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Correct option is E, The roots of Auxiliary equation are √6 and -√6.

The solution of an nth-order differential equation or difference equation depends on an algebraic equation of degree n called the characteristic equation (or auxiliary equation) in mathematics.

The auxiliary equation, however, lacks true roots. Consider the simple harmonic equation y + y = 0, which has solutions and was given that name because of how it relates to the vibration of a musical tone. Y2(t) = cos t and Y1(t) = sin t. r2 + 1 = 0 is the auxiliary equation for the straightforward harmonic equation.

The Auxiliary form of the given equation is m² - 6 = 0

Using identity a² - b² = (a - b) (a + b),

⇒ The equation becomes (m - √6)(m+√6) = 0

⇒ either m = √6 or m = -√6

Hence, The roots of Auxiliary equation are √6 and -√6.

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Note that in the above scenario, the original dimensions of the piece of metal that is rectangular are:

Let x equal the width of the box and x + 15 equal the length of the box.  

Subtract 6 (the amount cut out) from both of these to get the following terms: (x+9) and (x-6).  

These are the length and width of the box respectively.  We also know that the height of the box is 3.  Plugging these into the formula for the volume of a box gives you:

3(x+9)(x-6)=858

If the above is expanded, we have:

3x²+9x-162=858

Subtract 858 from both sides to make the quadratic equation equal zero.  Plug the a, b and c coefficients into the quadratic formula (or factor) to find x.

3x²+9x-162- 858 = 0

⇒ 3x²+9x -1020= 0

At the stage we solve for x:

3x2+9x−1020 = 0

Factor left side of the equation.

3(x−17)(x+20)=0

Set factors equal to 0.

x−17=0 or x+20=0

x=17 or x=−20

We can't get a negative side, so the answer that makes sense is x =17 recall that the metal is 15 inches longer than its width.

So if the Width is x and the Lenght is x +15

Then Width is 17 while the length is 17 + 15 = 32inces.

Thus the original dimensions of the metal are:

Length: 32 Inches

Width: 15 Inches

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D. Y= 4/3 (x)+4

The first step i did was finding C for the standard form equation of a linear function which is:

y=mx+c

Finding C is easy because C is simply the y-intercept(The point that crosses the y axis) which we can see from the graph is 4.

That allowed me to isolate possible answers of B and D because they contain a C which is equal to 4.

To determine which equation it was, i substituted an X coordinate to see if i got the correct Y coordinate from the graph.

Lets try B first and substitute the coordinate -3:

3/4 ×(-3)+4=1.75

As we see thats incorrect because the coordinate on the graph is (-3;0).

So we can conclude it is not b, however lets try D.

4/3 (-3)+4=0.

We see that this equation gives us the correct answer.

So therefore the correct equation is D

4/3 (x)+4

We want to get the equation on the graphed line.

The equation of the line is: y = 8*x - 9

Then the line is something like:

y = 8*x + b

To find the value of b, we can use one of the two points we got above.

The point (0, -9) means that when x = 0, y is equal to -9, then we can write:

-9 = 8*0 + b

-9 = b

Finally, the equation of the line is:

y = 8*x - 9

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The tomatoes will not likely occur in less than 30 and greater than 50 latitude regions.

When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.

Given that Tornadoes have occurred on every continent except Antarctica. They occur most often between latitudes of 30° and 50°.

The region between tomatoes occur is,

30< T < 50

The region in which the tomatoes are not occur is,

T < 30

T > 50

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On solving the provided question, we can say that = 9227446, depreciation  If round at hundredth, then, Number of tree = 9227400.

Identify the asset's cost. Get the asset's total depreciable value by deducting the projected salvage value from the asset's purchase price. Determine the asset's useful life. The annual depreciation amount is obtained by dividing the sum of steps (2) and (3) by the figure determined in steps (3).

Depreciation is the practice of deducting the cost of a company asset over a number of years as opposed to just one. There are four primary ways to depreciate assets: linear, twofold declining, sum of the years' digits, and manufacturing units.

As it is given, 1% of left tree is decaying every year.

Number of tree left after 8 years.

A = P{1-R/100}^ = 10M{1-1%/100}^  

= 10M *(0.99)^

= 9227446

If round at hundredth, then,

Number of tree = 9227400.

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Answer:

600 ml

Step-by-step explanation:

let x = amount of water needed

1/6 = 100/x

cross multiply to solve for x:

x = 600

Answer:AC is equal to 88

Step-by-step explanation:

C=11

A=8

B=2

The amount of blue paint and red paint that you use will be 3 quarts and 2 quarts, respectively.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

You mix 1/2 quart of blue paint for every 1/3 quart of red paint to make 5 quarts of purple paint.

Let 'x' be the amount of blue paint and 'y' be the amount of red paint. Then the equations are given as,

x / y = (1/2) / (1/3)

x / y = 3/2                        ...1

x + y = 5                          ...2

From equations 1 and 2, then we have

(3/2) y + y = 5

(5/2) y = 5

y = 2 quart

Then the value of 'x' is given as

x + 2 = 5

x = 3 quart

The amount of blue paint and red paint that you use will be 3 quarts and 2 quarts, respectively.

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The statements that are correct include the following:

B. The total number of students is > 160.

C. The percent as a part-to-whole ratio is 16/100.

E. There are 1600 students in the school.

In Mathematics, a percentage can be defined as any number that is expressed as a fraction of hundred (100). This ultimately implies that, a percentage indicates the hundredth parts of any given number.

For the total number of students, we have:

Total number of students = 10/100 × T = 160

0.1T = 160

T = 160/0.1

T = 1,600 students.

Note: Percentage as a part-to-whole ratio is 10/100.

In conclusion, we can reasonably infer and logically deduce that the total number of students at Park Junior High is equal to 1,600 students.

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Complete Question:

At park junior high 10% or 160 of the students play a musical instrument how many students attended the school?

Which statements are correct? check all that apply.

A.the total number of students is 160

B. The total number of students is > 160.

C.The percent as a part-to-whole ratio is 10/100

D.the percent as a part to whole ratio is 160/100

E.there are 1600 students in the school

F.there are 250 students in the school. ​

The equation to find the cost per slice is c=x/n  where x is the total prize of pizza and n is the total no. of slices.

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.

If x is the total price of pizza( which is $20 in this case) and if all the slices are equally cut, then the cost of each slice would be total price divided by total slices.

Hence, the equation is  c= x/n

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By evaluating the quadratic equation, we will see that the values of the expressions are:

f(-1) + f(4) = 27

f(-1) - f(4) =  -35

Here we need to work with the quadratic function:

f(x) = x^2 + 6x + 1

First, we want to find the value of the expression:

f(-1) + f(4)

To get that, we need to evaluate the quadratic equation:

f(-1) = (-1)^2 + 6*(-1) + 1

f(-1) = 1 - 6 + 1 = -4

f(4) = 4^2 + 6*4 + 1

f(4) = 16 + 24 + 1 = 31

Then the first expression is:

f(-1) + f(4) = -4 + 31 = 27

The second expression is:

f(-1) - f(4)

We already know these values, we can replace these to get:

f(-1) - f(4) = -4 - 31 = -35

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The value of a and b for the given quadrilateral to be a parallelogram is a = 16 and b = 4.

A quadrilateral having two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles that are additional to the transversal on the same side. 360 degrees is the sum of all interior angles.

A parallelepiped is a three-dimensional shape with parallelogram-shaped faces.

We know that the opposite sides of a parallelogram are equal in length so we can equate the two sides as follows:

For a:

2a + 6 = 3a -10

Isolate the value of a by moving all the values having a to one side:

2a + 6 = 3a -10

3a - 2a = 6 + 10

a = 16

For b:

6b - 3 = 5b + 1

Isolate the value of b by moving all the values having b to one side:

6b - 3 = 5b + 1

6b - 5b = 1 + 3

b = 4

Hence, the value of a and b for the given quadrilateral to be a parallelogram is a = 16 and b = 4.

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The number of years that is the investment period, given the account balance, is 7 years

You are given the equation, A = 5,000 ( 1 + 0.025t ) where t is the investment period.

Find the simple interest per year :

= 5, 000 x 2. 5 %

= $ 125

Then find the number of years as :

= ( Account balance - Opening deposit ) / Annual simple interest

= ( 5, 875 - 5, 000 ) / 125

= 7 years

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The value of B, the area of the base, is 1,540 m^2.

Value is the relative worth, merit, or importance of something. It can refer to an intangible concept such as a person's principles or a tangible object such as a rare coin. Values vary widely across cultures, societies, and individuals, and can even vary within the same individual over time. Value is often used to describe the worth of something in terms of its usefulness, beauty, or desirability. Value can also refer to the moral or ethical standards of a person or group.

The area of the base can be found by using the formula A = V/h where A is the area of the base, V is the volume, and h is the height. In this case, A = 26,214 / 17 = 1,540 m^2. Therefore, the value of B, the area of the base, is 1,540 m^2.

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Answer:

-2x + 9

Step-by-step explanation:

-2(x - 7) + (-5)

-2x + 14 -5

-2x + 9

By means of relationship between powers and roots and algebra properties we conclude that radical expression ∛y · [2 · y · ∛(8 · y²) - ∛(y⁵) - 4∛(8 · y²)] is equivalent to 2 · y · ∛(8 · y⁵) - ∛(y⁶) - 4 ∛(8 · y³).

In this problem we find a polynomial-like radical expression that must be simplified by algebra properties and by taking advantage of relationship between powers and roots. First, write the entire expression:

∛y · [2 · y · ∛(8 · y²) - ∛(y⁵) - 4∛(8 · y²)]

Second, apply distributive property:

2 · y · ∛(8 · y²) · ∛y - ∛(y⁵) · ∛y - 4∛(8 · y²) · ∛y

Third, use relationship between roots and powers:

2 · ∛(y³) · ∛(8 · y²) · ∛y - ∛(y⁵) · ∛y - 4∛(8 · y²) · ∛y

Fourth, apply multiplication of roots and multiplication of powers of equal base:

2 · y · ∛(8 · y⁵) - ∛(y⁶) - 4 ∛(8 · y³)

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The volume of the solid is 4π([tex]\frac{19}{6}[/tex]) and the integral of the f(x) , the line 2y=x, and x=2 is 3([tex]e^{4}[/tex] -1).

What is integral ?

An integral is a mathematical concept that is used to calculate the area under a curve or the total amount of change in a variable over a certain interval.

1)The first step is to find the intersection of the lines 2y=x and x=2. They will intersect at (2,1).

The second step is to find the intersection of the lines 2y=x and x=0. They will intersect at (0,0).

The third step is to find the integral over the region.

∫∫ (6e^(x^2))dxdy = ∫[0,2] (6e^(x^2))dx ∫[0,x/2] dy = ∫[0,2] (6e^(x^2))dx (x/2) = [3e^(x^2)] at [0,2] = 3e^(4) - 3e^(0) = 3(e^4 -1).

2)To find the volume of this solid, we can use the method of cylindrical shells.

we can find the function of the surface (1/3)x+(1/2)y+(1/4)z=1

by using cylindrical coordinates:

x = rcosθ , y = rsinθ , z = z

and substitute them in the equation (1/3)x+(1/2)y+(1/4)z=1

(1/3)rcosθ+(1/2)rsinθ+(1/4)z=1

(1/3)r(cosθ+2sinθ)+(1/4)z=1

z = 4(1-(1/3)r(cosθ+2sinθ))

V = ∫∫∫ 4(1-(1/3)r(cosθ+2sinθ)) rdrdθdz = 4π ∫[0,3] (1-(1/3)r(cosθ+2sinθ)) r^2 dr = 4π ∫[0,3] (1-r+(2/3)r^2) dr = 4π(3 - (3/2) + (2/3)) = 4π(19/6).

Hence , The volume of the solid is 4π([tex]\frac{19}{6}[/tex]) and the integral of the f(x) , the line 2y=x, and x=2 is 3([tex]e^{4}[/tex] -1).

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If f '(t) =20/1 + t2 f(1) = 0, f is f(x) = 20tan⁻¹(t) - 5π.

Integrals are the values of the function seen by the process of integration.

The procedure of getting f(x) from F (x) is called integration.

Given f '(t) = 20/ 1 + t2 and f(1) = 0

We know that by definition of the integral of tan⁻¹(x) = 1/1 + t2

f '(t) = 20 / 1 + t2 = 20(1/ 1 + t2)

Integrate on both sides, we get

f(t) = 20tan⁻¹(t) + c

f(1) = 0; f(1) = 20tan⁻¹(1) + c = 0

c = -20(π/4) = -5π.

Therefore, f(x) = 20tan⁻¹(t) -5π.

If f '(t) =20/1 + t2 f(1) = 0, f is f(x) = 20tan⁻¹(t) - 5π.

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The expression 3(x-2) is equivalent to 3x - 6.

When you distribute the 3 to the term inside the parentheses, you multiply 3 by x and by -2, resulting in 3x - 6.

This is because the distributive property states that for any real numbers a, b, and c, the expression a(b+c) is equivalent to ab + ac. In this case, we have a = 3, b = x, and c = -2, so we can apply the distributive property and get 3x - 6.

You can also think of it as 3 multiplied by (x-2)

3x - 32 = 3x - 6

Answer:

2,4,6,8,0

Step-by-step explanation:

A number is divisible by 2 if it ends in 2, 4, 6, 8 or 0. A number is divisible by 5 if it ends in 5 or 0. A number is divisible by 10 if it ends in a 0.

Answer: 3c+3

Step-by-step explanation:

Hector's homemade sandwich costs $2.00.

The correct option is B.

Combining objects and counting them as one big group is done through addition. In arithmetic, addition is the process of adding two or more integers together. Addends are the numbers that are added, and the sum refers to the outcome of the operation.

Given:

The total cost for the bread is $0.16.

And the other ingredients cost $1.84.

And the total cost of Hector's homemade sandwich,

= Total cost for the bread + The cost of the other ingredients.

Total cost of Hector's homemade sandwich,

= $0.16 + $1.84

Total cost of Hector's homemade sandwich,

= $2.00

Therefore, $2.00 is the cost for the sandwich.

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The complete question:

Using the calculations for each ingredient:

How much did Hector's homemade sandwich cost?

Hint:

Remember, the total cost for the bread is $0.16.

And the other ingredients cost $1.84.

A : 1.90

B : 2.00

C : 2.99

D : 1.74

The option A that is 2 days after being planted, the two plants will of the same height( Where the two lines intersect in a graph).

How to know if the lines intersect in the graph of a function?

All the points (and only those points) which lie on the graph of the function satisfy its equation.

Thus, if a point lies on the graph of a function, then it must also satisfy the function.

And after solving both equations if the values of x and y satisfies both equations then they intersect at that values of x and y.

Now,

On Monday, Eric planted a 3-centimeter-high tomato plant and a 5-centimeter-high tomato plant in his garden.

The graph models the growth of each plant since the day they were planted.

The intersection of the two lines represents the days after being planted were the two plants the same height.

The answer is that 2 days after being planted, the two plants are the same height.

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The answer is that 2 days after being planted, the two plants are the same height.

How to know if a point lies in the graph of a function?

The answer is that 2 days after being planted, the two plants are the same height.

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The difference between the two numbers -5-6 is -11 Option A

In basic arithmetic, difference implies the addition or two negative numbers or the subtraction of two positive numbers

The two given numbers we are asked to find the difference are -5-6

We can notice that these tow numbers are negative and negative. to this effect, we have to find the sum of the two negative integers to get

-5 + (-6)

This gives us -11

Conclusively the answer is -11

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Answer:

x = 1.28, y = -3.68

Step-by-step explanation:

y = -6x + 4

y = 1/4x - 4

-4= -6x - y

4 = 1/4x - y

4= 6x + y

4 = 1/4x - y

8= 6 1/4x

x = 1.28

y = -6(1.28) + 4

y = -7.68 + 4

y = -3.68

Therefore ,a s a result, the answer to the given equation problem is that a carpet costs $2072.

When all of the highest powers of the variables are equal to one, an equation is said to be linear. The parts are linked together. Another name for this kind of problem is a one-degree equation. The following paragraphs will teach you how to solve a linear equation in one or two variables using the usual form, formula, graph, and directions.

Here,

Given :

50a + 12h is the floor area of a square foot.

where h is the amount of hours.

Thus.

When t=6 and a=40

So ,

the price of carpeting

=> Price = 50a plus 12h

=> Cost = 50*40 + 12*6

=> Cost = 2000 + 72

=> Cost = 2072

As a result, the answer to the given equation's problem is that a carpet costs $2072.

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