Given the function value of the acute angle, find the other five trigonometric function values. cosα=√3/3 sinα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) tanα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cscα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) secα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cotα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

The multiplicative property of equality states that the same number can be added to or multiplied by both sides of an equation to obtain an equivalent equation. In this case, dividing both sides by 6 gives us u = 78/6 = 13.

The multiplicative property of equality states that if two numbers are equal, then multiplying both sides of the equation by the same number will also result in an equation that is still equal. In other words, if a=b, then ac=bc. We can use this property to solve for the variable u in the equation 78=6u.

To isolate the variable on one side of the equation, we can divide both sides by 6. This will give us:

78/6 = 6u/6

Simplifying the equation gives us:

13 = u

So the solution for u is 13.

In conclusion, the multiplicative property of equality with whole numbers was used to solve for the variable u in the equation 78=6u. The solution is u=13.

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The height of the building, given the angle of elevation and the distance from the top of the building, is such that he height of the building is 95 meter.

We shall assume that the building, the distance from the foot and the angle, are in a right - angled triangle.

This means that the height of the building is the opposite side and the given distance is the adjacent side.

The relevant operation would be Tan.

The height of the building would be:

Tan ( 60 ° ) = Height / 55 m

Height = Tan ( 60 ° ) x 55

Height = 95 meters

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The linear functions of the scenario are y = 12x and y = 24/2x

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

(1, 12) and (2,24)

From the question, we understand that the function is a linear function

A linear function is represented as

y = mx + c

Using the above as a guide, we have the following equations

m + c = 12

2m + c = 24

Subtract the equations

m = 12

Substitute 12 for m in m + c = 12

12 + c = 12

Evaluate

c =0

So, the equation is y = 12x

An equivalent equation is y = 24x/2

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Answer:

12/18, 2/3, and 18/27.

Step-by-step explanation:

In order to find ratios that are equivalent to a certain fraction they must have common divisibles.

In this case...
[tex]\frac{6}{9}[/tex]
If the common divisible is two....
[tex]6\times2=12[/tex]
[tex]9\times2=18[/tex]
[tex]=\frac{12}{18}[/tex]

You could also simplify the ratio:
[tex]\frac{6}{9} \div3=\frac{2}{3}[/tex]
[tex]=\frac{2}{3}[/tex]

If the common divisible is three:
[tex]6\times3=18[/tex]
[tex]9\times3=27[/tex]
[tex]=\frac{18}{27}[/tex]

The average rate of change of g(x) over the interval [2, 5] is -2.1.

The formula to calculate the slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is:

slope = (y₂ - y₁) / (x₂ - x₁)

Using this formula, we can calculate the slopes of the two secant lines for f(x) and g(x) over the interval [2, 5]. Let's start with f(x):

slope_f = (f(5) - f(2)) / (5 - 2)

= (-0.1(5)² - (-0.1(2)²)) / (5 - 2)

= (-0.1(25) + 0.1(4)) / 3

= (-2.5 + 0.4) / 3

= -2.1 / 3

= -0.7

Therefore, the average rate of change of f(x) over the interval [2, 5] is -0.7.

Now, let's calculate the average rate of change of g(x):

slope_g = (g(5) - g(2)) / (5 - 2)

= (-0.3(5)² - (-0.3(2)²)) / (5 - 2)

= (-0.3(25) + 0.3(4)) / 3

= (-7.5 + 1.2) / 3

= -6.3 / 3

= -2.1

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Complete Question:

how do the average rates of change for the pair of functions compare over the given interval

f(x)= -0.1x²

g(x)= -0.3x²

2≤x≤5

Answer:

hope this helps.

Step-by-step explanation:

To find out how many points Ava had at the end of round one, we need to work backwards from the end of the game using probability. Here are the steps to do so:

Step 1: At the end of the game, Ava had 50 points. This was after she lost 2/7 of her total points in the fifth round. Let's call the number of points she had before the fifth round X. So:

50 = X - (2/7)X

Step 2: Solve for X by combining like terms:

50 = (5/7)X

Step 3: Multiply both sides of the equation by 7/5 to isolate X:

X = 70

Step 4: Now we know that Ava had 70 points before the fifth round. This was after she increased her total points by 75% in the fourth round. Let's call the number of points she had before the fourth round Y using probability. So:

70 = Y + (75/100)Y

Step 5: Solve for Y by combining like terms:

70 = (175/100)Y

Step 6: Multiply both sides of the equation by 100/175 to isolate Y:

Y = 40

Step 7: Now we know that Ava had 40 points before the fourth round. This was after she decreased her point total by 3/5 in the third round. Let's call the number of points she had before the third round Z. So:

40 = Z - (3/5)Z

Step 8: Solve for Z by combining like terms:

40 = (2/5)Z

Step 9: Multiply both sides of the equation by 5/2 to isolate Z:

Z = 100

Step 10: Now we know that Ava had 100 points before the third round. This was after she increased her point total by 25% in the second round. Let's call the number of points she had before the second round A using probability. So:

100 = A + (25/100)A

Step 11: Solve for A by combining like terms:

100 = (125/100)A

Step 12: Multiply both sides of the equation by 100/125 to isolate A:

A = 80

Step 13: Now we know that Ava had 80 points before the second round, which means she had 80 points at the end of the first round.

Therefore, the answer is 80 points using probability.

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Answer:

32 cubic feet

Step-by-step explanation:

The formula for a cylinder is [tex]\pi r^{2} h[/tex].

The radius of the cylinder is equal to 1.5 feet, since it is [tex]\frac{diameter}{2}[/tex].

Plugging in, the cylinder's full volume is  [tex]6\pi 1.5^2[/tex] which is approximately 42.4 cubic feet.

To find the amount of water when it is 3/4 full, multiply 42.4 x .75, to get around 31.8, and 32 when rounded

Answer:

0.00265251989

Hope this helped.

The difference in area between a circle with a radius of 10 centimeters and a square inscribed in it is  114 cm².

The difference in area between a circle with a radius of 10 centimeters and a square inscribed in it can be found by calculating the area of the circle and the area of the square and then subtracting the two.

First, calculate the area of the circle using the formula

A = πr²,

where A is the area and r is the radius.

A = π(10)² = 100π ≈ 314.16 square centimeters

Next, calculate the area of the square. Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square. The diameter of the circle is 2r, or 20 centimeters.

Using the Pythagorean theorem, we can find the side length of the square:

s² + s² = (20)²

2s² = 400

s² = 200

s ≈ 14.14 centimeters

The area of the square is s² or (14.14)² ≈ 199.97 square centimeters.

Finally, subtract the area of the square from the area of the circle to find the difference:

314.16 - 199.97 ≈ 114.19 square centimeters

To the nearest whole, the difference in area is 114 square centimeters.

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Answer:

See explanation

Step-by-step explanation:

50 is 5*10, so multiply 3 by 5 also to get 5 and 15/50. First answer is 15

5 and 15/50 is also equal to 4 and 65/50. second answer is 65

carry the numerator on the second line to get 30 for the third answer.

finally, subtract 3 from 4 to get 1, and subtract 30 from 65 to get 35/50.

last two answers are 1, and 35.

The answer to this question is as follows The plaque measures 14 inches equation tall as a result.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.  

Because we are aware that the plaque has a parallelogram shape, we may determine its height using the formula for a parallelogram's area. The formula for a parallelogram's area is:

Base area x height

In this instance, we are aware that the area is 126 square inches, and the base (the bottom border) is 9 inches. Thus, we can enter those values into the formula to find the height:

126 = 9h

h = 126/9

h = 14

The plaque measures 14 inches tall as a result.

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The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Let's check all the options, then we have

A) y = 6x + 14, the equation is a linear equation because the degree is one.

B) y = x/4 + 1, the equation is a linear equation because the degree is one.

C) y = x³, the equation is a cubic equation because the degree is three.

D) y = 3/x  +2, the equation is a non-linear equation because the degree is negative one.

The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.

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Answer: B

Step-by-step explanation:

Answer: B -_-

Step-by-step explanation:

0 0=2

The correct statement regarding the monthly payments is given as follows:

D. Yes, because $320 is greater than the amount of interest he is

charged per year.

The monthly payment formula is defined by the equation as follows:

[tex]A = P\frac{\frac{r}{12}\left(1 + \frac{r}{12}\right)^n}{\left(1 + \frac{r}{12}\right)^n - 1}[/tex]

In which the parameters are listed as follows:

The parameter values for this problem are given as follows:

P = 2700, r = 0.12, n = 12.

Hence:

r/12 = 0.12/12 = 0.01.

Hence the monthly payment is calculated as follows:

A = 2700 x 0.01 x (1.01)^12/(1.01^12 - 1)

A = $240.

The interest is less than $320, hence he will manage to pay off the loan.

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Answer:

9/7 is smallerrrr

The limit of the given function can be evaluated by simplifying the expression and then substituting the value of x.

First, we can factor the numerator and denominator of the expression:
lim−(x−>9)(7x−63)/(x^2−2x−63)=lim−(x−>9)(7(x−9))/((x−9)(x+7))
Next, we can cancel out the common factor of (x-9) from the numerator and denominator:
lim−(x−>9)(7)/(x+7)
Now, we can substitute the value of x=9 into the expression:
lim−(x−>9)(7)/(x+7)=7/(9+7)=7/16
Therefore, the limit of the given function is 7/16.

As for the second part of the question, we can use the result of Example 1.3 to verify the formulas. According to Example 1.3, if x+y divides x^n+y^n for n≥0, then x+y also divides x^(n+1)+y^(n+1). Using this result, we can prove that x+y divides x^2n+1+y^2n+1 for n≥0 by induction.
Base case: n=0
x^2(0)+1+y^2(0)+1=x+y, which is divisible by x+y.
Inductive step: Assume that x+y divides x^2n+1+y^2n+1 for some n≥0. We need to show that x+y divides x^2(n+1)+1+y^2(n+1)+1.
x^2(n+1)+1+y^2(n+1)+1=(x^2n+1+y^2n+1)(x^2+y^2)
Since x+y divides x^2n+1+y^2n+1 by assumption, and x+y divides x^2+y^2 by Example 1.3, x+y also divides (x^2n+1+y^2n+1)(x^2+y^2) by the distributive property. Therefore, x+y divides x^2(n+1)+1+y^2(n+1)+1 for n≥0.

Finally, for the third part of the question, we can use the fact that n is a positive integer to show that (n^3) is also a positive integer. Since n is a positive integer, n^3 is also a positive integer because the product of three positive integers is a positive integer. Therefore, (n^3) is a positive integer.

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There are 8 possible ways for the first 2 finishers to complete the marathon with different genders. There are 20 possible ways for Chantal to finish the race before David.

There are a couple of different ways to approach this problem, but one common method is to use the multiplication principle, which states that if there are A ways to do one thing and B ways to do another, then there are A*B ways to do both. We can apply this principle to both parts of the question.

a) If the first 2 finishers must have different genders, then we can think about the possible combinations of boys and girls. There are 2 options for the first finisher (either a boy or a girl), and then there are 4 options for the second finisher (either a boy or a girl, but not the same gender as the first finisher).

So, using the multiplication principle, we can find the total number of ways for the first 2 finishers to complete the marathon with different genders:

2 * 4 = 8

Therefore, there are 8 possible ways for the first 2 finishers to complete the marathon with different genders.

b) If Chantal must finish the race before David, then we can think about the possible positions for Chantal and David. There are 5 possible positions for Chantal (first, second, third, fourth, or fifth), and then there are 4 possible positions for David (second, third, fourth, or fifth, but not before Chantal).

So, using the multiplication principle, we can find the total number of ways for Chantal to finish before David:

5 * 4 = 20

Therefore, there are 20 possible ways for Chantal to finish the race before David.

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Answer:

AB = 6.71

Step-by-step explanation:

The vertical leg of the right triangle

= absolute value of difference in y-coordinates between A(-1, 6) and B(5, 3)

= |6  - 3| = |3|
= 3

The horizontal leg of the right triangle
= absolute value of difference in x-coordinates between A(-1, 6) and B(5, 3)
= |- 1 - 5|
=|- 6|

= 6

By the Pythagorean theorem, the hypotenuse AB is related to each of these two legs by the formula

AB² = 3² + 6²
AB² = 9 + 36

AB² = 45

AB = √45

or

AB = 6.7082039324

= 6.71 significant to 3 significant figures

Significant figures means number of digits excluding leading and trailing zeros

Step-by-step explanation:

a = 55° because angles on a straight line (around a single point on one side of the line) add up to 180°.

b = 75° because alternate angles (with parallel lines) are equal.

c = 50°, because alternate angles (with parallel lines) are equal.

d = 50°, because corresponding angles (with parallel lines) are equal.

The final velocity of the jet flying in the north direction after accelerating for 15s is 473 m/s.

When observed from a specific point of view and as measured by a specific unit of time, velocity is the direction at which an item is moving and serves as a measure of the pace at which its position is changing. How quickly or slowly an object is travelling can be determined by its velocity and speed. Being a vector quantity, we need to define velocity in terms of both magnitude (speed) and direction. A body is considered to be accelerating if the magnitude or direction of its velocity changes.

Given,

The initial velocity u = 200 m/s

Acceleration of jet a = 18.2 m/s²

Time taken t = 15s

We are asked to find the final velocity v of the jet.

W can use the following formula to find the final velocity.

v = u+ at

  = 200 + (18.2) × 15

 = 473 m/s (north)

Therefore the final velocity of the jet flying in the north direction after accelerating for 15s is 473 m/s.

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After solving the equation, the solution if the equation is -2 and -11/5.

To solve the rational equation we first express the given statement in the form of mathematical equation. After that we solve the equation.

As the statement is 5 over X + 2. It can be written as 5/(x + 2)

Then the second statement is -11 over X to the 2nd + 2X which is written as -11/(x² + 2x).

There have equal sign between both expression. So the expression is

5/(x + 2) = -11/(x² + 2x)

Now solve it.

Multiply by (x² + 2x) on both side, we get

5(x² + 2x)/(x + 2) = -11

Multiply by (x + 2) on both side, we get

5(x² + 2x) = -11(x + 2)

Now simplify using the distributive property

5x² + 10x = -11x - 22

Add 11x on both side, we get

5x² + 21x = - 22

Add 22 on both side, we get

5x² + 21x + 22 = 0

Now we factor the equation

5x² + (11 + 10)x + 22 = 0

5x² + 11x + 10x + 22 = 0

x(5x + 11) + 2(5x + 11) = 0

(5x + 11)(x + 2) = 0

Now equating equal to 0.

5x + 11 = 0                     x + 2 = 0

5x = -11                          x = -2

x = -11/5

The solution if the equation is -2 and -11/5.

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The complete question is:

Solve the rational equation 5 over X + 2 equals -11 over X to the 2nd + 2X.

The solutions are x ≈ -2.744 and x ≈ -17.106.

To solve this equation, we need to get rid of the fractions by multiplying each term by the least common multiple (LCM) of the denominators. The LCM of x+6 and 5x+30 is (x+6)(5x+30).

Multiplying each term by the LCM gives us:
(5)(x+6)(5x+30)/(x+6) = (7)(x+6)(5x+30)/(5x+30) - 2(x+6)(5x+30)

Simplifying the fractions and distributing the terms gives us:
5(5x+30) = 7(x+6) - 2(x+6)(5x+30)

Expanding and simplifying the terms gives us:
25x + 150 = 7x + 42 - 10x^2 - 180x - 360

Combining like terms and rearranging gives us:
10x^2 + 198x + 468 = 0

Using the quadratic formula, we can find the values of x:
x = (-198 ± √(198^2 - 4(10)(468)))/(2(10))

Simplifying gives us:
x = (-198 ± √(39204 - 18720))/(20)

x = (-198 ± √20484)/(20)

x = (-198 ± 143.126)/(20)

The two solutions for x are:
x = (-198 + 143.126)/(20) ≈ -2.744
x = (-198 - 143.126)/(20) ≈ -17.106

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This construction is also known as the "perpendicular bisector" construction.

To construct a line through a given point that is perpendicular to a given line using a ruler and compass, you will need to follow these steps:

By following these steps, you have used the ruler and compass to construct a line through a given point that is perpendicular to a given line. This construction is also known as the "perpendicular bisector" construction.

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The standard form to this equation is x=2/3.

This equation is in the form of a linear equation in one variable, where the variable is x.

The equation is written as x=2/3, meaning that the value of x is equal to 2/3.

The equation can be interpreted as the ratio of two numbers, 2 and 3. The numerator, 2, represents the number of parts, and the denominator, 3, represents the total number of parts.

This equation can be used to solve for the fraction of the total number of parts represented by the numerator. In this case, the fraction is 2/3, or 2 parts out of a total of 3 parts.

The equation can also be interpreted as a proportion. If we make the numerator the unknown value, x, then the equation becomes x/3 = 2/3. This equation can be solved using the cross-multiplication method.

By multiplying the denominators together and setting them equal to each other, then solving for x, we get x = 2/3. This equation shows that the value of x is equal to 2/3 of the total number of parts.

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Bob did not commit tax fraud based on the given information.

To calculate Bob's net income, we need to start with his adjusted income, which is his income minus his expenses and deductible:

Adjusted Income = Income - Expenses - Deductible

Adjusted Income = $750 - $100 - $150

Adjusted Income = $500

Next, we need to determine the amount of tax that Bob owes based on his tax bracket information:

Tax Owed on first $200 = $200 * 0.10 = $20

Tax Owed on next $300 ($200 to $500) = $300 * 0.20 = $60

Tax Owed on remaining $0 ($500 and above) = $0 * 0.40 = $0

Total Tax Owed = $20 + $60 + $0 = $80

Now, we can calculate Bob's net income by subtracting his tax owed from his adjusted income:

Net Income = Adjusted Income - Tax Owed

Net Income = $500 - $80

Net Income = $420

Finally, we can compare Bob's returned amount to his tax owed to see if he committed tax fraud:

Returned Amount - Tax Owed = $120 - $80 = $40

Since Bob's returned amount is less than his tax owed, he did not commit tax fraud. His net income after taxes is $420.

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By following this process, we were able to evaluate each expression and get the answers of 590, 181, 25, and 6 respectively.

a) (fog)(4) = (f(g(4)) = f(7(4)^2+6) = 5(7(16)+6) = 5(118) = 590

b) (gof)(2) = (g(f(2)) = g(5(2)) = 7(5)^2+6 = 7(25)+6 = 181

c) (fof)(1) = (f(f(1)) = f(5(1)) = 5(5) = 25

d) (gog)(0) = (g(g(0)) = g(7(0)^2+6) = 7(0)+6 = 6

To find the expressions given, we need to first use the definition of a function composition. This means that when we have a function \(f\) composed with a function \(g\), this is represented by \(f\circ g\). When we use a function composition, we evaluate the function \(g\) first, then plug the result into the function \(f\).

For example, in part (a) we have the expression \(f\circ g(4)\). We need to first evaluate \(g(4)\), which gives us \(7(4)^2+6\). Then we plug this result into the function \(f\) to get \(f(7(4)^2+6)= 5(7(4)^2+6) = 590\). We can repeat this process for the rest of the parts. For part (b) we have \(g\circ f(2)\). Evaluating \(f(2)\) gives us \(5(2)\), which we then plug into \(g\) to get \(g(5(2))= 7(5)^2+6 = 181\).

For part (c) we have \(f\circ f(1)\), so evaluating \(f(1)\) gives us \(5(1)\) and then plugging this into \(f\) gives us \(f(5(1))= 5(5) = 25\). Finally, for part (d) we have \(g\circ g(0)\), so evaluating \(g(0)\) gives us \(7(0)^2+6\), which we plug into \(g\) to get \(g(7(0)^2+6) = 7(0)+6 = 6\).

In conclusion, we can use function composition to find the expressions given. We start by evaluating the inner function, then plugging the result into the outer function.

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In order for the system of equations to be consistent, k must not be equal to 31.6087.

In order for the system of equations to be consistent, the determinant of the coefficient matrix must not be equal to zero. The coefficient matrix is:

| -3  4  7 |
| -11 24 k |
| 2  -5 -8 |

The determinant of this matrix is:

(-3)(24)(-8) + (4)(k)(2) + (7)(-11)(-5) - (7)(24)(2) - (4)(-11)(-8) - (-3)(k)(-5)

Simplifying this expression gives:

576 + 8k + 385 - 336 - 352 + 15k = 0

Solving for k gives:

23k = 727

k = 727/23

k ≈ 31.6087

Therefore, in order for the system of equations to be consistent, k must not be equal to 31.6087.

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The profit is maximum when 40 lawns are mowed.

A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

Given is that Kieran is starting a lawn-mowing buisness in her neighborhood. She creates a graph to help her determine what to charge customers per lawn to maximize her profits. She uses {c} to represent the number of lawns she mows and {y} to represent her profit in dollars.

The profit is maximum when 40 lawns are mowed as at this point the the peak of the parabola occurs.

Therefore, the profit is maximum when 40 lawns are mowed.

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