Problem:
3/x^2 = x-4/3x^2 +2/x^2

If a/b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, the smallest value of the a+b is 2013.

A mathematician would tell you that there cannot be such a number since it would violate the principles of mathematics. There cannot be a number n/2 since n is already the smallest if you have a number n, where n is the smallest integer after 0. Mathematicians dislike this since it implies that division itself fails.

A computer will truly respond to your question. Computers don't have an endless number of numbers, unlike the physical world, because they couldn't all fit. Each memory register in a computer has a set number of bits that are used to store numbers. Imagine having just three digits. 999 is the largest number you may possibly portray.

The continued fraction representations of the limits of the interval are

0.0080120180265 = [0; 124, 1, 4, 2, 1, 463872, 1, 1, 12, 1, 1, 41]

0.0080120180275 = [0; 124, 1, 4, 3, 545777, 2, 13, 1, 1, 1, 1, 2]

The simplest continued fraction (and therefore also the simplest ordinary fraction!) in that interval

is

[0; 124, 1, 4, 3] 16 1997 = = 0.00801201802704056084...

and the sum of its numerator and denominator is 2013.

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Complete question:

If a/b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, what is the smallest possible value of a+b ?

Using the given information, the volume of the sphere is approximately 1521.88 cubic units, rounded to the hundredths place

From the question, we are to determine the volume of the sphere.

Let's start by using the formula for the circumference of a circle,

C = 2πr,

Where C is the circumference

and r is the radius.

Since the circumference of the great circle is given as 14π, we can solve for the radius as follows:

14π = 2πr

Dividing both sides by 2π, we get:

r = 7

Now, we can use the formula for the volume of a sphere,

V = (4/3)πr³

Where V is the volume

and r is the radius.

Substituting the value of r that we found, we get:

V = (4/3)π(7)³

V = (4/3)π(343)

V = 4/3 * 1141.45

V = 1521.88

Hence, the volume of the sphere is 1521.88 cubic units.

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Answer:       H=3feet

Step-by-step explanation:

Answer: h ≈ 3 ft

Step-by-step explanation: The formula for volume of a cylinder is v = pi times radius squared times height (πr^2h).

To solve this we need to use the formula h = v/πr^2

h = 37.68/π2^2

h = 2.99848 ft

h ≈ 3 ft

(note: ≈ means approximately so the answer is estimated as 3 ft but the actual answer is 2.99848)

Option A : The surface area of the larger sphere is 602.88 [tex]in^2[/tex].

Let's use the formula for the surface area of a sphere: A = 4π[tex]r^2[/tex], where r is the radius of the sphere.

Let's call the radius of the smaller sphere "r1" and the radius of the larger sphere "r2". We know that the surface area of the smaller sphere is 200.96 [tex]in^2[/tex], so we can set up the equation:

4π[tex]r1^2[/tex] = 200.96

Solving for r1, we get:

r1 = √(200.96 / 4π) ≈ 2.8

A2 / A1 = [tex](r2 / r1)^2[/tex]

A2 = [tex]A1 (r2 / r1)^2[/tex]

We know A1 = 200.96 [tex]in^2[/tex] and r1 ≈ 2.8, and we want to find r2, so we can use the formula for the volume of a sphere to relate r1 and r2:

(4/3)π[tex]r1^3[/tex] = (4/3)π[tex]r2^3[/tex]

Solving for r2, we get:

r2 = [tex]r1 (3)^(1/3)[/tex] ≈ 3.3

Now we can plug in the values for A1, r1, and r2:

A2 = [tex]200.96 (3.3 / 2.8)^2[/tex] ≈ 301.44

Therefore, the larger sphere has a surface area of roughly 301.44 in2, making answer choice C appropriate.

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The standard error to be used in the formula of a confidence interval for the difference in proportions is 0.07.

The estimated standard deviation of a statistical sample population is known as the standard error (SE) of a statistic.

By utilising standard deviation, the standard error is a statistical concept that assesses how well a sample distribution represents a population. In statistics, the difference between a sample mean and the population's actual mean is known as the standard error of the mean.

The standard error's primary function is to indicate how divergent the population mean will be from the sample mean. Because it aids in determining how well the standard data reflect the entire population, the standard error is significant. We may readily draw reliable conclusions by calculating the standard error in order to determine how representative our sample is of our population.

56 out of 100 boys = P1 = 0.56

43 out of 100 girls = P2 = 0.43

The formula for standard is given by,

[tex]S.E=\sqrt{\frac{P_1(1-P_1)}{n1} +\frac{P_2(1-P_2)}{n2} } \\=\sqrt{\frac{0.56(1-0.56)}{100} +\frac{0.43(1-0.43)}{100} } \\\\= \sqrt{\frac{0.2464}{100} +\frac{0.2451}{100} } \\\\=\sqrt{0.004915} \\=0.07[/tex]

Therefore, the standard error is 0.07.

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The area of the shaded region is equal to 280.6 ft².

Based on the diagram of these regular polygons, the area of the shaded region of one triangle that makes up the hexagon can be calculated by determining the difference between the area of each of the equilateral triangles formed.

In Mathematics and Geometry, the area of an equilateral triangle with known side length (l) can be calculated by using the following mathematical equation;

Area of equilateral triangle = √3/4 × l²

The area of the shaded region of the triangle = √3/4 × l² - √3/4 × l²

The area of the shaded region of the triangle = √3/4 × (12)² - √3/4 × 6²

The area of the shaded region of the triangle = √3/4 × 144 - √3/4 × 36

The area of the shaded region of the triangle = 62.35 - 15.59

The area of the shaded region of the triangle = 46.76 ft².

Since the regular polygon is a hexagon, the area of the shaded region is given by;

Area of the shaded region = 6(46.76)

Area of the shaded region = 280.6 ft².

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Answer:

Let $x$ be the number of weeks Robert has been on his diet. We know that he loses 2 pounds per week and that he currently weighs 205 pounds. We can write this as an equation:

$2x = 205$

Solving for $x$, we get:

$x = 102.5$

This means that Robert has been on his diet for 102.5 weeks.

We can also use this information to create a linear equation to model the situation. The equation would be:

$y = 2x$

Where $y$ is Robert's weight in pounds and $x$ is the number of weeks he has been on his diet.

We can plug in $x = 102.5$ to get:

$y = 2 \cdot 102.5 = 205$

This shows that the equation accurately models the situation.

Step-by-step explanation:

student can answer the questions on the test in 16,384 different ways if they answer every question.

To determine the number of ways a student can answer the questions on a multiple-choice test containing 7 questions with four possible answers for each question, we will use the multiplication principle. The multiplication principle states that if there are a number of independent choices, then the total number of possible outcomes is the product of the number of choices.
Identify the number of questions and possible answers. In this case, there are 7 questions and 4 possible answers for each question.
Calculate the number of ways to answer each question. Since there are 4 possible answers for each question, there are 4 ways to answer each of the 7 questions.
Apply the multiplication principle. To find the total number of ways to answer all 7 questions, multiply the number of ways to answer each question:
Total number of ways = (4 ways for question 1) x (4 ways for question 2) x ... x (4 ways for question 7)
Perform the calculation. Since there are 7 questions and 4 ways to answer each question, the total number of ways is:
Total number of ways = 4^7 = 16,384

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Answer:

(a) To find the probability that a trip will take at least 1/2 hour (30 minutes), we need to find the area under the normal distribution curve to the right of 30 minutes. We can standardize the distribution using the formula z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

z = (30 - 24) / 3.8 = 1.58

Using a standard normal distribution table or a calculator with a normal distribution function, we can find the probability that a trip will take at least 30 minutes is approximately 0.0571 or 5.71%.

(b) If the office opens at 9:00 a.m. and the lawyer leaves his house at 8:45 a.m. daily, he needs to arrive at the office before 9:00 a.m. to be on time. We can find the percentage of the time he is late for work by finding the area under the normal distribution curve to the right of 15 minutes (the difference between 8:45 a.m. and 9:00 a.m.), and then subtracting that value from 1 to get the percentage of the time he is on time or early.

z = (15 - 24) / 3.8 = -2.37

Using a standard normal distribution table or a calculator with a normal distribution function, we can find the probability that he is late for work is approximately 0.008 or 0.8%. Therefore, he is on time or early approximately 99.2% of the time.

The equation that will likely have the graph attached is

The first step will be checking the y intercepts of the equations. Considering the graph the y intercept is in the positive hence equation in d is eliminated

Another factor is vertex, solving for the vertex of the remaining equations would show the equation that has a vertex that would be in that position

Otherwise plot the graph of all the equations and match the likely equations to the graphs

Matching the equations to their graphs shows that graph of y = 3x² - 2x + 1 is attached

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Answer:

(1, 6)

Step-by-step explanation:

6x + 5y = 36 ----> 6x + 5y = 36

3x - 2y = -9 ----> -6x + 4y = 18

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

9y = 54

y = 6

3x - 2(6) = -9

3x - 12 = -9

3x = 3

x = 1

So the solution is (1, 6).

The answer is not listed in the choices given. None of the answer choices Match -20.

To find the third term in an arithmetic sequence, we need to use the recursive formula. The recursive formula for an arithmetic sequence is given as:

a(n) = a(n-1) + d

where a(n) represents the nth term in the sequence, a(n-1) represents the previous term, and d represents the common difference between terms.

Since we are looking for the third term, n = 3. We also need to know the value of a(2), which is the second term in the sequence. To find a(2), we use the recursive formula again:

a(2) = a(1) + d

We are not given the value of a(1), so we cannot directly calculate a(2). However, we are given the answer choices, so we can use them to work backwards.

If we assume that a(2) is equal to -5 (choice C), then we can find a(1) by using the recursive formula:

a(2) = a(1) + d
-5 = a(1) + d

If we assume that d = -3 (since the common difference between terms is the same), then we can solve for a(1):

a(1) = -5 + (-3)
a(1) = -8

Now that we know a(1) and d, we can use the recursive formula to find a(3):

a(3) = a(2) + d
a(3) = -5 + (-3)
a(3) = -8

However, none of the answer choices match -8. This means that our assumption for a(2) was incorrect. We can try the same process with the other answer choices to see if we get a matching answer.

If we assume that a(2) is equal to -14 (choice A), then we can find a(1) by using the recursive formula:

a(2) = a(1) + d
-14 = a(1) + d

If we assume that d = -3 (since the common difference between terms is the same), then we can solve for a(1):

a(1) = -14 + (-3)
a(1) = -17

Now that we know a(1) and d, we can use the recursive formula to find a(3):

a(3) = a(2) + d
a(3) = -14 + (-3)
a(3) = -17 - 3
a(3) = -20

Therefore, the answer is not listed in the choices given. None of the answer choices match -20.

In summary, to find the third term in an arithmetic sequence using the recursive formula, we need to know the previous term and the common difference between terms. If we are given answer choices, we can work backwards to find the missing information. However, we must check all answer choices to ensure that we have the correct solution.

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Answer: Any value of f that is less than 3 1/3 satisfies the inequality. The solutions to the inequality are all values of f that are less than 3 1/3.

Step-by-step explanation:

There is a 10.91% chance that the package will be refused.

The possibility that an event will occur is its probability, which is given as a number between 0 and 1.

Sample = 400 units

n = 400 units

If less than 10% of the units are the desired hue, the shipment will be rejected.

12% of the units are, in fact, the desired hue.

So, P = 12%

We can write it as

P = 0.12

Q = 1 - 0.12

Q = 0.88

σ = √PQ/n

Substitute the value

σ = √(0.12 × 0.88)/400

σ = √0.1056/400

σ = √0.000264

σ = 0.01625

Probability that the shipment will be rejected;

P(x < 10%) = P(x < 0.1)

P(x < 10%) = P([tex]Z_{0.1}[/tex])

[tex]Z_{0.1}[/tex] = (0.1 - 0.12)/0.01625

[tex]Z_{0.1}[/tex] = -0.02/0.01625

[tex]Z_{0.1}[/tex] = -1.231

P([tex]Z_{0.1}[/tex]) = 0.1091

P(x < 10%) = 0.1091

P(x < 10%) = 10.91%

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Answer:

[tex]9(x - 1)(x-3)[/tex]

Step-by-step explanation:

[tex]9 {x}^{2} - 36x + 27[/tex]

[tex]9x(x - 1) - 27(x - 1)[/tex]

[tex](x - 1)(9x - 27)[/tex]

9(x-1)(x-3)

The piecewise function that represent the graph is A)f(x)=[tex]\left \{{ x^3 ,if{x < 1} \atop x,if {x\ge1}} \right.[/tex].

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

In considering the definition of any piecewise function, the domain descriptions in the function definition must match the pieces shown in the graph.

Here, the right segment has no upper bound, so x > 1 is an appropriate description of its domain.

The left segment also has no upper bound , so x[tex]\ge[/tex] 1is an appropriate description of its domain.

The one answer choice that combines these domain descriptions is

A)f(x)=[tex]\left \{{ x^3 ,if{x < 1} \atop x,if {x\ge1}} \right.[/tex]

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Answer:

1 coat rack is $561.30

Step-by-step explanation:

8,419.50/15=561.30

The probability that the first spinner will land on 7 and the second spinner will land on C is 1/4. So correct option is D.

Probability is a branch of mathematics that deals with the study of random events or phenomena. It provides a framework for quantifying uncertainty and making predictions based on data and observations.

Probability is typically expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. For example, the probability of flipping a coin and getting heads is 0.5, or 50%, since there are two equally likely outcomes (heads or tails).

The probability of an event can be determined by calculating the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, the probability of rolling a 6 on a standard die is 1/6, since there is only one favorable outcome (rolling a 6) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).

The probability that the first spinner will land on 7 and the second spinner will land on C is 1/4.

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The character of Rue, who was a companion of Katniss from District 11 during the Hunger Games, is no longer alive at the beginning of the novel, but her legacy and her family have significance in the story's developments.

What is statement?

A statement is a declarative sentence that expresses a complete thought or idea. It is a sentence that makes a claim or expresses a fact, opinion, or belief about something.

At the end of the first book, The Hunger Games, Katniss Everdeen and her fellow tribute from District 12, Peeta Mellark, are the winners of the 74th Hunger Games. They return to District 12 as victors, but the Capitol is not pleased with their act of defiance during the Games, where they both chose to eat poisonous berries instead of killing each other, forcing the Gamemakers to declare them both winners.

As a result of their defiance, Katniss and Peeta have become symbols of hope for the oppressed districts, and the Capitol fears that their actions could spark a rebellion. To prevent this, the Capitol announces a special edition of the Hunger Games, called the Quarter Quell, which occurs every 25 years and has special rules.

At the start of Catching Fire, Katniss is dealing with the aftermath of the first Games and is struggling with post-traumatic stress disorder (PTSD) and survivor's guilt. She is also trying to navigate her complicated relationship with Peeta, who has publicly declared his love for her.

However, her world is turned upside down when the Capitol announces the rules of the Quarter Quell: this time, the tributes will be selected from the existing pool of Hunger Games victors. This means that Katniss and Peeta, as previous winners, are once again forced to compete in the Games.

As Katniss prepares for the Games, she is also dealing with the death of her younger sister Primrose, who dies in a bombing that destroys District 12. The bombing is believed to be the work of the Capitol, who are punishing District 12 for Katniss's and Peeta's defiance in the previous Games.

The character of Rue, who was a companion of Katniss from District 11 during the Hunger Games, is no longer alive at the beginning of the novel, but her legacy and her family have significance in the story's developments.

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The overlap of the doors is 20% of the width of the doorway.

There are 2 children who made 0 cards and 5 children who made 1 card.

Therefore, the total number of children who made fewer than 2 greeting cards is:
2 (children who made 0 cards) + 5 (children who made 1 card) = 7 children

From the given data, we can see how many children made a certain number of greeting cards:
- 2 children made 0 cards
- 5 children made 1 card
- 4 children made 2 cards
- 1 child made 3 cards
- 6 children made 4 cards
The question asks for the number of children who made fewer than 2 greeting cards. This includes children who made 0 or 1 card.

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Answer:

x=80

Step-by-step explanation:

Since, you do not know the percent you need to put a variable so we are going to named x and then we are going to cross multiply then we solv the equation.

Answer:

56 is 80% of 70

Step-by-step explanation:

[tex]\frac{56}{70} (100)=\frac{(56)(100)}{70} =\frac{5600}{70} =80[/tex]

Hope this helps.

This means that she is swimming with a speed of 4.27 feet per second in a direction that is 41.1° east of due north.

A vector is a mathematical quantity that has both magnitude and direction. Vectors are used to represent physical quantities that have both magnitude (such as speed, force, or displacement) and direction (such as north, east, up, or down). Vectors can be represented graphically as arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.

Here,

To find Erica's resultant speed and direction, we can use vector addition. We'll consider Erica's swimming speed as one vector and the current of the lake as another vector, and then find the vector sum of the two.

Let's denote Erica's swimming speed vector as A and the current vector as B.

Magnitude of A (Erica's swimming speed) = 7 feet per second

Direction of A = Due north, which can be represented as N or 0°

Magnitude of B (current of the lake) = 3 feet per second

Direction of B = S 75° W, which can be represented as 180° - 75°

= 105° in the clockwise direction from due north.

Now, we can add the two vectors A and B using vector addition.

To add vectors, we can break them down into their horizontal (x) and vertical (y) components, and then add the corresponding components separately.

A_x = A * cos(direction of A)

A_y = A * sin(direction of A)

B_x = B * cos(direction of B)

B_y = B * sin(direction of B)

Substituting the given values, we get:

A_x = 7 * cos(0°) = 7 * 1 = 7

A_y = 7 * sin(0°) = 7 * 0 = 0

B_x = 3 * cos(105°)

B_y = 3 * sin(105°)

Now, we can add the corresponding components:

Resultant x-component = A_x + B_x

Resultant y-component = A_y + B_y

Resultant x-component = 7 + 3 * cos(105°)

Resultant y-component = 0 + 3 * sin(105°)

Using a calculator, we can find the values of the x- and y-components. Let's assume the values to be:

Resultant x-component ≈ 3.23

Resultant y-component ≈ 2.97

Now, we can use these values to find the magnitude and direction of the resultant vector using trigonometry.

Magnitude of the resultant vector = √((Resultant x-component)² + (Resultant y-component)²)

Direction of the resultant vector = tan⁻¹(Resultant y-component, Resultant x-component)

Substituting the values, we get:

Magnitude of the resultant vector ≈ √((3.23)² + (2.97)²)

≈ 4.27 feet per second (rounded to two decimal places)

Direction of the resultant vector ≈ tan⁻¹(2.97, 3.23)

≈ 41.1° (rounded to one decimal place)

So, Erica's resultant speed is approximately 4.27 feet per second in the direction of 41.1° true bearing.

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Two lines intersect at (2,4). The last choice listed, "a coordinate grid with one line that passes through the points 0,0 and 1,2 and another line that runs through the points 0,-2 and 1,-1," is the graph.

The given system of linear equations is y = 2x and y = x + 2. To find the solution to this system, we can set the two equations equal to each other:

2x = x + 2

Subtract x from both sides:

x = 2

Substitute x = 2 into either equation:

y = 2x = 2(2) = 4

Therefore, solution to system of linear equations is (2, 4).

To check our answer, we can graph the two lines y = 2x and y = x + 2 on a coordinate grid. The intersection point of the two lines will be the solution to the system.

The line y = 2x passes through the points (0,0) and (1,2). The line y = x + 2 passes through the points (0,2) and (1,3). We can plot these points and draw the lines to get the following graph:

Linear equation graph is attached.

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For given Semi-circle shaped window,  the distance around the window is 38.55 inch.

What exactly is a circle?

A circle is a closed two-dimensional shape in which all points on the boundary are equidistant from a single point called the center. It is formed by taking all points in a plane that are a fixed distance (called the radius) away from the center. The distance around the circle's boundary is called the circumference, and the distance from the center to any point on the boundary is the radius.

Now,

As radius of semicircle = 7.5 in and

Circumference is given by = πr⇒3.14*7.5=23.55 inch

Perimeter of window= πr+D

D=diameter

P=23.55+15

P=38.55 inch

Hence,

         the distance around the window is 38.55 inch.

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