For a system to be closed under an operation, and when any two elements in the system are combined using the operation, the result must still be an element of that system. For instance, when any two natural numbers (1, 2, 3, . . .), are added together, the result will always be a natural number. When answering the following questions, consider the system of all polynomial equations and the system of all rational equations.

Part A
Which systems are closed under addition? If either system is not closed under addition, provide a counterexample
Part B
Which systems are closed under subtraction? If either system is not closed under subtraction, provide a counterexample.
Part C
Which systems are closed under multiplication? If either system is not closed under multiplication, provide a counterexample.
Part D
Which systems are closed under division? If either system is not closed under division, provide a counterexample.
Part E
Based on which operations the system of polynomial equations is closed under, which system of numbers is most similar to the system of polynomial equations? Explain your answer, and provide any needed counterexamples.
Part F
Based on which operations the system of rational equations is closed under, which system of numbers is most similar to the system of rational equations? Explain your answer, and provide any needed counterexample

Answer: 4*4= 16

Step-by-step explanation:

if Neha is 4 years older than her son, so she will be sixteen.

Answer:

10.19%

Step-by-step explanation:

1 gallon = 3.785 liters

350 gallons = 350 × 3.785 liters = 1324.75 liters

percent = part/whole × 100%

percent = 135 liters / 1324.75 liters × 100%

percent = 10.19%

The solution of the equation for g is (m+n^2)/5M. The solution has been obtained by solving the given equation for g.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal and joined by the equals sign "=". The two expressions need not be completely of variables i.e. one expression may contain variable and the other may not contain a variable.

We are given g: m+n^2 = g.5M

⇒m+n^2 = g.5M

⇒m+n^2 = 5gM

⇒5gM = m+n^2

⇒5gM/5M = (m+n^2)/5M      .... (Dividing both sides by 5M)

⇒g = (m+n^2)/5M

Hence, the solution of the equation for g is (m+n^2)/5M.

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Answer: The depth, d, of a lake is 500 m, rounded to the nearest integer. The error interval for d is 499.5 ≤ d < 500.5

When a value is rounded to the nearest integer, the possible error is half of the smallest unit of measurement. In this case, the smallest unit of measurement is 1 meter, so the possible error is 0.5 meters. Therefore, the error interval for d is from -0.5 meters to +0.5 meters around the rounded value of 500 meters.

This can be expressed in the form a ≤ d < b as follows:

a = 500 - 0.5 = 499.5

b = 500 + 0.5 = 500.5

So, 499.5 ≤ d < 500.5

Step-by-step explanation:

The six trigonometric functions of an angle θ in a right triangle are sine (sin θ), cosine (cos θ), tangent (tan θ), cotangent (cot θ), secant (sec θ) and cosecant (csc θ).

To find these functions for a given angle θ, we need to know the length of the sides of a right triangle with angle θ.

Given that the point (-5,12) is on the terminal side of θ, we can apply the Pythagorean Theorem to find the length of the sides of the triangle. The Pythagorean Theorem states that a² + b² = c² where a and b represent the lengths of the sides of the triangle and c is the length of the hypotenuse. Using the point (-5,12), we can solve for c, which is 13.

Therefore, the sides of the triangle have lengths 5, 12, and 13. We can now use these lengths to calculate the trigonometric functions of θ.

sin θ = 12/13

cos θ = 5/13

tan θ = 12/5

cot θ = 5/12

sec θ = 13/5

csc θ = 13/12

Therefore, the six trigonometric functions of θ are sin θ = 12/13, cos θ = 5/13, tan θ = 12/5, cot θ = 5/12, sec θ = 13/5 and csc θ = 13/12.

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The A's age  is 26 years and B's age is 24 years

Let the ages of A and B be x and y years respectively. Then, [tex]$x-y=\pm 2[/tex]Given

D's age [tex]$=2 x$[/tex]years and, C's age[tex]$=\frac{y}{2}$[/tex] years.

Clearly, D is older than C

[tex]$2 \mathrm{x}-\frac{\mathrm{y}}{2}=40 \Rightarrow 4 \mathrm{x}-\mathrm{y}=80$[/tex]

Thus, we have the following two systems of linear equations[tex]$x-y=2$[/tex].(i)

and, [tex]$4 x-y=80$[/tex](ii)

or

[tex]$x-y=-2$[/tex] (iii)

and, [tex]$4 x-y=80$[/tex]..(iv)

Subtracting equation (i) from equation (ii), we get [tex]$3 \mathrm{x}=78 \Rightarrow \mathrm{x}=26$[/tex]Putting [tex]$\mathrm{x}=26$[/tex] in equation (i), we get [tex]$\mathrm{y}=24$[/tex]or

Subtracting equation (iv) from equation (iii), we get [tex]$-3 \mathrm{x}=-82 \Rightarrow \mathrm{x}=\frac{82}{3}=27 \frac{1}{3}$[/tex]

Putting [tex]$\mathrm{x}=\frac{82}{3}$[/tex]in equation (iii), we get

[tex]$\mathrm{y}=\frac{82}{3}+2=\frac{88}{3}=29 \frac{1}{3}$[/tex]

Hence, A's age =26 years and B's age =24 years or

A's age =[tex]27 \frac{1}{3}$[/tex] years and B's age[tex]$=29 \frac{1}{3}$[/tex] years.

The complete question is,

A and B are friends and their ages differ by 2 years. A's father D is twice as old as A and B is twice as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.

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The total number of votes cast is 480

Ownership of more than 50% of voting shares generally gives the right of control and consolidation. In special cases, control is possible without having to own more than 50% of voting stock.

Let x represent total number of votes.

step1:

We have been given that Joel got 50% of the votes, which was 48 votes more than Sean.

50% of total votes:50/100[tex]x[/tex]=0.50[tex]x[/tex]

Since Joel got 50% of total votes, so Sean got 40% of total votes that is

40/100[tex]x[/tex]=0.40[tex]x[/tex]

Since Joel got 38 votes more than Sean, so we can represent this information in an equation as:

0.50[tex]x[/tex]=0.40[tex]x[/tex]+48

0.50[tex]x[/tex]-0.40[tex]x[/tex]=0.40[tex]x[/tex]-0.40[tex]x[/tex]+48

0.10[tex]x[/tex]=48

step2:

0.10[tex]x[/tex]/.10=48/.10

[tex]x[/tex]=480

Therefore, the total number of votes cast is 480

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Every one of a method's x-values, or entries, make up the domain, and all of a substring y-values, or outputs, make up the range. The domain of a vertex is every value in the structure, from left to right. The graph's range includes all integers from lower to higher.

The amount with simple interest is A = $ 16,380.92

Simple interest is a method of calculating interest that ignores the impact of compounding. While interest frequently compounds throughout the course of a loan's set periods, simple interest does not. Simple interest is calculated by multiplying the principal amount by the interest rate, times the number of periods.

Simple Interest = ( Principal Amount x Rate x Time Period ) / 100

Given data ,

Let the simple interest be represented as I

Now , the amount with simple interest = A

And , the principal P = $ 15,500

The rate of interest R = 6.2 %

The number of months n = 11 months

So , the number of years T = 11/12 years

Simple Interest I = ( Principal Amount x Rate x Time Period ) / 100

Substituting the values in the equation , we get

Simple Interest I = ( 15,500 x 6.2 x ( 11/12 ) ) / 100

On simplifying the equation , we get

Simple Interest I = $ 16,380.916987

Simple Interest I = $ 16,380.92

Hence , the future value if $ 16,380.92

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The area of the region is equal to the area of the septagon minus the area of the circumscribing circle. The area of the septagon = 7 x (5^2 x (1 + sqrt(3))/4). The area of the circumscribing circle = pi x (5^2/2).

The area of the region between the septagon and its circumscribing circle can be found by subtracting the area of the circumscribing circle from the area of the septagon. The side length of the septagon is 5 inches. The area of the septagon can be calculated using the formula A = 7 x (5^2 x (1 + sqrt(3))/4). This is equal to 58.82 inches squared. The area of the circumscribing circle can be calculated using the formula A = pi x (5^2/2). This is equal to 78.54 inches squared. Therefore, the area of the region between the septagon and its circumscribing circle is 58.82 inches squared minus 78.54 inches squared, which is equal to 19.72 inches squared.

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The volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 16π/3.

What is the bounded region?

A territory is said to be bounded if it has a boundary or other restrictions applied to it. In other words, a bounded shape is limited by a set of measurements or characteristics and cannot be an indefinitely huge area.

Here, we have

Given: x = 1+(y−2)², x=2

We have to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

Rotating with the cylinder method about the x-axis means it must be a dy integral. The given parabola equation is already solved for x. Find the intersections:

2 =  1 + (y−2)²

0 = (y -1)(y-3)

y = 1, 3

We will integrate from

y = 1 to 3

The radius of each cylinder will be r=y and the height will be (greater minus lesser, i.e. right minus left).

h = 2 - (1 + (y-2)²)

h = 1 - y² + 4y -4

h = - y² + 4y - 3

Volume = [tex]\int\limits^a_b {2πrh} \, dy[/tex]

= 2π∫₁³ y.( - y² + 4y - 3)dy

= 16π/3

Hence,  the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 16π/3.

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Answer:

14

Step-by-step explanation:

To find the value of 12 – 4 + 2h when h = 3, we need to substitute the value of h into the expression and simplify.

When h = 3, we can substitute this value into the expression:

12 – 4 + 2h = 12 - 4 + 2*3 = 12 - 4 + 6 = 12 - 4 + 6 = 8 + 6 = 14

So the value of 12 – 4 + 2h when h = 3 is 14.

Answer:

12 - 4 + 2h = 14

Step-by-step explanation:

Given expression,

→ 12 - 4 + 2h

Now we have to use,

→ h = 3

Let's simplify the expression,

→ 12 - 4 + 2h

→ 12 - 4 + 2(3)

→ 12 - 4 + 6

→ 8 + 6

→ 14 => {final value}

Hence, required value is 14.

n³ +2n is divisible by 3 is the statement that we want to prove and A) n³+2n=k³ +2k is divisible by 3 is not a valid statement.

Option (A) is correct.

What is mathematical induction?

Mathematical induction is a method of proof used to establish the truth of a statement for all positive integers (or natural numbers).

In step 3 of a proof by mathematical induction, you are trying to show that if the statement is true for some positive integer k, then it is also true for the next positive integer (k+1). So you would be trying to show that:

B) (k+1)³ +2(k+1) is divisible by 3

is true, assuming that

D) k³+2k is divisible by 3

is true.

In other words, you are trying to show that if the statement is true for some positive integer k, then it is also true for the next positive integer (k+1).

Hence, n³ +2n is divisible by 3 is the statement that we want to prove and A) n³+2n=k³ +2k is divisible by 3 is not a valid statement.

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Answer:

yes most are 12x12 inch tiles so yes

Step-by-step explanation:

Rolls the higher number wins; if they both roll the same number, neither wins.

The answer is-

a)They both start the game with an equal chance of winning, therefore 1/2

b) If she rolls a 3, Thomas must roll a 1 or 2 for her to win, making her chance of winning 1/3.

c) Thomas has a 50% chance of winning if she rolls a three because he has to roll a 4,5 or 6 to do so.

d) In order for Sarah to win, she had to roll a 2, 3, 4, or 6. If Thomas had rolled a 1, 2, 3, 4, or 5, he would have lost. Thus, there is a 1/5 chance that he will roll a 3.

e) If Sarah wins she must have rolled a 2,3,4,5 or 6. So the probability that she rolls a three is also 1/5.

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In  trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC

AB/DC = 5

given that

In American and Canadian English, a quadrilateral having at least one pair of parallel sides is referred to as a trapezoid (/traepzd/). It is referred to as a trapezium in British and other varieties of English (/trpizim/).

In Euclidean geometry, a trapezoid is a convex quadrilateral by definition. The bases of the trapezoid are the parallel sides. If the remaining two sides are parallel, they are referred to as the legs (or lateral sides); if not, the trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is one that lacks equal-sized sides.

In trapezoid ABCD we have AB parallel to DC

E as the midpoint of BC

F as the midpoint of DA.

The area of ABEF is twice the area of FECD.

AB+EF/2 = 2(DC+EF/2)

AB+EF/2 = DC+EF

AB+EF = 2DC+2EF

EF is the midline between AB and DC

AB + (AB+DC/2) = 2DC+2(AB+DC/2)

[tex]\frac{3}{2}[/tex]AB +DC/2 = 2DC +AB + DC

[tex]\frac{3}{2}[/tex]AB +DC/2 = 3DC+AB

[tex]\frac{3}{2}[/tex]AB - AB = 3DC - DC/2

AB/2 = 5DC/2

AB = 5DC

AB/DC = 5

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Answer: 1,500+7=2,200 Y=700 2.

Data handling is the process of ensuring that research data is stored, archived, or disposed of in a safe and secure manner during and after the conclusion of a research project.

The following are the given details:

Item  Leadtime   On hand Inventory Lot sizing criteria Schedulereceipts

A             2                0                                   L4L                  10 in week 2

B             1                 0                                   LAL                       0

C             1                 10                                  50                       0

D             2                0                                   50                       0

E             1                 50                                180               50 in week 1

F             1                180                                L4L               50 in week 1

The Complete MRP schedule can be seen in the attached images below:

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The gross requirements for each component are as follows:

b = 2 x a = 44 units in week 7

c = 3 x a = 66 units in week 7

d = 2 x e + 1 x c + 2 x a

= 110 units in week 3, 56 units in week 6, 44 units in week 7

e = 1 x b = 44 units in week 5

f = 2 x b + 2 x c = 88 units in week 5 and 112 units in week 6

Net requirements = Gross requirements - Projected on hand

Projected on hand = Planned order receipts - net requirements

Planned order receipts = net requirements

Planned order releases = net requirements adjusted as per the lead time

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The expression that represents Ana's age is 2 + w.

A mathematical expression is made up of a statement, at least two integers or variables, and one or more arithmetic operations. This mathematical operation enables the multiplication, division, addition, or subtraction of numbers. The following is the structure of an expression:

Expression: (Number/Variable, Math Operator, Math Operator)

Given:

let w represent Wanda's age.

let a represent Ana's age.

Also given Ana is 2 years older then Wanda

Ana's age can be calculated as Wanda's age plus 2

Hence the expression which represents Ana's age  is given below

Ana's age = 2 + Wanda's age

a = 2 + w

Hence Ana's age is 2 + w.

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Using the trigonometric ratios for the angle A-cosθ= Base/ Hypotenuse =11.8726

What is trigonometric?

Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure.

For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.

Here, if we consider the ∠A=θ, then AB is the base, BC is the perpendicular, and AC is the hypotenuse.

And it is given that, ∠A=θ=32°,AB=x, and C=14 units.

Now, using the trigonometric ratios for the angle A-cosθ= Base/Hypotenuse

cosA=AB/AC

cos30°=x/14

x=14×cos32°

=11.8726

≈11.9units

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To find the difference between the highest monthly expenses and the lowest monthly expenses, June has to find the range of the data.

Becuse there must be the same number of students in both halfs of the circle, we will see that there are 15 students on the circle.

We know that each student faces someone across the circle. If student 2 faces student 9.

Between student 2 and 9 there are:

9 - 2 = 7 students.

So we will have 7 students in both sides of the line formed between students 2 and 9.

And we know that student 1 is on the right side of student 2, so that is one of the seven students, and the other six ones are numbers larger than 9,

Then the total number of students is given by the addition:

9 +  6 = 15

There are 15 students in the circle.

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Answer: 146

Step-by-step explanation:

HK = GK x 2 because a rectangles diagonals are equal.

73 x 2 = 146

Answer:

146 that should be right

B) dx/dt=-2dy/dt is the relation of the rate of change for the conditon given in the question.

Rate of change is a measure of how quickly a variable is changing. It is often represented by the symbol "d/dt" or "∆y/∆x" and is calculated by finding the ratio of the change in one variable to the change in another variable.

Given :

x + 2y =k

where k is a constant

differentiating above equation with respect to time we get

=> dx/dt + 2dy/dt =dk/dt

=> dx/dt + 2dy/dt =0

=> dx/dt = -2dy/dt

so Option B) dx/dt=-2dy/dt is correct option for the above problem

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To form the image, the pre-image should move five units right and three units up.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.

Given, The triangles on the grid below represent a translation.

On a coordinate plane, triangle A B C is shifted 5 units to the right and 3 units up to form triangle A prime B prime C prime.

In geometry, figures in a plane can be transformed in a variety of ways, including shifts and scaling, to produce new shapes. The new (transformed) shapes are called images and the original, unaltered shapes are called preimages.

Therefore, The pre-image should travel five units right and three units up to form the image.

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To show that 11k+1 - 6 is divisible by 5 using mathematical induction, the first step is to assume that the statement is true for any given positive integer k.

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if N < 10 then the count of numbers will be N , if N / 10 < 10 then 9 , if

N / 100 < 10 then 9 + N – 99 , if N / 1000 < 10 then 9 + 900 , if N / 10000 < 10 then 909 + N – 9999.

in all other case 90909.

what is leading zeros?

Leading zeros are zeros that appear at the beginning of a number or numerical value. These zeros are used to align digits or to indicate the number of digits in a number, but do not affect the value of the number itself. For example, 007 and 7 are the same number but the leading zeros indicate that there are 3 digits in the number.

The above contions specifies the count of number having odd number of digits and without leading zeros.

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The value of baseball cards is increasing by 35% per year.

Option A is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 4 = 9 is an equation.

We have,

The value of a baseball card in dollars is given by,

0.35y + 0.15

Where y is the number of years since it was released.

This means,

The value of the baseball card is 0.15.

There is a yearly increase of 35%.

35% = 35/100 = 0.35

Thus,

The value of baseball card is increasing at 35% per year

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The result of this addition is a single tensor with 100 elements, which has the shape 100. Therefore, the shape of the resulting tensor is 100.

The expression is ones(100) + ones(100). This expression is valid, and the shape of the resulting tensor is a 100-dimensional vector.

We can calculate this by starting with the shape of each of the tensors. The first tensor, ones(100), is a 1-dimensional tensor with 100 elements. The second tensor, ones(100), is also a 1-dimensional tensor with 100 elements.

When we add these two tensors together, we are combining the elements of each tensor into a single tensor. In this case, we are combining the 100 elements from the first tensor with the 100 elements from the second tensor, resulting in a single tensor with 100 elements.

The result of this addition is a single tensor with 100 elements, which has the shape 100. Therefore, the shape of the resulting tensor is 100.

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