Use lagrange multipliers to find the minimum distance between the origin and the plane 3x y 2z = 28

The interpretation of x in the equation 75•x + 19•y = T, used by the geographer, based on the land area of Aruba, 75 m², and Bermuda, 19 m², is the population in a square mile of Aruba

The land area of Aruba = 75 m²

Land area of Bermuda = 19 m²

The total number of residents is given in the question by the equation;

75•x + 19•y = T

Therefore;

x = The number of people in a square mile in Aruba

y = The number of people in a square mile in Bermuda

Which gives;

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The percentage of such groups (of 400) would 150 or fewer improve is 16.60.

The percentage is a relative value indicating the hundredth component of any quantity. One percent (symbolized 1%) is the 100th element; as a result, 100 percent represents the entirety and 2 hundred percent specifies twice the given amount. percentage.

As finding 10% of a variety of means to divide through 10, it is not unusual to assume that to locate 20% of a variety you must divide through 20 and so on. Take into account, to discover 10% of the number of manner dividing through 10 because 10 is going into one hundred ten times. consequently, to discover 20% of various, divide via five due to the fact 20 is going into 100 five times.

In arithmetic, a percentage is a number or ratio expressed as a fraction of one hundred. it's far frequently denoted by the usage of the percentage signal, "%", although the abbreviations "pct.", "pct" and on occasion "computer" also are used. A percent is a dimensionless wide variety; it has no unit of size.

In what percentage of such groups (of 400) would

150

fewer improve?

i. e. PC X ≤ 150)

By using

-continuity correction we have to find.

PCX 150.5)= P "X-M1.6.150.5-160°

9.8

= p[ 2 < -0.97

By using the standard Normal table we get,

0.1660

P(X ≤150)

= 0. 1660 = 16.60 (

16.60 percent of such groups (of 400) would 150

fever improve.

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 45° only the possible values for x.

What is the meaning of trigonometric ratio?

The given equation can be presented as follows;

[tex]\frac{cos (2 . x )}{cosine(x) + sin(x)} = 0[/tex]

We have that cos(2·x) = cos²(x) - sin²(x)

Also, we have, by the difference of two squares, the following relation;

cos²(x) - sin²(x) = (cos(x) - sin(x))(cos(x) + sin(x))

Therefore, the given equation can be written as follows;

[tex]\frac{cos (2 . x )}{cosine (x) + sin(x) } = \frac{(cos(x) - sin(x) * (cos(x) + sin(x)}{(cos(x) + sin(x))} = 0[/tex]

Crossing the common term, (cos(x) + sin(x)), in the numerator and the denominator, we have-

[tex]\frac{(cos(x) - sin (x) * (cos(x) + sin(x)}{(cos(x) + sin(x))} = cos (x) - sin(x) = 0[/tex]

From cos(x) - sin(x) = 0, we have;

Adding sin(x) to both sides of the equation

cos(x) - sin(x) + sin(x) = 0 + sin(x)

cos(x) = sin(x)

Therefore, the opposite leg and the adjacent leg of the right triangle formed with reference to the angle x are equal

∴ x = 90/2 = 45° only for 0° ≤ x ≤ 180°

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The values of x are x₁ ≈ 4.8 and x₂ ≈ - 0.8.

Herein we have two circular sections of equal area, whose expressions are described by the following geometric equations:

Semicircle

A = 0.5π · x²     (1)

Half Semicircle

A = 0.25π · (x + 2)²      (2)

By equalizing (1) and (2):

0.5π · x² = 0.25π · (x + 2)²  

2 · x² = (x + 2)²  

2 · x² = x² + 4 · x + 4

x² - 4 · x - 4 = 0

x² - 4 · x + 4 = 8

(x - 2)² = 8

x - 2 = ±√8

x = 2 ± √8

x = 2 ± 2√2

x = 2 · (1 ± √2)

x = 2 · (1 ± 1.41)

x₁ = 2 · 2.41   ∨   x₂ = 2 · (- 0.41)

x₁ = 4.82   ∨   x₂ = - 0.82

The values of x are x₁ ≈ 4.8 and x₂ ≈ - 0.8.

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From the given information, a exists jointly proportional to b and c.

then [tex]$$a \propto b c$[/tex]

Therefore, the value of a exists 2.

Given, a exists jointly proportional to b and c.

[tex]$$a \propto b c$[/tex]

Taking K as the constant of proportionality.

a = K b c

For b = 8 and c = 9, the value of a = 4

4 = K(8)(9)

4 = 72 K

Dividing the equation by 72, we get

[tex]$K=\frac{1}{18}[/tex]

Using the value of K in the equation:

[tex]$a=\frac{b c}{18}[/tex]

Substitute the values of b = 2 and c = 18, in the above equation

[tex]$a=\frac{2 \times 18}{18} \\[/tex]

a = 2

Therefore, the value of a exists 2.

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The probability of Maria selecting 2 questions on Science is 1/ 5

From the information given, we have that there are;

The total questions = 2 + 4 + 4 = 10 questions

We have that Maria picks two questions at random

The probability of her selecting two questions on Science is given as

P(2 science) = number of Science questions/ total number of questions

P(2 science) = 2/ 10

P(2 science) = 1/ 5

The probability of picking two questions on Science is 1/5

Thus, the probability of Maria selecting 2 questions on Science is 1/ 5

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Answer:

1/45

Step-by-step explanation:

The percentage of the whole school that plays soccer is 35.5%

The percentage of the boys is given as

Percentage boys = 45%

This means that the percentage of the girls is given as

Percentage girls = 55%

Also, from the question, we have:

30% of the boys and 40% of the girls play soccer

So, the percentage of the whole school that plays soccer is

P = Boys * Percentage Boys + Girls * Percentage Girls

This gives

P = 45% * 30% + 55% * 40%

Evaluate the product

P = 13.5% + 22%

Evaluate the sum

P = 35.5%

Hence, the percentage of the whole school that plays soccer is 35.5%

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Answer:

A square matrix whose determinant is equal to zero is called singular matrix.

[tex]\huge \mathbb{ \underline{ANSWER:}}[/tex]

[tex]\leadsto[/tex] So the square matrix that has its det zero and its inverse is not defined is called a singular matrix.

[tex]\bold{Example:}[/tex]

[tex]\begin{bmatrix} \sf{3} & \sf{6} \\ \sf{2} & \sf{4}\end{bmatrix} \\ \\ \sf and \\ \begin {bmatrix} \sf{ 1} & \sf2 & { \sf2} \\ { \sf1} & { \sf2} & \sf{2} \\ \sf{3} & \sf {2} & \sf{1}\end{bmatrix}[/tex]

[tex]\huge \mathbb{ \underline{EXPLANATION:}}[/tex]

A singular matrix is a square matrix if its determinant is O. i.e., a square matrix A is singular if and only if det A = 0. The formula to find the inverse of a matrix is

▪ [tex] \bold{ {A}^{1} = \dfrac{1 \: }{det \:A } adj[A] }[/tex]

So if det A = 0 then the inverse of A will be not defined.

Answer:

age of son= X

Father's age =3x

After 10 years , father's age= 3x +10

Son's age = X+10

Operations:

3x + 10 =2(x + 10 )

3x + 10 = 2x + 20

X= 10

Therefore, present age of son= 10 years

Father's present age = 30 years

After 10 years son will be 20 years and the father 40 yes which is double

The current value of the stock is $53.413455.

To find the current price of stock:

Stock’s current price: 53.41 dollars

First, we determine the stock's value using the CAPM model.

Then, Ke = 0.085 + 1.3(0.045).

The current value of the future dividends must now be known:

The next dividends, which are at perpetuity will be solved using the dividend growth model:

In this case, dividends will be:

Return will be calculated using the CAPM and g = 7%.

plug this into the Dividend grow model,

Finally, it's crucial to remember that these statistics were calculated for the present year.

We must bring them to the present day using the present value of a lump sum:

We add them and get the value of the stock, 53.413455

Therefore, the current value of the stock is $53.413455.

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Domain and Range of f(x) is given by:

Domain: {x | all real numbers} ;

Range: {y | y > 0}

The sets of all the x-coordinates and all the y-coordinates of ordered pairs, respectively, are the domain and range of a relation. A function's range is its potential output, whereas its domain is the set of all possible input data.

The function can be written as :

[tex]f(x)=\sqrt[\frac{2}{3} ]{108^{2x} } \\f(x)=108^{\frac{3}{2}^{2x} }[/tex]

Now that x is an exponent, all real values are possible for it. Therefore, all real numbers make up its domain for f(x). But value of f(x) cannot be less than 1 because for x = 0 the value of f(x) is 1 and also for any values of x, the value of f(x) can never be less than 1

So, Range of f(x) is all real numbers greater than 0

Hence, Domain and Range of f(x) is given by:

Domain: {x | all real numbers} ;

Range: {y | y > 0}

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Check the picture below.

Answer:

Step-by-step explanation:

Comment

The semicircle comes out towards we the viewer, then it goes upwards, and what one gets is 1/2 a sphere. If one keeps on rotating in the direction of the arrow eventually winding up where one started, the result is a full sphere.

Answer: Sphere

Step-by-step explanation:

The volume of a cube is

[tex]a {}^{3} [/tex]

where a is the side length,

Note: If you want to remember this formula, know that a cube is basically a bunch of squares stacked on one another vertically and horizontally

The area of a square with side length a, is

[tex] {a}^{2} [/tex]

If we multiply that by the height of the cube, which is a.

[tex] {a}^{2} \times a = {a}^{3} [/tex]

That is the easy way to derive the formula of the volume of a cube.

Back on track, we know the volume so we must solve for a.

1.

[tex]125 = {a}^{3} [/tex]

Assuming you took algebra, to isolate the variable a, we must undo it being raised to the third power.

To do this, we take the cube root of both sides

[tex] \sqrt[3]{125} = \sqrt[3]{a {}^{3} } [/tex]

The cube root of 125 is 5 so

[tex]5 = a[/tex]

5 cm

2.

[tex]8 = {a}^{3} [/tex]

[tex] \sqrt[3]{8} = \sqrt[3]{ {a}^{3} } [/tex]

[tex]2 = a[/tex]

2 ft

3.

[tex]343 = {a}^{3} [/tex]

[tex]a = 7[/tex]

7 yd

4.

[tex]1000 = a {}^{3} [/tex]

[tex]10 = a[/tex]

10 mm

5.

[tex]1728 = {a}^{3} [/tex]

[tex]a = 12[/tex]

12 in. or 1 ft

6.

[tex]1 = {a}^{3} [/tex]

[tex]a = 1[/tex]

1 m

Answer:

A cube can be seen as a square that has been 'stretched'. We know that all four sides of a square are equal, thus, the sides of the faces of a cube are also equal.

In order to find the volume of a cube, the formula we apply is length x width x height (lwh).

Since all the sides are the same, another formula for the volume of the cube can be written as volume = x^3 in which x stands for the length of each side. When plugging in the formula:

1) V = 125 cm^3

   125 = x^3

   ∛125 = x

    5 = x

When finding the cube root, you can use a calculator or fingure out the answer yourself by multiplying any number by itself three times.

2) 8 = x^3

   ∛8 = x

      2 = x

3) 343 = x^3

 ∛343 = x

        7 = x

4) 1000 = x^3

 ∛1000 = x

         10 = x

5) 1728 = x^3

  ∛1728 = x

    12 = x

6) 1 = x^3

   ∛1 = x

      1 = x

Make sure to include the appropriate unit for each answer

Answer:

10

Step-by-step explanation:

[tex]\boldsymbol{\sf{\dfrac{19}{23}\times50 }}[/tex]

We express 19/25 x 53, as a single fraction as follows:

[tex]\boldsymbol{\sf{\dfrac{19\times50}{23} \ \ \longmapsto \ \ \ [Multiply] }}[/tex]

[tex]\boldsymbol{\sf{\dfrac{950}{23} }}[/tex]

Like the fraction 950/23, it cannot be further simplified, so it remains that way, therefore

                  [tex]\red{\boxed{\boldsymbol{\sf{Answer \ \ \longmapsto \ \ \ \frac{950}{23} }}}}[/tex]

The angle that corresponds to 3 is angle 5

Corresponding angles are angles that are at the same relative position between parallel lines and transversal

These angles are congruent

From the figure, we have:

Angle 3 and Angle 5 are in the same relative position

Hence, the angle that corresponds to 3 is angle 5

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See attachment for the graph of the functions y = x + 2, y = 2x and y = -x

Equation 14

The equation is given as:

y = x + 2

When x = 0, we  have

y = 0 + 2 = 2

When x = 1, we  have

y = 1 + 2 = 3

When x = 2, we  have

y = 2 + 2 = 4

So, the complete table is

x    y

0     2

1      3

2     4

See attachment for the graph of the function y = x + 2

Equation 15

The equation is given as:

y = 2x

When x = 0, we  have

y = 2 * 0 = 0

When x = 1, we  have

y = 2 * 1 = 2

When x = 2, we  have

y = 2 * 2 = 4

When x = 3, we  have

y = 2 * 3 = 6

So, the complete table is

x    y

0    0

1     2

2    4

3    6

See attachment for the graph of the function y = 2x

Equation 16

The equation is given as:

y = -x

When x = -3, we  have

y = 3

When x = -1, we  have

y = 1

When x = 1, we  have

y = -1

When x = 3, we  have

y = -3

So, the complete table is

x    y

-3    3

-1     1

1     -1

3    -3

See attachment for the graph of the function y = -x

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Answer:

Yes

Step-by-step explanation:

The given line is an average of the data points.

The given series is diverges according to the P Test.

According to the statement

we have given that the a series and we have to find that the those series is a converges or diverges.

So, For this purpose, we know that the

P-series test to say whether or not the series converges.

So, The given series is

[tex]1 + 1/\sqrt{32} + 1/\sqrt[3]{243 } + 1/\sqrt[4]{1024} + 1/\sqrt[5]{3125}[/tex]

And then

The value of  P becomes P = 5/6 < 1  when we find it using the p series test.

Hence, the p-series diverges

Because according to the p series test, The series converges if, and only if, the power satisfies p>1.

And diverges if, and only if, the power satisfies p<1.

So, The given series is diverges according to the P Test.

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Disclaimer: This question was incomplete. Please find the full content below.

Question:

Use Theorem 9.11 to determine the convergence or divergence of the p-series. 1 + 1/32 squareroot + 1/243 squareroot 3 + 1/1024 squareroot 4 + 1/3125 squareroot 5.

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Given the:

ln(x + 6) - ln(2x - 1) = 1

Using the logarithm rule:

We know,

Simplifying we get,

By cross multiplication we get,

Therefore, the value of x is 1.97.

The correct option in the exercise ln (x + 6) - ln (2x - 1) = 1 is alternative "D".

Answer:

MO = 12

PR = 3

Step-by-step explanation:

The perimeter of a triangle is calculated by adding all side lengths up:

The perimeter of triangle MNO is given as 48

The perimeter of triangle PQR is given as 12

The ratio is 1/4 (simplified ratio of 12/48)

we can write the following equation using this information:

[tex]\frac{x+2}{12x} = \frac{1}{4}[/tex]

cross multiply fractions

12x = 4x + 8

subtract 4x from both sides

8x = 8

divide both sides by 8

x = 1

Now to find the measure of side lengths in question we replace x with 1

for MO = 12x

12*1 = 12

for PR = x + 2

1 + 2 = 3

One fraction: 5/2.

Mixed-fraction: 2 1/2.

In the question, we are asked to assume that each circle shown below represents one unit, and are asked to express the shaded amount as a single fraction and a mixed fraction.

The given circle is divided into 8 equal parts.

In the first circle, all 8 parts are shaded, shown by the fraction, 8/8.

In the second circle, all 8 parts are shaded, shown by the fraction, 8/8.

In the third circle, 4 parts are shaded, shown by the fraction, 4/8.

Thus, the total shaded part can be calculated by adding:

8/8 + 8/8 + 4/8,

= (8 + 8 + 4)/8

= 20/8

= 5/2 {Cancelling off by 4}.

Thus, one fraction is 5/2.

To convert this to a mixed fraction, we divide 5 by 2, write the quotient as the whole part, the remainder as the numerator, and 2 as the denominator.

5÷2 gives quotient 2, and remainder 1.

Thus, the mixed-fraction is 2 1/2.

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Answer:

  (d)  (4, -10)

Step-by-step explanation:

We are asked to identify the point that falls on a line defined by parametric equations.

The parametric equation can be written in a number of forms One of these is a vector form, in which some multiple of a vector is added to a known point.

  (x(t), y(t)) = (1/2t +4, 2t -10) = (4, -10) +(1/2t, 2t)

  (x, y) = (4, -10) +(t/2)(1, 4)

Clearly, when t=0, one point on the line will be (4, -10) — the last of the offered choices.

Another form of the equation can be had by solving the x(t) and y(t) equations for t, then equating those values.

  x = 1/2t +4   ⇒   2(x -4) = t

  y = 2t -10   ⇒   (y +10)/2 = t

Equating these gives ...

  2(x -4) = (y +10)/2 . . . . t = t

  4x -16 = y +10 . . . . . . . multiply by 2

  4x -y = 26 . . . . . . . . . add 16-y to put in standard form

Dividing by 26 gives intercept form:

  x/6.5 +y/(-26) = 1 . . . . . x-intercept: 6.5, y-intercept: -26.

This tells you the value of y will be negative for all x-values less than 6.5. All of the offered points have x-values less than that, so the only viable answer choice is (4, -10).

Answer:

(4.-10)

Step-by-step explanation:

When t = 0:

x(t)=1/2t+4

x(0)=1/2(0)+4

x(0) = 4

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

y(t)=2t−10

y(0)=2(0)−10

y(0) = -10

(x,y) at t = 0 is (4, -10)

The simplified fraction expression results into [tex]\frac{1}{9}[/tex].

What is fraction?

A number that is stated mathematically as a quotient, where the numerator and denominator are divided. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

A number that represents a rational number is called a common fraction. The three main sorts of fractions are as follows. Proper fractions, incorrect fractions, and mixed fractions are these three types.

Given,

Expression = [tex]\frac{3^{-6}}{3^{-4}}[/tex]

                   = [tex]3^{-6-(-4)}[/tex]

                   = [tex]3^{-6+4}[/tex]

                   = [tex]3^{-2}[/tex]

                   = [tex]\frac{1}{3^2}[/tex]

                   = [tex]\frac{1}{9}[/tex]

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Using compound interest and a graphing calculator, it is found that it will take about 15 years for the loan to be paid off.

The amount of money earned, in compound interest, after t years, is given by:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which:

For this problem, the parameters are given as follows:

A(0) = 95000, r = 0.06, n = 12.

Hence the value of the loan after t years is:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

[tex]A(t) = 95000\left(1 + \frac{0.06}{12}\right)^{12t}[/tex]

[tex]A(t) = 95000(1.005)^{12t}[/tex]

You have monthly payments of $1,200, hence the amount paid after t years is:

P(t) = 12 x 1,200t = 14400t

Then we have to solve for:

A(t) = P(t)

[tex]14400t = 95000(1.005)^{12t}[/tex]

Which is solved in the graph below, meaning that it will take about 15 years for the loan to be paid off.

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Answer + Step-by-step explanation:

(a)

[tex]\overrightarrow{PQ} =a+\frac{3}{5} b[/tex]

(b)

we are given:

[tex]\left\vert \overrightarrow{RQ} \right\vert =\frac{3}{5} \left\vert \overrightarrow{OP} \right\vert[/tex]

Then

[tex]\left\vert \overrightarrow{WR} \right\vert =\frac{3}{5} \left\vert \overrightarrow{WO} \right\vert[/tex]

and we know that :

[tex]\overrightarrow{OW} = \overrightarrow{OR} + \overrightarrow{RW}[/tex]

Then

[tex]\overrightarrow{OW} = a +\frac{3}{5} \overrightarrow{OW}[/tex]

Then

[tex]\frac{2}{5} \overrightarrow{OW} = a[/tex]

Then

[tex]\overrightarrow{OW} = \frac{5}{2} a[/tex]