3sintheta=sintheta+1

The equation of the osculating plane is (-1/3)(x-0) + (1/3)(y+1) + (2/3)(z-8 π) = 0.

First, we find the point on the curve at

[tex]t = 2\pi[/tex]

[tex]x = sin(2(2\pi)) = 0 \\

y = -cos(2(2\pi)) = -1 \\

z = 4(2\pi) = 8\pi[/tex]

So the point on the curve is

[tex] (0, -1, 8\pi)[/tex]

To find the normal plane, we take the cross product of the partial derivatives:

[tex]r_t = (2cos(2t), 2sin(2t), 4) \\

r_t(2\pi) = (2cos(4\pi), 2sin(4\pi), 4) = (2, 0, 4) \\

r_tt = (-4sin(2t), 4cos(2t), 0) \\

r_tt(2\pi) = (0, -4, 0) \\

n = r_t × r_tt = (-16, -8, -8)[/tex]

So the equation of the normal plane is

[tex]-16(x-0) - 8(y+1) - 8(z-8\pi) = 0.[/tex]

To find the osculating plane, we first find the unit tangent and unit normal vectors:

[tex]T = r_t / ||r_t|| = ( \frac{1}{2})cos(2t)i + ( \frac{1}{2}) sin(2t)j + ( \frac{1}{4})k \\

N = n / ||n|| = ( \frac{2}{3})cos(2t)i + ( \frac{1}{3})sin(2t)j + ( \frac{1}{3})k[/tex]

Then the binormal vector is B = T × N, which is

[tex]( \frac{ - 1}{3})sin(2t)i + ( \frac{1}{3} )cos(2t)j + ( \frac{2}{3} )k[/tex]

Finally, the equation of the osculating plane is

[tex]( \frac{ - 1}{3})(x-0) + ( \frac{1}{3})(y+1) + ( \frac{2}{3})(z-8\pi) = 0[/tex]

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Answer: Decay

Step-by-step explanation:

The actual distance between M and μ is measured by the numerator of the z-score test statistic.

The distribution of the z-score test is normally distributed under the null hypothesis, making it a statistical test. The mean is tested for a known standard deviation using this method. The z score (z), which is determined by the following equation, is the difference between the sample mean and the population mean in units of standard error. The distribution of the z-score test is normally distributed under the null hypothesis, making it a statistical test. The mean is tested for a known standard deviation using this method. The z score (z), which is determined by the following equation, is the difference between the sample mean and the population mean in units of standard error. The equation is z = M - μ / σ/√n, where n is the number of samples in the population, is the mean, o is the standard deviation, and σ/√n is the standard error. The sample mean's (μ) distance from the population mean serves as the z score's numerator (M).

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89 centimeter is approximately equal to 35.0394 inches ( rounded to four decimal places )

To convert 89 cm to inches, we can use the following formula:

1 cm = 0.393701 inches

The conversion is the process of changing the unit of one quantity to another units  

The conversion factor is defined as the number that is used to change one unit to another units by multiplying or dividing

Therefore,

The length in inches = conversion factor × The length in centimeter

Substitute the values in the equation

89 cm = 89 x 0.393701

Multiply the numbers

= 35.0394 inches ( rounded to four decimal places )

Therefore, 89 centimeter is 35.0394 inches

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In general, whether or not an individual needs to file a tax return depends on a number of factors, including income, filing status, and age.

In the United States, for example, the IRS sets income thresholds each year, and individuals whose income exceeds these thresholds are generally required to file a tax return.

However, there are many exemptions and exceptions to this general rule, and individuals should consult the IRS website or a tax professional for more information specific to their situation.

Other countries may have different rules and requirements for filing tax returns, so it's important to check with the tax authority in your own country for specific guidance.

P.S Your question is incomplete, so I gave you a general overview to help you better understand filing tax returns.

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Line a could also be referred to as line EC.

In mathematics, a line can be named in a variety of ways depending on the context. One common way to name a line given a starting point and an ending point is to use the names of the two points.

For example, if the starting point is labeled A and the ending point is labeled B, we can refer to the line as "line AB" or "line BA" (the order doesn't matter, as the line extends in both directions).

In this case therefore, line a can be referred to as line EC which are the letters at the starting and ending points.

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The conversion of 21 km to miles is 13.04848 miles.

Kilometer (km) and mile are both units of measurement used to express distance or length.

Kilometer is a metric unit of distance commonly used in the International System of Units (SI) and is equal to 1000 meters.

To convert 21 kilometers (km) to miles, we can use the conversion factor of 1 km = 0.621371192 miles.

So, we can multiply 21 km by 0.621371192 miles/km to get the equivalent distance in miles:

21 km × 0.621371192 miles/km = 13.04848 miles (rounded to five decimal places)

Therefore, 21 km is approximately equal to 13.04848 miles.

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The converted value using standard formula from Fahrenheit to Celsius is equal to 45 Fahrenheit to 7.22222 degrees Celsius.

Standard formula which relates Fahrenheit and Celsius using the standard formula as mentioned below:

(°F − 32) × 5/9 = °C

We have to Substitute the value 45 Fahrenheit in the above mentioned Standard formula of the temperature to get into degrees Celsius ,

= ( 45°F − 32) × 5/9

= 13 × 5/9

= 65 / 9

= 7.22222 degrees Celsius

Therefore, the conversion value of 45 Fahrenheit into the Celsius is written as 7.22222 degrees Celsius .

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Roberto spent 41 more minutes doing homework than he did reading and making his lunch.

What is the basic algebraic expression?

An algebraic expression is when we use numbers and words in solving a particular mathematical question.

First, let's convert all the times to minutes:

Now, we can calculate the difference:

90 - (40 + 9) = 41

Hence, Roberto spent 41 more minutes doing homework than he did reading and making his lunch.

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The areas of the compound figures are following;

1). The area of the compound figure,

= 121.12 square feet.

2). The area of the compound figure = 128 square meters.

3). The area of the compound figure = 69 square centimeters.

4). The area of the compound figure,

= 53.12 square feet.

5). The area of the compound figure,

= 63 square meters.

6). The area of the compound figure,

= 108 square yards.

The area is the amount of area occupied by an object's flat (2-D) surface or shape.

Given:

The area of the compound shapes can be calculated by the following process,

1). The area of the compound figure,

= (12 x 8) + (1/2 x 3.14 x 4 x 4)

= 121.12 square feet.

2). The area of the compound figure,

= (13 x 8) + (1/2 x 8 x 6)

= 128 square meters.

3). The area of the compound figure,

= (11 x 4) + (5 x 5)

= 69 square centimeters.

4). The area of the compound figure,

= (1/2 x 3.14 x 4 x 4) + (1/2 x 8 x 7)

= 53.12 square feet.

5). The area of the compound figure,

= (7 x 7) + (1/2 x 7 x 4)

= 63 square meters.

6). The area of the compound figure,

= (12 x 6) + (6 x 6)

= 108 square yards.

Therefore, all the areas are given above.

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Answer:

7/12

Step-by-step explanation:

A fraction is a fragment of a whole number, used to define parts of a whole. The whole can be a whole object, or many different objects. The number at the top of the line is called the numerator, whereas the bottom is called the denominator.

An equation you can use to solve for this is:

Therefore, the difference of 8/12 - 1/12 is 7/12.

Answer:

Step-by-step explanation:

5/8

Answer: it equal 5 /8

The product of the fraction, 6 and 2/6 is greater.

A fraction is expressed as p/q, where q is equal to 0.

Proper fractions, in which the numerator is less than the denominator, and improper fractions, in which the numerator is higher than the denominator, are the two sorts of fractions.

Given, Are the product of two fractions, First one is 2 and 5/6 which is,

= (5/6)×2.

= 10/6.

= 5/3.

The second product is, 6 and 2/6 which is,

= 5×(2/6).

= 12/6.

= 2.

So, The product of 6 and 2/6 is greater.

Q. Which products are greater than 2 and 5\6 or 5 and 2/6?

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Answer: the answer is 48

Step-by-step explanation:

Answer: 48

Step-by-step explanation: A triangles formula is base x height divided by 2, therefore we will do first 8 x 12 which equals 96, and divide by 2, giving us 48.

The trigonometric ratios are presented in this answer.

The three trigonometric ratios are defined as follows:

The hypotenuse is the slanted segment connecting the horizontal segment and the vertical segment.

The adjacent side to an angle x is the side length between the angle x and the angle of 90º.

The opposite side to an angle x is the side length between the angle 90 - x and the angle of 90º.

The problem is incomplete, hence the important concepts of trigonometric ratios were presented.

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Answer: 16

Step-by-step explanation: Basic multiplication

4+4=8

4+4=8

8+8=16    [16!]

Answer: 29

Step-by-step explanation:

40 - 5 = 35

35 - 6 = 29

So, 29 + 6 = 5 less than 40

Answer: The number is 29 and the Sum is 35

Step-by-step explanation:

How you set up the equation 6+x = 40-5

Subtract 5 from 40 and you get 6+x = 35

Now subtract 6 from 35 to find your missing variable x = 35-6

x = 29

The final inequality  that represents how many blankets Elle can make in her workshop would be:

3x ≤ 52 and x ≥ 16

In mathematics, an inequality is a statement that compares two values or expressions using an inequality symbol such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).

An inequality indicates that one value or expression is smaller, greater, or not equal to the other value or expression.

 The inequality "y <= 10" states that the value of "y" is less than or equal to 10.

Let "x" be the number of blankets Elle can make in her workshop.

Since each blanket requires 3 ft. of material, the total amount of fleece required for "x" blankets would be 3x ft.

We know that Elle has 52 ft. of fleece in her workshop, so we can set up the inequality:

3x ≤ 52

This inequality states that the amount of fleece required for "x" blankets must be less than or equal to the amount of fleece Elle has available.

To satisfy the condition of making at least 16 blankets, we can add the requirement that "x" must be greater than or equal to 16.

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Answer:

To prove that AD is a bisector of BC, we need to show that AD divides BC into two congruent segments.

Since AB ≅ AC, we have ∠BAC ≅ ∠CAB by the isosceles triangle theorem.

Since BD ≅ CD, we have ∠CBD ≅ ∠CDB by the same reasoning.

Now, we can use the angle bisector theorem, which states that if a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides.

Applying the angle bisector theorem to triangle BCD with angle ∠BCD, we get:

BD/DC = AB/AC

Since AB ≅ AC and BD ≅ CD, we can substitute and simplify to get:

BD/CD = 1

This implies that BD ≅ CD.

Therefore, AD divides BC into two congruent segments, and AD is a bisector of BC.

Hope this helps!

m∠D = 40°, m∠F = 65°, and m∠E = 75°. Round to the nearest degree.

Angel is a geometric figure that can be described as the amount of rotation between two straight line or planes and angles is measured in degree with the full circle being 360°.

To determine the unknown angle measures of △DEF, we can use the fact that the sum of the angles of a triangle is 180°. Thus, if we know two angle measures in the triangle, we can use them to calculate the third.

Let's denote the two known angle measures as m∠D and m∠F. We can use the following equation to solve for the unknown angle measure:

m∠D + m∠F + m∠E = 180°

We can rearrange the equation to solve for m∠E:

m∠E = 180° - m∠D - m∠F

Therefore, once we know the measures of m∠D and m∠F, we can calculate the measure of m∠E.

For example, if m∠D = 40° and m∠F = 65°, then the measure of m∠E would be:

m∠E = 180° - 40° - 65° = 75°

Therefore, m∠D = 40°, m∠F = 65°, and m∠E = 75°. Round to the nearest degree.

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Answer: the answer is

m∠D = ✔ 46°

m∠F = ✔ 44°

Step-by-step explanation: I got it correct. it will be show below.

The tangent (tan) of an angle can be calculated using a calculator. The "tan" button on most scientific or graphing calculators can be used to calculate the tangent of an angle in degrees or radians.

Here's how to utilise a calculator's "tan" function:

To get the tangent of 45 degrees, for example, input "45" into the calculator and press the "tan" button. The calculator would then show the tangent value, which is around 1.00.

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95 kg is approximately equal to 209.4390 pounds ( rounded to four decimal places )

The conversion factor to convert kg to lbs is 2.20462

1 kg = 2.20462 pounds

The conversion is the process of changing the unit of one quantity to another units  

The conversion factor is defined as the number that is used to change one unit to another units by multiplying or dividing

The mass in lbs = The mass in kg × conversion factor

Substitute the values in the equation

1 kg = 2.20462 pounds

95 kg = 95 × 2.20462

Multiply the numbers

= 209.4390 pounds  ( rounded to four decimal places )

Therefore, 95 kg is 209.4390 pounds

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The true statements are: The Chinese Remainder Theorem guarantees a unique solution to any system of modular equations.

Fermat's Little Theorem allows us to conclude that ap^−1≡1(mod p) for any integer a and any prime p.

The first formula given in the question for computing x using CRT is also true.

Statement 3, "If b≥p, then a^b−p+1≡a^b (mod p)", is not necessarily true. Fermat's Little Theorem states that if p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). However, this does not extend to arbitrary exponents b.

Statement 4, "If b ≡ c (mod p), then a^b≡a^c (mod p)" is true by Fermat's Little Theorem, which implies that a^(p-1) ≡ 1 (mod p), we can write b = kp + c for some integer k, then

a^b = a^(kp+c) = (a^p)^k * a^c ≡ a^c (mod p)

since a^p ≡ a^(p-1) * a ≡ 1 * a ≡ a (mod p) by Fermat's Little Theorem.

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Does the graph represent a function: B. No.

In Mathematics, a function can be defined as a mathematical expression which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair.

This ultimately implies that, a function is typically used in mathematics for uniquely mapping an input variable to an output variable.

Based on the ordered pairs (-5, 7) and (-2, 7), we can reasonably infer and logically deduce that the relation represented by this graph does not represent a function because the are not uniquely mapped.

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Answer:

a, d=90 degre

and b, b=90degree

Answer:

Step 1: Given AD || BC and AB || CD.

Step 2: Angle A and Angle C are alternate interior angles, so they are congruent.

Step 3: Angle B and Angle D are alternate interior angles, so they are congruent.

Step 4: Angle B + Angle D = 180 degrees (because Angle A + Angle B + Angle C + Angle D = 360 degrees for a quadrilateral)

Step 5: Since Angle B and Angle D are congruent and their sum is 180 degrees, they must each be 90 degrees.

Step 6: Triangle ABZ and Triangle CDZ are both right triangles, since Angle B and Angle D are each 90 degrees.

Step 7: By the Angle-Angle Similarity Theorem, triangles ABZ and CDZ are similar.

Step 8: Since triangles ABZ and CDZ are similar, corresponding sides are proportional.

Step 9: Therefore, ZB/ZD = AB/CD, since ZB corresponds to AB and ZD corresponds to CD.

Step 10: Given that AB || CD, we know that AB/CD = AZ/CZ (the corresponding sides of similar triangles are proportional).

Step 11: Therefore, ZB/ZD = AB/CD = AZ/CZ.

Step 12: Since Angle A and Angle C are congruent (Step 2), we know that AZ = CZ (since they are opposite sides of an isosceles triangle).

Step 13: Therefore, ZB/ZD = AZ/CZ = 1, which means that ZB = ZD.

Thus, we have proved that ZB = ZD!

- Dante

The function family based on the information is a linear function.

A linear function is a type of mathematical function that has the form:

f(x) = ax + b

where a and b are constants, and x is a variable. The variable x is often called the independent variable, and the variable f(x) is called the dependent variable.

The value of a determines the slope of the line, while the value of b determines the intercept of the line with the y-axis. The slope of the line represents the rate of change of the dependent variable with respect to the independent variable, and the intercept represents the value of the dependent variable when the independent variable is zero.

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This inequality is true, the given statement "The point (2,5) is inside the region defined by |a - 3| + |b - 4| < 5" is true. Therefore, the answer is True.

To determine whether the given statement is true or false, we need to substitute the values of a and b into the inequality and simplify:

|a - 3| + |b - 4| < 5

Substituting a = 2 and b = 5, we get:

|2 - 3| + |5 - 4| < 5

|-1| + |1| < 5

1 + 1 < 5

2 < 5

Since this inequality is true, the given statement "The point (2,5) is inside the region defined by |a - 3| + |b - 4| < 5" is true. Therefore, the answer is True.

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As per the information provided, a = 4√3/3 + 4, b = 4 - 4√3/3 the answer can be calculated with optimization method. it will be as follows:

Sum of a and b is 8, we get

a+b=8

b=8−a

Now, let x=a−b

Then we get,

[tex]x=a−(8−a) \\ x=2a−8 \\ x+8=2 \\ 1 \div 2x+4=a

[/tex]

we use this to answer to solve for b

[tex]b=8−a \\ =8−(1 \div 2x+4) \\ =4−1 \div 2x

[/tex]

Now we use the product of two numbers and its difference. This can be expressed as:

[tex]a⋅b⋅x=(1 \div 2x+4)(4−1 \div 2x) \\ x=2 {x}^{2} − \frac{1}{4} {x}^{3} +16x−2x^{2} \\ =−14x^{3} +16x

[/tex]

Thus, this expression that we need to maximize. Take the derivative, set it equal to zero, and solve for x

[tex]−3 \div 4x ^{2} +16=0 \\ 16 =3 \div 4 x ^{2} \\ 643=x \\ 28√3 \div 3=x[/tex]

Now for us to check that this is a maximum, we have to note that the second derivative is

[tex]−3 \div 2x

\\ At \\

x=8√3 \div 3

[/tex]

the second derivative is −4√3. Since this number is negative, the point is a maximum.

Now we must find the values of a and b for this x. We have to use the relationship

[tex]a=1 \div 2x+4[/tex]

[tex]a=1 \div 2 \times 8√3 \div 3+4 \\ =4√3 \div 3+4[/tex]

now we use the relationship b=8−a

[tex]b=8−(4√3 \div 3+4) \\ =4−4√3 \div 3[/tex]

The first step in determining a function's maximum or minimum value is differentiating it. Then, set this derivative to zero and conduct the computation.

x. This will reveal the location of a function's maximum or minimum, but it won't reveal which.

Take the second derivative to get more details. A local maximum occurs when both the first and second derivatives are negative. You have a local minimum when both the first derivative and the second derivative are zero.

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