question 2 theorem: if r and s are rational numbers, then the product of r and s is a rational number. which facts are assumed in a direct proof of the theorem?

In response to the stated question, we may state that As a result, the Pythagorean theorem angle of depression from the helicopter to Landing Pad B is around 16.7 degrees.

The Pythagorean theorem is a fundamental mathematical principle that describes the connection between the sides of a right triangle. It asserts that the sum of the squares of the other two sides' lengths equals the square of the hypotenuse's length (the side opposite the right angle). The mathematical theorem is as follows: c2 = a2 + b2 Where "c" denotes the hypotenuse length and "a" and "b" indicate the lengths of the other two sides, known as the legs.  

We can calculate the angle of depression from the helicopter to Landing Pad B using trigonometry.

Let we make a diagram:

            B

              /|

             / |

            /  |  8 miles

        14 /   |

          /θ  |

         /    |

        /     |

       /______|______ A

             20 miles

Using trigonometry, we can deduce:

tan(θ) = opposite / adjacent

The opposing side in this example is 14 - 8 = 6 miles (since the helicopter is 14 miles from Landing Pad A and 8 miles from Landing Pad B, and the pads are 20 miles apart). The distance between the two sides is 20 miles (the distance between the landing pads).

tan(θ) = 6 / 20

tan(θ) = 0.3

θ = arctan(0.3)

θ ≈ 16.7 degrees

As a result, the angle of depression from the helicopter to Landing Pad B is around 16.7 degrees.

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The probability of randomly drawing a participant with a score of 34 or less is 0.2088 or approximately 20.88%.

The probability of randomly drawing a participant with a score of 34 or less from a normal distribution with a mean of 37 and a standard deviation of 3.7 can be calculated as follows:

First, we must convert the score of 34 or less to a z-score:z = (X - μ) / σ = (34 - 37) / 3.7 = -0.81. The probability of obtaining a score of 34 or less can be determined using the standard normal distribution table or a calculator that has the capability to compute probabilities of the standard normal distribution.Using the standard normal distribution table, the area to the left of -0.81 is 0.2088. Therefore, the probability of randomly drawing a participant with a score of 34 or less is 0.2088 or approximately 20.88%.

The standard deviation (SD) is a measure of the degree of variability or deviation of a distribution's scores from the mean or average value of the distribution. The standard deviation is expressed in the same units as the original distribution's scores, whether they are income, test scores, or some other measure of performance. It can be considered a statistic that quantifies the variability of data values, and the deviation is the amount by which a value differs from the mean or average value.

The probability of obtaining a specific score or range of scores from a distribution can be computed using the mean and standard deviation of that distribution by converting the score(s) to a z-score(s) and looking up the appropriate area on a standard normal distribution table or using a calculator that has the capability to compute probabilities of the standard normal distribution.

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The length for which the volume of Box A equals the volume of Box B, obtained by resolving the cubic equation is 4 inches. The correct option is therefore;

A cubic equation is an equation of the form; a·x³ + b·x² + c·x + d, where a is a non zero number.

The x represent the width of Box A, we get;

The length of Box A + 4 = x

The length of Box A = x - 4

Height of Box A = x - 4 - 1 = x - 5

Volume of Box A = x·(x - 4)·(x - 5) = x³ - 9·x² + 20·x

Length of Box B = 2 × (x - 4)

Width of Box B = (2 × (x - 4))/2 - 2 = x - 4 - 2 = x - 6

Height of Box B = (2 × (x - 4))/2 + 2 = x - 4 + 2 = x - 4

The volume of Box B = 2 × (x - 4) × (x - 6) × (x - 4) = 2·x³ - 28·x² + 128·x - 192

The volume of Box A is equal to the volume of Box B when we get the following cubic equation;

x³ - 9·x² + 20·x = 2·x³ - 28·x² + 128·x - 192

The above cubic equation can be factored using an online tool to get;

(x - 4)·(x² - 15·x + 48) = 0

Therefore; (x - 4) = 0, and x² - 15·x + 48 = 0

x = 4, x = (15 + √(33))/2, and x = (15 - √(33))/2

Therefore, one of the lengths for which the volume of Box A is the same as Box B ; x = 4 inches. The correct option is option A.

A. x = 4 inches

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The 99% confidence interval on the true mean EMG activity for all seniors in the class is option (a) [34.7296, 36.8704]

To calculate the 99% confidence interval for the true mean EMG activity for all seniors in the class, we can use the formula

CI = X ± Zα/2 (σ/√n)

where

X = sample mean = 35.8

Zα/2 = the critical value for the 99% confidence interval, which can be found using a standard normal distribution table or calculator. For a two-tailed test, α/2 = 0.005, so Zα/2 = 2.576.

σ = sample standard deviation = 2.5

n = sample size = 40

Substituting these values into the formula, we get

CI = 35.8 ± 2.576 (2.5/√40) = [34.7296, 36.8704].

Therefore, the correct option is (a) [34.7296, 36.8704]

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The total number of t-shirts principal ordered in $1520 for the school store is given 214.

Let us consider the number of t-shirts the principal ordered be equal to 'x'.

The cost of each t-shirt is equal to $7.

⇒ The total cost of 'x' t-shirts =  7x dollars.

Shipping cost is a flat rate for any order, regardless of the number of t-shirts =  $22

Total cost of the order (including shipping) =  $1520.

Required equation is equal to,

7x + 22 = 1520

Subtract 22 from both sides of the equation we get,

⇒ 7x = 1498

Divide both sides of the equation by 7 we get,

⇒ x = 214

Therefore, the number of t-shirts ordered by  principal for the school store is equal to  214.

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Answer:

To find the explicit formula for the sequence, we need to determine the common ratio between each term.

The common ratio is found by dividing any term by the previous term. For example:

The common ratio between the second and first terms is 150/30 = 5.

The common ratio between the third and second terms is 750/150 = 5.

Since the common ratio is the same for all terms, we can use it to find any term in the sequence.

Let's call the first term a₁, and let r be the common ratio. Then we have:

a₁ = 30

a₂ = a₁ * r = 30 * 5 = 150

a₃ = a₂ * r = 150 * 5 = 750

a₄ = a₃ * r = 750 * 5 = 3750

a₅ = a₄ * r = 3750 * 5 = 18750

We can see that the explicit formula for the sequence is:

aₙ = a₁ * r^(n-1) = 30 * 5^(n-1)

Therefore, the explicit formula for the sequence is aₙ = 30 * 5^(n-1).

The length of the minute hand of the clock is given as follows:

10.6 inches.

Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

The ordered pairs for this problem are given as follows:

(0,0) and (7.5, 7.5).

Hence the distance between the two points, representing the length of the minute hand, is given as follows:

d = sqrt(7.5² + 7.5²)

d = 10.6 inches.

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Using the distance formula of coordinate geometry the expression represents the distance between point J and point K where J= (6, -2) and K= (6, -9) is  7 units.

To find the distance between point J and point K, we use the distance formula, which involves the coordinates of the two points in a coordinate plane.

Identify the coordinates of point J and point K. In this case, we are given that J has coordinates (6, -2) and K has coordinates (6, -9).

Substitute the values of the coordinates into the distance formula, which is d = √[(x2 - x1)² + (y2 - y1)²].

For x1 and y1, use the coordinates of point J, which are x1 = 6 and y1 = -2.

For x2 and y2, use the coordinates of point K, which are x2 = 6 and y2 = -9.

Simplify the formula by substituting the values and solving:

d = √[(6 - 6)² + (-9 - (-2))²]

d = √[0² + (-7)²]

d = √[49]

d = 7

Therefore, the distance between point J and point K is 7 units.

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The median of the numbers 8.5, 1.2, 7.6, 3.8, 8.1, 9.4, 8.9, 0.7, 2.7  is 7.6.

The median of numbers is the value that's exactly in the middle of a dataset when it is ordered.

Therefore, before we find the middle value of the dataset, we have to arrange the numbers in ascending or descending order.

Hence, let's find the median of 8.5, 1.2, 7.6, 3.8, 8.1, 9.4, 8.9, 0.7, 2.7

Firstly, let arrange the numbers in ascending order.

Hence,

0.7, 1.2, 2.7, 3.8, 7.6, 8.1, 8.5, 8.9, 9.4

Therefore,

median = 7.6

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a) There are 1,000 small cubes in the large one.

b) if each side has 3 times more cubes, then the total number of cubes will be 27 times larger.

So we know that the large cube has 10 small cubes in each side.

Then each one of the faces of the cube will have:

10*10 = 100 small cubes on it.

Now, if we look at one of the faces, you can see 10 of these 100 square arrays on it, then the total number of small cubes is:

10*100 = 1000 small cubes.

b) If there are 3 times as many cubes per side, then each side has 30 cubes now, and the total number of cubes needed will be:

30*30*30 = 27,000

This is 27 times the amount of small cubes that we had before,  then here the answer is no.

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Answer:

The slope is "m"

m= -5

Step-by-step explanation:

x-- Axis interception points of x- 2x - 4x +8 ( 8 over 5, 0)

y-- Axis interception points of x - 2x - 4x +8 (8,0)

x-2x-4x+8

y=-5x+8

it will take approximately 22.56 years for the amount to triple.

The given information for this problem is that there is an initial investment of $600 in an account with an annual interest rate of 4%. The task is to determine after how many years the amount will triple.Using the compound interest formula, we can find the amount in the account after t years:A = P(1 + r/n)nt Where,A = final amount in the account, P = initial amount in the account r = annual interest rate ,n = number of times the interest is compounded per year ,t = time in years.

From the problem statement, we know that the initial amount, P, is $600 and the annual interest rate, r, is 4%. Let's assume that the interest is compounded annually, i.e., n = 1.Substituting these values in the formula, we get:A = $600(1 + 0.04/1)1t Simplifying this expression,A = $600(1.04)t.

Taking the ratio of the final amount to the initial amount, we get: 3P = $600 × 3 = $1800. Therefore,A/P = 3 = (1.04)t.Dividing both sides by P, we get:3 = (1.04)t ln(3) = ln(1.04)t. Using the logarithmic property, we can bring down the exponent to the front:ln(3) / ln(1.04) = t Using a calculator, we get ≈ 22.56. Therefore, it will take approximately 22.56 years for the amount to triple.

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The ratio of QR to ST in the triangle is 5/7

Three angles and the three sides of a triangle are equal to the corresponding angles and the corresponding sides of another triangle, then both the triangles defined as congruent.

To establish that two triangles are congruent, not all six matching elements of either triangle must be located. There are five condition for two triangles to be congruence,  according to  the trials. The congruence qualities are SSS, SAS, ASA, AAS, and RHS.

PQR and PST are congruent

So PQ/PS=QR/ST

We know that

PQ=10

PS=14

QR/ST=10/14

QR/ST=5/7

hence, the ratio of QR to ST in the triangle is 5/7.

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As per the surface area, the height of the cylindrical gallon tub of ice cream is approximately 2.625 inches. (option d).

In this problem, we have been given the diameter of the tub, which is 8 inches. The radius of the tub can be calculated by dividing the diameter by 2, which gives us a radius of 4 inches.

We have also been given the total surface area of the tub, which is 232.36 square inches. Using the formula for surface area, we can write:

232.36 = 2π(4)² + 2π(4)h

232.36 = 100.48 + 25.12πh

Subtracting 100.48 from both sides, we get:

131.88 = 25.12πh

Dividing both sides by 25.12π, we get:

h = 131.88 / (25.12π)

Using the value of π as 3.14, we can simplify this expression to:

h = 131.88 / 79.104

h ≈ 2.6256 inches

Hence the correct option is (d).

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a. The equation to represent the situation is y = -12x + 120, where x is the number of movies and y is the amount of money remaining on the gift card.

Money is a medium of exchange that is widely accepted as a way to pay for goods and services or to settle debts. Money also serves as a store of value, providing a way for people to save for the future. Money is generally created through government-backed fiat currencies, such as the U.S. dollar, which are issued and regulated by central banks. Money can also be created in the form of crypto-currencies, such as Bitcoin, which are not issued by any single government or central bank. Money is essential for economic growth and stability, as it allows for efficient exchanges of goods and services. Money can also be a source of financial security, providing people with a way to manage their finances and plan for the future.

b. The slope of -12 indicates that for every movie that Elyse sees, she will spend $12 from her gift card. The y-intercept of 120 indicates that if Elyse has not seen any movies, she will have $120 remaining on her gift card.

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The situation is,

a) The line equation is [tex]y = -3x - 6[/tex]

b) The line's y-intercept is -6, which indicates that when the amount of movies x = 0 , the amount on gift y = -6

c) The slope of the line is -3, indicating that as the number of movies x increases, the rate of change of the amount on the gift is declining.

a). The equation provides the line's slope.

Slope,

[tex]m=\frac{(y_2-y_1)}{(x_2-x_1)}[/tex]

Changing the numbers indicated in the slope equation,

Slope,

[tex]m=\frac{(6-12)}{(4-2)}[/tex]

Slope m = -6/2

Slope m = -3

The slope is -3

The equation of the line is,

[tex]y - y_1 = m ( x - x_1 )[/tex]

Substitute the given values in the equation,

[tex]y - 12 = -3 ( x - 2 )[/tex]

Simplify the equation,

[tex]y - 12 = -3x + 6[/tex]

Adding 12 on both sides

[tex]y = -3x - 6[/tex]

The equation of line is [tex]y = -3x - 6[/tex]

b). The y-intercept of the equation of line [tex]y = -3x - 6[/tex] is [tex]-6[/tex], when [tex]x=0[/tex]

c). The slope of the line [tex]y = -3x - 6[/tex] is [tex]m = -3[/tex] and the value is decreasing

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The completr question and graph attached below,

c. What is the slope, and what does it mean in the context of the situation?

Men will have completed oil changes in hours Therefore, Will and Gabriel will each have done 12 oil changes after 4 hours.

Hours is a unit of time measurement. It is used to measure a specific amount of time and is usually denoted by the symbol “h”. There are 24 hours in a day, 60 minutes in an hour and 60 seconds in a minute. Hours are used to measure both short and long periods of time. Commonly, hours are used to measure the length of a workday, the length of a school day, or the length of a movie. Hours are also used to measure how long a person has been alive, how long an event has been going on, or how long an item has been in use.

Let W be the total number of oil changes Will has completed, and G be the total number of oil changes Gabriel has completed.

System of equations:

W = 8 + 2t

G = 3t

Since they will be tied at some point during the day, W = G.

Substituting W into G's equation:

8 + 2t = 3t

Solving for t:

2t = 8

t = 4

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The solution to the system of equations is (x, y) = (2, 1).

A system of equations is a set of two or more equations that are to be solved simultaneously, meaning that the values of the variables that satisfy each equation in the system must be found. The solution to a system of equations is the set of values for the variables that satisfy all the equations in the system.

To solve the system of equations:

8x - 5y = 11 ...(1)

4x - 3y = 5 ...(2)

We can use the elimination method to eliminate one of the variables. We want to eliminate the variable "y", so we need to multiply equation (2) by -5/3, which will give us:

-5/3(4x - 3y) = -5/3(5)

-20x/3 + 5y = -25/3 ...(3)

Now we can add equations (1) and (3) to eliminate "y":

8x - 5y + (-20x/3 + 5y) = 11 - 25/3

Combining like terms, we get:

(24x - 15y - 20x + 15y)/3 = 8/3

Simplifying, we get:

4x/3 = 8/3

Multiplying both sides by 3, we get:

4x = 8

Dividing both sides by 4, we get:

x = 2

Now we can substitute x = 2 into equation (1) or (2) to find y. Let's use equation (1):

8x - 5y = 11

8(2) - 5y = 11

16 - 5y = 11

Subtracting 16 from both sides, we get:

-5y = -5

Dividing both sides by -5, we get:

y = 1

Therefore, the solution to the system of equations is (x, y) = (2, 1).

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Complete question:

Given the system of equation:

8x - 5y = 11

4x - 3y = 5

Find the value of x and y.

The required medians of the triangle intersect at the point [tex]\left(4,\frac{10}{3}\right)$[/tex].

The medians of a triangle intersect at a point known as the centroid. To find the centroid of a triangle, we need to find the average of the x-coordinates and the average of the y-coordinates of its vertices.

Let the vertices of the triangle be A(0,0), B(5,0), and C(7,5). Then the midpoint of AB is

[tex]\left(\frac{0+5}{2},\frac{0+0}{2}\right) = (2.5,0)$[/tex],

the midpoint of BC is [tex]\left(\frac{5+7}{2},\frac{0+5}{2}\right) = (6,2.5)$[/tex], and the midpoint of CA is

[tex]\left(\frac{0+7}{2},\frac{0+5}{2}\right) = (3.5,2.5)$[/tex]

Therefore, the centroid of the triangle is:

[tex]$\begin{align*}\left(\frac{0+5+7}{3},\frac{0+5+5}{3}\right) &= \left(\frac{12}{3},\frac{10}{3}\right) \&= \left(4,\frac{10}{3}\right)\end{align*}[/tex]

So the medians of the triangle intersect at the point [tex]\left(4,\frac{10}{3}\right)$[/tex].

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The angle that satisfies the necessary requirements is  1041° - 2(360°) = 321°. Coterminal angles are those with the same terminal point in the standard position.

To understand, follow the 2 basic steps:

Step 1: Find the specified angle.

Step 2: Finding a coterminal angle.

Tally up or tally down a multiple of 360.

θ = 1041°

Every multiple of 360 degrees added or subtracted results in an angle that is coterminal with the provided angle.

Hence, 1041° - 2(360°) = 321° is an angle that satisfies the necessary requirements.

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The chance that we picked 1 red and 1 blue is 0.2448 or 24.48%.

When we choose two balls at random without replacement, there are 50C2 = (50*49)/(2*1) = 1225 ways to choose two balls from 50 balls. Using the conditional probability formula, P(A and B) = P(A) * P(B|A), where A is the event of choosing a red ball and B is the event of choosing a blue ball.

P(A) = (20/50) = 0.4 (Probability of choosing a red ball at random from the bag)

P(B|A) = (15/49) = 0.306 (Probability of choosing a blue ball at random from the bag after choosing a red ball in the first trial)

Similarly, P(B) = (15/50) = 0.3 (Probability of choosing a blue ball at random from the bag)

P(A|B) = (20/49) = 0.408 (Probability of choosing a red ball at random from the bag after choosing a blue ball in the first trial)

Therefore, P(choosing one red and one blue) = P(A and B) + P(B and A) = P(A) * P(B|A) + P(B) * P(A|B)

= 0.4 * 0.306 + 0.3 * 0.408

= 0.1224 + 0.1224

= 0.2448 or 24.48%

Therefore, the chance that we picked 1 red and 1 blue is 0.2448 or 24.48%.

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An algebraic expression for representing the Nora's income in form of Tonya's income is given by  y = ( x / 4 ) .

Let us consider 'x' represents the Tonya's income.

And variable 'y' represents the Nora's income.

Tonya income is equal to four times of Nora's income.

This implies,

Nora's income is equal to one fourth times of Tonya's income.

⇒ y = ( x / 4 )

Rewrite an algebraic expression to represents Tonya's income in terms of Nora's income we have,

Simplify by multiplying both the sides of the algebraic expression by 4 we get,

⇒ x  = 4y

Therefore, an algebraic expression to represents the Nora's income in terms of Tonya's income is equal to y = ( x / 4 ).

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During bracket off, when 6x⁴ * 6x⁴= 36x⁸ instead of 36x¹⁶ using Perfect square method.

When 4y² * 4y² = 16y^4 instead of 25y⁴⁴ gotten in the question. To correct the error, using the perfect square method, it can be corrected.

The answer will be (6x⁸ - 5y²)(6x⁸ + 5y²)The last error found is sign rule-4y^2  x -4y^2 = +16y⁴ instead of -4y² * +4y² = -16y⁴

Perfect square is  is a number that is generated by multiplying two equal integers by each other.

Given 36x¹⁶ - 25y⁴

The rule = A difference of two perfect squares, A² - B²²  can be factored into (A+B) x (A-B)36x¹⁶ is a perfect square = 6₂ * x⁸2)= (6x⁸)^225y⁴ is also a perfect square= 5² * x²(2)= (5x²)²

Using the rule for Perfect square,

When A² - B² = (A + B) x (A - B)

So,(6x⁸)₂ - (5x²)²

=(6x⁸ + 5x²) * (6x⁸ - 5x²)

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De acuerdo con la información, podemos inferir que el volumen total de la figura sería 1,476.58 cm³

To find the volume of the figure we must divide it into different sections because it has cones, pyramids, hemispheres, cylinders, and rectangular cubes. Below is the procedure:

Volume of rectangular segments:

Volume of the pyramids:

First we must calculate the height of the pyramids.

Cone volume:

First we need to calculate the height of the cone.

Cylinder volume:

Volume of the hemisphere:

Finally we just have to add all the values and we will obtain the total volume of the figure:

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The error made by the student is in simplifying the expression inside the square root.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The student made an error in simplifying the expression inside the square root.

The expression should be simplified as follows:

25 + 4(2)(-12) = 25 - 96 = -71

So the correct expression inside the square root is -71, not 121.

Therefore, the correct solution using the Quadratic Formula is:

x = (-(-5) ± √((-5)² - 4(2)(-12)))/(2(2))

x = (5 ± √(25 + 96))/4

x = (5 ± √121)/4

x = (5 ± 11)/4

Hence, the solutions are x = -3 and x = 4/2, which simplifies to x = 2.

Therefore, the error made by the student is in simplifying the expression inside the square root.

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Answer: 1.5

Step-by-step explanation:

Answer:

5a² + 4ab = a(5a+4b)

6p - pq - 5pr = p(6-q-5r)

8st +9su + 7sv = s(8t+9u+7v)

8pq² +9qr+ 7pqr = q(8p²+7pr+9r)

2a²b + ab² - 6ab = ab(2a+b-6)

6t²u +8tu²y + 5tuw = tu(6t+8uy+5w)

18abc²-42a²bc + 18ab²c = 6abc(3c-7a+3b)

24pqrs +56q²rs - 40qrs = 8qrs(3p+7q-5)

The heights of the tips of the propeller blades at π/6, 5π/6, and 3π/2 radians are 15.5 feet, 11.5 feet, and 10 feet, respectively.

What is the equation of a circle in parametric form?

The equation of a circle in parametric form is given by x=acosθ,y=asinθ.

We can use the equation for a circle in standard form:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

In this case, the center of the propeller is (0, 13) and the radius is 3. So the equation of the circle is:

x² + (y - 13)² = 9

We can use this equation to find the x and y coordinates of the tip of each propeller.

For the blade at π/6 radians, we can use the parametric equations for a circle:

x = r cos(t) + h

y = r sin(t) + k

where t is the angle in radians.

Substituting r = 3, h = 0, k = 13, and t = π/6, we get:

x = 3 cos(π/6) + 0 = 3√3/2

y = 3 sin(π/6) + 13 = 13 + 3/2 = 15.5

So the height of the tip of the propeller blade at π/6 radians is 15.5 feet.

Using the same method for the blades at 5π/6 and 3π/2 radians, we get:

Blade at 5π/6 radians:

x = 3 cos(5π/6) + 0 = -3√3/2

y = 3 sin(5π/6) + 13 = 13 - 3/2 = 11.5

Height of tip: 11.5 feet

Blade at 3π/2 radians:

x = 3 cos(3π/2) + 0 = 0

y = 3 sin(3π/2) + 13 = 10

Height of tip: 10 feet

Therefore, the heights of the tips of the propeller blades at π/6, 5π/6, and 3π/2 radians are 15.5 feet, 11.5 feet, and 10 feet, respectively.

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Answer:

Step-by-step explanation:

If we use the substitution u = 4x - 3, then du = 4 dx. Solving for dx, we get dx = du/4. Also, we can solve for x in terms of u: x = (u + 3)/4.

Substituting these values into the integral, we get:

∫ x (4x - 3)^10 dx = ∫ [(u + 3)/4] * u^10 * (du/4) = (1/16) ∫ (u + 3) * u^10 du = (1/16) ∫ (u^11 + 3u^10) du = (1/16) [(u^12)/12 + (3u^11)/11] + C = (1/192) * u^12 + (3/176) * u^11 + C

Substituting back for u = 4x - 3, we get:

(1/192) * (4x - 3)^12 + (3/176) * (4x - 3)^11 + C

So the integral of x(4x - 3)^10 dx is equivalent to (1/192) * (4x - 3)^12 + (3/176) * (4x - 3)^11 + C.

The integral of the function ∫x ( 4x - 3 )¹⁰ is given by A=( 1/16 ) ∫(u¹¹ + 3u¹⁰ )du

The mathematical procedure known as the integral of a function depicts the accumulation of a quantity over a period of time or the region beneath a curve. It is used in calculus and is represented by the symbol ∫

In math, the integral of a function f(x) with respect to x is written as f(x) dx, where f(x) is the function being integrated and dx is the change in the variable x that is infinitesimally small. The antiderivative or integral of the function is the outcome of the integration.

Given data ,

Let the value of the function be f ( x ) = ∫x ( 4x - 3 )¹⁰

On substituting the value of u = 4x - 3 , we get

du/dx = 4

So , On replacing x with (u + 3)/4, and dx with du/4 in the integral:

∫x(4x - 3)¹⁰ dx = ∫((u + 3)/4)(4(u - 3))¹⁰ du

On further simplifying , we get

∫((u + 3)/4)(4(u - 3))¹⁰ du = ∫(1/4)(4(u + 3))(u - 3)¹⁰ du

Hence , the integral is  A = ( 1/16 ) ∫(u¹¹ + 3u¹⁰ )du

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The complete question is attached below :

Using the substitution u = 4x - 3, the integral of x ( 4x - 3 ) ^ 10 dx is equivalent to which of the following?

Answer:

To solve this problem, we can use trigonometry and differentiation. Let θ be the angle of elevation formed by the lines from the top and bottom of the building to the tip of the shadow. Then, we have:

tan(θ) = height of building / length of shadow

Differentiating both sides with respect to x, we get:

sec^2(θ) dθ/dx = (-1 / length of shadow^2) (d length of shadow / dx) height of building

Substituting the given values, we get:

sec^2(θ) dθ/dx = (-1 / x^2) (d x / dx) 235

At x = 286, we have:

length of shadow = x + 235 tan(θ)

Differentiating this expression with respect to x, we get:

d length of shadow / dx = 1 + 235 sec^2(θ) dθ/dx

Substituting this into the previous equation and simplifying, we get:

dθ/dx = - x^2 / (235 (x + 235 tan(θ)))

At x = 286, we have:

length of shadow = 286 + 235 tan(θ)

tan(θ) = height of building / length of shadow = 235 / (286 + 235 tan(θ))

Solving for tan(θ), we get:

tan(θ) = 235 / (286 + 235 tan(θ))

tan(θ) (286 + 235 tan(θ)) = 235

235 tan^2(θ) + 286 tan(θ) - 235 = 0

Using the quadratic formula, we get:

tan(θ) = 0.470835 or -1.00084

Since the angle of elevation is positive, we take:

tan(θ) = 0.470835

Substituting this into the expression for dθ/dx, we get:

dθ/dx = - 286^2 / (235 (286 + 235 (0.470835)))

Simplifying this expression, we get:

dθ/dx ≈ -0.00074675 radians per foot (rounded to five decimal places)

Therefore, the rate of change of the angle of elevation at x = 286 feet is approximately -0.00074675 radians per foot.

Answer:

Choice B:  distance from home to library is length of HL

HL = 75      see calculations below

Step-by-step explanation:

use points (0,0) and (2.1, 7.2) to find HL:

HL = √(2.1 - 0)² + (7.2 - 0)² = √4.4 + 51.8 = √56.25 = 7.5