(Drill) Vectors ar (3, -1,2), b = (0,2,1), and c= (1,1, 2) are given. (a) Find (2a + 3b).c. · (b) Which of a, b, and c are mutually perpendicular (if any)? Justify.

To find the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ), we need to first plot the curve on a polar graph.

From the graph, we can see that the curve has two loops - one larger loop and one smaller loop. The larger loop encloses the smaller loop.

To find the area inside the larger loop and outside the smaller loop, we can use the formula:

Area = 1/2 ∫[a,b] (r2 - r1)2 dθ

where r2 is the equation of the outer curve (larger loop) and r1 is the equation of the inner curve (smaller loop).

The limits of integration a and b can be found by setting the angle θ such that the curve intersects itself at the x-axis. From the graph, we can see that this occurs at θ = π/2 and θ = 3π/2.

Plugging in the equations for r1 and r2, we get:

r1 = 1/2 + cos(θ)
r2 = 1/2 - cos(θ)

So the area inside the larger loop and outside the smaller loop is:

Area = 1/2 ∫[π/2, 3π/2] ((1/2 - cos(θ))2 - (1/2 + cos(θ))2) dθ

Simplifying and evaluating the integral, we get:

Area = 3π/2 - 3/2 ≈ 1.07

Therefore, the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ) is approximately 1.07. Note that this area is smaller than the total area enclosed by the curve, since it excludes the area inside the smaller loop.
To find the area inside the larger loop and outside the smaller loop of the limaçon given by the polar equation r = 1 + 2cos(θ), follow these steps:

1. Find the points where the loops intersect by setting r = 0:
1 + 2cos(θ) = 0
2cos(θ) = -1
cos(θ) = -1/2
θ = 2π/3, 4π/3

2. Integrate the area inside the larger loop:
Larger loop area = 1/2 * ∫[r^2 dθ] from 0 to 2π
Larger loop area = 1/2 * ∫[(1 + 2cos(θ))^2 dθ] from 0 to 2π

3. Integrate the area inside the smaller loop:
Smaller loop area = 1/2 * ∫[r^2 dθ] from 2π/3 to 4π/3
Smaller loop area = 1/2 * ∫[(1 + 2cos(θ))^2 dθ] from 2π/3 to 4π/3

4. Subtract the smaller loop area from the larger loop area:
Desired area = Larger loop area - Smaller loop area

After evaluating the integrals and performing the subtraction, you will find the area inside the larger loop and outside the smaller loop of the given limaçon.

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Answer:

A part of a line with a start point but no end point (it goes to infinity) Try moving points "A" and "B": line

Step-by-step explanation:

its just straight forward

Answer:

\(\textsf{A)} \quad \cos (2\theta)=-\dfrac{11}{25}\)

\(\textsf{B)} \quad \sin \left(\dfrac{\theta}{2}\right) = \sqrt{\dfrac{5+\sqrt{7}}{10}\)

Step-by-step explanation:

Given:

\(\sin \theta=\dfrac{3\sqrt{2}}{5}, \quad \quad \dfrac{\pi}{2} < \theta < \pi\)

\(\boxed{\begin{minipage}{4 cm}\underline{Trigonometric Identities}\\\\$\cos (2 \theta)=\cos^2 \theta - \sin^2 \theta\\\\\sin^2 \theta + \cos^2 \theta=1$\\\end{minipage}}\)

Using the trigonometric identities, rewrite cos(2θ) using sinθ only:

\(\begin{aligned}\implies \cos (2 \theta) & =\cos^2 \theta - \sin^2 \theta\\&=1-\sin^2 \theta - \sin^2 \theta\\&=1 - 2 \sin^2 \theta\end{aligned}\)

Substitute the given value of sinθ into the rewritten cos double angle identity:

\(\begin{aligned}\implies \cos (2 \theta) & =1 - 2 \sin^2 \theta \\ & =1 - 2 \left(\dfrac{3\sqrt{2}}{5}\right)^2 \\& = 1-2\left(\dfrac{18}{25}\right)\\& = 1-\dfrac{36}{25}\\& = -\dfrac{11}{25}\end{aligned}\)

\(\boxed{\begin{minipage}{5.5cm}\underline{Trigonometric Identities}\\\\$ \sin \left(\dfrac{\theta}{2}\right)=\pm \sqrt{\dfrac{1-\cos \theta}{2}} $\\\\\sin^2 \theta + \cos^2 \theta=1\\\end{minipage}}\)

Rewrite the second identity to isolate cosθ:

\(\begin{aligned}\sin^2 \theta+\cos ^2 \theta & =1\\\cos ^2 \theta & =1-\sin^2 \theta\\\cos \theta & = \pm \sqrt{1-\sin^2 \theta}\end{aligned}\)

As θ is obtuse, cosθ < 0.  Therefore:

\(\begin{aligned}cos \theta & =- \sqrt{1-\sin^2 \theta}\\& =- \sqrt{1-\left(\dfrac{3\sqrt{2}}{5}\right)^2}\\& =- \sqrt{1-\left(\dfrac{18}{2}\right)}\\& =- \sqrt{\dfrac{7}{25}}\\& =- \dfrac{\sqrt{7}}{5}\end{aligned}\)

Substitute this into the Half Angle Identity, using the positive root as sinθ is positive:

\(\begin{aligned}\sin \left(\dfrac{\theta}{2}\right) & = \sqrt{\dfrac{1-\cos \theta}{2}}\\& = \sqrt{\dfrac{\dfrac{5+\sqrt{7}}{5}}{2}}\\& = \sqrt{\dfrac{5+\sqrt{7}}{10}}\\\end{aligned}\)

Answer:

its c

Step-by-step explanation:

because i multuply

Answer:

320 cubic centimeters

Step-by-step explanation:

Multiply.

--

20 x 4 x 4 = 320

--

Finding Volume

Length x Width x Height = Volume

--

Answer:

1 and 3

Step-by-step explanation:

where the dots on the graph are, you can read that in 5 minutes, 3 pages are edited. in 10 minutes, 6 pages.
so statement 1 is correct.
when every 10 minutes 6 pages are edited, divide both by 10 to get what is edited per minute. -> 6/10 pages are edited per minute.

The answer is C, 1179 grams. To calculate the total weight of the bunch of 9 bananas, we need to find the average weight of the bunch of 8 bananas.

It is calculated by taking the sum of all the weights during the period, then dividing it by the number of weights recorded.

To calculate the total weight of the bunch of 9 bananas, we need to find the average weight of the bunch of 8 bananas.

To do this, we add the weights of each banana in the bunch

(138 + 118 + 121 + 164 + 140 + 115 + 129 + 123 = 1048)

and divide by the total number of bananas (8).

This gives us the average weight of the bunch of 8 bananas, 131 grams. Adding 131 grams to the weight of the 9th banana gives us

= 131+1048

= 1179

Total weight of 1179 grams, which is the closest to the given choices.

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A 2.5-foot-radius sand mandala's surface area is 19.63 square feet.

A Tibetan Buddhist practise known as "sand mandala" involves building and tearing down mandalas composed of different coloured sand. The ritualistic breakdown of the sand mandala is followed by ceremonies and viewing when it is finished to represent Buddhism's doctrine on the fleeting nature of material life.

Here is how to calculate a circle's area:

Circle area is equal to  πr² (where r is the radius of the circle)

We can use the radius value that has been provided to solve for the area using the following formula:

The sand mandala's size is πr²= π(2.5)²= π(6.25)= 19.63 square feet.

Hence, the sand mandala has a 19.63 square foot surface area.

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The arithmetic mean is commonly used as a measure of central tendency because it is intuitive, easy to calculate, and suitable for most data types.

The arithmetic mean, or simply the mean, is the sum of all data values divided by the number of values. It is commonly used as a measure of central tendency because it is intuitive and easy to calculate. However, the mean can be influenced by extreme values, known as outliers, which may distort its interpretation. In contrast, the geometric mean is useful for dealing with multiplicative quantities, such as growth rates or ratios, but it is less commonly used as a measure of central tendency.

To calculate the population standard deviation and variance, assuming equal weights, you first find the mean of the data set. Then, for each data point, subtract the mean and square the result. Sum up all the squared differences, divide by the total number of data points, and take the square root to obtain the standard deviation. Variance is obtained by taking the average of the squared differences from the mean, without taking the square root.

Standard deviation measures the average distance between each data point and the mean. A larger standard deviation indicates a greater spread or variability in the data. Variance is similar to standard deviation but lacks the square root, so it represents the average of the squared differences. These measures are commonly used in statistical analysis to describe the dispersion of data points, assess the uncertainty or variability in a sample or population, and make comparisons between different data sets. They play a crucial role in hypothesis testing, confidence intervals, and inferential statistics.

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Answer:

x=5

Step-by-step explanation:

4x-x = 7 +8

3x = 15

x = 15/3

x = 5

Answer:

5

Step-by-step explanation:

4x - 8 - x =7

3x - 8 =7

3x =7 +8

3x =15

x =15 :3

x =5

The given system of equations  intersect in quadrant II at (-6,2)

What is linear equation ?

linear equation can be defined as the equation in which highest power is one.

Given system of equations,

y = (-1/3) x -4

y = (2/3) x + 2

In the first equation ,

y = (-1/3)x- 4

the 1st line slopes up with y intercept = -4

y = (2/3) x + 2

the 2nd line slopes down with y intercept = 2  

By equating the equations , we get

(-1/3) x -4 =  (2/3) x + 2

-4-2 = (2/3) x + (1/3) x

-6 = x

x = -6

By putting x = -6 in y = (-1/3) x -4

we get ,

y= (-1/3 ) -6 - 4

y = 2-4

y = -2

Hence, The given system of equations  intersect in quadrant II at (-6,2).

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to get the equation of any straight line, we simply need two points off of it, let's use those in the picture below.

\((\stackrel{x_1}{0}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{3}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{3}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{0}}} \implies \cfrac{ -2 }{ 3 } \implies - \cfrac{ 2 }{ 3 }\)

\(\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{5}=\stackrel{m}{- \cfrac{ 2 }{ 3 }}(x-\stackrel{x_1}{0})\implies {\Large \begin{array}{llll} y=-\cfrac{2}{3}x+5 \end{array}}\)

The proof that c = 3a - b + 6. is given below:

We are given the expression ax³ + bx² + 2x + c and the condition that the remainder when the expression is divided by x - 1 is twice the remainder when the expression is divided by x + 1.

We are to show that c = 3a - b + 6.

To find the remainder when the expression is divided by x - 1, we can use the Remainder Theorem, which states that the remainder of a polynomial f(x) when divided by x - r is f(r). So we will find the value of the expression at x = 1.

Remainder when divided by x - 1: R₁ = a(1)³ + b(1)² + 2(1) + c = a + b + 2 + c

Now we will find the remainder when the expression is divided by x + 1. According to the Remainder Theorem, we will find the value of the expression at x = -1.

Remainder when divided by x + 1: R₂ = a(-1)³ + b(-1)² + 2(-1) + c = -a + b - 2 + c

According to the given condition, the remainder when the expression is divided by x - 1 is twice the remainder when it is divided by x + 1.

R₁ = 2R₂

Substituting R₁ and R₂ from our earlier calculations:

a + b + 2 + c = 2(-a + b - 2 + c)

Simplify and solve for c:

a + b + 2 + c = -2a + 2b - 4 + 2c

3a - b + 6 = c

Thus, we have shown that c = 3a - b + 6.

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To show that w = u × v and v = w × u, we need to demonstrate that the cross product of vectors u and v yields the same result as the cross product of vectors w and u.

Given that u = v × w, let's calculate the cross product of w and u:

w × u = (u × v) × u

Since the cross product is not associative, we need to use the vector triple product identity:

w × u = u × (v × u) - (u · u) v

Since u is orthogonal to v, the dot product (u · v) is zero. Therefore, we can simplify the equation:

w × u = u × (v × u)

Next, let's calculate the cross product of v and u:

v × u = (u × v) × v

Using the vector triple product identity:

v × u = v × (u × v) + (v · v) u

Again, since u is orthogonal to v, the dot product (v · v) is zero:

v × u = v × (u × v)

Thus, we have shown that w = u × v and v = w × u, indicating that the cross products are equivalent in both directions.

In summary, when u, v, and w are orthogonal unit vectors, the cross product of u and v yields the same result as the cross product of w and u, as well as the cross product of v and w.

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Answer:

(3,6.3), (7, 14.7)

Step-by-step explanation:

For this question, you want to find the values where 2.1×x is the same as y. It's mostly just trial and error.

x is always first in the bracket, so by multiplying it by 2.1, you would get y (the second value) if it applies.

3*2.1=6.3, so the first one is true.

6.3*2.1=13.23, so the second one is false.

2.1*2.1=4.41, so the third one is false.

7*2.1=14.7, so the fourth one is true.

10.5*2.1=22.05, so the fifth one is false.

14.7*2.1=30.87, so the sixth one is false.

This leaves the first one (3,6.3) and the fourth one (7, 14.7) as true.

** This is mostly algebra, so you may want to look at that. If this question was supposed to be done without a calculator, look at multiplying decimals too. I'm always happy to help!

Answer:

They are (3, 6.3) and (7, 14.7)

Answer:

312

Step-by-step explanation:

120x0.4=48

48x3=144

120x0.7=84

84x2=168

168+144=312

Answer:

Hope this helped.

Answer:70,030 I hope u get a grade for what ever it

Spherical harmonics are an integral part of quantum mechanics. They describe the shape of the orbitals, which electrons occupy in atoms. Moreover, the spherical harmonics provide the angular distribution of a wave in spherical coordinates. In 3D, the spherical harmonics can be written as:

Ylm(θ, φ) = √(2l + 1)/(4π) * √[(l - m)!/(l + m)!] * Plm(cosθ) * e^(imφ)

Here, l and m are known as the angular quantum numbers. They define the shape and orientation of the orbital. Plm(cosθ) represents the associated Legendre polynomial, and e^(imφ) is the exponential function. The spherical harmonics have various forms, including:

Y1,1 = -Y1,-1 = 1/2 √(3/2π) sinθe^(iφ)

Y1,0 = 1/2 √(3/π)cosθ

Y2,2 = 1/4 √(15/2π)sin²θe^(2iφ)

Y2,1 = -Y2,-1 = 1/2 √(15/2π)sinθcosφ

Y2,0 = 1/4 √(5/π)(3cos²θ-1)

Y0,0 = 1/√(4π)

The spherical harmonics have various applications in physics, including quantum mechanics, electrodynamics, and acoustics. They play a crucial role in understanding the symmetry of various systems. Hence, the spherical harmonics are an essential mathematical tool in modern physics. Thus, this is how one can show another form for spherical harmonics.

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Answer:

147+64=c

Step-by-step explanation:

you would add the two numbers together to get the original.

Answer:

5(m+n)

Step-by-step explanation:

\(\textit{exponential form of a logarithm} \\\\ \log_a(b)=y \qquad \implies \qquad a^y= b \\\\[-0.35em] ~\dotfill\\\\ \log_2(x)=-5\implies 2^{-5}=x\)

Answer:

y=mx+b

Step-by-step explanation:

hey! I would do this:

5a + 6 = -6a

+6a        +6a

11a + 6 = 0

     - 6    -6

11a = -6

/11     /11

a = -6/11

Complementary angles, Supplementary angles, Vertical angles, Adjacent angles and Linear pairs these are the  5 Pre algebra angle.

To summarize 5 angle relationships in pre-algebra, we can discuss the following:

1. Complementary angles: These are two angles that add up to 90 degrees. To find a complementary angle, subtract the given angle from 90 degrees. For example, if the given angle is 30 degrees, its complementary angle is 60 degrees (90 - 30 = 60).

2. Supplementary angles: These are two angles that add up to 180 degrees. To find a supplementary angle, subtract the given angle from 180 degrees. For example, if the given angle is 120 degrees, its supplementary angle is 60 degrees (180 - 120 = 60).

3. Vertical angles: These are angles that are opposite each other when two lines intersect. Vertical angles are always equal. For example, if one angle measures 40 degrees, its vertical angle will also be 40 degrees.

4. Adjacent angles: These are angles that share a common side and vertex but do not overlap. Adjacent angles can be complementary, supplementary, or neither, depending on their degree measures.

5. Linear pairs: These are adjacent angles that add up to 180 degrees,

forming a straight line. Both angles in a linear pair are supplementary.

When working with angle relationships in pre-algebra, remember these

five concepts to help you find missing angle measurements and solve problems.

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Answer:

40 kilometers

Step-by-step explanation:

the hypotenuse is a² + b² = c²

so first we have to get 32 and 24 into ²

            32² = 1024       (32 × 32)

            24² = 576         (24 × 24)

1024 + 576 = 1600

now we need to find the square root of 1600

√1600 = 40     (calculator)

therefore, the hypotenuse is 40 kilometers

goodluck, hope this helped,

-cheesetoasty

Answer:

6p^5 q^3

Step-by-step explanation:

download the app cymath for these types of questions

this (^) means exponents

-----

Number 5:

2p^2qx3p^3q^2

Take out the constants (2x3)p^2p^3qq^2

Simplify 2x3 to 6, 6p^2p^3qq^2

6p^2+^3q^1+^2

simplify 2+3 to 5, 6p^5q^1+^2

simplify 1+2 to 3

6p^5 q^3

Answer:

Length = 21 yards

Width = 15 yards

Step-by-step explanation:

Let

Length = x

Width = x - 6

Perimeter ≤ 71 yards

Perimeter of a rectangle = 2 (length + width)

72 ≤ 2(x + x-6)

72 ≤ 2 (2x - 6)

72 ≤ 4x - 12

72 + 12 ≤ 4x

84 ≤ 4x

x ≤ 84/4

x ≤ 21

Length = x = 21 yards

Width = x - 6

= 21 - 6

= 15 yards

Answer:

B. 7−10; 11−14; 15−18; 19−22

Step-by-step explanation:

Answer:

60 pairs of shoes

Step-by-step explanation:

To find the answer, multiply 4x15. The words four times as many mean to do the operation of multiplication. 4x15= 60 So finally, each have 60 pairs of shoes.

i hope this helped:)

Answer:

7.5

Step-by-step:

Julie: ?

Her brother: ?

Total number: 15

So: 15 divided to julie and her brother-

There for 15 divide by 2 is 7.5

7.5+7.5=15