You are building an entertainment center with shelves that are x in. deep by x in. long. The height of the unit will be twice the depth. If the volume of the unit will be 8,192 in³, what is the height, in inches, of the entertainment center?

The solution of x , y , z are 3 , -3 , 9 respectively using matrix.

We have given three equations -x+2 y+z=0 , y =-2 x+3 , z = 3 x

This can be written as AX=B

Where [tex]A=\left[\begin{array}{ccc}-1&2&1\\2&1&0\\-3&0&1\end{array}\right][/tex]    and [tex]X=\left[\begin{array}{ccc}x\\y\\z\end{array}\right][/tex]    [tex]B=\left[\begin{array}{ccc}0\\3\\0\end{array}\right][/tex]  

A is the coefficient matrix, X the variable matrix and B the constant matrix.

       [tex]|A|=-1(1)-2(2)+1(3)\\|A|=-1-4+3=-2[/tex]

Since , [tex]|A|\neq 0[/tex]

Hence, the system of equations is consistent and has a unique solution

A inverse is given by [tex]\frac{adjA}{|A|}[/tex]     .....(2) and adj(A) is given by [tex]AdjA=C^T[/tex]   ...(3)

Minor of matrix A will be

                          [tex]M_{11}=1\\M_{12}=2\\M_{13}=3\\M_{21}=2\\M_{22}=2\\M_{23}=6\\M_{31}=-1\\M_{32}=2\\M_{33}=-5[/tex]

Then Co - factor will be

           [tex]C_{11}=1\\C_{12}=-2\\C_{13}=3\\C_{21}=-2\\C_{22}=2\\C_{23}=-6\\C_{31}=-1\\C_{32}=-2\\M_{33}=-5[/tex]

Matrix C will be represented by

   [tex]C=\left[\begin{array}{ccc}1&-2&3\\-2&2&-6\\-1&-2&-5\end{array}\right][/tex]

Transpose of matrix C will be represented by

[tex]C^T=\left[\begin{array}{ccc}1&-2&-1\\-2&2&-2\\3&-6&-5\end{array}\right][/tex]

Then , putting this value in equation (3) , we get

     [tex]adjA=C^T\\adjA=\left[\begin{array}{ccc}1&-2&-1\\-2&2&-2\\3&-6&-5\end{array}\right][/tex]

Putting the value of |A|=-2 and adjA in equation (2) , we get

   [tex]A^{-1}=\frac{1}{-2}\left[\begin{array}{ccc}1&-2&-1\\-2&2&-2\\3&-6&-5\end{array}\right][/tex]

Putting the value of A inverse in equation (1) , we get

         [tex]X=\frac{1}{-2}\left[\begin{array}{ccc}1&-2&-1\\-2&2&-2\\3&-6&-5\end{array}\right]\left[\begin{array}{ccc}0\\3\\0\end{array}\right]\\\\\\\left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\frac{-1}{2} \left[\begin{array}{ccc}-6\\6\\-18\end{array}\right] \\\\\\\left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}3\\\ -3 \\9\end{array}\right][/tex]

On comparing , we get x = 3 , y = -3 , z = 9

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The solution to the system of equations using substitution is: x = 3 and y = -4.

Given the system of equations:

-7x - 2y = -13 -----> equation 1

x - 2y = 11 -----> equation 2

Rewrite equation 2

x - 2y = 11

x = 11 + 2y

Substitute x = (11 + 2y) into equation 1

-7(11 + 2y) - 2y = -13

-77 - 14y - 2y = -13

-77 - 16y = -13

-16y = -13 + 77

-16y = 64

y = -4

Substitute y = -4 into equation 2

x - 2(-4) = 11

x + 8 = 11

x = 11 - 8

x = 3

Therefore, the solution to the system of equations using substitution is: x = 3 and y = -4.

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The inverse of the function y=4 x²+5  is y = √((x-5)/4).

Inverse functions are those that cancel one another out. In other words, if g(x) is f's inverse, then f(g(x)) = g(f(x)) = x. (x). Given an equation for which we want the inverse, we have a fantastic method we can utilize to do this.

Not every function has a counterpart. If f(x) has an inverse, it is clear to see that it will never again have the same value. We give this resource a special name.

A function (f(x)) is said to be one-to-one if each member of the range corresponds to exactly one element of the domain.

Given,

We have a function,

y=4 x²+5

subtracting 5 from each side.

⇒ y - 5 = 4 x²

Divide by 4 and then either

⇒ x² = (y - 5) / 4

⇒ x = √((y-5)/4)

Just change the labels now, x to y.

⇒ y = √((x-5)/4)

So, the inverse of given function is y = √((x-5)/4)

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(-2, -3) represents the coordinate of point B from the given expression.

The formula for calculating the midpoint of coordinates is expressed as shown below:

Midpoint = {(x₁+x₂)/2, (y₁+y₂)/2}

Given the following parameters

Midpoint = (3, -4)

A = (8,-5)

Required

Coordinate point B

Substitute the given parameters

3 = (8+x₂)/2

3(2) = 8+x₂

6 = 8+x₂

x₂ = -2

Similarly

-4 = (-5+y₂)/2

-4(2) = -5+x₂

-8 = -5 +y₂

y₂ = -3

Hence the coordinate of point B from the given expression is (-2, -3)

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Both 3 is an extraneous solution.

We have the equation x - 6 = [tex]\sqrt{3x}[/tex]

To find the value of x we need to square on both side of the equation

[tex](x-6)^{2}[/tex] = [tex](\sqrt3{x} )^{2}[/tex]

[tex]x^{2}[/tex] - 12x + 36 = 3x

[tex]x^{2}[/tex] -15x + 36 = 0

[tex]x^{2}[/tex] - 12x - 3x + 36 = 0

x ( x- 12 ) - 3( x - 12) = 0

(x -12 ) ( x - 3) = 0

Therefore we get x = 12 , 3

Now we will put both the value of x in the equation

x - 6 = [tex]\sqrt{3x}[/tex]

Let put 12 first

12 - 6 = [tex]\sqrt{3 * 12}[/tex]

6 = 6

Now put 3

3 - 6 =[tex]\sqrt{3 * 3}[/tex]

- 3 [tex]\neq[/tex] 3

Therefore 3 is an extraneous solution .

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Answer:

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The domain for g (x) + f (x) = x² + x - 20 will be all real numbers when f (x) = x - 4 and g (x) = x² - 16.

We are given the two functions as:

f (x) = x - 4

g (x) = x² - 16

Now, first, we need to add f (x) from g (x).

So, we get that:

g (x) + f (x) = x² - 16 + (x - 4)

g (x) + f (x) = x² - 16 + x - 4

g (x) + f (x) = x² + x - 20

Now, the domain of the function will be all real numbers as it will be defined for all the real numbers.

Therefore, we get that the domain for g (x) + f (x) = x² + x - 20 will be all real numbers when f (x) = x - 4 and g (x) = x² - 16.

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Answer:

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Pakistan

The dimensions of the fencing is 28 ft and 37 ft respectively.

He use 130 feet of fencing to go around a rectangular pool and patio.

The length of the area is 9 more than the width.

The dimension of the fence can be found as follows:

Therefore,

perimeter of the rectangular fence = 2(l + w)

where

Hence,

l = 9 + w

perimeter of the rectangular fence = 2(w + 9 + w)

perimeter of the rectangular fence = 2(2w + 9)

130 = 4w + 18

130 - 18 = 4w

112 = 4w

w = 112 / 4

w = 28 ft

l = 9 + 28 = 37 ft

Therefore, the width and length of the fencing is 28 ft and 37 ft respectively.

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The velocity of the spacecraft is 7.7025 km/s as calculated with the help of mass ratio.

Given:

To find: Velocity of the spacecraft (v) = ?

Finding:

Here, c = 2.7 km/s, t = 30s, R = 20

On substituting the values of 'R', 't', 'v', we get:

v = -0.0098 ( 30) + 2.7 ( ln (20))

=> v = - 0.294 + 2.7 (2.995)

=> v = 7.7025 km/s

Hence, the velocity of the spacecraft is 7.7025 km/s as calculated with the help of mass ratio.

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One pair of vertical angles are ∠1 and ∠4.

One pair of angles that are congruent are ∠3 and ∠6.

One pair of angles that are congruent are ∠3 and ∠6.

Vertical angles are the angles opposite each other when two lines cross.

Vertical angles are congruent. This means vertical angles are equal to each other.

Therefore, one pair of vertical angles are ∠1 and ∠4.

Linear pair of angles are formed when two lines intersect each other at a single point.

Therefore, one pair of linear pair angles are ∠6 and ∠8.

Angles are congruent when they are the same. Congruent angles includes corresponding and alternate angles.

Therefore, one pair of angles that are congruent are ∠3 and ∠6.

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Answer: Yes, all horizontal lines have slope of 0

Reason:

Recall that slope = rise/run

The rise tells us how far to go up or down. A horizontal line does neither. There's no change in y value.

So we have slope = rise/run = 0/run = 0, where "run" is any positive number you want. The run is how far to the right we go each time moving up or down.

Answer: Yes

Step-by-step explanation:

When you have a horizontal line it is flat and has no slope at all

____________________________________

So the slope = 0

Answer:

Odd

Step-by-step explanation:

The answer to the equation is 1,859, making the answer odd.

Answer:

Step-by-step explanation:

upset

Answer:

here to try to help!

Step-by-step explanation:

346.96 why? Kerry was not payed at the overtime for extra hours hope it helped!

Answer:0.16=16%

Step-by-step explanation:    [tex]\frac{0.16}{100}[/tex]

the 100 is directly under the point so the zero will cancel out leaving= [tex]\frac{16}{100}[/tex]

And  [tex]\frac{16}{100}[/tex] equals to 16%

if we had a value "z", and we'd like to know how much is 16% of "z", well

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{16\% of z}}{\left( \cfrac{16}{100} \right)z}\implies 0.16z[/tex]

so we can use the "factor" of 0.16, to get the 16% of any value,

since 0.16   is the decimal representation of the percentage of 16.

hmmm what is 16% of 50 anyway?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{16\% of 50}}{\left( \cfrac{16}{100} \right)50}\implies 8[/tex]8 out of 50

how much is 16% of 25 anyway?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{16\% of 25}}{\left( \cfrac{16}{100} \right)25}\implies 4[/tex]4 out of 25

Solution of this polynomial equation is x = 3

In mathematics, polynomial equation can be defined as the equation which consist of coefficient, variables and exponents. A polynomial equation has positive numbers as their exponent which is also known as its degree. All the equations such as quadratic equation, cubic equation which has non  negative number as their powers are part of polynomial equation.

Types of polynomial equations :

Some examples of polynomial equation are :

According to question,

3x = 9

=> 3x / 3 = 9 /3

=> x = 3

Thus solution for given equation is x = 3.

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Answer:

- 2(x - 3zy)

Step-by-step explanation:

The solution to the given equation -1 = -1 + (p/8) is p = 0.

Given the data in the question;

Solve for p to determine the solution to the equation.

-1 = -1 + (p/8)

Rewrite the equation as;

-1 + (p/8) = -1

Next, move all terms containing p to the left side and all constants to the right side of the equation.

-1 + (p/8) = -1

(p/8) = -1 + 1

Add -1 and 1

p/8 = -1 + 1

p/8 = 0

p = 0 × 8

p = 0

The solution to the given equation -1 = -1 + (p/8) is p = 0.

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2.89 is the value of x in equation [tex]e^x[/tex] = 18 solved by using natural logarithms.

We have been asked to find x in [tex]e^x[/tex] = 18 using natural logarithms

⇒  [tex]e^x[/tex] = 18

⇒  x = ㏑ (18)

⇒ x = 2.89

A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.

Typically, the natural logarithm of x is denoted by the symbols ln x, loge x, or, if the base e is implicit, just log x. When adding parentheses for clarity, the values ln(x), loge(x), or log (x). In order to avoid ambiguity, this is notably done when the argument to the logarithm is not a single symbol.

The power to which e would need to be raised in order to equal x is the natural logarithm of x.

For instance, e2.0149... = 7.5, thus ln 7.5 equals 2.0149.

Since e1 = e, the natural logarithm of e itself, or ln e, is equal to 1, while the natural logarithm of 1 is 0, since e0 = 1.

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Use natural logarithms to solve each equation.

[tex]e^x[/tex] = 18

The problem that can be defined by the function is:

A cat breeding company has 20 cats initially. The number of cats increases by 10% each year. Write a function for the exponential growth of the number of cats over the years. How many cats will the company have after 10 years?

A quantity must vary by a specific percentage each time period in order for growth or decay to be exponential.

With the function displayed to the right, you may represent exponential growth or decay.

A(x) = a( 1 + r)ˣ

Where A is the amount after x time periods, a is the initial amount and x is the number of time periods.

Consider the modeled function y = 20(1.1)ˣ.

Now, y = 20 × ( 1 + r)ˣ

f(x) = 20 × ( 1 + r)ˣ

Comparing the above equation with A(x) = a( 1 + r)ˣ

The initial amount is 20 and it increases 1 + r = 1.1.

Therefore, r = 0.1 = 10 %

Let's say we are starting a cat breeding company. Initially, we have 20 cats.

So, the problem for the function is:

A cat breeding company has 20 cats initially. The number of cats increases by 10% each year. Write a function for the exponential growth of the number of cats over the years. How many cats will the company have after 10 years?

So, a = 20, x = 10, and r = 10% = 0.1

Therefore,

A(x) = a( 1 + r)ˣ

y = a( 1 + r)ˣ

y = 20( 1 + 0.1 )ˣ

y = 20( 1.1)¹⁰

y = 20 × 2.594

y = 52

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Answer:

65%

Step-by-step explanation:

To find percentage, we take the fraction of the total and multiply it by 100.

[tex]\frac{325}{500} \times=\frac{325}{5}=65[/tex]

Answer:

[tex]153.84\%[/tex]

Step-by-step explanation:

[tex] 1. \: \frac{500}{3.25} \\ 2. \: 153.84\%[/tex]

The function f(x) = ( x + 7)³ - 8 will shift left 7 units along x axis and will shift down 8 units along y axis to its parent function g(x) = x³.

By translation, we can stretch, shrink and shift graphs along the x-y coordinate.

We know a parent function is a function that is in its simplest form without any translation.

Given f(x) = (x + 7)³ - 8 and its parent function is g(x) = x³.

Now let us break this function f(x),

f(x) = ( x+ 7)³ implies x + 7 = 0 ⇒ x = -7.

So, x will shift 7 units to the left along the x-axis.

f(x) = ( x + 7)³ - 8, This will shift the function down 8 units along y-axis.

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Answer:

Comparamos los dos números decimales anteriores, y podemos ver que 2.5 es mayor que 1.75. Por lo tanto, 5/2 es mayor que 7/4

Step-by-step explanation:

Answer:

7/4 because if you divide them both, 7/4's decimal will be greater                                                                      

According to the question, to calculate the relationship model of the given equations which are as follows:

[tex]z=30[/tex] when [tex]x =3 ; y=2[/tex]

And the condition is z varies jointly with x and y which can be represented as: [tex]z = \frac{k}{xy}[/tex]

Substituting the values to get the final expression:

[tex]30 = \frac{k}{(3)(2)} =\frac{k}{6}[/tex]

[tex]k=180[/tex]

Substitute the value of 'k' in the original equation. And the final expression is: [tex]z = \frac{180}{xy}[/tex]

What is function?

In mathematics, function can be defined as an expression or the rule or the law. It defines the relationship between dependent variable and the independent variable.

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If the baker has a total of 1/2 bags of flour, the total weight of the bags of flour is 1.625

What is the weight of a bag of flour in pounds?

The fact that each bag of flour weighs 3 and 1/4 pounds means that each bag of flour is 3.25 pounds, in essence, the total weight of flour that the baker has can be determined as the weight of a bag of flour multiplied by the total of 1/2 bags of flour.

total weight of bags of flour=weight of a bag of flour*total weight of baker's flour

weight of a bag of flour=3.25 pounds

total weight of baker's flour=1/2 bag

total weight of bags of flour=3.25bags*1/2

total weight of bags of flour=1.625 pounds

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10 order of small popcorn Should be made to keep  the cost same at both the theaters

The theater P charges Admission fee of $10 and small popcorn order of $4.55

The theater Q charges Admission fee of $8 and small popcorn order of $4.75

We need to find the number of order of small popcorn so that the cost remain same

Let the number of small popcorn order be x

Then we can form the algebraic equation as

10 + 4.55x = 8 + 4.75x

now we should simplify the equation

10- 8 = 4.75x-4.55x

2= .2x

x = 2/0.2

x = 10

Hence , 10 order of small popcorn Should be made to keep  the cost same at both the theaters

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The length of the diagonal (D) of this rectangular prism is equal to 9.5 feet.

Pythagorean theorem can be defined as a mathematical expression in Euclidean geometry that can be used to solve for any of the three (3) side lengths of a right triangle.

In order to determine the length of the diagonal (D) of this rectangular prism, we would first of all determine the diagonal of the base by applying Pythagorean theorem:

d² = l² + b²

d² = 4² + 7²

d² = 16 + 49

d² = 65

d = √65

Now, we can determine the length of the diagonal (D) of this rectangular prism:

D² = d² + h²

D² = √65² + 5²

D² = 65 + 25

D² = 90

D = √90

Diagonal, D = 9.5 feet.

Therefore, the length of the diagonal (D) of this rectangular prism is equal to 9.5 feet.

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