Please Help! Multiple Choice

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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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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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.

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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Answer:Well, we already see that the number of girls is greater than the number of boys, to find the result of girls, just do the division that is 30:2= 15 and boys is 30:3= 10

i hope i have helped you

Step-by-step explanation:

Answer:

8 boys, 22 girls

To solve, you must isolate the variable. This means moving the variable and it's coefficient to the other side of the equation with only the same variable on the same side with all the constants on the other side.

If you multiply, divide, add, subtract, square root, exponent both sides of the equation by the same value, that end value stays the same.

Q)

Lets set the amount of boys to x.

The amount of girls is 3x-2 as the amount of boy is x.

Set up an equation.

x+3x-2 = 30

30 is the amount of members.

Add 2 on both sides.

x+3x+2-2=30+2

Simplify by combining like terms.

4x = 32

Divide 8 on both sides.

[tex]\frac{4x}{4} =\frac{32}{4}[/tex]

x = 8

There are 8 boys in the class.

Plug back into the equation or subtract from 30 to solve for the amount of girls.

30 - 8 = 22

8(3) - 2 = 22

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

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: 3c+3

Step-by-step explanation:

The rectangular box dimensions are 21m ,12m and 10m.

What is volume of the rectangular box?

The area of the base times the height of the rectangular prism equals the volume of the object. As a result, the formula for a rectangular prism's volume is given. A rectangular prism's volume is equal to its length, width, and height in cubic units.

Here the dimensions of a box is 2x+5 , x+4 and x+2 meters and volume of the box is 2520 [tex]m^3[/tex].

Volume of rectangular box = w×h×l [tex]unit^3[/tex]

=> 2520 = (2x+5)(x+4)(x+2)

=> 2520 = [tex](2x^2+13x+20)(x+2)[/tex]

=> 2520 = [tex]2x^3+4x^2+26x+13x^2+20x+40[/tex]

=> [tex]2x^3-16x^2+33x^2-264x+310x-2480=0[/tex]

=> [tex]2x^2(x-8)+33x(x-8)+310(x-8)=0[/tex]

=> [tex](x-8)(2x^2+33x+310)=0[/tex]

Here x-8=0 and [tex]2x^2+33x+310=0[/tex]

Then x=8 and x ∈ R.

Now the dimensions 2x+5=2(8)+5=16+5=21m.

=> x+4 = 8+4=12 m.

=> x+2= 8+2 = 10m.

Hence the rectangular box dimensions are 21m ,12m and 10m.

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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

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 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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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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The number of months it will take for both cities to have the same population is; 36 months

We are given;

Current population of city of rock hill = 70000

Rate at which people move into  city of rock hill = 100 people per month

Rate at which people move out of city of rock hill = 150 people per month

Net average increase a month = 100 - 150 = -50 people per month

Current population of City of Greenville = 61000

Rate at which people move into  City of Greenville = 200 people per month

If the number of months it will take for both cities to have the same population is x, then it means that;

70000 - 50x = 61000 + 200x

Rearrange to get;

200x + 50x = 70000 - 61000

250x = 9000

x = 9000/250

x = 36 months

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Yes, the graph of linear equation is always a straight line.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

Consider the equation y = mx + c to define a linear function with a straight line as its graph.

We are aware that the rate of change, m, for a linear function is constant. On the graph, shifting by 1 always causes you to go up by m as shown in attached image.

The graph thus resembles a staircase. It is a straight line because it always rises in equal-sized steps.

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Answer:AC is equal to 88

Step-by-step explanation:

C=11

A=8

B=2

a) The total cost of purchasing the car with the harmonized sales tax (HST) is $36,160.

b) If the customer leases the car instead of buying it, they will save $14,760.

c) After the customer returns the leased car, their lease options include:

A lease is a financing arrangement for capital assets, that enables the lessee to use the asset for a determined period while making periodic payments to the lessor.

Leases are classified into operating and finance (capital) leases.

Value of the car = $32,000

Harmonized Sales Tax (HST) = 13%

Cost of the car with HST = $36,160 ($32,000 x 1.13)

Down payment = $1,000

Financing part = $35,160 ($36,160 - $1,000)

Installment payments = 48

Periodic payments = $425

Total lease cost = $21,400 ($1,000 + 48 x $425)

Savings from leasing = $14,760 ($36,160 - $21,400)

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On solving the provided question we can say that in the circle the radius, r = 4 and center, c = -3, -1, so area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.

Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. At every angle, it is also rotationally symmetric about the center. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.

here,

in the circle

the radius, r = 4

and center, c = -3, -1

area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.

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A summary of 200 factory workers' monthly sick days is provided. From this, we can tell that about 15 workers took more than 10 sick days per month.

The number of times an event or observation occurred during an experiment or research is referred to as its frequency in statistics. It can alternatively be described as a straightforward count of a specific occurrence. Relative frequency and cumulative frequency are the two main types of frequency seen in statistics.

The given table contains the number of sick days per month and the frequency of factory workers on sick days. Then, the number or frequency of workers who took more than 10 sick days per month is,

number of workers = frequency for 11-15 days + frequency for 16-20 days

=14+1

=15

The required answer is 15 workers.

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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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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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Answer:

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

We observe that, the sequence is symmetric and all the terms are perfect squares:

This is neither arithmetic nor geometric.

The zero term is 16 and the fourth term is 0 so the nth term would be:

Answer:

a)  Neither

b)  t(n) = n² - 8n + 16

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

As the sequence has neither a constant difference or a constant ratio, the sequence is neither arithmetic or geometric.

Work out the differences between the terms until the differences are the same:

First differences

[tex]16 \underset{-7}{\longrightarrow} 9 \underset{-5}{\longrightarrow} 4 \underset{-3}{\longrightarrow} 1 \underset{-1}{\longrightarrow} 0 \underset{+1}{\longrightarrow} 1 \underset{+3}{\longrightarrow} 4[/tex]

Second differences

[tex]-7 \underset{+2}{\longrightarrow} -5 \underset{+2}{\longrightarrow} -3\underset{+2}{\longrightarrow} -1\underset{+2}{\longrightarrow} 1\underset{+2}{\longrightarrow} 3[/tex]

As the second differences are the same, the sequence is quadratic and will contain an n² term.  The coefficient of n² is always half of the second difference.  Therefore, the coefficient of n² = 1.

Write out the numbers in the sequence n² and determine the operation that takes n² to the given sequence:

[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 &0& 1& 4& 9&16 & 25&36 \\\cline{1-8} \sf operation&+16& +8&+0&-8&-16&-24&-32\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]

As the operation is not constant, work out the differences between the operations:

[tex]16\underset{-8}{\longrightarrow} 8\underset{-8}{\longrightarrow} 0\underset{-8}{\longrightarrow} -8\underset{-8}{\longrightarrow} -16\underset{-8}{\longrightarrow} -24\underset{-8}{\longrightarrow} -32[/tex]

As the differences are the same, the second operation in the sequence is -8n.  Write out the numbers in the sequence with both operations and and determine the operation that takes (n² - 8n) to the given sequence:

[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 -8n&-0&-7&-12&-15&-16&-15&-12\\\cline{1-8}\sf operation &+16&+16&+16&+16&+16&+16&+16\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]

As the operation is constant, the final operation in the sequence is +16.

So the equation for the nth term is:

[tex]\implies t(n)=n^2-8n+16[/tex]

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 given differential equation by undetermined coefficients are y(x)=c1cos(31/2 x)+c2sin(31/2x)+(-4x2+4x-4/3)e3x

A differential equation in mathematics is an equation that includes one or more functions and their derivatives. The rate of change of a function at a place is determined by the derivatives of the function. It is mostly employed in disciplines like physics, engineering, biology, and others. The study of solutions that satisfy the equations and the characteristics of the solutions is the main goal of the differential equation.

y'' + 3y=0

λ2+3 =0

λ 1=-i31/2,

λ 2=i31/2.

y0(x)=c1cos(31/2 x)+c2sin(31/2x),

yp(x)=(Ax2+Bx+C)e3x

yp'(x)=(3Ax2+(3B+2A)x+3C+B)e3x

yp''(x)=(9Ax2+(9B+12A)x+2A+6B+9C)e3x

x2e3x:3A+9A=-48,

xe3x:3B+9B+12A=0,

e3x:3C+2A+6B+9C=0.

=>A=-4,B=4,C=-4/3.

yp(x)=(-4x2+4x-4/3)e3x,

y(x)=y0(x)+yp(x)

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(-1, -2)is the best solution of  the system of equations.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

A system of equations is graphed on this coordinate grid.

we need to find the ordered pair which is the best estimate of the solution of the system of equations.

(-1, -2) is the best solution of this graph because both the lines are intersecting at that point

Hence, (-1, -2)is the best solution of  the system of equations.

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The correct rule of the transformation is,

⇒ Dilation of 0.5.

A transformation that occurs when a figure is moved from one location to another location without changing its size or shape is called translation.

We have to given that;

The coordinate of CDE are,

⇒ C (4, - 5), D (3, - 1), E (5, - 3)

And, After transformation the coordinate of image C'D'E' are,

⇒ C' (2, - 2.5) , D' (1.5, - 0.5), E' (2.5, - 1.5)

Now, We have;

⇒ C (4, - 5) = C' (2, - 2.5)

Let a dilation of the points = k

So, We can formulate;

⇒ 4 × k = 2

⇒ k = 2/4

⇒ k = 1/2

⇒k = 0.5

Hence, The correct rule of the transformation is,

⇒ Dilation of 0.5.

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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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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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