence

A function to define the area of the triangle is A(b,h) = (1/2)*b*h

A function is a process or a relation that associates each element 'a' of a non-empty set A , at least to a single element 'b' of another non-empty set B. A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math. f = {(a,b)| for all a ∈ A, b ∈ B}.

1) A relation is said to be a function if every element of set A has one and only one image in set B.

2) A function is a relation from a non-empty set B such that the domain of a function is A and no two distinct ordered pairs in f have the same first element.

A function from A → B and (a,b) ∈ f, then f(a) = b, where 'b' is the image of 'a' under 'f' and 'a' is the preimage of 'b' under 'f'.

If there exists a function f: A → B, the set A is called the domain of the function f, and the set B is called its co-domain.

Let us assume that the base of the triangle is b and the height of the triangle is h.

So, a function to define the area of the triangle is

A(b,h) = (1/2)*b*h

This function is a function in two variables.

Thus, a function to define the area of the triangle is A(b,h) = (1/2)*b*h

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The answer to the given  the vector subtraction of these vector question is →=→−→=−11ˆ 1ˆ −7ˆ.

The vector subtraction is a mathematical operation that subtracts two vectors. It is denoted by the symbol →=→−→, where → and → are two vectors. The formula for vector subtraction is →−→=→−(−→).

For the given question, let us consider the vectors →=−6ˆ 8ˆ−4ˆ and →=5ˆ 7ˆ−3ˆ. Applying the formula, the vector subtraction of these vectors can be calculated as follows: →−→=(−6ˆ 8ˆ−4ˆ)−(5ˆ 7ˆ−3ˆ) = (−6ˆ−5ˆ) 8ˆ−(7ˆ−3ˆ)−4ˆ = −11ˆ 1ˆ −7ˆ.

Therefore, the answer to the given question is →=→−→=−11ˆ 1ˆ −7ˆ.

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The solution to the function is 289

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

f(x) = (1 + 2x)^2

Substitute the known values in the above equation, so, we have the following representation

f(8) = (1 + 2 * 8)^2

Evaluate the product

f(8) = (1 + 16)^2

So, we have

f(8) = 17^2

This gives

f(8) = 289

Hence, the value of the function is 289

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Answer:

The equation that represents the value y (in dollars) of the ring after x years is y = 500x + 1000. b. The y-intercept of the graph of the equation is 1000, which is the initial value of the ring. c. The constant rate of increase in the value of the ring is 500 dollars per year. d. The constant rate of increase in the value of the ring is not expected to continue forever. As the value of the ring increases, the rate of increase will likely slow down as it approaches its maximum value.

Step-by-step explanation:

The lateral area of the cone is approximately 401 cm²

Let's call the radius of the circular base of the cone "r". We know that the area of the circular base is 16π cm², so:

r² * π = 16π

r² = 16

r = 4

Now that we know the radius, we can find the slant height. The slant height is 4 times the radius, so:

4 * r = 4 * 4 = 16

The lateral area of a cone is equal to π times the slant height times the radius of the base, so:

lateral area = π * r * slant height = π * 4 * 16 = 128π

Using π ≈ 3.14

we can estimate the lateral area of the cone:

lateral area ≈ 128 * 3.14 = 401.12

Rounding to the nearest whole number, the lateral area of the cone is approximately 401 cm².

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Answer: 201 cm  is the answer

Step-by-step explanation: i took the quiz

Answer: 48 plastic bandages

Step-by-step explanation:

1. Divide the ratio of 3 fabric bandages to the amount of fabric bandages in the kit

16/3 = 5.33333333 (repeating) or 5 1/3

2. Multiply that amount (5.33333) by 9

5 1/3 x 9 = 48

3. Check your work!

I hope this helped. Leave a comment if there is something wrong! Give me a Thanks if this helped. Thank u!

Step By Step:

First, you look at the 6. 22 can't go into 6, so we combine the first digit with the second and put an x above the 6 (if you're on paper. If not, just imagine a blank space there.) Now we have 69. 22 can go into 69. As you did it, 22 can only go in three times. So we add the 3 to the top and put 66 on the bottom, as 22*6 = 66. Now, we subtract 69 and 66. This gives us 3. Now, we bring down the 1. So instead of 3 as you have it, it'd be 31. 22 can go into 31 one time. So, we put our 1 up top and put our 22 on the bottom, as 22*1 = 22. Then, we finish by subtracting 31 from 22, which equals 9.

Hopefully, this made sense. If not, I would be happy to provide further in depth explanation or examples.

The area of the top surface of the d) Vanilla cake is 50.24 square inches and the Chocolate cake is 78.5 square inches.

The formula for the area of a circle is π times the radius squared. To find the top surface area of the cakes, we first need to find the radius of each cake by dividing the diameter by 2.

The radius of the vanilla cake is 4 inches and the radius of the chocolate cake is 5 inches. Then, we can use the formula: π times the radius squared to find the area of each cake. Using π = 3.14, we get:

Vanilla cake: π * 4^2 = 50.24 square inches

Chocolate cake: π * 5^2 = 78.5 square inches

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Complete question:

A vanilla cake has a diameter of 8 inches. A chocolate cake has a diameter of 10 inches. What is the area of the top surface of the two cake?

Ue 3. 14 for p. Repone

vanilla cake=100. 48in2

and chocolate cake=314in2

vanilla cake=100. 48in quared and chocolate cake=314in quared

vanilla cake= 25. 12in2

and chocolate cake=62. 8in2

vanilla cake= 25. 12in quared and chocolate cake=62. 8in quared

vanilla cake= 50. 24in2

and chocolate cake=31. 5in2

vanilla cake= 50. 24in quared and chocolate cake=31. 5in quared

vanilla cake= 50. 24in2

and chocolate cake=78. 5in2

All the pair of angles are congruent except ∠5 and ∠3 are not congruent.

Lines in a plane that are consistently spaced apart are known as parallel lines. Parallel lines don't cross each other.

Given:

Two parallel lines cut by a transversal line.

Then,

∠1 and ∠5 = Corresponding angles.

∠2 and ∠8 = Same-side exterior angles.

∠4 and ∠5 = Alternate interior angles.

From the given choices:

∠5 and ∠3 are not congruent.

Therefore, all the required values are given above.

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the volume of the solid under the surface z = 9x + 5y² and above the region bounded by x = y² and x = y³ is: 89/210

Solid, liquid, and gas are the three different states of matter. Solids have a distinct volume and form. Although they have a specific capacity, liquids adopt the form of the container. Gases lack a distinct volume or form. Solids do not adapt to the shape of the container in which they are stored since they have a distinct shape, as well as mass and volume. In addition to having a high density, solids also feature closely packed particles. The volume of space that what a solid covers is assessed by its volume. It is calculated using the volume of unit cubes required to completely fill the solid.

Firstly, we will find bounds:

given that,

x = y²

x = y³

so, y² = y³

thus, y = 0, y = 1

now, we can set up integral for volume:

V=  [tex]\int\limits^1_0 {x} \, dx \int\limits^{y_{2}}_{y_{3}} {9x + 5y^2} \, dx[/tex]

Firstly, we will solve innermost integral:

V = [tex]\int\limits^{y_{2}}_{y_{3}} {9x + 5y^2} \, dx[/tex]

V = [tex]\int\limits^{y_{2}}_{y_{3}} {5y^2} \, dx + \int\limits^{y_{2}}_{y_{3}} {9x } \, dx[/tex]

V = 5y⁴ - 5y⁵ + 9 (y⁴/2 - y⁶/2)

V = -9/2 y⁶ - 5y⁵ + 19y⁴/2

now, we can find outermost integral:

= [tex]\int\limits^1_0 {-9/2y^6 - 5y^5 + 19y^4/2} \, dx[/tex]

= -9/14 - 5/6 + 19/10

=  89/210

Thus, the volume of the solid under the surface z = 9x + 5y² and above the region bounded by x = y² and x = y³ is: 89/210

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The solution of the expression is -97 / 12. The correct option is A.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the expression is -12¹/₈ ÷ 1¹/₂. The expression will be solved as:-

E =  -12¹/₈ ÷ 1¹/₂

E = 97/ 8 ÷ 3/2

E = ( 97 / 8 ) x ( 2 / 3 )

E = -97 / 12

Therefore, the solution of the expression is -97 / 12. The correct option is A.

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a) The solution of equation [tex]4^{x+1} = 5^{3x}[/tex] is x = log4/(3log5 - log4).

b) The extraneous roots of equation [tex]\log_{2}x + \log_{2} (x+2) = 3[/tex] is -4 and 2.

a) The first equation is [tex]4^{x+1} = 5^{3x}[/tex].

Now solving the equation.

To solve the equation we take log on both side.

[tex]\log(4^{x+1}) = \log(5^{3x})[/tex]

Using the formula [tex]\log m^n = n\log m[/tex].

Now the equation is

(x+1) log4 = 3x log5

x log4 + log4 = 3x log5

Subtract x log4 on both side, we get

3x log5 - x log4 = log4

Taking common x

x(3log5 - log4) = log4

Divide by (3log5 - log4) on both side, we get

x = log4/(3log5 - log4)

b) The second equation is [tex]\log_{2}x + \log_{2} (x+2) = 3[/tex]

Now solving the equation.

We can write the equation as

[tex]\log_{2}(x(x+2)) = 3[/tex]

Now eliminating the log

x(x+2) = (2)^3

x(x+2) = 8

Now simplify the bracket

x^2 + 2x = 8

Subtract 8 on both side, we get

x^2 + 2x - 8 = 0

Now factor the equation

x^2 + (4-2)x - 8 = 0

x^2 + 4x - 2x - 8 = 0

x(x + 4) -2(x + 4) = 0

Now the factor of the equation is

(x + 4)(x - 2) = 0

Now equating the factor equal to zero.

x + 4 = 0                                 x - 2 = 0

x = -4                                      x = 2

Now the solution of equation is -4 and 2.

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the equality holds for n = k+1, and by mathematical induction, it holds for all n ≥ 2.

Proof by induction:

a. Base case: For n = 2, the left-hand side evalutes to 1/2 and the right-hand side evaluates to 1/2, so the equality holds.

Inductive step: Assume that the equality holds for n = k. We must show that it holds for n = k+1.

The left-hand side evaluates to

∑ ― 1 = 1 ( 1 i(i 1)) = 1 ― 1/k+1

The right-hand side evaluates to

1 ― 1/k+1

Therefore, the equality holds for n = k+1, and by mathematical induction, it holds for all n ≥ 2.

b. Base case: For n = 2, the left-hand side evaluates to 1/2 and the right-hand side evaluates to 1/2, so the equality holds.

Inductive step: Assume that the equality holds for n = k. We must show that it holds for n = k+1.

The left-hand side evaluates to

∑ = 1 ( 1 i(i 1)) = 1 i!k+1

The right-hand side evaluates to

1 i!k+1

Therefore, the equality holds for n = k+1, and by mathematical induction, it holds for all n ≥ 2.

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Yes, the description corresponds to an observational study.

An observational study is used in subjects such as epidemiology, social sciences, psychology, and statistics to draw conclusions from a sample to a population where the independent variable is not within the researcher's control due to ethical concerns or practical restrictions.

Cohort studies, case-control studies, and cross-sectional studies are three types of observational research.

An observational study examines how participants do various behaviours or activities without instructing them which methods or actions to use. An observational research, for example, may be chosen by a scientist to investigate how the amount of water persons drink impacts their diets. An observational research involves measuring or surveying members of a sample without attempting to influence them.

In a controlled experiment, we divide people or items into groups and administer a treatment to one of the groups while leaving the other group alone.

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Answer: 32a^5b^3√b

Step-by-step explanation:

Simplify the radical by breaking the radicand up into a product of known factors assuming positive real numbers

The length of the base of the triangle is 16 m and the height of the triangle is 10 m

A polygon having three edges and three vertices is called a triangle. It is one of the fundamental geometric forms. Triangle ABC is the designation for a triangle with vertices A, B, and C. In Euclidean geometry, any three points that are not collinear produce a distinct triangle and a distinct plane.Use the equation area = 1/2 * base * height to determine a triangle's surface area.

The basic Properties of a triangle :

The length of the base = b m

The height of the triangle = b - 6 m

The area of the triangle = 80 m²

Therefore,

(1/2)*b*(b-6) = 80

⇒ b² - 6b = 160

⇒ b² - 6b - 160 = 0

⇒b² - 16b + 10b - 160 = 0

⇒b(b-16) + 10(b- 16) = 0

⇒(b-16)(b =10) =0

The value of b = 16 and b = -10

The value of b is 16 as -10 is negetive and cannot be the length of the triangle.

The height of the triangle = b-6 = 10 m

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The equation used to find the coordinates of B is given as follows:

B - A = 5/8(C - A).

The coordinates of B are found using the proportions in the context of this problem.

The ratio of AB to B C is 5 to 3, hence we have that point B is 5/(5 + 3) = 5/8 of the distance from point A to point C.

Considering the distance, the proportional equation used to find the coordinates of B is given as follows:

B - A = 5/8(C - A).

This equation is used to find both the x-coordinate and the y-coordinate of B.

The problem is incomplete, hence the answer was given in general terms.

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The trigonometri Identity :

LHS  =  1- 2cos²x  = 1 - 2(1 - sin²x) = 1 + 2sin²x - 2 = 2sin²x - 1 = RHS (proved)

Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions. There are several distinctive trigonometric identities that relate a triangle's side length and angle.

Trigonometry equations known as trig identities are used often to comprehend various mathematical aspects and to solve trigonometry and geometry-related issues. Knowing essential trig identities aids in remembering and comprehending crucial mathematical concepts as well as in resolving a variety of mathematical issues.

The Pythagorean trigonometric identity, commonly known as the Pythagorean identity, is an identity that uses trigonometric functions to represent the Pythagorean theorem. It is one of the fundamental relationships between the sine and cosine functions, along with the sum-of-angles equations.

We know

sin²x + cos²x = 1

LHS ,

=  1- 2cos²x

= 1 - 2(1 - sin²x)

= 1 + 2sin²x - 2

= 2sin²x - 1 = RHS

Therefore we can see that LHS = RHS (proved)

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To calculate the sample relative risk, we compare the proportion of individuals in each group who get lung cancer.where  the sample relative risk is 0.4.

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