for what mileages will company a charge less than company b? use for the number of miles driven, and solve your inequality for .

The probability that the waves are turbulent given that the water is cold is approximately 0.112 (rounded to the nearest millionth).

Given that the water is cold, the probability that the waves are turbulent needs to be found. Let's apply the Bayes' Theorem to find the probability of the waves being turbulent given that the water is cold.

The formula for Bayes' Theorem is as follows:

P(B|A)={P(A|B)\ P(B)} * {P(A)}

Where, P(B|A) is the probability of event B given that event A has occurred, P(A|B) is the probability of event A given that event B has occurred, P(A) is the probability of event A, and P(B) is the probability of event B. The question requires us to find the probability of the waves being turbulent given that the water is cold. So, the conditional probability we need to find is P(T|C), where T represents the event of waves being turbulent and C represents the event of the water being cold. The given probabilities are:

P(T)=0.36

P(T^C)=0.64

P(C|T)=0.21

P(C|T^C)=0.51

By the complement rule, P(C)=1-P(C^C)

P(C)=1-0.15=0.85

Let's substitute the given probabilities into the Bayes' Theorem formula and solve for

P(T|C) :

(T|C)={P(C|T)\ P(T)} * {P(C|T)\P(T)+P(C|T^C)\ P(T^C)}={0.21\ 0.36} * {0.21\0.36+0.51\ 0.64} = 0.112

Therefore, the probability that the waves are turbulent given that the water is cold is approximately 0.112 (rounded to the nearest millionth).

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Thus, the value of y for the given set of parallel lines is found as:  y = (115)⁰.

Co-interior angles are located on the same sides of a transversal and between two lines. The two defined angles in each diagram are referred to as co-interior angles.

Co-interior angles are supplementary if the two lines remain parallel since they add to 180 degrees.

Given data:

AB || CD

EF || GH

(3x - 4) and (5x) are the linear co-interior angles.

Thus,

(3x - 4) + (5x) = 180 (supplementary angles).

8x - 4 = 180

8x = 184

x = 23

Then,

(5x)⁰ = (5*23)⁰ = (115)⁰

Now,

(5x)⁰ = y = (115)⁰ (corresponding angles)

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Answer:

x = 4

Step-by-step explanation:

if a line is parallel to a side of a triangle and it intersects the other 2 sides, then it divides those sides proportionally.

line EF is parallel to side CD of the triangle and intersects the other 2 sides , then

[tex]\frac{CE}{EG}[/tex] = [tex]\frac{DF}{FG}[/tex] ( substitute values )

[tex]\frac{6}{x+8}[/tex] = [tex]\frac{5}{2x+2}[/tex] ( cross- multiply )

6(2x + 2) = 5(x + 8) ← distribute parenthesis

12x + 12 = 5x + 40 ( subtract 5x from both sides )

7x + 12 = 40 ( subtract 12 from both sides )

7x = 28 ( divide both sides by 7 )

x = 4

Plates that slip and slide past each other form a transform boundary.

At a divergent boundary, two tectonic plates move away from each other. In the case of a mid-ocean ridge, the boundary occurs between two oceanic plates. As the plates move apart, magma from the mantle beneath rises up to fill the gap created by the separation. This magma cools and solidifies, adding new material to the ocean floor and creating a continuous chain of underwater mountains, known as a mid-ocean ridge.

Mountains often form at convergent boundaries because the plates are pushed together, causing the crust to fold and buckle, leading to the formation of mountain ranges.

Evidence that would help determine the type of plate boundary that exists between tectonic plates on Earth includes the topography of the surrounding areas, the presence of volcanoes and earthquakes, and the direction and speed of plate motion.

When the mantle rock is heated, the particles begin to move faster and spread apart, so its density decreases. This change in density causes the hot mantle rock to rise. As the mantle rock moves farther away from the core towards the crust, it begins to cool. The density of the mantle rock will increase because it is cooler. This causes the mantle rock to sink. The mantle rock moves back towards the core and the cycle restarts.

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Image transcribed:

Plates that slip slide past each other form a __________ boundary.

10. Explain why a mid-ocean ridge can form at a divergent boundary.

11. Why do mountains often form at convergent boundaries?

12. What evidence would help you determine the type of plate boundary that exists between tectonic plates on Earth?

13. Use these words to complete the paragraph: decreases, rise, density, crust, cool, sink, increase.

When the mantle rock is heated, the particles begin to move faster and spread apart, so its density ________. This change in density causes the hot mantle rock to ________. As the mantle rock moves farther away from the core towards the _______, it begins to _________. The _______ of the mantle rock will _________ because it is cooler. This causes the mantle rock to ______. The mantle rock moves back towards the core and the cycle restarts.

14. Complete the grid.

Boundary Type | Describe the Plate Motion | What Happens on the Surface of Earth He-

Transform |  _____________________  |  ______________________

Divergent |  _____________________  |  ______________________

The dimensions of the pool are either (21.55, 16.45) or (28.45, 7.55).

The perimeter of the bottom of a rectangular swimming pool is 60 meters, and the area is 221 square meters.

To find the dimensions of the pool, we can use the formula P = 2(l + w) where P is the perimeter and l and w are the length and width of the pool respectively. We can rearrange this formula to solve for either l or w, so we'll solve for l.

2(l + w) = 60
2l + 2w = 60
2l = 60 - 2w
l = (60 - 2w)/2

Now, we can use the formula A = l*w to solve for w.

221 = l*w
221 = (60 - 2w)/2 * w
221 = (60w - 2w^2)/2
442 = 60w - 2w^2
2w^2 - 60w + 442 = 0

Using the quadratic formula, we get w = 16.45 or 7.55.

Now that we know the width, we can use the formula l = (60 - 2w)/2 to find the length. For w = 16.45, l = 21.55, and for w = 7.55, l = 28.45.

Therefore, the dimensions of the pool are either (21.55, 16.45) or (28.45, 7.55).

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There are six possible ways a person can choose two balls at random from an urn containing five balls numbered 1, 2, 2, 6, and 6.

To understand how to calculate the number of combinations, let's break it down into two parts: selecting the first ball, and then selecting the second ball. Since there are five balls in the urn, the first ball can be chosen in five possible ways. Then, when selecting the second ball, the number of possible selections is reduced to four because one ball has already been selected. This can be expressed mathematically as: n(A)=5x4.

To calculate the total number of combinations, we then multiply the two numbers: 5x4=20. This result is then divided by two because the order of selection does not matter. As a result, 20/2=10, which means that there are ten possible combinations when choosing two balls at random from the urn. In conclusion, if an urn contains five balls numbered 1, 2, 2, 6, and 6, there are six possible ways a person can choose two balls at random. This can be expressed mathematically as n(A)=6 or n(A)=5x4/2.

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Tim's fair share is 175 rupees and Tom's fair share is 125 rupees.

To find the fair share of both Tim and Tom depending on the work done, follow these steps:

Step 1: Find the total number of windows cleaned by both Tim and Tom.

Total number of windows cleaned = number of windows on ground floor + number of windows on first floor

Total number of windows cleaned = 7 + 8 = 15

Step 2: Find the ratio of the number of windows cleaned by Tim to the number of windows cleaned by Tom.

Ratio of Tim's work to Tom's work = number of windows cleaned by Tim / number of windows cleaned by Tom

Ratio of Tim's work to Tom's work = 7 / 8 = 0.875

Step 3: Divide the total money paid by the master (300 rupees) in the same ratio as the work done by Tim and Tom.

Tim's share = (0.875 / (0.875 + 1)) * 300 = 175 rupees

Tom's share = (1 / (0.875 + 1)) * 300 = 125 rupees

Step 4: Check the shares to ensure they add up to the total amount paid by the master.

Tim's share + Tom's share = 175 + 125 = 300 rupees

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The intersection of two sets is the set of elements that are common to both sets, while the union of two sets is the set of elements that are in either of the two sets. Intersection and union are two of the most important operations in set theory and are used for a variety of purposes.

The purpose of the intersection operation is to find common elements between two sets, such as the common elements in two groups of people. The intersection operation produces a new set, which contains only the elements that are in both sets.
The purpose of the union operation is to join two sets together. The union operation produces a new set, which contains all the elements that are in either of the two sets. This is useful when you want to combine two different sets of data.

In summary, the intersection operation finds the common elements between two sets, while the union operation combines two sets of data into one. Both operations are important in set theory and are used for a variety of purposes.

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Answer:

True

Step-by-step explanation:

The given statement "John Graunt is said to be the first to employ quantitative methods to describe population vital statistics" is true as he was interested in how various factors affected mortality rates and population growth.

Quantitative methods are used to quantify and assess data. These techniques are employed by statisticians and scientists in a variety of fields, including psychology, economics, and public health. Quantitative data can be transformed into statistical analyses to make sense of patterns, patterns, and relationships.

John Graunt was a London haberdasher who lived from 1620 to 1674. He is known as the father of vital statistics, and he was the first to employ quantitative methods to describe population vital statistics.

Graunt was interested in how various factors affected mortality rates and population growth, and he used statistical methods to investigate these issues.

In summary, John Graunt is said to be the first to employ quantitative methods to describe population vital statistics.

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The rocket will be 30 feet from the ground approximately 0.4 seconds or 9.6 seconds after it is launched.

To solve this problem, we need to find the value of t for which the height of the rocket is 30 feet. We can use the given equation to set up an equation for this as follows:

-16t² + 160t + 90 = 30

Simplifying and rearranging, we get:

-16t² + 160t + 60 = 0

Dividing by -4, we get:

4t² - 40t - 15 = 0

Using the quadratic formula, we get:

t = [40 ± √(40² - 4(4)(-15))] / (2×4)

t = [40 ± √(1840)] / 8

t = [40 ± 2×√(460)] / 8

t = 5 ± (1/2)×√(460)

Therefore, the rocket will be 30 feet from the ground at t = 5 - (1/2)×√(460) seconds or t = 5 + (1/2)×√(460) seconds. These are the two possible solutions to the equation. We can simplify them as follows:

t = 0.4 seconds or t = 9.6 seconds

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The circumference of Circle C is found as 53π.

In the given figure:

AD = 28 and DB = 45

∠ADB = 1/2 *∠ACB

∠ADB = 90

Applying Pythagorean theorem in ΔABD.

AB² = AD² + DB²

AB² = 28² + 45²

AB = √2809

AB = 53

Radius = 53/2

circumference of Circle C:

C = 2πr

C = 2*π* (53/2)

C =  53π

Thus, the circumference of Circle C is found as 53π.

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The probability that either of spin two wheel lands on an even value is 1.

Define the term probability?

Probability is a way to gauge how likely something is to happen. Even though no event can be predicted with absolute certainty, probability can be used to express how likely it is that it will happen.

For the stated question-

Total number marked on the wheel (1 to 8) = 8 numbers.

Total even number (2,4,6,8) = 4numbers

Probability of getting an even number on one wheel = 4/8 = 1/2.

Let the probability of getting an even number on one wheel P(A).

Then, the Probability of getting an even number on the second wheel P(B).

Then,

Only only wheel have the even number,

P(A∩B) = 0.

Thus,

Probability that either wheel lands on an even value;

P(A∪B) = P(A) + P(B) - P(A∩B)

P(A∪B) = 1/2 + 1/2 - 0

P(A∪B) = 1

Thus, probability that either wheel lands on an even value is 1.

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Answer:

The theoretical probability of being dealt exactly two 2's in a 5-card hand from a standard 52-card deck is:

[tex]= \dfrac{2162}{32487}\\\\\approx 0.06655 \text{ in decimnal}[/tex]

Step-by-step explanation:

The problem can be solved by using the combinatorics formula. The number of ways of drawing a subset of r items from a population of n items is given by

[tex]^nC_r = \dfrac{n!}{r! (n-r)!}[/tex]

where n! is the factorial of n, r! the factorial of r and (n-r)! = factorial of (n-r)

The general formula for k! = k x (k - 1) x (k - 2) x ..... x 3 x 2 x 1

The number or ways in which you can get two 2's in a deal of 5 cards is given by

[tex]^5C_2 = \dfrac{5!}{ 2! (5 - 2)! } \\\\= = \dfrac{5!}{2! \times 3! }\\\\= 10[/tex]

Once we have been dealt 2 2's we have to compute how many ways we can get the remaining 3 cards. Since we are looking for exactly two 2's we cannot draw another 2

The number of cards left that we can draw the remaining three cards = 52(total cards) -2(two 2's already drawn) - 2(two 2's that cannot be drawn)

= 48 cards

We can draw 3 cards from 48 cards in [tex]^{48}C_3[/tex]  ways
[tex]^{48}C_3 = \dfrac{48!}{ 3! (48 - 3)! }\\\\\\= \dfrac{48!}{3! \times 45! }\\\\= 17296[/tex]

Therefore the total number of ways of drawing exactly two 2's
= 10 x 17296 = 172960

The number of ways in which we can draw 5 cards from 52 cards is given by
[tex]^{52}C_5 = = \dfrac{52!}{5! (52 - 5)! }\\\\= \dfrac{52!}{5! \times 47! }\\\\= 2598960[/tex]

P(exactly two 2's in a 5-card hand)
[tex]= \dfrac{172960}{2598960}\\\\ \\= \dfrac{2162}{32487}\\\\[/tex]

or, in decimal
[tex]\approx 0.06655[/tex]

Answer:

p = -2

Step-by-step explanation:

slope = 0.75 = Δy/Δx = (1 - p) / (5 - 1) = (1 - p) / 4

4(0.75) = 1 - p

-p = 3 - 1 = 2

p = -2

A(-5,-3); B(-6,-1); C(-3,-1) ; D(-2,-3) , are the cοοrdinates οf the image οf pοint A after a reflectiοn acrοss the y-axis.

Mathematical statements knοwn as equatiοns have twο algebraic expressiοns οn either side οf the equals (=) sign.

It indicates a relatiοnship οf equality between the expressiοns printed οn the left and right sides.

LHS = RHS (left hand side = right hand side) can be fοund in any mathematical equatiοn.

d the value οf a variable with nο knοwn value that stands in fοr a thing withοut knοwn value.

An assertiοn is nοt an equatiοn if there is nο "equal tο" symbοl.

A mathematical equatiοn includes the symbοl "equal to" between the twο expressiοns when their values are equal.

Accοrding tο οur questiοn-

A'(-5,+3) ==>A"(-2,3)

B'(-6,+1) ==>B"(-3,1)

C'(-3,+1) ==>C"(0,1)

D'(-2,+3) ==>D"(1,3)

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a) Xiao is 13 hours short of his sleep goal for the week.

b) After 4 nights, Xiao is 5 hours short of his sleep goal.

a) Given,Xiao's goal is to get 8 hours of sleep per weeknight, which means he aims to get 40 hours of sleep (8 hours per night x 5 weeknights) during the week. To find the difference between the amount of sleep he got and his goal, we need to calculate the total amount of sleep he got during the week and subtract it from his goal:

Total sleep = [tex]6 \frac{1}{2} + 7 \frac{1}{2} + 5 \frac{3}{4} + 8 \frac{1}{4} = 27[/tex]

Difference from goal = 40 - 27

= 13

Therefore, Xiao is 13 hours short of his sleep goal for the week.

b. After 4 nights, Xiao has slept for a total of:

Total sleep = 27

Since he aims to get 8 hours of sleep per weeknight, he would have gotten a total of 32 hours of sleep (8 hours per night x 4 weeknights) if he had met his goal. To find out how much he is ahead or behind in his sleep goal, we need to calculate the difference between the total amount of sleep he got and his goal:

Difference from goal = 32 - 27 = 5

Therefore, after 4 nights, Xiao is 5 hours short of his sleep goal.

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Roll a 5: There is only one way to roll a 5 out of six possible outcomes, so the probability of rolling a 5 is 1/6.

Roll a 6: Similarly, there is only one way to roll a 6 out of six possible outcomes, so the probability of rolling a 6 is 1/6.

Roll a c or 4: There are two ways to roll a 4 (rolling a 4 or rolling a 3) and one way to roll a 3, so there are three ways to roll a 4 or c out of six possible outcomes. Therefore, the probability of rolling a 4 or c is 3/6, simplifying it to 1/2.

Roll an odd number: There are three odd numbers (1, 3, 5) out of six possible outcomes, so the probability of rolling an odd number is 3/6, simplifying to 1/2.

Roll an even number: There are three even numbers (2, 4, 6) out of six possible outcomes, so the probability of rolling an exact number is 3/6 or 1/2.

Roll a number greater than 3: There are three numbers greater than 3 (4, 5, 6) out of six possible outcomes, so the probability of rolling a number greater than 3 is 3/6, which simplifies to 1/2.

Roll an even number less than 5: There is only one number less than 5 (2) out of six possible outcomes, so the probability of rolling an actual number less than 5 is 1/6.

Roll a multiple of 2 (2, 4, 6): There are three multiples of 2 out of six possible outcomes, so the probability of rolling a multiple of 2 is 3/6, simplifying to 1/2.

Roll a factor of 6 (1, 2, 3, 6): There are four factors of 6 out of six possible outcomes, so the probability of rolling a factor of 6 is 4/6, which simplifies to 2/3.

So the probabilities for each event expressed as fractions are:

Roll a 5: 1/6

Roll a 6: 1/6

Roll a 4 or c: 1/2

Roll an odd number: 1/2

Roll an even number: 1/2

Roll a number greater than 3: 1/2

Roll an even number less than 5: 1/6

Roll a multiple of 2: 1/2

Roll a factor of 6: 2/3

Answer:

∠ FZG = 38°

Step-by-step explanation:

∠ HZJ and ∠ FZG are vertically opposite angles and are congruent , then

∠ ZFG = ∠ HZJ = 38°

The probability using the demand distribution of a stockout is given as 0.6293.

Out-of-stock situations, commonly referred to as stockouts, occur when a company runs out of a product that a client is prepared to purchase. Stockouts have a direct influence on customer satisfaction and are a major reason for bad reviews, additional expenses, and lost revenue or, worse, customers.

Let X = no of demand for the toy

μ =20,000

P(10000 < X < 30000) = 0.90

X follows  normal distribution

z=x−μ​/σ

P(10000 < X < 30000)=0.90

P((10000−20000​)/σ) < (x−20000​)/σ < (30000−20000​)/σ )=0.90

as probability is 0.90, the z-score =1.645.

(30000−20000)​/σ =1.645

10000​/σ=1.645

σ=10000​/1.645

σ=6079

1.P(X>15,000)

=P(Z> (15000−20000​)/6079)

=P(Z> −0.82)

=P(Z< 0.82)

=0.5+0.1293

=0.6293

2. P(X> 24000)

=P(Z> (24000−20000​)/6079)

=P(Z> 0.66)

=0.5−0.2454

=0.2546

3.Expected profit = (profit per unit )*(no of sold units) + (loss per unit )*(of units unsold)

profit = $8

loss = -$11

a. profit if 15000 units are sold =(8)(15000) + (-11)(0) = $120,000

b. if there is demand for 15,000 units but orders only 18000 units profit is (8)(15000) + (-11)(3000) = $87,000

c. if there is demand for 18,000 units but orders only 24000 units profit is (8)(18000)+(-11)(6000)= $78,000.

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Complete question:

(a) What is the expected profit if all 15,000 units are sold? type your answer...

(b) Suppose there is demand for 15,000 units, but Specialty Toys orders 18,000 units? What is the expected profit? type your answer...

(c) Suppose there is demand for 18,000 units, but Specialty Toys orders 24,000 units? What is the expected profit? type your answer... 15 3 points Based on what you know about the distribution of demand, the probability of stock-outs, and the expected profit, how many units of Weather Teddy do you recommend that Specialty Toys purchase?

The number of seconds that the ball was in the air is approximately 5.11 seconds

To find the time the ball was in the air, we need to find the time when the ball hits the balcony, which is located 218 feet above the ground.

Let's set h = 218 in the equation h = -16t^2 + 48t + 506 and solve for t

218 = -16t^2 + 48t + 506

Simplifying, we get

16t^2 - 48t - 288 = 0

Dividing both sides by 16, we get

t^2 - 3t - 18 = 0

Using the quadratic formula, we can solve for t:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -3, and c = -18

t = (3 ± sqrt(3^2 - 4(1)(-18))) / 2(1)

t = (3 ± sqrt(105)) / 2

We can ignore the negative solution because time cannot be negative. Therefore, we have

t = (3 + sqrt(105)) / 2

t ≈ 5.11 seconds

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The given question is incomplete, the complete question is:

A person throws a ball upward from a 506 foot tall building the balls height h in feet after t seconds is given by the equation h= -16t^2+48t+506 the ball lands on a balcony that is 218 feet above the ground how many seconds was it in the air

The inverse function is as follows: f-1(x) = (x - 2) with y > 0 as the domain as limit the range of the inverse function to y > 0 .

A function is a mathematical rule that relates every element in one set, known as the domain, to a single element in another set, known as the range. A function, then, is a relationship between two sets in which each input in the domain has a single output in the range. Equations, tables, graphs, and symbols can all be used to express it. Consider the function f(x) = 2x + 1 as an illustration. This function multiplies an input value x by 2, adds 1, and returns the result.

given

The steps listed below can be used to determine the inverse of the function f(x) = x2 + 2:

Step 1: Substituting y for f(x)

[tex]y = x^2 + 2[/tex]

Switch x and y in step 2:

[tex]x = y^2 + 2[/tex]

Step 3: Calculate y

[tex]x - 2 = y^2[/tex]

y = ± √(x - 2) (x - 2)

Since we want the inverse to be a function as well, we may limit the range of the inverse function to y > 0 because the original function's domain is x > 0.

The inverse function is as follows: f-1(x) = (x - 2) with y > 0 as the domain.

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According to the question, "Suppose we randomly select 5 cell-phone owners. What is the probability that all 5 of them find themselves checking their phone for messages, alerts, or calls even when they don't notice their phone ringing or vibrating?"The formula for finding the probability of an event is:P(E) = number of favorable outcomes/number of total possible outcomes.The given data in the question states that 67% of cell-phone owners find themselves checking their phone for messages, alerts, or calls even when they don't notice their phone ringing or vibrating.

Therefore, the probability of any cell-phone owner checking their phone when it is not ringing or vibrating is:P(E) = 67/100 = 0.67To find the probability of 5 cell-phone owners checking their phones when they are not ringing or vibrating, we have to use the multiplication rule of probability. [tex]P(E₁∩E₂∩E₃∩E₄∩E₅) = P(E₁) × P(E₂) × P(E₃) × P(E₄) × P(E₅)P(E₁∩E₂∩E₃∩E₄∩E₅) = 0.67 × 0.67 × 0.67 × 0.67 × 0.67P(E₁∩E₂∩E₃∩E₄∩E₅) = 0.2197[/tex]

Therefore, the probability that all 5 of them find themselves checking their phone for messages, alerts, or calls even when they don't notice their phone ringing or vibrating is approximately 0.2197 or 22%.

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We know that the workers are required to pay 4 and a half of their salaries into an education fund.

If the salary is S = 120.00, the educational fund receive from that worker:

[tex]F=4.5\times S=4.5 \times120=540[/tex]

Answer: he pays GH 540.00 into the education fund.

The value of angle x in the given figure is 79°.

The highest and lοwest walls οf a skew are used tο split the curved lines that make up its ends using Cartesian measurements. A cοllisiοn between twο pοles at an intersectiοn is a pοtential. Angle is anοther οutcοme οf twο things interacting. They resemble dihedral fοrms the mοst clοsely. A twο-dimensiοnal curve can be created by placing twο line beams in variοus cοnfiguratiοns between their ends.

In ΔBFG, ∠FBG = 46°

So according to angle sum property

46° + ∠BFG + BGF = 180°

Here ΔBFG is an isosceles triangle

46° + 2∠BFG = 180°

2∠BFG = 180° - 46°

2∠BFG  = 134

∠BFG = 134/2

∠BFG = 67° = ∠BGF

Using Vertically opposite angle

∠AFJ =  67°

With that in mind ∠BFA = ∠JFG

2∠JFG + 2∠AFJ = 360°

2∠JFG  = 360° - 2∠AFJ

2∠JFG  = 360° - 2(67)

2∠JFG  = 360° - 134

2∠JFG  = 226

∠JFG  = 226/2

∠JFG = 113°

In ΔFDC, ∠JFG  + ∠JDI + ∠HCG = 180°

113° + 33° + ∠HCG = 180°

∠HCG = 180° - (113° + 33°)

∠HCG = 34°

Also, ∠HGC = ∠BGF = 67°

Now, In ΔCGH, ∠HGC + ∠HCG + x = 180°

67° + 34° + x = 180°

x = 180° - (67° + 34°)

x = 79°

Thus, the value of angle x in the given figure is 79°.

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The 90% confidence interval for the population standard deviation of the thicknesses of all linoleum tiles in this factory (3.27, 7.78)

To construct the 90% confidence interval for the population standard deviation of the thicknesses of all linoleum tiles in the factory, we need to use the chi-square distribution since the population standard deviation is unknown.

1. Identify the sample size (n) and sample standard deviation (s):
n = 13 (13 randomly selected linoleum tiles)
s = 4.76 (standard deviation of the thicknesses)

2. Determine the degrees of freedom (df):
df = n - 1 = 13 - 1 = 12

3. Find the chi-square values for the 90% confidence interval using a chi-square table or calculator:
The lower tail value (with 5% in the lower tail) is χ² = 4.404
The upper tail value (with 5% in the upper tail) is χ² = 21.026

4. Calculate the confidence interval for the population standard deviation:
Lower limit = √[(n - 1) * s² / χ²_upper] = √[(12 * (4.76)²) / 21.026] ≈ 3.27
Upper limit = √[(n - 1) * s² / χ²_lower] = √[(12 * (4.76)²) / 4.404] ≈ 7.78

The 90% confidence interval for the population standard deviation is approximately (3.27, 7.78).

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It is important to understand that the force of gravity acting on an object does not violate Newton's second law. It is an external force that acts on an object and produces an acceleration in it. Therefore, the acceleration due to gravity does not violate Newton's second law.

According to the Newton's second law, a force acting on a body of mass m produces an acceleration a in the body which is directly proportional to the force and inversely proportional to the mass. For instance, If a body of mass m is acted upon by a force F, then the acceleration produced in the body is given by F/m.As per the law, a larger force will produce a larger acceleration in the object.

When a body falls under gravity, it is said to experience weight. The weight of an object is the force of gravity acting on the object. According to Newton's law of gravitation, every object attracts every other object with a force proportional to their masses and inversely proportional to the square of their distance. In general, the weight of an object can be given by the formula weight = mass x acceleration due to gravity.

Therefore, as the acceleration due to gravity is the same for all objects, the weight of an object is proportional to its mass. It implies that a heavier object has more weight, while a lighter object has less weight.As per the above facts, a heavier object would require a larger force to move it than a lighter object. However, when they fall under gravity, both objects fall at the same rate. It appears as a contradiction, but the reason for it is the difference between weight and mass.

The mass of an object is a measure of the amount of matter in it, while the weight of an object is the force exerted on it by gravity.The acceleration produced in an object by an external force is not the same as the acceleration due to gravity. A force F produces an acceleration F/m, whereas the acceleration due to gravity is the same for all objects.

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Thus, the distance of ladder from the building is found to be 11.53 ft.

Exactly single right angle measure 90 degrees characterises a right triangle.

The hypotenuse of the a right triangle's square is equal to the sum of its other two sides, according to the Pythagorean Theorem. It is written as a²+b²=c² in equation form.

Pythagoras is in the form of;

a²+b²=c²

Let the distance of ladder from the building be 'x'.

Height of ladder from ground h = 6 ft.

Length of ladder l = 13 ft.

Here, using the Pythagorean theorem;

x² + 6² = 13²

x²  = 13² -  6²

x²  = 169 - 36

x²  = 133

x = √133

x = 11.53

Thus,  the distance of ladder from the building is found to be 11.53 ft.

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The rate of decrease of thickness of the ice when it is 2 inches thick is 8π/min.

The spherical iron ball 10 inches in diameter is coated with a layer of ice of uniform thickness. When the thickness of the ice is 2 inches, the rate of decrease of thickness can be calculated as follows:

Let r be the radius of the ball and h be the thickness of the ice.

The volume of ice = Volume of sphere = 4/3 π r3. When h = 2, 4/3 π r3 = 8π.

The rate of decrease of thickness of the ice when it is 2 inches thick is given by,

Rate of decrease = 15 in3/min = 8π/min

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The surface area οf the cylinder is  170π & the lateral area οf the cylinder is apprοximately 376.99 square  centimetres.

In geometry, a cylinder is one of the basic three-dimensional shapes with two distant parallel circular bases.

Tο find the surface area and lateral area οf the given shape, we need tο first identify the shape. Based οn the given image, it appears tο be a right circular cylinder.

The surface area οf a right circular cylinder is given by the fοrmula:

Surface Area = 2πr² + 2πrh

where r is the radius οf the base and h is the height οf the cylinder.

Frοm the image, we can see that the radius οf the cylinder is 5 cm and the height is 12 cm.

Consequently, the cylinder's surface area is as follows:

Surface Area = 2π(5)² + 2π(5)(12)

Surface Area = 2π(25) + 2π(60)

Surface Area = 50π + 120π

Surface Area = 170π

Sο, the surface area οf the cylinder is apprοximately 533.15 square centimetres.

The lateral area οf the cylinder is the area οf the curved surface, which is given by: Lateral Area = 2πrh.

Substituting the values οf r and h, we get:

Lateral Area = 2π(5)(12)

Lateral Area = 120π

Sο, the lateral area οf the cylinder is apprοximately 376.99 square centimetres.

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