Algebra
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1. Evaluate -5p² - 8p when p=-3

The equation that gives d as a function of h is d = 7200/h

The apparent height of the tower would be 3.6 inches

From the question, we have the following parameters that can be used in our computation:

Apparent height (h) = 800 feet

Distance (d) = 9 inches

Variation = inverse variation

The inverse variation implies that

k = dh

Where k is the constant of variation

Substitute the known values in the above equation, so, we have the following representation

k = 800 * 9

k = 7200

So, we have

7200 = dh

Make d the subject

d = 7200/h

Here, we have

She was standing 2000 feet away from the tower

This means that

d = 2000

So, we have

2000 = 7200/h

Make h the subject

h = 7200/2000

Evaluate

h = 3.6

This means that the height is 3.6 inches

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The car wash made $600 since "there will be 8 teams" meaning 8 * $75.

The length of the car wash will be five and a half hours, with two 15-minute breaks for each team.

Thus, each team works for five hours.

Two automobiles will be washed per hour by each team.

Ten vehicles per squad, then.

[5*2]

Given that "there will be 8 teams," 80 vehicles will be washed (assumed to be in a continuous line of vehicles) [10*8].

The charge must be $600/80=$7.50 each car in order to make $600.

Check (extremely important): "each crew will wash two cars per hour," meaning that each team will earn $15 per hour; "the car wash will run five and a half hours," and "each team will take two 15-minute breaks." As a result, five times $15 equals $75 earned each team.

The car wash made $600 since "there will be 8 teams" meaning 8 * $75.

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Answer:

Step-by-step explanation:

104 x 4 = 416

The approximation for f(−2.2) found by using the line tangent to the graph of F at x= -2 is f(-2.2) = 11

Linear approximation is the procedure of using derivatives to find the value of a function.

Given that F be a differentiable function such that f(-2) = -10 and f'(-2)= -5, and we desire f(2.2). The linear approximation is given as

f(x + Δx) = f(x) + f'(x)Δx

Since we require f(-2.2) = f(-2 - 0.2).

So,

So, substituting these into the equation, we have

f(-2 - 0.2) = f(-2) + f'(-2)(-0.2)

Given that

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

f(-2 - 0.2) = f(-2) + f'(-2)(-0.2)

f(-2 - 0.2) = -10 + (5)(-0.2)

f(-2 - 0.2) = -10 - 1

f(-2.2) = -11

So, the approximation of f(-2.2) = -11

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P(both are greater than 15) is 3/16. The first tile is odd and the second tile is less than 25 is 7/20. The first tile is a multiple of 6 and the second tile is a multiple of 4 is 3/80 and The first tile is less than 15 and the second tile is even or greater than 25 is 21/50.

Given that,

Total number of outcomes = 20

1. P(both are greater than 15)   =  (5/20) (15/20)  =  (1/4) (3/4)  =  3/16

2. The first tile is odd and the second tile is less than 25  

so, (10/20) (14/20)  =  (1/2) (7/10) = 7/20

3. The first tile is a multiple of 6 and the second tile is a multiple of 4  

so, (3/20) ( 5/20) = (3/20) (1/4) = 3/80

4.  The first tile is less than 15 and the second tile is even or greater than 25

So, Number ( even or > 25)  = Number(even) + Number (.>25) - Number (even and > 25) = 10+5-3 = 12

Thus, (14/20) ( 12/20)  = (7/10) (3/5) = 21 / 50

Complete question:

Tiles numbered 1 through 20 are placed in a box. Tiles numbered 11 through 30 are placed in a second box. The first tile is randomly drawn from the first box. The second tile is randomly drawn from the second box. Find each probability.

1. P(both are greater than 15)

2. The first tile is odd and the second tile is less than 25.

3. The first tile is a multiple of 6 and the second tile is a multiple of 4.

4.  The first tile is less than 15 and the second tile is even or greater than 25.

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Answer:

Step-by-step explanation:

The answer is d

So the equation x² + 2y² + 3z² + 2xy + 2yz - 2zx can be put into canonical form, (x + y + z)²/2.

The quadratic form you've provided can be put into canonical form by completing the square. The process involves a few steps:

Rearrange the terms so that all of the x, y, and z terms are on one side of the equation, with a constant term on the other side. In this case, the equation is already in this form.

Divide every term by 2, to make the next step more manageable. This gives us x²/2+ y²/2 + z²/2 + xy + yz - zx

For each variable, add and subtract the square of half of the coefficient of that variable's corresponding x,y,z coefficient. In this case, x²/2, y²/2, z²/2 are all squares so no action need to be done

Group like terms: (x²/2 + xy/2 + xz/2) + (y²/2 + yz/2 + yx/2) + (z²/2 + zx/2 + zy/2)

Each of the group in step 4 is a perfect square, and can be rewritten as (x/√2 + y/√2 + z/√2)² so that the equation becomes (x + y + z)²/2

So the equation x² + 2y² + 3z² + 2xy + 2yz - 2zx can be put into canonical form, (x + y + z)²/2. The conanical form is a useful representation of a quadratic form because it allows you to see the symmetry in the equation and also it allow us to see the level set of the form.

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For the given quadratic form equation x² + 2y² + 3z² + 2xy + 2yz - 2zx has the canonical form as [tex]\frac{ (x + y + z)^2}{2}[/tex]

Here we have to find the quadratic form you've provided can be put into canonical form by completing the square.

And the process involves a few steps,

First, we have to rearrange the terms so that all of the x, y, and z terms are on one side of the equation, with a constant term on the other side.

Now we have to divide every term by 2, to make the next step more manageable,

[tex]= > \frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + yz - zx[/tex]

Now, we have to add and subtract the square of half of the coefficient of that variable's corresponding x,y,z coefficient, then we get the following,

[tex]= > \frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + yz - zx[/tex]

And then we have to group like terms, then we get the equation like the following,

[tex]= > (\frac{x^2}{2} + \frac{xy}{2} + \frac{xz}{2} ) + (\frac{y^2}{2} + \frac{yz}{2} + \frac{yx}{2} ) + (\frac{z^2}{2} + \frac{zx}{2} + \frac{zy}{2} )[/tex]

In order to convert each of the group in step 4 is a perfect square, and then we have to rewritten as [tex](\frac{x}{\sqrt{2} } + \frac{y}{\sqrt{2} }+ \frac{z}{\sqrt{2} } )^2[/tex]

Then we get the canonical equation as

[tex]= > \frac{ (x + y + z)^2}{2}[/tex]

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The average value of function f(x,y,z)=x+z^2 is 3.

To find the average value of f(x, y, z) = x + z^2 on the truncated cone z^2 = x^2 + y^2, with 1 ≤ z ≤ 16, we need to evaluate the integral of f(x, y, z) over the region defined by the truncated cone, and divide by the volume of that region.

The volume of the truncated cone is given by 1/3 * π * (R^2 + r^2 + R*r), where R and r are the radii of the top and bottom circles of the cone respectively. In this case R = 16 and r = 1, so the volume is 1/3 * π * (256 + 1 + 16) = 85.33...

The integral of f(x, y, z) over the region defined by the truncated cone can be found by converting the problem to cylindrical coordinates and then evaluating the integral.

After this, The solution will be

∫∫∫ (x + z^2) dV

= ∫∫∫ (rcos(θ) + z^2) rdzdθdr

= ∫∫∫ (rcos(θ) + r^2) dzdθdr

= ∫∫ (r^2cos(θ) + r^3/3) dθdr

= (r^3/3) * ∫∫ cos(θ) dθdr

after evaluating this, the final answer will be

(1/3) * (r^3) * (∫ cos(θ) dθ) * (∫ dr)

= (1/3) * (1^3) * (sin(θ)) * (r^2)

= (1/3) * (1^3) * (sin(θ)) * (16^2 - 1^2)

= (1/3) * (1) * (sin(θ)) * (255)

= (255/3) * (sin(θ))

and the average value is (255/3) * (sin(θ)) / 85.33... = 3

Therefore, The average value of function f(x,y,z)=x+z^2 is 3.

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Answer:

Step-by-step explanation:

a =  number of apples

"four times as many apples" written as an algebraic expression:

4a

The receiver should be placed 9 feet away from the base of the dish, along its axis of symmetry.

This was found by using the equation for a vertical parabola, y = (¹/₄a)(x - h)² + k,

where (h, k) are the coordinates of the center.

As the parabola is at the center, (h, k) = (0, 0) and the equation simplifies to y = ¹/₄ax² .

Using the given information that the dish is 12 feet across at its opening and 2 feet deep at its center, the coordinate of the given point is (12,4) and by substituting this information into the equation,

the value of a is found as 9 ft which represents the distance between the vertex and the focus.

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The area of square will be;

⇒ 16/81 cm²

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

The width of square = 4/9 centimeters

Now,

We know that;

The area of square = Side × Side

Here, The width of square = 4/9 centimeters

Hence, The area of square with side 4/9 cm is,

⇒ The area of square = Side × Side

⇒ The area of square = 4/9 × 4/9

⇒ The area of square = 16/81 cm²

Thus, We get;

⇒ The area of square = 16/81 cm

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A dot plot is a graph of each data value plotted as a point.

A dot plot, which also known as a strip plot or dot chart, is a simple form of data visualization which consists of data points plotted as dots on a graph with an x- and y-axis. A dot plot is used to encode data in a small circle or a dot. The dot plot is presented on a number line which displays the distribution of numerical variables where a value is described by each dot.

Dot plots are used for continuous, quantitative, univariate data. Data points may be marked if there are few of them. Dot plots are one of the simplest statistical plots, and are fit for small to moderate sized data sets. Dot plots are useful for highlighting the clusters and gaps, as well as the outliers.

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The solution of the initial value problem: y" + 164 = 5(t – 2), y(0) = 3, y'(0) = 8, by using the Laplace transform is equals to 3/s - 8/s² - 166/s³ + 5/s⁴ .

We have, a intial value problem as present below,

y" + 164 = 5(t – 2) ---(1)

and y(0) = 3, y'(0) = 8

We have to solve (1) using laplace transform. Taking Laplace Transform both sides in (1), L[y" + 164] = L[5(t – 2)]

Since, Laplace transform is linear nature,

L [ y" ] + L [ 164] = L[5(t – 2)]

Now, determine the Laplace Transform of left hand side, L [ y" ] + 164 L [1]

= s²L [ y(t)] - s y(0) + y'(0) + 164L [ 1]

= s²L [ y(t)] - 3s + 8 + 164/s

( since, y(0) = 3, y'(0) = 8 , L(1) = 1/s )

Consider right hand side, L[5(t – 2)]

= L( 5t ) - L(2) = 5L(t) - 2 L(1)

= 5 (1!/s²) - 2/s = 5/s² - 2/s

Now, s²L [ y(t)] - 3s + 8 + 164/s = 5/s² - 2/s

=> s²L [ y(t)] = 3s - 8 - 164/s + 5/s² - 2/s

=> s²L[y(t) ] = 5/s² + 3s - 166/s - 8

=> L[ y(t)] = 5/s⁴ + 3/s - 166/s³ - 8/s²

=> L[ y(t)] = 3/s - 8/s² - 166/s³ + 5/s⁴

So, the Laplace Transform of Initial value problem (1), Y(s) is 3/s - 8/s² - 166/s³ + 5/s⁴

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The team that had the best overall record for the season is C. Eagles; they have a larger mean value of about 22 points Falcons; they have a larger mean value of about 20.9 points

The best measure of center to compare the overall records of the two teams is the mean (average) value of the points earned in each game. This is because the mean is a commonly used measure of center in statistics and provides a good overall summary of the data set.

In this case, the Eagles have a larger mean value of about 22 points (calculated by summing the points and dividing by the number of games) compared to the Falcons' mean value of about 20.9 points. So, the correct answer would be Eagles; they have a larger mean value of about 22 points

It's worth noting that using the median value in this case is not the most accurate, because this will give you a more robust representation of the center of the dataset in cases where data have outliers.

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Answer:

  y = -5/3x -7

Step-by-step explanation:

You want the slope-intercept equation of the line that goes through points (-3, -2) and (0, -7).

The slope is given by the formula ...

  m = (y2 -y1)/(x2 -x1)

  m = (-7 -(-2))/(0 -(-3)) = -5/3 . . . . . . use the given point coordinates

The y-intercept is the point on the y-axis, where the y-value is -7.

The slope-intercept form of the equation is ...

  y = mx +b . . . . . . . line with slope m and y-intercept b

  y = -5/3x -7 . . . . . . line with slope -5/3 and y-intercept -7

y = -3x - 4 is the figure's reflection, which is y = -3x + 2. An object with only one dimension, a line has length but no width .

A line is a straight, thickness-free, one-dimensional object in geometry that extends indefinitely in both directions. An object with only one dimension, a line has length but no width. A line is made up of several points that are endlessly stretched in the opposing directions. Collinear points are described as two points that are on the same line. An endlessly long, two-directional line is referred to as a line. It just has one, and that is length. Collinear points are those that are situated along the same path. A line is drawn with an arrowhead as it is defined by two points as illustrated below.

given

We know that the line y = 3x + 2y is reflected in the line y = -1

Observing the coordinate system, we conclude that the lines y = 3x + 2 and y=−1 intersect at the point whose coordinates are (-1,-1).

The line y = 3x + 2y intersect y-axis at the point whose coordinates are (0,-4).

So, the image must pass trough these two points (-1,-1) and (0,-4).

Now, we will substitute these two points in the equation of the line.

[tex]\frac{y - y2}{y2 - y1} = \frac{x - x2}{x2 - x1}[/tex]

x1 = -1

x2 = 0

y1 = -1

y2 = -4

[tex]\frac{y - ( -4 )}{-4 - ( -1 )} = \frac{x - 0}{0 - ( -1 )}[/tex]

[tex]\frac{y + 4}{3} = \frac{x}{1}[/tex]

y = -3x - 4

y = -3x - 4 is the figure's reflection, which is y = -3x + 2 .

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In the complex number 4 + 2i, 4 is the real part. In the complex number 4 + 2i, 2 is the imaginary part.

Complex numbers are those that are represented as a+ib, where a and b are actual numbers and i is an imaginary number termed a "iota."

Imaginary numbers are those that don't exist in reality. An imaginary number produces a negative value when squared. It is shown as Im (). Example: The numbers 2, 7, and 11 are all fictitious.

The two parts of the complex number are called the real part and the imaginary part. The imaginary part is identified by its multiplier of i.

In the given number, the 2 is multiplied by i, so 2 is the imaginary part. The other part, 4, is the real part.

Therefore, In the complex number 4 + 2i, 4 is the real part. In the complex number 4 + 2i, 2 is the imaginary part.

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In the complex number 4+2i, the real part is 4 and the imaginary part is 2. In the complex number 4+2i, 4 is the real part and 2 is the imaginary part.

In general, a complex number can be written as a + bi, where a is the real part and b is the imaginary part. For example, in the complex number 3 + 4i, the real part is 3 and the imaginary part is 4.

The real part of a complex number is the coefficient of the real number (in this case, 3) in the complex number. The imaginary part of a complex number is the coefficient of the imaginary unit i in the complex number.

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Each exponential expression is matched with its expanded form as;

-2⁴. Option E

2³. Option C

(-2)⁴. Option A

2². 2³. Option B

A index form can be described as a mathematical method or way of writing numbers or values that re too small or too large in simpler  and comprehensible forms.

Other forms for index forms are standard forms or scientific forms.

From the information given, we have that;

-2⁴

This is represented as;

-2 × 2 × 2 × 2

2³

The index form is written as;

2 × 2 × 2

(-2)⁴, written as

-2 ×-2 × -2 ×-2

Also, 2². 2³ is written in the form;

(2 × 2)× ( 2× 2 × 2)

Hence, index forms simplifies expressions

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Fractions that are written as a numerator and denominator, including proper, improper, mixed, unit, complex, reciprocal, and equivalent.

Fractions are a mathematical concept written as a fraction, which is a combination of a numerator and denominator. There are 7 types of fractions: proper fractions, improper fractions, mixed fractions, unit fractions, complex fractions, reciprocal fractions, and equivalent fractions. Proper fractions have a numerator that is less than the denominator. Improper fractions have a numerator that is greater than the denominator. Mixed fractions are a combination of a whole number and a proper fraction. Unit fractions have a numerator of 1 and a denominator that can be any number. Complex fractions have both a numerator and denominator that are fractions. Reciprocal fractions are the inverse of a fraction, with the numerator and denominator reversed. Understanding these 7 types of fractions is essential for solving math problems and understanding mathematical concepts.

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Proper, improper, mixed, unit, complex, reciprocal, and equivalent fractions are all written with a numerator and a denominator.

Combining a full number and a legal fraction results in a mixed fraction. In unit fractions, the denominator, which can be any number, has a numerator of 1. Numerator and denominator of complex fractions are also fractions. With the numerator and denominator switched around, reciprocal fractions are the inverse of a fraction.

As a combination of a numerator and a denominator, a fraction is a mathematical term that expresses fractions. There are seven different kinds of fractions: unit fractions, complex fractions, reciprocal fractions, equivalent fractions, and proper and improper fractions. Numerators that are less than the denominator are required for proper fractions. Numerators that are larger than the denominator are indicative of improper fractions.

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The values rewritten into an equivalent form as fractions include the following:

In Mathematics, a fraction simply refers to a numerical quantity which is not expressed as a whole number. This ultimately implies that, a fraction is simply a part of a whole number.

In Mathematics, a fraction comprises two (2) main parts and these include the following:

For instance, 5 by 8 should be read as "five eighths" and it simply means that five (5) parts out of eight (8) equal parts in which the whole part is divided.

Similarly, -3 by 2 should be read as "negative three halves" and it simply means that two (2) parts out of negative three (3) equal parts in which the whole part is divided. Additionally, the number 3 represents the numerator while the number 2 represents the denominator.

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The iterated integral evaluates to zero when converted to polar coordinates.

The iterated integral can be converted to polar coordinates using the following equation:

0 0 4x2y dx dy = 2π ∫0 a r dr dθ

This can be further simplified by integrating with respect to r:

2π∫0 a (a2−r2) dr

The integral can now be evaluated by calculating the definite integral between 0 and a:

2π [a3/3 - r3/3] |0 a

The evaluated integral is then simplified to:

2π[a3/3 - a3/3]

Finally, the evaluated integral is multiplied by 2π to give the final result:

2π(0)

Therefore, the answer is 0.

The iterated integral can be converted to polar coordinates using the equation

0 0 4x2y dx dy = 2π ∫0 a r dr dθ.

This can then be further simplified by integrating with respect to r, resulting in

2π∫0 a (a2−r2) dr.

The integral can be evaluated by calculating the definite integral between 0 and a, which is

2π [a3/3 - r3/3] |0 a.

This evaluated integral is simplified to

2π[a3/3 - a3/3]

and is then multiplied by 2π to give the final result; 2π(0). Therefore, the answer is 0.

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The value of x is 2.

An expression, rule, or law in mathematics that establishes the relationship between an independent variable and a dependent variable (the dependent variable). In mathematics and the sciences, functions are fundamental for constructing physical relationships.

Given

g(x) = 4x + 6.....g(x) = 14

14 = 4x + 6

subtract 6 both sides

14 - 6 = 4x

8 = 4x

divide by 4 both sides

8/4 = x

2 = x

Hence value of x is 2.

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The expression that represents the total number of pages is as follows:

number of printed pages = 10p + 12r

A teacher prints off review packets for her classes. Each 6th-grade packet is 10 pages long and each 7th-grade packet is 12 pages long.

Therefore, the expression that represents the total number of pages printed if there are p 6th-grade packets printed and r 7th-grade packets printed is as follows:

Hence,

number of printed pages = 10p + 12r

where

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Answer: 90 miles

Step-by-step explanation:

67+23=90

=

Answer:

[tex]\textsf{13.}\quad(x+1)^2-\dfrac{y^2}{3}=1[/tex]

[tex]\textsf{14.}\quad\dfrac{x^2}{16}-\dfrac{y^2}{65}=1[/tex]

[tex]\textsf{15.}\quad y^2-\dfrac{x^2}{9}=1[/tex]

[tex]\textsf{16.}\quad\dfrac{x^2}{36}-\dfrac{y^2}{64}=1[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a vertical hyperbola}\\\\$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center.\\ \phantom{ww}$\bullet$ $(h,k\pm a)$ are the vertices.\\\phantom{ww}$\bullet$ $(h,k\pm c)$ are the foci where $c^2=a^2+b^2$\\\phantom{ww}$\bullet$ $y =k\pm\left(\dfrac{a}{b}\right)(x-h)$ are the asymptotes.\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a horizontal hyperbola}\\\\$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center.\\ \phantom{ww}$\bullet$ $(h\pm a,k)$ are the vertices.\\\phantom{ww}$\bullet$ $(h\pm c,k)$ are the foci where $c^2=a^2+b^2$\\\phantom{ww}$\bullet$ $y=k\pm\left(\dfrac{b}{a}\right)(x-h)$ are the asymptotes.\\\end{minipage}}[/tex]

Given:

As the y-values of the foci and vertices are the same, the hyperbola is horizontal (opening left and right).

The center (h, k) is the midpoint of the vertices.

Therefore, the center is (-1, 0) and so:

Use the formula for the vertices (h±a, k) to determine the value of a:

[tex]\begin{aligned}\implies h\pm a&=-2\\-1\pm a&=-2\\\pm a&=-1\end{aligned}[/tex]                [tex]\begin{aligned}\implies h\pm a&=0\\-1\pm a&=0\\\pm a&=1\end{aligned}[/tex]

Therefore:

Use the formula for the foci (h±c, k) to determine the value of c:

[tex]\begin{aligned}\implies h\pm c&=-3\\-1\pm c&=-3\\\pm c&=-2\end{aligned}[/tex]                [tex]\begin{aligned}\implies h\pm c &=1\\-1\pm c&=1\\\pm c &=2\end{aligned}[/tex]

Therefore:

To find b² use c² = a² + b² and the found values of a and c:

[tex]\begin{aligned}\implies c^2&=a^2+b^2\\2^2&=1^2+b^2\\4&=1+b^2\\b^2&=3\end{aligned}[/tex]

Substitute the found values of h, k, a² and b² into the formula to create an equation of the hyperbola:

[tex]\implies\dfrac{(x+1)^2}{1}-\dfrac{(y-0)^2}{3}=1[/tex]

[tex]\implies(x+1)^2-\dfrac{y^2}{3}=1[/tex]

Given:

As the y-values of the foci and vertices are the same, the hyperbola is horizontal.

The center (h, k) is the midpoint of the vertices.

Therefore, the center is (0, 0) and so:

Use the formula for the vertices (h±a, k) to determine the value of a:

[tex]\begin{aligned}\implies h\pm a&=\pm4\\0\pm a&=\pm4\\\pm a&=\pm4\end{aligned}[/tex]

Therefore:

Use the formula for the foci (h±c, k) to determine the value of c:

[tex]\begin{aligned}\implies h \pm c&=\pm9\\0\pm c&=\pm9\\\pm c&=\pm9\end{aligned}[/tex]

Therefore:

To find b² use c² = a² + b² and the found values of a and c:

[tex]\begin{aligned}\implies c^2&=a^2+b^2\\9^2&=4^2+b^2\\81&=16+b^2\\b^2&=65\end{aligned}[/tex]

Substitute the found values of h, k, a² and b² into the formula to create an equation of the hyperbola:

[tex]\implies\dfrac{(x-0)^2}{16}-\dfrac{(y-0)^2}{65}=1[/tex]

[tex]\implies\dfrac{x^2}{16}-\dfrac{y^2}{65}=1[/tex]

Given:

As the x-values of the vertices are the same, the hyperbola is vertical (opening up and down).

The center (h, k) is the midpoint of the vertices.

Therefore, the center is (0, 0) and so:

Use the formula for the vertices (h, k±a) to determine the value of a:

[tex]\begin{aligned}\implies k \pm a&=\pm1\\0\pm a&=\pm1\\\pm a&=\pm1\end{aligned}[/tex]

Therefore:

Use the formula for the asymptotes to determine the value of b:

[tex]\begin{aligned}\implies k\pm\left(\dfrac{a}{b}\right)(x-h)&=\pm\dfrac{1}{3}x\\\\0\pm \left(\dfrac{1}{b}\right)(x-0)&=\pm\dfrac{1}{3}x\\\\\pm\dfrac{1}{b}x&=\pm\dfrac{1}{3}x\\\\\pm b&=\pm3\end{aligned}[/tex]

Therefore:

Substitute the found values of h, k, a² and b² into the formula to create an equation of the hyperbola:

[tex]\implies\dfrac{(y-0)^2}{1}-\dfrac{(x-0)^2}{9}=1[/tex]

[tex]\implies y^2-\dfrac{x^2}{9}=1[/tex]

As the y-values of the vertices are the same, the hyperbola is horizontal.

The center (h, k) is the midpoint of the vertices.

Therefore, the center is (0, 0) and so:

Use the formula for the vertices (h±a, k) to determine the value of a:

[tex]\begin{aligned}\implies h \pm a&=\pm6\\0\pm a&=\pm6\\\pm a&=\pm6\end{aligned}[/tex]

Therefore:

Use the formula for the asymptotes to determine the value of b:

[tex]\begin{aligned}\implies k \pm\left(\dfrac{b}{a}\right)(x-h)&=\pm\dfrac{4}{3}x\\\\0\pm \left(\dfrac{b}{6}\right)(x-0)&=\pm\dfrac{4}{3}x\\\\\pm \dfrac{b}{6}x&=\pm\dfrac{4}{3}x\\\\ \pm b&=\pm8\end{aligned}[/tex]

Therefore:

Substitute the found values of h, k, a² and b² into the formula to create an equation of the hyperbola:

[tex]\implies \dfrac{x^2}{36}-\dfrac{y^2}{64}=1[/tex]

The term "sine" is derived from the Latin word sīnus, which means "bay". Sine is a mathematical function used to define a ratio between the length of a side of a right triangle and its opposite side.

It is also used to describe waveforms in trigonometry, electricity, and other sciences. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. The sine of a given angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.

This ratio can be expressed as a number between -1 and 1, with 0 representing a right angle. Sine is also used to describe waveforms in trigonometry, electricity, and other sciences. In addition, sine is used to solve problems involving triangles, circles, and various quadratic equations.

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Reporting frequencies of sample data, such as eye color, ethnicity, grade point average, and height is done by using (Option C.) descriptive statistics.

Descriptive statistics are used to summarize, organize, and describe sample data, such as eye color, ethnicity, grade point average, and height. These statistics are used to analyze data and make inferences about larger populations.

Descriptive statistics are used to calculate the mean, median, mode, and range of the data, as well as to provide visual representation of the data through charts and graphs. Descriptive statistics are one of the most commonly used forms of statistical analysis.

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the centroid of the region is (x, y) = (10, 1000/3).

(x, y) = (10, 1000/3)

1. Set up the equation for the centroid formula: x = (1/A)∫y dx and y = (1/A)∫x dy

2. Find the area of the region: A = ∫(y2 - y1) dx

3. Calculate the integral: ∫y dx = x4/4 + C and ∫x dy = xy + C

4. Substitute the boundaries into the integrals and solve for C: x4/4 + C = 30x and xy + C = 0

5. Substitute the solutions for C in the centroid formula: x = (1/A)∫y dx = (1/A)(30x - x4/4) and y = (1/A)∫x dy = (1/A)(xy - 0)

6. Substitute the boundaries into the area equation and solve for A: A = ∫(y2 - y1) dx = ∫(30x - x4/4 - 0) dx = 30x2/2 - x5/5 + C

7. Substitute the solutions for C in A: A = 30x2/2 - x5/5 + C = 30(30)2/2 - (30)5/5 + C = 27000/2 - 27000 + C = 13500 + C

8. Substitute the solutions for C in the centroid formula and solve for x and y: x = (1/13500 + C)(30x - x4/4) and y = (1/13500 + C)(xy - 0)

9. Substitute the boundaries into the centroid formula and solve for x and y: x = 10 and y = 1000/3

Therefore, the centroid of the region is (x, y) = (10, 1000/3).

the complete question is :

Find the centroid of the region bounded by the given curves. y = x3, x + y = 30, y = 0 (x, y) = (10, 1000/3)

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Answer:

Step-by-step explanation:

300

The corresponding angles are equal angles. Then the correct option is D.

The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360°.

Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.

Corresponding angle - If two lines are parallel then the third line. The corresponding angles are equal angles.

Let's check all the options:

The sum of corresponding angles is 180°. The statement is false.

The angles in a vertical pair are acute. The statement is false.

Corresponding angles are supplementary. The statement is false.

Corresponding angles are congruent. The statement is True.

The corresponding angles are equal angles. Then the correct option is D.

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