2 ( 2 n - 1 ) = - ( n + 7 )

By using the unit conversion method, 2 million seconds converted to days are equal to 23 days.

A unit conversion expresses the same aspect as another unit of measurement. For example, time can be expressed in minutes rather than hours, and distance can be represented from miles to kilometers, feet, or any other unit of measurement.

In this situation, you need to convert between seconds and days:

A day consists of 24 hours. Therefore, the fraction 24 hours/day is equal to 1. There is no value change when multiplying a number by 1. The same holds true for the conversion factors 1 hour/60 minutes and 1 minute/60 seconds.

If you multiply 2,000,000 seconds (2 million seconds) by these conversion factors, you will change the seconds to days:

2,000,000 sec. x (1 min/60 sec.) x (1 hour/60 min) x (1 day / 24 hours)

Therefore, all that is required is to divide 2,000,000 by the product of 60, 60, and 24.

(2,000,000) / (60*60*24) = 2,000,000 / 86,400 = 23.148 days ≈ 23 days

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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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1a) An equation for the water level, L, after D days is L = 35 - 1.5D.

1b) Using the above equation, in 10 days, the water level will be 20 feet.

2a) An equation for the cost, C, after H hours of babysitting by Tina is C = 3 + 5H.

2b) The slope for this problem represents the variable charge per hour while the y-intercept represents the total charge for H hours of babysitting, C.

2c) Based on the above equation, if Tina babysits for 5 hours, she will make $28.

3a) An equation in slope-intercept for the cost, C, after H hours of service by the electrician is C = 25 + 50H.

3b) The electrician's total cost for 8 hours of work is $425.

3c) If an electrician earned $225, the repair lasted 4 hours.

An equation is a statement describing two mathematical expressions as equal or equivalent using the equation symbol (=).

1) Water level of a lake = 35 feet

Receding rate per day = 1.5 feet

Let L = water level and D = days

For the water level, L, to become 20 feet, it will take 10 days (D)

L = 35 - 1.5D

20 = 35 - 1.5D

1.5D = 15

D = 10

2) Fixed fee = $3

Variable fee per hour = $5

Let the charge of babysitting for H hours = C

C = 3 + 5H

For 5 hours of babysitting, Tina will make C = 3 + 5(5) = $28

3) Fixed charge for a service call = $25

Variable cost per hour of service = $50

An Equation in slope intercept:

C = 25 + 50H

For 8 hours of work, the total cost, C = 25 + 50(8) = $425

Total earnings = $225

The number of repair hours = 4 ($225 - $25/$50)

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Answer:

Step-by-step explanation:

300

Answer: 90 miles

Step-by-step explanation:

67+23=90

=

The slope intercept form of the line is y=mx+c where m is the slope of the line and c is the y-intercept is +3

What is meant by slope?

Change the equation x−y=3 in slope intercept form:

x + y = 3

⇒y = x - 3

⇒y= + x−3

Hence, the slope of the line x + y= -3 is m= - 1 and the y+intercept is +3.

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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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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 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:

[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  McCandless method has four steps divide the stage, assign lights, use color and fill the empty spaces.

Setting up the McCandless method for a thrust stage involves several steps.

To light a thrust stage using the McCandless method, divide the space into separately illuminated sections as the first step. The size of the stage will determine how you do it, but there should be enough areas for uniform lighting throughout.

Assign lights to each part of your stage once you've divided it into lighting segments. Three lights per section is the traditional norm, although you should use as many or as few as necessary to provide a consistent effect.

Use color: After placing your lights in each region, you may use gels to balance out their hues. To balance out the shadows on the performers' faces, the McCandless technique recommends a mix of warm and cold hues.

Fill in the blanks: At this stage, your lighting setup should be clear enough for you to understand how it affects your actors and see any potential issues. Use smaller lights, such as Fresnels, to tone down any unnaturally lit areas and blend the performers' looks.

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The process of McCandless method is explained below.

According to the theory of "Into the Wild", the process of McCandless method calls for a combination of warm and cool colors to balance out the shadows on the actors' faces.

And here we know that for a double McCandless system, then you can achieve this by making two of the lights warm and the other two cool.

Here you can also have all your lights be similar, neutral colors.

Apart from this, we have the following steps to do it:

We have to choose the right type of visualization.

And then we have to declutter your visualization.

And then we have to focus your audience's attention.

And the final steps is to think Like a Designer.

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Two single receptacles can be installed in a two-gang box made up of two single-gang boxes ganged together.

A two gang-box is a square electrical box, also called a double-gang box, houses two devices. The two gang-box have a combination switch/outlet or a pair of switches/outlets inside that can control two lighting circuits.

A two-gang box made up of two single-gang boxes which are ganged together can install two single receptacles- receptacles refer to the openings in the gang box into which electronics can be plugged.

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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:

  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

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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Answer:

28

Step-by-step explanation:

step 1: put into fraction form (but it's a ratio)

[tex]\frac{6s}{7b} =\frac{24s}{Xb}[/tex] -> ( s = soccer ) (b = basketball)

step 2: find out the ratio between 6 soccer balls and 24 soccer balls

(the difference is 24 divided by 6, with is 4)

step 3: multiply the basketballs by that ratio

(7b * 4 = 28b)

Therefore, the answer is 28 basketballs.

Answer: I think that you will have to times each side by 2 that whould equal 16.

Step-by-step explanation:

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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The correct option regarding which bus has the least spread among the travel times is given as follows:

Bus 14, with an IQR of 6.

First we consider the dot plot, which shows the number of times that each observation appears in the data-set.

Then we consider the interquartile range, which gives the difference between the third quartile and the first quartile of the data-set.

The interquartile range is a better measure of spread compared to the range of a data-set, as it does not consider outliers.

For groups of 15 students, we have that:

The quartiles for Bus 14 are given as follows:

Hence the IQR is of:

IQR = Q3 - Q1 = 18 - 12 = 6.

The quartiles for Bus 18 are given as follows:

Hence the IQR is of:

IQR = Q3 - Q1 = 16 - 9 = 7.

Hence Bus 14 is the more consistent bus, due to the lower IQR.

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Answer: i dont know  why my previous question was deleted for being unclear but lets forget it, The correct answer one be option 2 or B.

Step-by-step explanation:

i had got it right on the test

Answer:

2

Step-by-step explanation:

Point c is at (1, 2)

point d is at (2, 4)

rate of change formula

(f(d) - f(c))/2 - 1

f(2) = 4

f(1) = 2

4-2/1 = 2

Explore all sim

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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Answer:

slope = 2

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, 3) and (x₂, y₂ ) = (4, 7) ← 2 points on the line

m = [tex]\frac{7-3}{4-2}[/tex] = [tex]\frac{4}{2}[/tex] = 2

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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The equation that can be used to find the number of snowballs Jeffrey and his friends make after lunch is = 153 snowballs - 85 snowballs

The number of snowballs that Jeffrey and his friends made after lunch, can be found by the formula :

= Number of total snowballs made - Number of snowballs made before the boys went to lunch

Number of total snowballs made = 153 snowballs

Number of snowballs made before the boys went to lunch = 85 snowballs

The equation for the number of snowballs Jeffrey and his friends make after lunch is :
= 153 snowballs - 85 snowballs

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Answer:

[tex]21[/tex]

Step-by-step explanation:

[tex]\textrm{If y varies directly with x we can write this relationship as}[/tex]

[tex]y = k \cdot x[/tex]

[tex]\textrm{k is referred to as the constant of proportionality.}[/tex]

[tex]\textrm{Substituting these values into the above equation we get}[/tex]

[tex]3 = k \cdot 2[/tex]

[tex]So\; k = \dfrac{3}{2}[/tex]

[tex]\mathrm{When }[/tex] [tex]x = 14,[/tex]

[tex]y = \dfrac{3}{2} \cdot 14[/tex]

[tex]y = 21[/tex]

For a regular Hexagon , the formula for Area of hexagon is  [tex]\frac{3\sqrt{3} s^{2} }{2}[/tex] and Perimeter of Hexagon is  [tex]6s[/tex] .

What is a Regular Hexagon ?

A regular hexagon is defined as a closed shape polygon that  has six equal sides and six equal angles.

The area of the regular hexagon is the space that is enclosed by all of the six  sides .

let the side length of the regular polygon be = "s" ;

So , the Area will be denoted by = [tex]\frac{3\sqrt{3} s^{2} }{2}[/tex] ;

and the perimeter of regular hexagon is represented as = the sum of all six sides , that is "6s" .

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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 solid is a rectangular prism with a triangular cut-out.

The volume of the solid is given by the iterated integral 1 0 1− x 0 3 − 3z dy dz dx 0. The integral is taken over a three-dimensional rectangular region with x-coordinates between 0 and 1-x, y-coordinates between 0 and 3, and z-coordinates between 0 and 3-3z.

This describes a rectangular prism with length 1-x, width 3 and height 3-3z. However, the triangular cut-out is defined by the equation of the plane z = y/3. It's a triangular region in the zy-plane, with a height of 3 and a base of 3x. This triangular cut-out is "slicing" the rectangular prism from the corner (x,0,0) to the corner (x,3,3x/3) in the xyz-plane.

The final shape is a rectangular prism with a triangular cut out from the corner, with a base on the xy-plane and the vertex on the z-axis. The height, width and length of the rectangular prism are 3, 3 and 1-x respectively.

This type of iterated integral is useful to calculate the volume of a solid with complex boundaries, it's also easy to visualize the shape of the solid by looking at the limits of integration.

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