A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
a) Write an expression for the volume V in terms of both x and y.
b) Use the given information to write an equation that relates the variables.
c) Use part (b) to write the volume as a function of only x.
d) Finish solving the problem by finding the largest volume that such a box can have.

Answer:

answer is option A

Step-by-step explanation:

2x+6≤10or2x+8>20

2x≤10-6 or 2x>12

x≤4/2 or x>2/12

x≤2 or x>6

By translating, stretching and contracting, or reflecting, two functions from the same function family can become one another. Any function from a function family can be obtained by applying one or more transformations to a parent function.

By altering the graph of the parent absolute value function, many other absolute value functions can be created. The resulting absolute value functions take the following form:

[tex]y = |x-h| + k[/tex], where h and k are real numbers.

Absolute value functions can be created using the same kinds of operations that produce new linear functions. They also have a similar impact on absolute value functions. The transformations may, however, appear differently graphically due to some important discrepancies between linear functions and absolute value functions.

Here are some of the transformations:

Vertical Transformations

Horizontal Transformations

This is how we graph the transformation of an absolute value function.

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The number that should be added in the polynomial [tex]x^{3}-6x^{2} +11x+8[/tex]  , so that it is completely divisible by  [tex]x^{2} -3x+2[/tex] is  -14  .

What are Polynomials ?

A polynomial is defined as a type of algebraic expression where the exponents of all the variables is a whole number.

The exponents of the variables in any polynomial is a non-negative integer. A polynomial consists of constants and variables.

the polynomial expression is ⇒ [tex]x^{2} -3x+2[/tex], which can be written in factor form  as  : [tex](x-1)(x-2)[/tex] ,

that means the polynomial is divisible by both x = 1 and x = 2 ;

So , on substituting [tex]x=1[/tex] in the polynomial [tex]x^{3}-6x^{2} +11x+8[/tex] ,

we get , [tex]1^{3}-6\times1^{2} +11\times 1+8[/tex] = 14 ;

on substituting [tex]x=2[/tex] in the polynomial [tex]x^{3}-6x^{2} +11x+8[/tex] ,

we get , [tex]2^{3}-6\times2^{2} +11\times 2+8[/tex] = 14 ;

So , the remainder is 14  .

Therefore , -14 should be added in the given polynomial .

The given question is incomplete , the complete question is

What should be added in the polynomial x³ -6x² +11x +8 so that it is completely divisible by x² -3x +2 ?

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The smallest positive integer that is both a multiple of 7 and a multiple of 4, is 28 .

To find the smallest positive integer that is both a multiple of 7 and a multiple of 4, first find the multiples of both 4 and 7 to see which multiples are common between them and which are the lowest.

The multiples of 4 and 7 are:

Multiples of 4 :

4, 8, 12, 26, 20, 24, 28, 32, 36, 40,

Multiples of 7 :

7 , 14 , 21 , 28 , 35 , 42 , 49

The smallest positive integer that is both a multiple of 7 and a multiple of 4 is therefore 28.

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The term "revolution" is often used to refer to a dramatic and sudden change in a situation, especially  number of a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another.

Revolution is a term used to refer to a dramatic and sudden change in a situation, especially a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another, thus making it a revolution.

The term "revolution" is often used to refer to a dramatic and sudden change in a situation, especially number of a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another.

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Therefore , the solution of the given problem of tractable problem comes out to be Undecidable troubles are those for which there will never be a computer solution.

a technique that completes a manageable task in a polynomial period of time. Utilizing the polynomial upper bound is advantageous. A problem is said to be intractable if it cannot be solved in a polynomial amount of time. The bound has a minimum that is exponential.

Here,

Intractable problems would be those in which there is significant proof that, while they can be solved by a desktop, they cannot be solved quickly enough to be actually useful in practice.

Undecidable troubles are those for which there will never be a computer solution, whereas undecidable difficulties are those for which there will never be a computer solution.

Therefore , the solution of the given problem of tractable problem comes out to be Undecidable troubles are those for which there will never be a computer solution.

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The value of linear equations are  f(x) = -17 and f(x) = -73 .

What is linear equation ?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y).

Equation with two variables of degree one is referred to as a linear equation with two variables. Formal expression: ax+by+c=0, where a, b, c R & a, b 0.

A)

   f (x) = -2*12 + 7

          = -24 + 7

          = -17

       f(x) = 5*-14 - 13

             = -60 -13

             = -73

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Answer: To find an equation for a parabola that passes through three specific points, we can use the method of algebraic manipulation.

Since the parabola is symmetric about the y-axis, the equation of the parabola will have the form y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.

Given three points (-2,7), (1, 10), and (2,27), we can use the vertex form of the equation and substitute the x, y values of the three points to find the values of a, h and k

y = a(x-h)^2 + k

Since it is passing through (-2,7), we can substitute the values of x and y in the equation and get

7 = a(-2 - h)^2 + k

It passing through (1, 10), we can substitute the values of x and y in the equation and get

10 = a(1 - h)^2 + k

It passing through (2,27), we can substitute the values of x and y in the equation and get

27 = a(2 - h)^2 + k

Now we have three equations with three variables, we can solve it using any of the techniques such as substitution, elimination or matrix. But the final equation will be in the form of y= a(x-h)^2 + k

Here, x= (-2, 1, 2) and y = (7, 10, 27) . Therefore it has a unique parabola passing through these points.

Step-by-step explanation:

The number that should be added to the polynomial 2x³ - 3x² - 8x is 12x

When a polynomial is divided by a linear factor, the remainder is equal to the constant term of the polynomial multiplied by the reciprocal of the linear factor.

In this case, the polynomial is 2x³ - 3x² - 8x, and the linear factor is 2x-1.

To find the remainder when this polynomial is divided by 2x-1, we need to multiply the constant term of the polynomial, which is -8, by the reciprocal of 2x-1, which is 1/(2x-1).

So the remainder should be -8*(1/(2x-1)) = -8/(2x-1) = -4/(x-0.5)

To make the remainder 12, we will add 12x to the polynomial so that the new constant term is 12.

The polynomial 2x³ - 3x² - 8x + 12x will have a remainder of 12 when divided by 2x-1.

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The value of x is 7 and value of y is 8

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Given data set is {4, 6, x, y, 10}

The mean and the median are the same value.

The median is the middle value, which is x, and the mean is:

M = (4 + 6 + x + y + 10)/5

Then we have that:

(4 + 6 + x + y + 10)/5 = x

Now let's solve this:

(4 + 6 + x + y + 10) = 5x

20 + x + y = 5x

20 + y = 4x

And x and y are whole numbers. Then we can take x = 7 to get:

20 + y = 4*7

y = 28 - 20 = 8

Then we have x = 7 and y = 8.

Hence, the value of x is 7 and value of y is 8 in the data set 4 6 x y 10

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Answer:

Step-by-step explanation:

Rewrite the function as an equation.

y

=

3

x

−

4

Use the slope-intercept form to find the slope and y-intercept.

Tap for fewer steps...

The slope-intercept form is

y

=

m

x

+

b

, where

m

is the slope and

b

is the y-intercept.

y

=

m

x

+

b

Find the values of

m

and

b

using the form

y

=

m

x

+

b

.

m

=

3

b

=

−

4

The slope of the line is the value of

m

, and the y-intercept is the value of

b

.

Slope:

3

y-intercept:

(

0

,

−

4

)

Using the mean value theorem, the least possible value of f(3) is - 2

The mean value theorem states that for a function f(x) on the interval (a, b), there exists a value c such that

f'(c) = f(b) - f(a)/(b - a)

Since ƒ(−1) = 10 and ƒ'(x) ≥−3 for all values of x, we require the least possible value of f(3).

Using the mean value theorem where

We choose f'(c) = 3 since f'(x) = 3 is the minimum value of f'(x).

So, substituting the values of the variables into the equation, we have that

f'(c) = f(b) - f(a)/(b - a)

f'(x) = f(3) - f(-1)/(3 - (-1))

f'(x) = f(3) - f(-1)/(3 + 1)

f'(x) = f(3) - f(-1)/4

So, making f(3) subject of the formula, we have that

f(3) = f(-1) + 4f'(x)

Since

Substituting the values of the variables into the equation, we have that

f(3) = f(-1) + 4f'(x)

f(3) = 10 + 4(-3)

f(3) = 10 - 12

f(3) = - 2

So, the least possible value is - 2

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The greatest poible value of b is (72-14)/(14-(-2))×(x+2)+14.

Possible value is the potential for something to have a particular outcome, result, or consequence. It is the amount of beneficial or desirable result that can be achieved, given the available resources. Possible value is often determined by assessing various factors such as cost, risk, and time. It is an important concept in economics, finance, and business, as it helps to identify the most beneficial course of action.

The greatest possible value of b is found by solving the equation of the line through A and B, which is y = (72-14)/(14-(-2))×(x+2)+14.

Setting the y-value to 0 and solving for x gives x = -14, which means b = 14. Therefore, the greatest possible value of b is 14.

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The common difference is 0.42. The comparison of each ratio's successive values over time for a single firm.

Now, According to the question:

What is meant by ratio?

Therefore, simplifying the equation then we get,

if you take the ranges (y) and subtract one from the other you get 0.42.

(22.84 - 22.42)/(2 - 1) = 0.42/1 = 0.42

(23.26 - 22.84)/(3 - 2) = 0.42

(23.68 - 23.26)/(4 - 3) = 0.42

Hence, The common difference is 0.42

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Answer:

L' = (-1,2)

M' = (1,2)

L'M' = 2

The slope of the original segment and dilated segment are both zero

Step-by-step explanation: just did it on edge!

"Thickness of two boards = Thickness of first board + Thickness of second board" would be the closely estimated equation.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, separated by the 'equal' sign.

x ax2 + bx + c = 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. a 0. It has at least one solution since it is a second-order polynomial equation, which is guaranteed by the algebraic basic theorem.

The thickness of the two boards together can be represented by the sum of the thickness of each individual board. Therefore, the equation that most closely estimates the thickness of the two boards stacked together is:

Thickness of two boards = Thickness of first board + Thickness of second board

So in this case, the equation would be:

Thickness of two boards = inch + inch

This equation gives the total thickness of two boards stacked together and it's the closest estimation.

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Answer:

Step-by-step explanation:

A half of a cup is greater than 1/3 of a cup. This is because if you convert the fractions to the nearest common denominator (6) you can see that 1/2 = 3/6 and 1/3 = 2/6, so 1/2 of a cup is greater.

I hoped this helped!!

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members of the sequence is a constant value. The general form of an arithmetic sequence is:

a, a + d, a + 2d, a + 3d, ...

where "a" is the first term of the sequence, and "d" is the common difference between successive terms.

Given the function f(n) = 2n - 1, we can see that the function outputs a sequence of integers, with a common difference of 2. So f(n) is not an arithmetic sequence.

Arithmetic sequence is a list of numbers in which each number is the sum of the previous number and a constant number.

Example: 1,3,5,7,9 is an arithmetic sequence because the constant difference is 2.

If you have any other question feel free to ask.

The movie need to earn $ 1,512,500 in the next 7 weeks.

The unitary technique involves first determining the value of a single unit,

followed by the value of the necessary number of units.

For Example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

A company spent 762500 to make a movie.

The total movie will be earned

= 762500 + 2000000

= 2,762,500

First week the movie earned 1, 250,000.

So, in 7 weeks the movie needs to earned

= 2,762,500 - 1, 250,000

= 1,512,500

Hence, the movie earn 1,512,500 more.

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Answer:

Step-by-step explanation:

factoring

e =

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

= (a+1)(-2a-4)

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

24 cars lacked power steering but did have air conditioning and an automatic gearbox.

According to the question,

No. of cars with Air conditioning = 88

No. of cars with Automatic transmission = 96

No. of cars with power steering = 76

All three = 4

None of these extras = 19

Only air conditioning = 23

Only automatic transmissions = 61

Only power steering = 28

Power steering and automatic transmission both = 11

The calculation for missing values:

11 - 4 = 7

96-61-4-7 = 24

88-24-4-23=37

It is obvious that 24 cars lacked power steering but did have air conditioning and an automatic gearbox.

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The radius of convergence R = 1/b

The radius of convergence for a power series is given by the formula:

1/R = lim sup |bn|^(1/n)

In this case, the radius of convergence is given by:

1/R = lim sup |bn|^(1/n) = lim sup b^(1/n) = b^(1/n) as b > 0.

So R = 1/b

As for the interval of convergence, it is the set of values of x for which the series converges. The series converges when |x-a| < R, and diverges when |x-a| > R. So the interval of convergence is (a-R, a+R) which is (a-1/b, a+1/b)

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On solving the provided proportionality question we got that, U = 7.72, is rounded off nearest to tenth digit.

When two quantities or variables are linearly related in mathematics, the term "proportional" is used. Both quantities increase by two times when one increases by two. Each variable decreases in proportion to the other when it falls to 1/100th of its previous value.We must determine the ratio of the two quantities for each value supplied in order to determine if two quantities are proportionate. The proportionate link between them is evident if their proportions are equal. Their connection is not proportionate if all the ratios are out of balance.

Here,

U/5 = 17/11

U = 17/11 X 5

U = 7.72

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Answer: k = 11

Step-by-step explanation:

[tex]\frac{11}{3k} = \frac{1}{3} \\\\[/tex]

Cross multiply:

[tex]11 * 3 = 1*3k[/tex]

[tex]33 = 3k[/tex]

[tex]k = 11[/tex]

Using the linear function given, the tree will have a height of 5.25 feet in 3 years

Linear functions are mathematical functions where the output values are in direct proportion to the input values. A linear function has the form

y = mx + c, where m is the slope of the line and c is the y-intercept.

In this problem, the function given already models y = mx + c

Since y = 3/4x + 3;

The height of the tree in 3 years can be solved as;

y = 3/4(3) + 3

y = 9/4 + 3

y = 5.25

The height of the tree in 3 years is 5.25 feet

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The slope of the line that passes through the points (-5, 8)and (−5,4) is undefined.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

Given:

A line that passes through the points (-5, 8)and (−5,4).

If a line passes through two points (x₁ ,y₁) and (x₂, y₂),

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

m = (4-8) / (-5 + 5)

m = -4/0

The slope is undefined.

Therefore, the slope is undefined.

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Answer: Zasha spent $6 on packages of gum, which means that she bought 6 / $1.20 = 5 packages of gum.

If she has a $20 bill, she has $20 - $6 = $14 left to spend on gum.

Zasha can buy an additional 14 / $1.20 = 11.6666666666666666666666667 packages of gum with this money.

Rounded down, Zasha can buy an additional 11 packages of gum.

Therefore, Zasha can buy a total of 5 + 11 = 16 packages of gum.

Step-by-step explanation:

The value of a is [tex]-\frac{12}{5}[/tex] if 2 is a zero of the polynomial [tex]$4 x^2-2 x+5 a$[/tex].

As per the given data, the polynomial is p(x) = [tex]$4 x^2-2 x+5 a$[/tex]

The polynomial is zero polynomial if x is 2

The places where a polynomial becomes zero overall are known as the zeros of the polynomial. The term "zero polynomial" refers to a polynomial with a value of zero (0).

Replace x with 2 we get

Therefore [tex]4(2)^2-2(2)+5 \mathrm{a}=0[/tex]

[tex]\Rightarrow[/tex] 4 (4) - 2 (2)+5 (a) = 0

[tex]& \Rightarrow[/tex] 16 - 4 + 5 (a) = 0

[tex]& \Rightarrow[/tex] 12 + 5 (a) = 0

Add -12 on both sides we get

12 + 5 (a) - 12 = 0 - 12

5 (a) = -12

Divided by 5 into both sides we get

a = [tex]-\frac{12}{5}[/tex]

Therefore the value of a is [tex]-\frac{12}{5}[/tex]

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Answer:

$18

Step-by-step explanation:

To find how many times she has saved money, you would do 27/3=9. This means that she worked 9 times. Since she worked 9 times, she earned $45 in total. To find how much she saved you would do 45-27=$18