completed the questions below

Using a double integral, the area of the region is [tex]\frac{\pi}{12}[/tex].

Consider the following polar equation:

r = 8cos3[tex]\theta[/tex]

This is the process for measuring the intensity of light. Such intensity varies according to how close or far you are from LED light bulbs. Engineers use polar curves to plot the level of the intensity at any given point. In other words, a polar curve tells an engineer how LED light bulbs spread light across a room or surface.

The objective is to find the area of the region using double integral.

The sketch of the curve attached at the end of solution.

The intersection points of the curve with horizontal axis are given by,

8cos3[tex]\theta[/tex] = 0

cos3[tex]\theta[/tex] = 0

[tex]3 \theta & =\pm \frac{\pi}{2}[/tex]

[tex]\theta & =\pm \frac{\pi}{6}[/tex]

Therefore, the loop is given by the region,

[tex]D=\left\{(r, \theta) \mid-\frac{\pi}{6} \leq \theta \leq \frac{\pi}{6}, 0 \leq r \leq \cos 3 \theta\right\}[/tex]

Let A be the area of the curve.

The area of the loop is given by,

[tex]& A(D)=\iint_D d A \\[/tex]

[tex]& =\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \int_0^{\cos 3 \theta} r d r d \theta \\[/tex]

[tex]& =\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left[\frac{r^2}{2}\right]_0^{\cos 3 \theta} d \theta \\[/tex]

[tex]& =\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{6}} \cos ^2 3 \theta d \theta \\[/tex]

[tex]& =\frac{1}{2} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(\frac{1+\cos 6 \theta}{2}\right) d \theta \\[/tex]

[tex]& =\frac{1}{4}\left[\theta+\frac{1}{6} \sin 6 \theta\right]_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \\[/tex]

[tex]& =\frac{1}{4}\left[\left(\frac{\pi}{6}+\frac{1}{6} \sin \pi\right)-\left(-\frac{\pi}{6}+\frac{1}{6} \sin (-\pi)\right)\right] \\[/tex]

[tex]& =\frac{1}{4}\left(\frac{\pi}{3}\right) \\[/tex]

[tex]& =\frac{\pi}{12} \\[/tex]

Hence, the area of the one loop of the rose [tex]r = 8 \cos(3\theta)[/tex] is [tex]\frac{\pi}{12}[/tex].

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Applying the definition of the median of a triangle, the length of AO is calculated as: 14 units.

The median of a triangle is a line segment that joins the midpoint of a side of a triangle and the opposite vertex to the side.

There are three medians in a triangle, and the three of them intersect at a point inside the triangle. This point of intersection is known as the centroid.

According to the centroid theorem of a triangle, the length from the vertex to the centroid is two-third of the length of a median.

Thus, the figure shows that triangle ABC has three medians, BL, AM, and CK, which intersect at point O. This means the centroid of the triangle is point O.

By implication, it means that:

Length of AO = 2/3 * (length of AM)

Plug in the value of AM:

Length of AO = 2/3 * (21) [centroid theorem]

AO = 2 * 7

AO = 14

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Mr. Kenny can complete five projects without running out of clay.

The number of projects that can be completed will be calculated using division.

1/6 of a pound of clay is required for number of project = 1

7/8 pounds of clay will be required for number of project = 7/8 ÷ 1/6

Rewriting the fraction with numerator and denominator

Number of projects = (7×6)÷(8×1)

Performing multiplication in numerator and denominator

Number of projects = 42/8

Performing division

Number of projects = 5.25

Rounding off, we get number of projects = 5

Mr. Kenny can complete 5 projects.

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Answer:

720 / 6 = 120

120 * 4 = 480

P(1) = 120/720

P(2) = 120/720

P(3) = 120/720

P(4) = 120/720

P(5) = 120/720

P(6) = 120/720

P(1 + 2 + 3 + 4) = 480/720

Step-by-step explanation:

Answer: x = 19

Step-by-step explanation:

Let's start with what we have, we'll let x = the number:

4x - 12 = 64

Then we add 12 to both sides:

4x = 76

Divide by 4 from both sides:

x = 19

From 1750 to 1900, Europe's strength and dominance over the world grew dramatically.

New western political concepts developed strong, centralized states that directed empires all around the world, while industrialization created unprecedented wealth.

The following are the reasons and impacts of the various revolutions that occurred between 1750 and 1900:

The globe became really autonomous as a result of industrialization.

They were able to push for changes to their terrible working conditions through the formation of unions.

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Refer to the photos taken.

The value of function  f(2) if  f'(x)= cos(x²), and f(3) = 7 is  [tex]\sqrt{\frac{\pi }{2}}\mathbb{C}\left(\sqrt{\frac{4}{\pi }}\right)+C[/tex].

The relationship between a set of potential outputs and a set of potential inputs is called a function, and it has one relationship per output between each input. This indicates that a function f will map an object x to exactly one object f(x) in the set of potential outputs if it is present in the set of inputs, also known as the domain (called the co-domain).

Given:

f'(x )= cos(x²),

f(3) = 7

Calculate the integration of the following function as shown below,

∫ f'(x) = 3∫cos(x²)

The integral is the Fresnel integral function, C(x)+C

[tex]\sqrt{\frac{\pi }{2}}\mathbb{C}\left(\sqrt{\frac{2}{\pi }}x\right)+C[/tex]

f(3) = 7

[tex]\sqrt{\frac{\pi }{2}}\mathbb{C}\left(\sqrt{\frac{2}{\pi }}3\right)+C[/tex]

Calculate the value of  f(2) as shown below,

f(2) = [tex]\sqrt{\frac{\pi }{2}}\mathbb{C}\left(\sqrt{\frac{2}{\pi }}\times 2\right)+C[/tex]

f(2) = [tex]\sqrt{\frac{\pi }{2}}\mathbb{C}\left(\sqrt{\frac{4}{\pi }}\right)+C[/tex]

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an equation showing a way to estimate to the neasrt inch the total rainfall for the two day 2.45 + 1.72 ≈ 4.2 inches

1. Add the two amounts of rainfall together: 2.45 + 1.72

2. Round the answer to the nearest inch: 4.2 inches

3. The equation to estimate the total rainfall for the two days is 2.45 + 1.72 ≈ 4.2 inches.

This equation provides a simple way to estimate the total rainfall for two days. By adding the two amounts of rainfall, we can then round the answer to the nearest inch to obtain the estimated total rainfall. This equation can be used to quickly and accurately estimate rainfall totals for any two-day period.

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Answer:

Step-by-step explanation:

m=riserun=ΔyΔx

m=y2−y1x2−x1

m=0−2−2−2

m=−2−4

m=12

The answer is :

m = 0.5

Step-by-step explanation:

since f(x) = 2×x + 1

therefore f(2.99) = 2×2.99 + 1 = 6.98 ≈ 7

therefore f(2.999) = 2×2.999 + 1 = 6.998 ≈ 7

therefore f(3.01) = 2×3.01 + 1 = 7.02 ≈ 7

therefore f(3.001) = 2×3.001 + 1 = 7.002 ≈ 7

now for the what conclusion can you draw from these results i dont 100% know but i think he wants that

the point x = 3 and the point y = 7

since f(x) = y

and since we have the approximate of x 3

Answer: C

Step-by-step explanation:

There are 324 miles can Mario drive with 9 gallons of gas.

Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

Given that;

He can drive 18 miles on ½ a gallon of gas.

Now,

Since, He can drive 18 miles on ½ a gallon of gas.

Let the number of miles cover with 9 gallons of gas = x

So, By definition of proportion, we get;

⇒ 18 / (1/2) = x / 9

⇒ 18 × 2 = x / 9

⇒ 36 × 9 = x

⇒ x = 324 miles

Thus, The number of miles cover with 9 gallons of gas = 324 miles

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Answer: 96 crayons

Step-by-step explanation:

Take 182 subtract 86 = 96 crayons

96 crayons are not broken

A child of a parent with a mutation in collagen and displaying OI will have 50% probability of also having OI.

Osteogenesis imperfecta (OI) is an inherited or genetic bone disorder which is present at birth. It is also known as brittle bone disease. A child born with OI may have soft bones which fracture easily, bones which are not formed normally, and other problems. Signs and symptoms may range from mild to severe.

Life expectancy for OI babies vary greatly depending on OI type. Babies with Type II generally die soon after birth. Children with Type III may live longer, but mostly only until around age 10.

About 90% of OI cases are caused by autosomal dominant mutations in the type I collagen genes. Mutations in one or the other of these genes cause the body to make either abnormally formed collagen or too little collagen. Mutations in these genes cause OI Types I through IV.

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Answer:

x= -28

y= 6

Step-by-step explanation:

-x-7y = -14  

-4x-14y=28

Let's find x first! First, we time the first equation by -2

2x + 14y = 28

-4x -14y = 28

Add them; you will get

-2x = 56

x = -28

Then put -28 back in the x, and you will get

y= 6

Answer:Y=10X-16

Step-by-step explanation:

y = x3- 2x 2,( 2, 4)( Given)

Find the first outgrowth, calculate the pitch and find the equation.

We should first know the pitch of the wind at(,4)

pitch at any point on the wind is written by its first outgrowth as

dy/ dx = 3x2- 2

pitch at x = 2

m = 3( 2) 2- 2 = 12- 2 = 10

Equation of a digression is given as

c mx = y

Where m is the pitch

Substituting the values

c 10 × 2 = 4

On farther computation

c 20 = 4

c = 16

The equation of the digression is y = 10 x- 16

Thus, the equation of the digression line to the wind is y = 10x-16

Expreion have a quotient of 6 is 0. 48 ÷ 0. 08.

What is quotient?

In mathematics, the quotient is the result of performing division operations on two integers. It is essentially the outcome of the division procedure. In arithmetic division, the terms divisor, dividend, quotient, and remainder are employed in four different ways.Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication.

0. 48 ÷ 0. 8

=0.06

The solution for Long Division of [tex]$\frac{4.8}{8}$[/tex] is 0.6

The solution for Long Division of [tex]$\frac{0.48}{0.08}$[/tex] is 6.

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Answer: 2/12= 16.67% four sig fig

Step-by-step explanation:

Using (mass of sugar)/(total mass)

and total mass equals (mass of sugar) + (mass of water)

get a total mass of 12kg

Therefore percentage is 2/12=1/6=16.666666%

Answer:

The percentage of sugar in the syrup is 16.66...%

Step-by-step explanation:

lmk

Answer:

The correct answer is C: Y = -1/2x - 4.The slope of the given line is -1/2, so any line that is parallel to it will have the same slope. In order to find the equation of a line with the same slope that passes through the point (2, -5), we can use the point-slope form of the equation of a line.The point-slope form of the equation of a line is:y - y1 = m(x - x1)where m is the slope of the line, and (x1, y1) is a point on the line.In this case, we can plug in the values m = -1/2, x1 = 2, and y1 = -5 to get:y + 5 = -1/2(x - 2)Solving for y, we get:y = -1/2x - 4Therefore, the equation of the line that is parallel to the given line and passes through the point (2, -5) is Y = -1/2x - 4.

The equations are 2r + b = 8.40 and 3r + b = 9.35. Then the cost of refill and egg breakfast will be $0.95 and $6.50, respectively.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The eatery charge you and your companion got was not organized. You requested 2 beverage tops off and the egg breakfast, for a sum of $8.40. Your companion requested 3 tops off and the egg breakfast for $9.35.

Let 'r' be the cost of refill and 'b' be the cost of egg breakfast. Then the equations are given as,

2r + b = 8.40            ...1

3r + b = 9.35            ...2

From equations 1 and 2, then we have

3r - 2r = 9.35 - 8.4

r = $0.95

Then the cost of the egg breakfast will be given as,

2 (0.95) + b = 8.40

b = $6.50

The equations are 2r + b = 8.40 and 3r + b = 9.35. Then the cost of refill and egg breakfast will be $0.95 and $6.50, respectively.

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Franklin has 10 granola bars and 6 pieces of dried fruit. The greatest number of snack bags Franklin can make is 2 bags.

The factor shared by at least two numbers is known as the common factor. Each of the given numbers is divided exactly by two or more numbers.

Here given granola bar = 10 and dried fruit = 6.

Then, the common factors of 10 and 6 are

1, 2, 3, and 6 are used to represent factors of 6.

1, 2, 5, and 10 are used to represent factors of 10.

2 is discovered to be the biggest common factor between 6 and 10.

Then, the greatest number of snack bags Franklin can make is 2 bags.

The complete question -

Franklin wants to create a snack bag for a trip he is going on. He has 10 granola bars and 6 pieces of dried fruit. If the snack bag should be identical without any food left over, what is the greatest number of snack bags Franklin can make?

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The axis of symmetry of the parabola would be x = 2

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

Vertex = (2, 13)

Rewrite the vertex as follows

So, we have the following representation

(x, y) = (2, 13)

Remove the y-coordinates

This gives

x = 2

This represents the axis of symmetry

Hence, the axis of symmetry is x =2

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Answer: 1/20

Step-by-step explanation:

....................................

Answer: 22%

Step-by-step explanation:

The equation of the line would be y = (-3/4)x + 5.

What is the slope-point form of the line?

For the line having slope "m" and the point (x1, y1) the equation of the line passing through the point (x1, y1) having slope 'm' would be

                      y - y1 = m(x - x1)

The given equation is [tex]y=-\frac{3}{4}x-17[/tex]

The required line is parallel to the given line.

and we know that the slopes of the parallel lines are equal so the slope of the required line would be m = -3/4

And the required line passes through (8, -1)

so by using slope - point form of the line,

      y - (-1) = (-3/4)(x - 8)

      y + 1 = (-3/4)x - (-3/4)8

       y + 1 = (-3/4)x + 24/4

       y = (-3/4)x + (12/2 - 1)

       y = (-3/4)x + 5

Hence, the equation of the line would be y = (-3/4)x + 5.

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The position of mailbox after 10s is equal to sum displacements is 14m.

The principle is that the slope of the line on a velocity-time graph reveals useful information about the acceleration of the object. If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line).

Given velocity time graph understand in three parts.

1. first 6s mailbox have constant velocity 3m/s

the displacement covered by it = area under v-t graph(square shape)

s1=3 x 6 =18m

2. the mailbox at rest at 6-8 s.

the displacement covered by it s2=0

3. the mail returned with constant velocity 2m/s

Displacement is defined to be the change in position of an object. It can be defined mathematically with the following equation:

Displacement=Δx=[tex]x_{f} - x_{0}[/tex]

[tex]x_{f}[/tex]=refers to the value of the final position.

[tex]x_{0}[/tex]= refers to the value of the initial position.

Δx is the symbol used to represent displacement

the displacement covered by it s3 =(10-8) (-2) =-4m

the position of mailbox after 10s is equal to sum displacements=s1+s2+s3 =18+0-4=14m the displacement - time graph.

Therefore, the position of mailbox after 10s is equal to sum displacements is 14m.

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Draw the mail carrier's position-versus-time graph. Assume that x=0m at t=0s

What is the position of the mailbox?

Answer:

[tex] r = \sqrt{\dfrac{V}{\pi h}} [/tex]

Step-by-step explanation:

V = πr²h

Switch sides.

πr²h = V

Divide both sides by πh.

r² = V/(πh)

Take the square root of both sides.

r = √[V/(πh)]

[tex] r = \sqrt{\dfrac{V}{\pi h}} [/tex]