Q6) The diagram shows a pyramid. The apex of the pyramid is V.
Each of the sloping edges is of length 6 cm.
A
6 cm
2 cm
B
2 cm с
F
6 cm
The base of the pyramid is a regular hexagon with sides of length 2 cm.
O is the centre of the base.
B
2 cm
E
2 cm C
Calculate the height of V above the base of the pyramid.
Give your answer correct to 3 significant figures.

the matrix [cosθ sinθ; -sinθ cosθ] represents the rotation of a vector by an angle θ in the counterclockwise direction.

The matrix you have provided is the 2x2 rotation matrix for rotating a vector in the two-dimensional plane by an angle of θ in the counterclockwise direction.

To understand how this matrix is derived, let's consider a vector v = [x, y] in the two-dimensional plane. We want to rotate this vector by an angle of θ in the counterclockwise direction.

We can represent the vector v as a column matrix [x, y] and then multiply it by a rotation matrix R(θ) to obtain the rotated vector. The rotation matrix R(θ) is defined as follows:

cos(θ) -sin(θ)

sin(θ) cos(θ)

Multiplying the vector v by the rotation matrix R(θ) gives:

[x', y'] = [x, y] * R(θ)

[x', y'] = [x cos(θ) - y sin(θ), x sin(θ) + y cos(θ)]

Thus, the rotated vector v' is given by:

v' = [x', y'] = [x cos(θ) - y sin(θ), x sin(θ) + y cos(θ)]

We can see that the elements of the rotation matrix R(θ) correspond to the cosine and sine of the rotation angle θ. The matrix element in the first row and first column (cosθ) gives the amount of x that is rotated into the x' direction, while the matrix element in the first row and second column (-sinθ) gives the amount of y that is rotated into the x' direction. Similarly, the matrix element in the second row and first column (sinθ) gives the amount of x that is rotated into the y' direction, while the matrix element in the second row and second column (cosθ) gives the amount of y that is rotated into the y' direction.

Thus, the matrix [cosθ sinθ; -sinθ cosθ] represents the rotation of a vector by an angle θ in the counterclockwise direction.

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The description of the ensembles would be: 1. Students who attend both recitals, 2. Students who attend one of the two recitals, 3. students who attend both recitals, 4.The difference between the set P and U, that is, 3 students.

To graph the information in a Venn diagram we must take into account the information in the statement. In this case we must place 6 students outside the circles because there are 6 who do not attend any activity.

So, there would be 34 remaining students, of which 20 go to the piano recital and 23 to the voice recital. So, we can establish that 20 go to both recitals, 3 go to the voice recital, and 1 go to the piano recital.

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Answer:

D?

Step-by-step explanation:

I'm not completely sure, but there are 2 different 180 degree lines that y falls on. On one line (y+70=180) it is for sure 110 degrees

When measured on the other line it is (y+70+x=180) I'm not too familiar with this??

Answer:

Boy = 40%

Girl = 66.6%

Step-by-step explanation:

1) Work out probability that student is a boy who wears cap

8 boys out of 20 wears a cap so to find the probability we have to do 8 divided by 20

2) Work out probability for girls who doesn't wear spectacles

To find the probability of girls who doesn't wear spectacles we have to do 8 divided by 12

Hope this helps, have a lovely day! :)

Answer:

1 - [tex]\frac{2\sqrt{2} }{3}[/tex]

Step-by-step explanation:

[tex]\frac{\sqrt{3}-\sqrt{6} }{\sqrt{3}+\sqrt{6} }[/tex]

rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

the conjugate of [tex]\sqrt{3}[/tex] + [tex]\sqrt{6}[/tex] is [tex]\sqrt{3}[/tex] - [tex]\sqrt{6}[/tex]

= [tex]\frac{(\sqrt{3}-\sqrt{6})(\sqrt{3}-\sqrt{6}) }{(\sqrt{3}+\sqrt{6})(\sqrt{3}-\sqrt{6}) }[/tex] ← expand numerator/ denominator using FOIL

= [tex]\frac{3-\sqrt{18}-\sqrt{18}+6 }{3-\sqrt{18}+\sqrt{18}+6 }[/tex]

= [tex]\frac{9-2\sqrt{18} }{3+6}[/tex]

= [tex]\frac{9-2(3\sqrt{2}) }{9}[/tex]

= [tex]\frac{9-6\sqrt{2} }{9}[/tex]

= [tex]\frac{9}{9}[/tex] - [tex]\frac{6\sqrt{2} }{9}[/tex]

= 1 - [tex]\frac{2\sqrt{2} }{3}[/tex]

Answer:

The correct answer is:

315 ≤ t ≤ 385

This is because the oven temperature display can vary up to 35 degrees higher or lower than the actual temperature being produced. Therefore, the actual temperature (t) could be 35 degrees higher or lower than the displayed temperature (350 degrees), resulting in a range of possible temperatures from 315 to 385 degrees.

The values of a such that p({z > 1.35}) or {z > 2.34}) is 0.0981.

Let z be a standard normal random variable.

The standard normal distribution is a probability distribution that is symmetric around a mean of zero and has a standard deviation of one. It is also referred to as the Gaussian distribution or the normal distribution. This distribution is commonly used in the field of statistics to measure the likelihood of certain outcomes.

We have to determine the values of a such that p({z > 1.35}) or {z > 2.34}).

p(z > 1.35) + p(z > 2.34) = p(z > a)

After using the standard normal distribution table.

0.0885 + 0.0096 = p(z > a)

p(z > a) = 0.0981

p(z > 1.29) = 0.0981

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The value of the line integral is 1/2.

To evaluate this line integral, we need to parameterize the curve C that consists of straight-line segments joining (1, 0, 0) to (0, 1, 0) to (0, 0, 1).

Let's parameterize the first segment from (1,0,0) to (0,1,0) using t as the parameter

r(t) = (1-t) * <1,0,0> + t * <0,1,0>

= <1-t, t, 0>

where 0 ≤ t ≤ 1.

Now, let's parameterize the second segment from (0,1,0) to (0,0,1) using s as the parameter

r(s) = (1-s) * <0,1,0> + s * <0,0,1>

= <0, 1-s, s>

where 0 ≤ s ≤ 1.

We can then use the parameterization of C to evaluate the line integral:

∫_c yz dx + xz dy + xy dz

= ∫_0^1 yz dx/dt dt + xz dy/dt dt + xy dz/ds ds

= ∫_0^1 (t(1-t)) 0 dt + ((1-t)t) 0 dt + ((1-t)t) ds

= ∫_0^1 (1-t)t ds

= 1/2

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The given question is incomplete, the complete question is:

∫_c yz dx + xz dy + xy dz, where c consists of straight-line segments joining (1, 0, 0) to (0, 1, 0) to (0, 0, 1)  Evaluate Line Integrals

Samir's new balance is $5,044.17.

To calculate Samir's new balance, add the previous balance, subtract the payment, add the new transaction, and multiply by the interest rate for one period. The following formula can be used to calculate the interest for a single period:

balance * APR / 12 = interest

where APR stands for annual percentage rate and 12 represents the number of months in a year.

When we apply this formula to Samir's balance and APR, we get:

5336.22 * 0.29 / 12 = 128.95 in interest

As a result, the total new balance is:

5336.22 - 607 + 186 + 128.95 = 5044.17

We get the following when we round to the nearest hundredth:

$5,044.17

As a result, Samir now has a balance of $5,044.17.

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Answer:

$31,142

Step-by-step explanation:

To convert a percentage into a decimal, you move the decimal two places to the left. 1354% converted into a decimal is 13.54.

$23,000 * 13.54 = $31,142

Answer:  Sure! Here's how you can rewrite -6xy²(8y^6-3x^4+4) without parentheses:

Start by distributing the -6xy² to the terms inside the parentheses:

-6xy²(8y^6) - (-6xy²)(3x^4) - (-6xy²)(4)

Simplify each term using the product rule of exponents:

-48x y^8 + 18x^5 y^2 + 24xy²

So the final answer, without parentheses and simplified, is:

-48x y^8 + 18x^5 y^2 + 24xy²

Enjoy!

Step-by-step explanation:

The equivalent z-score for a raw score of 115 under a normal distribution with a mean of 100 and a standard deviation of 15 is 1.

Under the normal distribution, if the mean of the distribution of raw scores is equal to 100 and the standard deviation is known, we can use the z-score formula to calculate the equivalent z-score for any given raw score.

The z-score formula is given by:

z = (x - μ) / σ

where x is the raw score, μ is the mean of the distribution, σ is the standard deviation of the distribution, and z is the corresponding z-score.

Since the mean of the distribution is 100, we have μ = 100. To calculate the z-score, we need to know the standard deviation of the distribution or have information about the distance of the raw score from the mean in terms of standard deviations.

If we assume that the standard deviation is 15, which is a common value used in educational testing, and the raw score is 115, then the corresponding z-score would be:

z = (115 - 100) / 15 = 1

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Answer:

y = (-1/3)x^2 - 2x + 5/3

Step-by-step explanation:

To find the quadratic equation that passes through these points, we can start by using the standard form of a quadratic equation:

y = ax^2 + bx + c

We know that the graph passes through the point (-2,2), so we can substitute these values into the equation:

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

Simplifying this equation, we get:

4a - 2b + c = 2

We also know that the graph has x-intercepts at (-1,0) and (6,0). This means that when x = -1 and x = 6, the value of y (i.e., the height of the graph) is 0. We can use these two points to write two more equations:

0 = a(-1)^2 + b(-1) + c

0 = a(6)^2 + b(6) + c

Simplifying these equations, we get:

a - b + c = 0

36a + 6b + c = 0

Now we have a system of three equations:

4a - 2b + c = 2

a - b + c = 0

36a + 6b + c = 0

We can solve for a, b, and c using any method of solving systems of equations. One way is to use substitution:

From the second equation, we get:

c = b - a

Substituting this into the other two equations, we get:

4a - 2b + (b - a) = 2

36a + 6b + (b - a) = 0

Simplifying these equations, we get:

3a - b = 1

35a + 7b = -b

Multiplying the first equation by 7 and adding it to the second equation, we eliminate b:

21a = -7

a = -1/3

Substituting this value of a into the first equation, we can solve for b:

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

-4/3 - b + b + 1/3 = 2

b = -2

Finally, we can use either of the first two equations to solve for c:

c = a - b = (-1/3) - (-2) = 5/3

So the quadratic equation in standard form is:

y = (-1/3)x^2 - 2x + 5/3

The z-scores that represent the middle 50% of the standard normal distribution are between -0.6745 and 0.6745, the lowest 5% of the standard normal distribution is -1.645, and the 90% of the standard normal distribution is between -1.645 and 1.645.

The mean and variance of a standard normal distribution are both 0. A z distribution is another name for this.

Yes, here are the z-scores for the given standard normal areas:

A. Middle 50%: The area between the 25th and 75th percentiles corresponds to the middle 50% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:

-0.6745 is the z-score corresponding to the 25th percentile.

The 75th percentile z-score is 0.6745.

As a result, the z-scores representing the middle 50% of the standard normal distribution range between -0.6745 and 0.6745.

B. Lowest 5%: The area to the left of the 5th percentile corresponds to the lowest 5% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-score that corresponds to that percentile as follows:

The z-score associated with the fifth percentile is -1.645.

As a result, the z-score representing the bottom 5% of the standard normal distribution is -1.645.

C. Middle 90%: The area between the 5th and 95th percentiles corresponds to the middle 90% of the standard normal distribution. Using Excel's NORM.S.INV() function, we can calculate the z-scores that correspond to those percentiles as follows:

The z-score associated with the fifth percentile is -1.645.

The z-score associated with the 95th percentile is 1.645.

As a result, the z-scores representing the middle 90% of the standard normal distribution range between -1.645 and 1.645.

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The fig's right side of total surface = 42yds.

A solid object's surface area is a measurement of the total area that the surface of the object takes up.

The definition of arc length for one-dimensional curves and the definition of surface area for polyhedra (i.e., objects with flat polygonal faces), where the surface area is the sum of the areas of its faces, are both much simpler mathematical concepts than the definition of surface area when there are curved surfaces.

A smooth surface's surface area is determined using its representation as a parametric surface, such as a sphere.

This definition of surface area uses partial derivatives and double integration and is based on techniques used in infinitesimal calculus.

Henri Lebesgue and Hermann Minkowski at the turn of the century sought a general definition of surface area.

According to our question-

(17+17+8)yds

42yds

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As the triangles are similar to each other, using congruent theorem, we get the value of side JK = 63.8.

Comparable triangles are those that resemble one another but may not be precisely the same size. Comparable items are those that share the same shape but differ in size.

This shows that when shapes are amplified or demagnified, they superimpose one another. This feature of similar shapes is often known as "similarity".

As per the triangles,

Let JK be = x.

GF/GH = JI/JK

⇒ 11/18 = 36 /x

⇒ x = 36 × 18/11

⇒ x = 63.8.

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Answer:

Pair 1 Pair 2 Pair 3

a^6b^4 a^2b^2 a^-562

a^5b^3 a^-36-1 ab^-4

b^3 95 a^-46-2 a^-263

a^-362 66            ཚ

Similar shapes are shapes whose lengths are in equivalent ratio. [tex]\Delta ABC[/tex] and [tex]\Delta DE[/tex][tex]F[/tex] are similar because the side lengths of both triangles are in equivalent ratio (refer to attachment)

Step 1: Draw a random triangle [tex]\Delta ABC[/tex]

The lengths of two sides and an angle are:

[tex]AB=10[/tex]

[tex]BC=6[/tex]

[tex]\angle C=90^o[/tex]

Step 2: Draw and measure the length of DE

[tex]DE=15[/tex]

Step 3: Calculate the ratio

The corresponding line segment to DE is line segment AB.

So, the ratio (k) is:

[tex]k=\dfrac{DE}{AB}[/tex]

[tex]k=\dfrac{15}{10}[/tex]

[tex]k=1.5[/tex]

Step 4: Multiply the ratio by the other line segment in step 1

In (1), we have:

[tex]BC=6[/tex]

So:

[tex]EF=k\times BC[/tex]

[tex]EF=1.5\times6[/tex]

[tex]EF=9[/tex]

Step 5: Draw a circle with center F and radius EF  

The center of the circle is point F and the radius of the circle is 9 units

Step 6: Draw a ray from the center (i.e. point F) to DE

Refer to the attached image for

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The answer of the given question based on the limits the answers are as follows, (a)  lim f(x) = 1 , (b)  lim f(x) = 3 , (c)  lim f(x) = 3.

A graph is  visual representation of data that shows the relationship between two or more variables. Graphs can be used to display  wide variety of information, including numerical data, functions, and networks. The most common types of graphs like line graphs, bar graphs, scatter plots, and pie charts.

Graphs are widely used in many fields, like science, economics, engineering, and social sciences, to help people understand and analyze complex data. They are  powerful tool for visualizing trends, patterns, and relationships, and are often used to communicate findings to wider audience.

a) The limit of f(x) as x approaches 2 from the left:

We can see from the graph that as x approaches 2 from the left, f(x) approaches 1. Therefore, we can write:

lim f(x) = 1

x→2-

b) The limit of f(x) as x approaches 2 from the right:

Similarly, as x approaches 2 from the right, f(x) approaches 3. Therefore:

lim f(x) = 3

x→2+

c) The limit of f(x) as x approaches 2:

Since the limit from the left and the limit from the right exist and are equal, we can say that the limit of f(x) as x approaches 2 exists and equals the common value of the left and right limits. Therefore:

lim f(x) = 3

x→2

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The area of the composite shape using the area formula for the different shapes is 460.48ft².

The area of composite shapes refers to the space occupied by any composite shape. A composite shape is a shape that is made by connecting a few polygons to form the required shape.

These figures or shapes can be built from a wide range of shapes, such as triangles, squares, quadrilaterals, etc. Divide a composite item into basic forms such a square, triangle, rectangle, or hexagon to get its area.

Now in the question,

First let us find the area of the semi-circle.

Area of semi-circle = πr²/2

= [3.14 × (16/2) ²]/2

= (3.14 × 8²)/2

= 200.96/2

= 100.48ft²

Now coming to the rectangle,

area of the rectangle = l × b

= 20 × 15

= 300ft²

Now for calculating the area of the triangle,

area = 1/2 × b × h

= 1/2 × 12 × 10

= 60ft²

Therefore, the area of the total figure = 100.48 + 300 + 60 = 460.48ft².

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The numbers of ways in which we can select the letters without replacement are: a. order relevant - 6 ways. b. order not relevant- 3 ways.

Both permutations and combinations count the number of possible ways to select a particular number of things from a bigger collection of options. The distinction is that whereas combinations ignore the order of the components, permutations do. In other words, combinations don't care about the order in which the elements are selected, but permutations require arranging the components in a certain sequence.

Given that, we want to choose 2 letters, without replacement.

a. When the order is relevant we have the following choices:

3 options for the first letter and 2 options for the second letter, that is:

3 x 2 = 6 ways

b. If the order is not relevant we use combinations:

n C r = n! / (r! (n-r)!)

3 C 2 = 3! / (2! (3-2)!) = 3

Hence, the numbers of ways in which we can select the letters are: a. order relevant - 6 ways. b. order not relevant- 3 ways.

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The complete question is:

Answer:

This table represents a linear relationship that is also proportional:

x 0 1 2

y −4 0 4

Answer:

The second table represents a linear relationship that is also proportional.

To check if a relationship is proportional, we need to see if the ratio of y to x is constant for all values of x and y. In other words, if we divide any y value by its corresponding x value, we should get the same number for all values.

Let's check the ratio for each table:

Ratio for the first table:

-3/2 = -1.5

0/3 = 0

3/4 = 0.75

The ratio is not constant, so this relationship is not proportional.

Ratio for the second table:

-2/4 = -0.5

-1/2 = -0.5

0/0 = undefined

The ratio is constant (-0.5), so this relationship is proportional.

Ratio for the third table:

0/(-2) = 0

1/1 = 1

2/4 = 0.5

The ratio is not constant, so this relationship is not proportional.

Ratio for the fourth table:

-4/0 = undefined

0/1 = 0

4/2 = 2

The ratio is not constant, so this relationship is not proportional.

Therefore, the second table is the only one that represents a linear relationship that is also proportional.

Answer:

[tex]\sqrt{122}[/tex] units

Step-by-step explanation:

To find the distance between points A(1,3) and D(2,-8), we can use the distance formula:

distance = [tex]\sqrt{(x2 - x1)^2 + (y2 - y1)^2}[/tex]

Plugging in the coordinates, we get:

distance = [tex]\sqrt{(2 - 1)^2 + (-8 - 3)^2}[/tex]

distance = [tex]\sqrt{1^2 + (-11)^2}[/tex]

distance = [tex]\sqrt{122}[/tex]

Therefore, the distance between points A(1,3) and D(2,-8) is [tex]\sqrt{122}[/tex] units.

The value of the constant c that makes the following function are c = 0.

Constant is a term used to describe a value that remains unchanged or fixed throughout a program or process. It can be a numeric value, a character value, a string, or a Boolean (true/false) value. Common examples of constants include physical constants, mathematical constants, and programming-language keywords.A constant is a value that does not change, regardless of the conditions or context in which it is used. Common examples of constants include mathematical values such as pi (3.14159), physical constants such as the speed of light (299,792,458 m/s), and other constants such as the universal gravitational constant (6.67408 × 10−11 m3 kg−1 s−2).

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Therefore, c must equal 0 in order for the two sides of the function to be equal. and The one with the greater absolute value is b = 10.

A function is a block of code that performs a specific task. It is a subprogram or a set of instructions that can be used multiple times in a program.

27. For the function to be continuous at x = 7, the limit of the function as x approaches 7 from the left must equal the limit of the function as x approaches 7 from the right.
This means that the value of y as x approaches 7 must be the same on both the left and right sides of the point.
Since the left side of the function is y = c*y + 3, the right side of the function must also be equal to y = c*y + 3.
Therefore, c must equal 0 in order for the two sides of the function to be equal.

28. In order for the function to be continuous at x = 5, the value of y at x = 5 must be the same on both the left and right sides of the point.
Since the left side of the function is y = b - 2x, the right side of the function must also be equal to y = b - 2x.
Therefore, b must equal 10 in order for the two sides of the function to be equal.

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Complete Question:

TRUE. Combinations and permutations are used in probability calculations to determine the number of possible outcomes in a sample space.

When order is important, permutations are utilized, and when order is not important, combinations are used. The formula P(n,r) = n!/(n-r)!, where n! signifies the factorial of n, and the expression C(n,r) = n!/r!(n-r)!, give the number of permutations of n items taken r at a time and the number of combinations of n objects taken r at a time, respectively.

It is possible to count the number of ways that a subset of items can be chosen from a larger set using both permutations and combinations. These ideas are crucial for estimating probabilities because they let us know how many different outcomes could satisfy a certain condition.

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Therefore, the average rate of change between the points (3,9) and (5,15) is 3.

Coordinates are a set of values that locate the position of a point in space. In mathematics, coordinates are used to represent the position of points on a plane or in space, using a set of numerical values that correspond to the distance along each axis from an origin point. In two-dimensional Cartesian coordinate systems, for example, a point is represented by two numbers (x, y) that indicate its position relative to the x and y axes. In three-dimensional Cartesian coordinate systems, a point is represented by three numbers (x, y, z) that indicate its position relative to the x, y, and z axes.

Here,

The average rate of change between the points (3,9) and (5,15) is the slope of the line passing through those two points. We can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) = (3,9) and (x2, y2) = (5,15).

slope = (15 - 9) / (5 - 3)

= 6 / 2

= 3

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Answer: We can simplify the expression inside the absolute value first:

|(sqrt(2) + 3 - 5)|

= |(sqrt(2) - 2)|

Since sqrt(2) > 2, we know that sqrt(2) - 2 is negative. Therefore, we can rewrite the absolute value as a negative:

|sqrt(2) - 2| = -(sqrt(2) - 2)

So the expression without absolute value is:

-(sqrt(2) - 2)

Step-by-step explanation:

The length of the curve of intersection, correct to four decimal places, is 27.7128 units.

To find the length of the curve of intersection, we need to first parameterize the curve.

From the equation of the cylinder, we have:

4x² + y² = 4

Dividing both sides by 4, we get:

x²/1 + y²/4 = 1

This is the equation of an ellipse centered at the origin with semi-major axis a=1 and semi-minor axis b=2.

From the equation of the plane, we have:

x + y + z = 6

Solving for z, we get:

z = 6 - x - y

So the equation of the curve of intersection is:

r(t) = <t, 2sin(t), 6-t-2sin(t)>

where 0 <= t <= 2π.

To find the length of the curve, we need to integrate the magnitude of its derivative with respect to t:

L = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt

Taking the derivative of r(t), we get:

r'(t) = <1, 2cos(t), -1-2cos(t)>

So the magnitude of r'(t) is:

|r'(t)| = √(1² + (2cos(t))² + (-1-2cos(t))²) = √(9+9cos(t)²)

Therefore, the length of the curve is:

L = ∫_0⁽²π⁾ √(9+9cos(t)²) dt

This integral can be solved using the substitution u = tan(t/2), which gives:

L = 18∫_0∞ 1/√(1+u²) du

This integral can be evaluated using trig substitution, letting u = tanθ, which gives:

L = 18∫_0⁽π/²⁾ secθ dθ = 18 ln|sec(π/4)+tan(π/4)| = 18 ln(1+√2)

Using a calculator, we get:

L ≈ 27.7128

Therefore, the length of the curve of intersection, correct to four decimal places, is 27.7128 units.

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Answer: y = -x

Step-by-step explanation:

The equation of the hyperbola is x² - y² = 16.

Taking the derivative of the equation with respect to x, we get:

2x - 2yy' = 0

y' = x / y

At the point (2,-2), we have x = 2 and y = -2. Thus, the slope of the tangent is:

y' = 2 / (-2) = -1

So, the equation of the tangent through the point (2,-2) is:

y + 2 = -1(x - 2)

y + 2 = -x + 2

y = -x