A 5-inch candle burns down in 10 hours. After how many hours will it have burned 3 1/4 ​inches?

Since the probability of success is 0.74, the probability of failure is 1 - 0.74 = 0.26. This means that there is a 26% chance of failure in the given binomial trial. Option C

In a binomial trial, the probability of success, denoted by "p," represents the likelihood of the desired outcome occurring. The probability of failure, denoted by "q," represents the complement of the probability of success, i.e., the likelihood of the desired outcome not occurring.

In this case, the probability of success is given as 0.74. To find the probability of failure, we subtract the probability of success from 1, since the sum of the probabilities of success and failure must equal 1.

Probability of failure = 1 - Probability of success

Therefore, the probability of failure = 1 - 0.74 = 0.26.

Hence, the correct answer is C. 0.26.

It's important to understand that in a binomial distribution, there are only two possible outcomes: success and failure. The probabilities of these outcomes must add up to 1. Therefore, if the probability of success is known, the probability of failure can be obtained by subtracting the probability of success from 1.

Option C

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The key differences between the graphs of a nonmember's and a member's yearly cost lie in the slope, y-intercept, and the overall rate of increase in cost as the number of game tokens purchased increases.

The given equations represent the yearly cost in dollars for a member and a nonmember at a video game arcade based on the total game tokens purchased.

For a member:

y = (1/10)x + 60x + 60

For a nonmember:

y = (1/5)x

To understand how the graph of a nonmember's yearly cost differs from a member's yearly cost, let's analyze the equations and their characteristics.

Slopes:

The slope of the member's equation is (1/10), indicating that for every unit increase in the number of game tokens purchased (x), the yearly cost (y) for a member increases by 1/10 of a dollar. This means that the member's yearly cost increases at a slower rate compared to the nonmember's yearly cost.

The slope of the nonmember's equation is (1/5), which means that for every unit increase in the number of game tokens purchased, the yearly cost for a nonmember increases by 1/5 of a dollar. Therefore, the nonmember's yearly cost increases at a faster rate compared to the member's yearly cost.

y-intercepts:

For the member's equation, the y-intercept is 60, which represents the fixed cost component for being a member of the arcade. This means that even without purchasing any game tokens (x = 0), a member incurs a yearly cost of $60.

For the nonmember's equation, there is no additional fixed cost component. The y-intercept is 0, indicating that a nonmember has zero yearly cost if no game tokens are purchased (x = 0).

Overall cost:

The member's equation includes both a fixed cost component and a variable cost component, whereas the nonmember's equation only includes the variable cost component. This means that for any given number of game tokens purchased, the member's yearly cost will be higher than the nonmember's yearly cost.

Graphically, the member's equation will result in a line with a positive slope that intersects the y-axis at 60. The nonmember's equation will yield a line with a steeper positive slope that intersects the origin (0,0). The graph of the nonmember's yearly cost will rise more quickly than the graph of the member's yearly cost.

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Trey must pay $2420 in income tax for last year.

Based on the tax rates, we'll divide his taxable income into two halves.

The first portion is the minimal amount of Trey's taxable income between $9500 and $20,000 that is subject to a 10% tax rate. Consequently, the first component is $9500.

The second portion is the amount over $9500, which is $20,000 - $9500 = $10,500.

Now, let's calculate the tax for each portion:

Tax on the first portion (10% rate) = $9500 * 0.10 = $950.

Tax on the second portion (14% rate) = $10,500 * 0.14 = $1470.

We sum up the taxes on both portions to find Trey's total income tax:

Total income tax = Tax on the first portion + Tax on the second portion = $950 + $1470 = $2420.

So, Trey must pay $2420 in income tax for last year.

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Let [tex]x[/tex] be the number of pounds of Nigerian coffee and [tex]y[/tex] be the number of pounds of Bolivian coffee.

We can set up a system of equations to represent the given information:

So we have the following system of equations:

[tex]\qquad\quad\begin{aligned} 17x + 18y &= 437.50 \\ x + y &= 50 \end{aligned}[/tex]

Solving this system of equations, we get:

[tex]\qquad\qquad\quad\begin{aligned} x &= 12.5 \\ y &= 37.5 \end{aligned}[/tex]

[tex]\therefore[/tex] We need 12.5 pounds of Nigerian coffee and 37.5 pounds of Bolivian coffee for the mixture.

[tex]\blue{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]

(ノ^_^)ノ [tex]\large\qquad\qquad\qquad\rm 06/21/2023[/tex]

The inequality shown by the graph is m ≤ 1. This means that the values of m are less than or equal to 1. Any value of m that is equal to or smaller than 1 satisfies the inequality. However, any value of m that is greater than 1 does not satisfy the inequality.

Inequalities can be represented graphically using number lines.

The inequality m ≤ 1 means that all values of m that are less than or equal to 1 are solutions to the inequality.

The solution set is represented by a closed circle on the number line at the point where m = 1, and a line segment extending to the left of this point.
If we choose a value of m from the shaded region on the graph, such as m = 0, the inequality m ≤ 1 is satisfied because 0 is less than 1.

If we choose a value of m from the unshaded region, such as m = 2, the inequality is not satisfied because 2 is greater than 1. Therefore, the inequality shown by the graph is m ≤ 1.

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The domain and the range of the function are (-∝, ∝) and (-∝, -1), respectively

From the question, we have the following parameters that can be used in our computation:

The graph

The above graph is an quadratic function

The rule of an function is that

The domain is the set of all real values

In this case, the domain is (-∝, ∝)

For the range, we have

Range = (-∝, -1)

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Answer:

A:(-1,0)

B:(0,5)

C:(2,9)

D:(5,0)

Step-by-step explanation:

count the x axis to get x and y axis to get y

(x,y)

Answer:

See below

Step-by-step explanation:

A is located at (-1,0) which is an x-intercept

B is located at (0,5) which is the y-intercept

C is located at (2,9) which is the vertex

D is located at (5,0) which is an x-intercept

As we can see, this is the equation of line `g(x)` that passes through the points `(-4,0)` and `(-3,-2)`. Therefore, the value of `k` is `2`.

The correct answer to the given question is option d.

We have the function `g(x) = k⋅f(x)`. The values of `f(x)` and `g(x)` are given as follows:

Line `f(x)` passes through points `(-4,0)` and `(-3,1)`.

Line `g(x)` passes through points `(-4,0)` and `(-3,-2)`.

Now, we have to determine the value of `k`.

Formula to find slope of a line is given by:(y2 - y1)/(x2 - x1)

Here, (x1, y1) = (-4, 0) and (x2, y2) = (-3, 1) for line f(x).

So, slope of line `f(x)` is given by:(1 - 0)/(-3 - (-4)) = 1

So, equation of line `f(x)` is given by:

y - y1 = m(x - x1) ⇒ y - 0 = 1(x - (-4)) ⇒ y = x + 4

Also, (x1, y1) = (-4, 0) and (x2, y2) = (-3, -2) for line g(x).

So, slope of line `g(x)` is given by:(-2 - 0)/(-3 - (-4)) = 2

So, equation of line `g(x)` is given by: y - y1 = m(x - x1) ⇒ y - 0 = 2(x - (-4)) ⇒ y = 2x + 8

Now, we can substitute the value of `k` and the equation of line `f(x)` to find the equation of line `g(x)`.

Let `k = 2`.

Then, `g(x) = k⋅f(x) = 2(x + 4) = 2x + 8`.

As we can see, this is the equation of line `g(x)` that passes through the points `(-4,0)` and `(-3,-2)`.

Therefore, the value of `k` is `2`. Hence, option (d) is the correct answer: `2`.

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Answer:

30: 100

31: -13

32: -45

33: 14

34: -32

35: -22

36: 17

Answer:5 drops

Step-by-step explanation:

Every 100 nanograms is 0.1 microgram so 500 nanograms + 0.5 micrograms so 5 drops because everydrop is 100 nanograms

Answer:

y = -2x+10

Step-by-step explanation:

Slope-intercept form of the equation for a line is

y = mx+b  where m is the slope and b is the y intercept

We know the slope is -2

y = -2x+b

Substituting the point in for x and y and solving for b.

8 = -2(1) + b

8 = -2+b

10 =b

Now

y = -2x+10

Answer:

y = - 2x + 10

Step-by-step explanation:

The equation of a line in slope-intercept form is given by:

y = m x + c

Here,

m → slope of the line

c →  y-intercept

Using the point-slope form of a line, we have:

( y - y₁ ) = m ( x - x₁ )

Substituting the coordinates ( x₁, y₁ ) = ( 1, 8 ), and the slope m = -2:

( y - 8 ) = - 2 ( x - 1 )

Expanding and rearranging the equation:

y - 8 = - 2x + 2

Adding 8 to both sides:

y = - 2x + 10

Answer:

4x²=2(x-4)

2x²=x-4

2x²-x+4=0​

x=1+√31​/ 4, 1-√31​/4

Step-by-step explanation:

1. Cancel log on both sides

2. Divide both sides by 2

3. Move all terms to one side

4. Use the quadratic formula

The distance between points C and D is given as follows:

5 units.

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 coordinates for this problem are given as follows:

(-4, -6) and (1, -6).

Hence the distance is given as follows:

[tex]D = \sqrt{(-4 - 1)^2 + (-6 - (-6))^2}[/tex]

D = 5 units.

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[tex]~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+5}x\stackrel{\stackrel{c}{\downarrow }}{+6}=0 \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ x= \cfrac{ - (5) \pm \sqrt { (5)^2 -4(1)(6)}}{2(1)} \implies x = \cfrac{ -5 \pm \sqrt { 25 -24}}{ 2 } \\\\\\ x= \cfrac{ -5 \pm \sqrt { 1 }}{ 2 }\implies x=\cfrac{-5\pm 1}{2}\implies x= \begin{cases} -2\\ -3 \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{-3}x\stackrel{\stackrel{c}{\downarrow }}{-2}=0 \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ x= \cfrac{ - (-3) \pm \sqrt { (-3)^2 -4(2)(-2)}}{2(2)} \implies x = \cfrac{ 3 \pm \sqrt { 9 +16}}{ 4 } \\\\\\ x= \cfrac{ 3 \pm \sqrt { 25 }}{ 4 }\implies x=\cfrac{3\pm 5}{4}\implies x= \begin{cases} 2\\ -\frac{1}{2} \end{cases}[/tex]

The sum  in scientific notation is  2.88 × 10⁰

To add numbers in scientific notation, we need to make sure the exponents are the same. Let's add 5.8 × 10⁴ and 2.3 × 10⁵.

First, we need to adjust the numbers so that they have the same exponent. We can do this by moving the decimal point.

5.8 × 10⁴ can be written as 0.58 × 10⁵ (moving the decimal point one place to the right).

Now, we have 0.58 × 10⁵ + 2.3 × 10⁵. Since the exponents are the same, we can add the coefficients:

0.58 + 2.3 = 2.88

The sum of the coefficients is 2.88. To express this in scientific notation, we need to adjust the decimal point and exponent.

Since we moved the decimal point one place to the right in 0.58 × 10⁵, we need to move it one place to the left in 2.88.

2.88 can be written as 0.288 × 10¹.

Therefore, the sum of 5.8 × 10⁴ and 2.3 × 10⁵ is 0.288 × 10¹.

In scientific notation, this can also be expressed as 2.88 × 10⁰ or simply 2.88.

So, the sum of 5.8 × 10⁴ and 2.3 × 10⁵ is 2.88.

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The median cost of a home in 2014 is $480

First, we need to know that the median of a given set of data is expressed as the middle number determined with the data set is arranged in an order from least to greatest or in an ascending order.

Also, note that the median is one of the measures of central tendency.

From the information given, we have that;

The median cost of a home in 2014 traced from the point year to the cost on the graph is;

$480

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The ratio of the length of the shortest diagonal to the length of the longest diagonal in a regular hexagon is [tex]$\sqrt{3}/2$[/tex]. This can also be written as [tex]$\frac{\sqrt{3}}{2}$[/tex].

A hexagon is a six-sided polygon with all angles equal to 120 degrees. In a regular hexagon, all sides are equal in length and all angles are equal. The diagonals of a hexagon are line segments that connect non-adjacent vertices of the hexagon.

A regular hexagon has nine diagonals. The shortest diagonal of a regular hexagon is the one that connects opposite vertices and is equal to the length of a side of the hexagon. The longest diagonal of a regular hexagon is the one that connects opposite vertices and passes through the center of the hexagon.

To find the ratio of the length of the shortest diagonal to the length of the longest diagonal, we need to find the length of the longest diagonal in terms of the length of the shortest diagonal. We know that the length of the shortest diagonal is equal to the length of a side of the hexagon.

We can draw the longest diagonal and form an equilateral triangle by connecting the center of the hexagon to the two endpoints of the longest diagonal. The length of the side of this equilateral triangle is equal to the length of the longest diagonal of the hexagon.

The ratio of the length of the shortest diagonal to the length of the longest diagonal is then equal to the ratio of the side length of this equilateral triangle to the length of the side of the hexagon, which is [tex]$\frac{\sqrt{3}}{2}$[/tex].
Therefore, the ratio of the length of the shortest diagonal to the length of the longest diagonal in a regular hexagon is $\sqrt{3}/2$. This can also be written as [tex]$\frac{\sqrt{3}}{2}$[/tex].

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Reason:

0.5% = 0.5/100 = 5/1000

or you could say

5/100 = 0.05 = 5%

The ordered pair which is not a solution to the inequality y - 3x < 10 is (-6,0).

Given the inequality in the question:

y - 3x < 10

Given the ordered pairs: (0,-4), (0,-1), and (-6,0). To determine which ordered pair is not a solution, we need to plug the values into the inequality and check.

For (0,-4):

y - 3x < 10

Plug in x = 0 and y = -4

-4 - 3(0) < 10

-4 < 10

True: -4 is less than 10.

For (0,-1):

y - 3x < 10

Plug in x = 0 and y = -1

-1 - 3(0) < 10

-1 < 10

True: -1 is less than 10.

For (-6,0):

y - 3x < 10

Plug in x = -6 and y = 0

0 - 3(-6) < 10

18 < 10

False: 18 is Not less than 10.

Therefore, (-6,0) is not a solution to the inequality.

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Answer:

∠ MON = 51°

Step-by-step explanation:

∠ LON is composed of the 2 angles LOM and MON , that is

∠ LOM + ∠ MON = ∠ LON

42° + ∠ MON = 93° ( subtract 42° from both sides )

∠ MON = 51°

Answer:

<MON= 51°

Step-by-step explanation:

Look at the diagram and locate LON. You can see that LON is the angle of the complete line. Now LOM is given which is the angle of a part of the lines. So that means that to find MON we can minus LON with LOM.

<MON= <LON - <LOM

= 93-42

<MON= 51°

Feel free to ask any doubt you have!

Answer: the correct answer would option (C).

Step-by-step explanation:7x²+16x+25  is corresponds to the equivalent expression of (x²+15x+65) + (2x - 5) (3x +8).

7x²-3x+8 is corresponds to the equivalent expression of (4x + 1)(3x - 4) - (5x²-10x-12)

11x²+6x-6 is corresponds to the equivalent expression of (8x²+19x+4) + (3x + 2)(x - 5).

11x²-5x+13 is corresponds to the equivalent expression of (6x + 1)(3x - 7) - (7x²-34x-20)

What is expression?

Expressions is the defined as mathematical statements that have a minimum of two terms containing variables or numbers.

Given expression no. 1 as :

⇒ (x²+15x+65) + (2x - 5) (3x +8).

⇒ x²+15x+65 + 6x²-15x+16x-40

⇒ 7x²+16x+25

Hence, the correct answer would be option (B).

Given expression no. 2 as :

⇒ (4x + 1)(3x - 4) - (5x²-10x-12)

⇒ 12x²+3x-16x-4 - 5x²+10x+12

⇒ 12x²-13x-4 - 5x²+10x+12

⇒ 7x²-3x+8

Hence, the correct answer would be option (D).

Given expression no. 3 as :

⇒ (8x²+19x+4) + (3x + 2)(x - 5).

⇒ (8x²+19x+4) + 3x²+2x-15x-10

⇒ 8x²+19x+4 + 3x²-13x-10

⇒ 11x²+6x-6

Hence, the correct answer would be option (A).

Given expression no. 4 as :

⇒ (6x + 1)(3x - 7) - (7x²-34x-20)

⇒ 18x²+3x-42x-7- 7x²+34x+20)

⇒ 11x²-5x+13

Hence, the correct answer would be option (C).

Answer:

2.4 yd

Step-by-step explanation:

Let the width of the walkway = x.

total length = x + 7.5

total width = x + 4.5

total area = (x + 7.5)(x + 4.5)

total area = 68.31 yd²

(x + 7.5)(x + 4.5) = 68.31

x² + 4.5x + 7.5x + 33.75 - 68.31 = 0

x² + 12x - 34.56 = 0

x = [-12 ± √(12² - 4(1)(-34.56)]/(2 × 1)

x = [-12 ± √(144 + 138.24)]/(2 × 1)

x = [-12 ± 16.8]/2

x = 2.4 or x = -14.4

Answer: 2.4 yd

The correlation coefficient (r) between the number of plays and the number of passes completed by the top 30 college quarterbacks, using Excel, the correlation coefficient is approximately 0.83.

To calculate the correlation coefficient (r) between the number of plays and the number of passes completed by the top 30 college quarterbacks, you can use Excel. Here's a step-by-step guide:

Open Excel and enter the data for the number of plays in one column and the number of passes completed in another column. Ensure that the data is entered consistently, with the same row representing the same quarterback's data in both columns.

Select an empty cell where you want to display the correlation coefficient.

Use the CORREL function in Excel to calculate the correlation coefficient. The syntax for the CORREL function is: =CORREL(array1, array2). In this case, array1 represents the range of cells containing the number of plays, and array2 represents the range of cells containing the number of passes completed. For example, if the number of plays is in column A (A2:A31) and the number of passes completed is in column B (B2:B31), the formula would be: =CORREL(A2:A31, B2:B31).

Press Enter to calculate the correlation coefficient. The result will be displayed in the cell you selected.

Round the correlation coefficient to two decimal places using the ROUND function. The syntax for the ROUND function is: =ROUND(number, num_digits). In this case, number represents the correlation coefficient, and num_digits represents the number of decimal places to round to. For example, if the correlation coefficient is in cell C1, the formula would be: =ROUND(C1, 2).

By following these steps, you can use Excel to calculate the correlation coefficient (r) between the two data sets for the top 30 college quarterbacks.

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hmmm let's reword it

what is the Decay equation for sunlight, decaying at 69% at every interval of 10 mm of thickness for "t"?

[tex]\textit{Periodic/Cyclical Exponential Decay} \\\\ A=(1 - r)^{\frac{t}{c}}\qquad \begin{cases} A=\textit{current amount}\\ r=rate\to 69\%\to \frac{69}{100}\dotfill &0.69\\ t=thickness\\ c=period\dotfill &10 \end{cases} \\\\\\ A=(1 - 0.69)^{\frac{t}{10}}\implies A=0.31^{\frac{t}{10}}\hspace{5em}\boxed{S=0.31^{\frac{t}{10}}}[/tex]

The best description of the graph is option B) It is a line which intersects the y-axis at (2, 2). This is because the line has a y-intercept of 2 and intersects the y-axis at the point (0, 2). Option B

To determine the best description of the graph of the equation x + y = 2, we can rearrange the equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

Rearranging the equation, we have:

y = -x + 2

From this equation, we can see that the slope is -1, meaning that for every unit increase in x, y decreases by 1. The y-intercept is 2, indicating that the line intersects the y-axis at the point (0, 2).

Therefore, the best description of the graph is option B) It is a line which intersects the y-axis at (2, 2). This is because the line has a y-intercept of 2 and intersects the y-axis at the point (0, 2).

Option A) It is a line which intersects the x-axis at (2, 2) is incorrect because the line intersects the x-axis at the point (2, 0), not (2, 2).

Option C) It is a line joining the points whose x- and y-coordinates add up to 2 is incorrect because the equation represents a line and not a set of points.

Option D) It is a line joining the points whose x- and y-coordinates add up to 4 is incorrect because the equation represents a line where the x- and y-coordinates add up to 2, not 4.

Option B

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Answer:

x^2-5x-25 doesnt factorise im very sorry, maybe check ur question

We can conclude that the function f(x) = x² is increasing when x > 1.The given function is f(x) = x². You need to determine whether this function is increasing or decreasing when x > 1.

To do this, we can find the derivative of the function and evaluate it for x > 1.If the derivative is positive, then the function is increasing, and if it is negative, then the function is decreasing.

The derivative of the function f(x) = x² is given by:f '(x) = 2x

We can see that the derivative is always positive when x > 1, as 2x is always positive for x > 0.Therefore, we can conclude that the function f(x) = x² is increasing when x > 1.

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The probability that both events B and C will occur is:

P(B and C) = 9/20

Probability is the likelihood of an event to occur. It is expressed as a number in the range from 0 to 1.

The probability of an impossible event is 0, that of an event that is certain to occur is 1.

We have:

The probability that event B will occur, P(B) = 3/4

The probability that event C will occur, P(C) = 3/5

The probability that both events B and C will occur is:

P(B and C) = P(B) × P(C)

P(B and C) = 3/4 × 3/5

P(B and C) = 9/20

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The expression (sinx + cosx)^2 + (sinx - cosx)^2 simplifies to

To simplify the expression (sinx + cosx)^2 + (sinx - cosx)^2 using trigonometric identities, we can expand and simplify the expression.

Expanding the squared terms

(sin^2x + 2sinxcosx + cos^2x) + (sin^2x - 2sinxcosx + cos^2x)

Using the trigonometric identity sin^2x + cos^2x = 1, we can simplify further:

(1 + 2sinxcosx + 1) + (1 - 2sinxcosx + 1)

Simplifying the expression, we have:

2 + 2sinxcosx + 2

Combining like terms, we get:

4 + 2sinxcosx

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that is physics ...

but yes, applied math. we need to know the formulas though.

(1)

in general, since there are 2 cables supporting in an equal way.

that means each cable is responsible for 600/2 = 300 pounds to bring and hold up.

a cable or rope at an angle has to handle a combined tension force : horizontally (Fx) and vertically (Fy).

the tension force (Ftens) on the rope is a combination of both.

we know

Fx = Ftens × cos(theta)

Fy = Ftens × sin(theta)

from the problem we do know Fy (the vertical = up/down force), as this is the force needed to lift and keep the 300 pound weight up there.

and that is Fgravity, the force needed to counteract gravity.

Fgravity = mass × g

g being the constant gravitational acceleration of Earth = 9.8 m/s²

forces are described in Newton.

1 N ≈ 0.225 pounds (lifting on Earth)

so, to lift 1 pound requires 1/0.225 ≈ 4.44822 N

to lift 300 pounds requires

4.44822 × 300 ≈ 1334.47 N

that is what Fy is for one of the 2 cables.

the tension on one of the cables is then given by

Fy = Ftens × sin(60)

Ftens = Fy / sin(60) = 1334.47 / sin(60) =

= 1,540.913227... N = 346.41107515867... pounds

≈ 346 pounds per cable.

(2)

3cos(t) + 4 = 2

3cos(t) = -2

cos(t) = -2/3

cosine is negative in the 2nd and 3rd quadrant.

so, for t > pi/2 and t < 3pi/2.

because the given interval is [0, pi), we are only looking at the 2nd quadrant (pi/2, pi).

t = 131.8103149...° = 2.300523983... rad

(3)

well, that are the numbers

1/2

sqrt(2)/2 = 1/sqrt(2)

sqrt(3)/2

1

they are getting bigger and bigger, all positive, so they indicate larger and larger angles

1/2 is :

sin(30° or pi/6 or 150° or 5pi/6)

cos(60° or pi/3 or 300° or 5pi/3)

1/sqrt(2) is :

sin(45° or pi/4 or 135° or 5pi/4)

cos(45° or pi/4 or 315° or 7pi/4)

sqrt(3)/2 is :

sin(60° or pi/3 or 120° or 2pi/3)

cos(30° or pi/6 or 330° or 11pi/6)

1 is :

sin and csc(90° or pi/2)

cos and sec(0° or 0pi or 360° or 2pi)

tan and cot(45° or pi/4 or 225° or 5pi/4)

(4)

the height moves between 4 meters and 70 meters in a circle.

the circumference of the circle is 2pi×r or pi×d, so in our case : 66pi meters.

it takes 5 minutes to move along these 66pi meters.

let's say, when the height is 4 meters (starting position), the angle is 0 and the arc is 0.

after a quarter trip the angle is 90° or 66pi/4, and the height is 4 + 66/2 = 37 meters

and at 70 meters the angle is 180° or 66pi/2.

the function of the height based on the current angle is then for the first half-circle

height(theta) = 4 + (theta/360)×2×66

or

height(theta) = 4 + (theta/(2pi))×2×66

now we need to find the angle theta for which we reach the height of 47 meters :

47 = 4 + (theta/360)×132

43 = (theta/360)×132

theta/360 = 43/132

theta = 360×43/132 = 117.2727272...°

= 2.046795214... rad

so, after starting at the lowest position at 4 meters we reach the height of 47 meters at an angle of about 117°.

then we get and stay above 47 meters until we get to

360 - theta = 242.7272727...°

= 4.236390093... rad

when going down again on the second half-circle of the trip.

that means we are at and above 47 meters for

(360 - theta) - theta = 360 - 2×theta = 125.4545455...°

= 2.18959488... rad

of the whole trip of 360° or 2pi. which takes 5 minutes.

the time we spend there is then

5 × (360 - 2×theta)/360 = 1.742424242... minutes

= 1 minute 44.54545454... seconds