To indirectly measure the distance across a river, Madeline stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Madeline draws the diagram below to show the lengths and angles that she measured. Find PR, the distance across the river. Round your answer to the nearest foot.

The area of the region inside the curve formed by the given complex number is  28.26 sq. units.

What are complex numbers?

A complex number in mathematics is part of a number system that includes an element with the symbol i, often known as the imaginary unit, to expand the real numbers. The formula a + bi, where a and b are real numbers, can be used to express any complex number.

Given a complex number z.

Let z = a +bi

Then,

[tex]\frac{1}{z} = \frac{1}{a+bi} =\frac{a -bi}{(a+bi)(a-bi)} = \frac{a -bi}{(a^2+b^2)}[/tex]

The real part of the above equation is a / (a²+b²)

This value is given as 1/6.

a / (a²+b²) = 1/6

 (a²+b²) = 6a

 a² - 6a + 9 - 9 + b² = 0

(a - 3)² + b² = 9

This is an equation of a circle with a centre of (3,0) and a radius of 3.

This is the curve formed.

Therefore for the given complex number the area of the region inside the circle = πr² = 3.14 * 3² = 28.26 sq. units

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In this case, 2x - 2 < g(x) < x^2 + 2x - 3 for all x in some interval.

The Sandwich Theorem states that if f(x) < g(x) < h(x) for all x in an interval, then g must have a value that is equal to either f or h at some point in that interval.

In this case, 2x - 2 < g(x) < x^2 + 2x - 3 for all x in some interval.

Hence, there exists a constant c such that g(c) = x^2 + 2x - 3.

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Given that2x - 2 < g(x) < x^2 + 2x - 3, use the Sandwich Theorem to prove that there exists a constant c such that g(c) = x^2 + 2x - 3.

The height of the school is given by the equation H = 13.274 m

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the height of the building be represented as ED = H

Let the distance of mirror from the building be CD = 11.15 m

The distance walked extra from the mirror CB = 1.05 m

The distance of Fawzia's eyes to the ground AB = 1.25 m

Now , let the triangles be represented as ΔABC and ΔCED ,

where both the triangles are similar and have a common angle

So , the corresponding sides of similar triangles are in the same ratio

And , ED / AB = CD / CB

Substituting the values in the equation , we get

ED / 1.25 = 11.15 / 1.05

Multiplying by 1.25 on both sides of the equation , we get

The height of the school building ED = ( 11.15 x 1.25 ) / 1.05

On simplifying the equation , we get

The height of the school building ED = 13.2738 m

Hence , the height of the school is 13.274 m

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If a is a set of real numbers that is bounded above and b is a set of real numbers that is bounded below, there is at most one real number that can exist in both sets.

A real number is any number that can be expressed as a decimal or fraction and exists on the number line.

If a set of real numbers, represented by a, is bounded above, it means that there exists a real number, represented by M, such that all the numbers in the set are less than or equal to M. Similarly, if a set of real numbers, represented by b, is bounded below, it means that there exists a real number, represented by m, such that all the numbers in the set are greater than or equal to m.

Now, let's consider a real number, represented by x, that exists in both sets a and b. If x exists in a, it must be less than or equal to M and if x exists in b, it must be greater than or equal to m. Hence, x must satisfy both conditions: M >= x >= m.

From these conditions, it can be deduced that M and m must be equal to x. In other words, there can only be one real number that is simultaneously the greatest value in a and the smallest value in b.

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The length of the third side of the given right triangle is 6.32.

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The formula for the same is;

c² = a² + b², where a, b, c are sides of a right triangle.

Given that, a right triangle, with two sides 7 and 3, since, the figure is unavailable, let us assume that 7 is the hypotenuse and 3 is the other side, let the unknown side be x,

According to the Pythagoras theorem,

7² = 3² + x²

x² = 7² - 3²

x² = 49-9

x² = 40

x = √40

x = 6.32

Hence, the length of the third side of the given right triangle is 6.32.

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Answer: x = 15

Step-by-step explanation:

Use the Pythagorean theorem

The missing 7th step is ΔDGH ≅ ΔFEH.

The correct option is A.

A special form of quadrilateral called a parallelogram has both pairs of opposite sides parallel and equal.

Given:

Quadrilateral DEFG is a parallelogram.

We have to prove: GH ≅ EH

DH ≅ FH

We drew the diagonals in the DEFG.

In triangle HGD and HEF:

By the alternate interior angle property,

∠HGD ≅ ∠HEF                          

∠HDG ≅ ∠HFE                      

From the definition,

DG ≅ EF    

By ASA criterion,

ΔDGH ≅ ΔFEH (step  7).

Since corresponding sides of congruent triangles are congruent,

So,

GH ≅ EH                    

DH ≅ FH                    

Hence, the required step is ΔDGH ≅ ΔFEH.

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The complete question:

Statement Reason

1. Quadrilateral DEFG is a parallelogram. given

2.

definition of a parallelogram

3. Draw and . These line segments are

transversals cutting two pairs of parallel lines:

and and and . drawing line segments

4. Place point H where and intersect. defining a point

5. ∠HGD ≅ ∠HEF

∠HDG ≅ ∠HFE

6. DG ≅ EF Opposite sides of a parallelogram are congruent.

7. ASA criterion for congruence

8. GH ≅ EH

DH ≅ FH Corresponding sides of congruent triangles are congruent.

1

What is the missing statement for step 7 in this proof?

A.

ΔDGH ≅ ΔFEH

B.

ΔGHF ≅ ΔEHD

C.

ΔDGF ≅ ΔFED

D.

ΔDEF ≅ ΔEDG

The general form of the equation of the hyperbola with the given information is [tex]9x^2 - 16y^2 - 36x + 160y - 508 = 0[/tex]. The equation can be determined by first finding the center of the hyperbola, which is (2, 5), and the distance between the foci, which is 4.

The general form of the equation of a hyperbola can be determined from the given information. The vertices of the hyperbola are given as (-2, 5) and (6, 5). The foci of the hyperbola are given as (-3, 5) and (7, 5). The first step in finding the equation of the hyperbola is to determine the center of the hyperbola. The center of the hyperbola can be calculated by taking the average of the x-coordinates of the vertices and then the average of the y-coordinates of the vertices. The center of the hyperbola is then (2, 5). The distance between the foci of the hyperbola is 4. This distance can be calculated by subtracting the x-coordinates of the foci and then subtracting the y-coordinates of the foci. The standard equation for a hyperbola can then be formed by substituting the center coordinates and the distance between the foci into the equation. The resulting equation is [tex]9x^2 - 16y^2 - 36x + 160y - 508 = 0[/tex]. This equation is the general form of the equation of the hyperbola with the given

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So Isaiah has a total of 6 gallons of paint and each bookcases takes 1/3 gallon. How many bookcases can he paint?

Alright to solve this problem, we can simply take the total gallons of paint Isaiah has and divide that by 1/3

So 6 divided by 1/3 should give us the answer.

But how do we divide fractions? Remember this useful tip for all throughout your school life when you don't have a calculator: Keep, Change, Flip

Keep the first fraction, change the sign to a multiplication sign, and flip the second fraction.

This should give us 6 x 3/1 or 3, which is 18.

Isiah will be able to paint 18 bookcases.

Hope this helped!

The amount of cardboard needed for the cone shaped hat is  753.6 inches².

Julie is making 8 cone-shaped party hats for her sisters birthday party from cardboard. Each party hat has a radius of 6 inches and a slant height of 5 inches. Th amount of the carboard Julie needs is the lateral area of the cardboard.

Therefore,

lateral area of each hat = πrl

where

Therefore,

lateral area of each hat = 3.14 × 6 × 5

lateral area of each hat = 3.14 × 30

lateral area of each hat = 94.2 inches²

Therefore,

amount of cardboard needed = 94.2(8) = 753.6 inches²

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The advantages of bar charts over pie charts while graphing the given data are discussed below.

A bar chart or bar graph is a visual representation of categorical data that uses rectangular bars with heights or lengths proportional to the values they represent. The possibility of a bar plot, both vertical and horizontal, exists. Vertical bar graphs are also referred to as column charts.

A circular statistical image known as a pie chart uses slices to represent numerical proportions. Each slice's arc length in a pie chart varies depending on the amount it represents.

Any numbers can be chosen for the numeric value axis in a bar chart.

Pie charts can only be used if the sum of the various parts equals a meaningful whole because they are intended to illustrate how each portion contributes to the whole.

Hence bar chart is more advantageous over pie chart when more information is to be presented through charts.

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The range of values for the width w of the garden is 15 ≤ w ≤ 19

Given the following parameters:

Length of garden = 14 feets

Perimeter must be atleast 58 but no more than 66

The range of value for the width ;

Perimeter = 2 length + 2 width

Perimeter = 2(14) + 2w

If perimeter = 58

58 = 28 + 2w

58 - 28 = 2w

30 /2 = w

w = 15

If perimeter = 66

66 = 28 + 2w

66 - 28 = 2w

38 = 2w

w = 38 / 2

w = 19

Range of the width should be at least 15 and not more Than 19

Hence, the range of the garden is 15 ≤ w ≤ 19

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The domain of f(g(x)) is the set of all real numbers x for which g(x) is defined and lies in the domain of f(x).

The function is defined as a mathematical expression that defines a relationship between one variable and another variable.

Given f(x) = 1/(x + 1) and g(x) = 1/(x+2)

First, let's find the domain of g(x). The function g(x) = 1/(x+2), so the denominator of this fraction must be non-zero. This means that x + 2 must be non-zero, or x must not equal -2. Thus, the domain of g(x) is all real numbers except for x = -2.

Next, let's find the domain of f(x). The function f(x) = 1/(x + 1), so the denominator of this fraction must also be non-zero. This means that x + 1 must be non-zero, or x must not equal -1. Thus, the domain of f(x) is all real numbers except for x = -1.

So, for f(g(x)) to be defined, g(x) must be in the domain of f(x), or g(x) must not equal -1. Since g(x) = 1/(x+2), this means that x + 2 must not equal 0, or x must not equal -2.

Combining these two conditions, the domain of f(g(x)) is all real numbers except for x = -2 and x = -1.

Therefore, the domain of f(g(x)) is all real numbers except for x = -2 and x = -1.

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Hey, for this type of question, go on desmos it a calculator. but I hope this is right. Sorry if wrong

The restriction on the variable for the given equation is x ≠ 0

A restricted variable is a variable whose values are confined to some only of those of which it is capable.

Given is an equation, -3/x + 8 = 3/4,

In the given equation, -3/x + 8 = 3/4

We see that, x is in denominator, therefore, x ≠ 0,

Therefore, the restriction is x ≠ 0

Finding the value of x,

-3/x + 8 = 3/4

-3/x = 3/4 - 8

-3/x = -29/4

3/x = 29/4

x/3 = 4/29

x = 12/29

Hence, the restriction on the variable for the given equation is x ≠ 0

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Answer:

Step-by-step explanation:

58

Answer:0.628571428571- or 63% and in fraction form it would be 5/8

Step-by-step explanation:

Answer: 22/35 (Result in decimals: 0.62)

Step-by-step explanation:

13/15 - 5/21

= 13 x 7/15 x 7 - 5 x 5/21 x 5

= 91/105 - 25/105

= 91 - 25/105

= 66/105

= 66 divided by 3 / 105 divided by 3

= 22/35

well, the actual cost of it was 12.5% of 45 less, hmmm what's 12.5% of 45?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{12.5\% of 45}}{\left( \cfrac{12.5}{100} \right)45} ~~ \approx ~~ 5.63~\hfill \stackrel{45~~ - ~~5.63}{\approx\text{\LARGE 39.37}}[/tex]

Answer: 20

Step-by-step explanation:

6 (1/2) + 17

3 + 17 = 20

Answer:

20

Step-by-step explanation:

6(1/2) + 17

6(1/2) = 3

3 + 17 = 20

The value of cosθ is -0.54.

What is a trigonometric function?

The right-angled triangle's angle and the ratio of its two side lengths are related by the trigonometric functions, which are actual functions. They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.

Here, we have

Given: If sin( θ - π/2) = 0.54. find the value of cos(- θ).

According to cofunction inequalities

sin( θ - π/2) = -cosθ

cos(- θ) = cosθ

sin( θ - π/2) = 0.54

-cosθ = 0.54

cosθ = -0.54

Hence, the value of cosθ is -0.54.

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Answer:

Step-by-step explanation:

We can solve this problem by finding all the perfect squares with square roots that can be expressed in the form a√b, where a and b are integers, and n is greater than or equal to 10, and b is as small as possible.

The first few perfect squares with roots that can be expressed in this form are:

10^2 = 100 = 10√1

13^2 = 169 = 13√1

17^2 = 289 = 17√1

19^2 = 361 = 19√1

23^2 = 529 = 23√1

There are 5 perfect squares in the range n <= 500 with roots that can be expressed in the form a√b. So, the answer is 5.

Quadrilateral ABCD is congruent to quadrilateral A'B'C'D'. Therefore, the correct answer option is: a. Congruent.

In Mathematics, a transformation can be defined as the movement of a point from its original (actual) position to a final (new) location such as the following:

In Mathematics, a translation can be defined as a type of transformation which moves every point of the object in the same direction, as well as for the same distance.

In conclusion, we can reasonably infer and logically deduce that quadrilateral ABCD and quadrilateral A'B'C'D' are congruent because quadrilateral ABCD was translated to form quadrilateral A'B'C'D'.

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Complete Question:

Fill in the blank in the sentence given below.

Quadrilateral ABCD is _____ to quadrilateral A'B'C'D'.

a. Congruent

b. Similar

c. Neither

Answer:

  A.  1008 square inches

Step-by-step explanation:

You want the lateral area of a triangular prism with base edge lengths 10 in, 10 in, and 8 in, and a height of 36 in.

The lateral area of the prism is the area of the three rectangular sides. The area of each rectangle is the product of its length and width. All have the same length, so we can sum the widths before multiplying:

  LA = Pl

  LA = (10 in + 10 in + 8 in)(36 in) = 28·36 in² = 1008 in²

The lateral surface area of the prism is 1008 square inches.

The probability of drawing two blue card is 1/16

A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.

Probability = sample space / total outcome

The total number of cards = 12

number of blue cards = 3

Probability of drawing blue card = 3/12

= 1/4

Since the blue card is replaced,

the probability to draw blue card in the second draw is 3/12 = 1/4

probability of drawing two blue cards = 1/4× 1/4

= 1/16

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Answer:

In order in which the boxes appear:

[tex]\boxed{a\sqrt{2}}[/tex]

[tex]\boxed{a\sqrt{2}}[/tex]

[tex]\boxed{a\sqrt{2}}[/tex]

Step-by-step explanation:

At any coordinate (x , y) the distance from the origin (0, 0) is computed by the distance (Pythagorean formula) as:

d = [tex]\sqrt{x^2+y^2}[/tex]

Since x = y = a for both ants, the distance is

[tex]d = \sqrt{a^2+a^2}\\\\d = \sqrt{2a^2}\\\\d = \sqrt{a^2}\cdot \sqrt{2}\\\\d = a\sqrt{2}[/tex]

It does not matter whether both coordinates are positive or both negative since we are taking the squares of the coordinates and distance is always positive

The volume when the pressure is 400 mm Hg is  1600  cubic centimeters

From the information given, we have that The volume of a gas V held at a constant temperature in a closed container varies inversely with its pressure P

This is represented as;

V ∝ 1/P

Find the constant value

V/P = K

Substitute the values

800/200 = k

K = 400

when the pressure is 400 mm Hg, the volume would be; substitute the values, we get;

V/400= 400

cross multiply the values

V = 400(400)

Multiply

V = 1600  cubic centimeters

Hence, the value is  1600  cubic centimeters

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The equations of each line are listed below:

Case 5: y = - (1 / 2) · x + 5

Case 6: y = 7

Case 7: y = (1 / 3) · x

Case 8: y = (1 / 3) · x + 2 / 3

Case 9: y = - (5 / 2) · x - 1

Case 10: y = 6 · x - 2

Case 11: x = - 4

Case 12: y = - (3 / 8) · x

In this problem we find eight cases of equations of the line, whose form is described below:

y = m · x + b

Where:

We need to determine the slope and intercept of each line to write the resulting line. Slope of the line is determine by secant line formula:

m = Δy / Δx

Now each line equation is determined below:

Case 5

Slope

m = (0 - 9) / [10 - (- 8)]

m = - 9 / 18

m = - 1 / 2

Intercept

b = y - m · x

b = 5

Equation of the line

y = - (1 / 2) · x + 5

Case 6

Slope

m = 0 (Horizontal line)

Intercept

b = y - m · x

b = 7 - 0 · 0

b = 7

Equation of the line

y = 7

Case 7

Slope

m = [3 - (- 2)] / [9 - (- 6)]

m = 5 / 15

m = 1 / 3

Intercept

b = y - m · x

b = 0 - (1 / 3) · 0

b = 0

Equation of the line

y = (1 / 3) · x

Case 8

Slope

m = (1 - 0) / [1 - (- 2)]

m = 1 / 3

Intercept

b = y - m · x

b = 0 - (1 / 3) · (- 2)

b = 2 / 3

Equation of the line

y = (1 / 3) · x + 2 / 3

Case 9

Slope

m = (- 1 - 9) / [0 - (- 4)]

m = - 10 / 4

m = - 5 / 2

Intercept

b = y - m · x

b = - 1 - (- 5 / 2) · 0

b = - 1

Equation of the line

y = - (5 / 2) · x - 1

Case 10

Slope

m = [- 2 - (- 8)] / [0 - (- 1)]

m = 6

Intercept

b = y - m · x

b = - 2 - 6 · 0

b = - 2

Equation of the line

y = 6 · x - 2

Case 11

x = - 4 (Vertical line)

Case 12

Slope

m = (- 3 - 0) / (8 - 0)

m = - 3 / 8

Intercept

b = y - m · x

b = 0 - (- 3 / 8) · 0

b = 0

Equation of the line

y = - (3 / 8) · x

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The value of x in this proposition would be the first option i.e. [tex]-13\frac{1}{4}[/tex]

4/11= -33/x+5,

to get the value of x we need to get the x to the left hand side,

4(x+5)= -33,

4x+20 = -33.

subtracting 20 from both the sides,

4x= -33-20

4x = -53.

dividing both the sides by 4,

x= -53/4

x= [tex]-13\frac{1}{4}[/tex] ,

which is option A in the given question

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