Instructions: Find the value of the trigonometric ratio. Make sure to simplify the fraction if needed.

Sin A= Answer

(a) The result of the long division is a - b + c.

(b) The result of the long division is 3 · m² + 2 · m + 1 + (12 · m³ - 16 · m + 6 · m²) / (m⁴ - 3 · m² + 4).

Long division is an algebraic procedure with which we can find the result of a division of two polynomials. Now we proceed to present the entire procedure explained:

(a) a² - b² + 2 · b · c - c² divided by a + b - c.

1) Multiply a + b - c by a and subtract it from a² - b² + 2 · b · c - c²: Result: a / Remainder: - b² - a · b + 2 · b · c + a · c - c²

2) Multiply a + b - c by - b and subtract it from the result of 1): Result: a - b / Remainder: b · c + a · c - c²

3) Multiply a + b - c by c and subtract it from the result of 2): Result: a - b + c / Remainder: 0

The result of the long division is a - b + c.

(b) 3 · m⁷ - 11 · m⁵ + m⁴ + 18 · m³ - 8 · m + 3 · m² + 4 divided by m⁴ - 3 · m² + 4.

1) Multiply m⁴ - 3 · m² + 4 by 3 · m³ and subtract it from 3 · m⁷ - 11 · m⁵ + m⁴ + 18 · m³ - 8 · m + 3 · m² + 4: Result: 3 · m³ / Remainder: 2 · m⁵ + m⁴ + 6 · m³ - 8 · m + 3 · m² + 4.

2) Multiply m⁴ - 3 · m² + 4 by 2 · m and subtract it from the result of 1): Result: 3 · m³ + 2 · m / Remainder: m⁴ + 12 · m³ - 16 · m + 3 · m² + 4.

3) Multiply m⁴ - 3 · m² + 4 by 2 · m and subtract it from the result of 2): Result: 3 · m² + 2 · m + 1 / Remainder: 12 · m³ - 16 · m + 6 · m².

The result of the long division is 3 · m² + 2 · m + 1 + (12 · m³ - 16 · m + 6 · m²) / (m⁴ - 3 · m² + 4).

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The value of f(a)=4-2a+6[tex]a^{2}[/tex], f(a+h) is [tex]6a^{2} +6h^{2} -2a-2h+12ah[/tex] , [f(a+h)-f(a)]/h is 6h+12a-2 in the function f(x)=4-2x+6[tex]x^{2}[/tex].

Given a function f(x)=4-2x+6[tex]x^{2}[/tex].

We are told to find out the value of f(a), f(a+h) and [f(a+h)-f(a)]/hwhere h≠0.

Function is like a relationship between two or more variables expressed in equal to form.The value which we entered in the function is known as domain and the value which we get after entering the values is known as codomain or range.

f(a)=4-2a+6[tex]a^{2}[/tex] (By just putting x=a).

f(a+h)==[tex]4-2(a+h)+6(a+h)^{2}[/tex]

=4-2a-2h+6([tex]a^{2} +h^{2} +2ah[/tex])

=4-2a-2h+6[tex]a^{2} +6h^{2} +12ah[/tex]

=[tex]6a^{2} +6h^{2}-2a-2h+12ah[/tex]

[f(a+h)-f(a)]/h=[[tex]6a^{2} +6h^{2}-2a-2h+12ah[/tex]-(4-2a+6[tex]a^{2}[/tex] )]/h

=[tex](6a^{2} +6h^{2} -2a-2h+12ah)/h[/tex]

=[tex](6h^{2} -2h+12ah)/h[/tex]

=6h+12a-2.

Hence the value of function f(a)=4-2a+6[tex]a^{2}[/tex], f(a+h) is [tex]6a^{2} +6h^{2} -2a-2h+12ah[/tex] , [f(a+h)-f(a)]/h is 6h+12a-2 in the function f(x)=4-2x+6[tex]x^{2}[/tex].

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The rate of the slower jet is 672 miles per hour.

The rate of the slower jet is 752 miles per hour.

Two jets leave an air base at the same time and travel in opposite directions.

One jet travels 80 mi/h faster than the other.

let

x = rate of the slower jet

faster jet = 80 + x

Therefore,

rate = distance / time

The two jets are 11392 miles apart.

(80 + x)8 + 8x = 11392

640 + 8x + 8x = 11392

640 + 16x = 11392

16x = 11392 - 640

16x = 10752

divide both sides by 16

x = 10752 / 16

x = 672 miles per hour

rate of the faster jet = 80 + 672 = 752 miles per hour

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Based on the amount that Steve Weatherspoon wants to withdraw every year beginning in June 30, 2024, and the interest rate, the balance on June 30th 2023 should be $45,203.

The fact that Steve Weatherspoon wants to be able to withdraw a particular amount every year, this makes this amount an annuity.

The value in 2023 would therefore be the present value of the annuity that will then accrue to the required amounts as the years go by.

The present value of an annuity is:

= Annuity amount per year  x Present value interest factor of an annuity, 11%, 3 years between 2024 and 2027

Solving gives:

= 13,126.25 x 3.44371

= $45,203

In conclusion, the balance on the fund in 2023 should be $45,203 in order for Steve Weatherspoon to achieve his objectives.

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Answer:

Step-by-step explanation:

Here's the solution

f^(-1)(x)=(\sqrt(5(x-10)))/(5), -(\sqrt(5(x-10)))/(5)

Using the probability concept, we have that:

A probability is given by the number of desired outcomes divided by the number of total outcomes.

The total number of students is given by:

26 + 3 + 9 + 10 + 6 + 5 + 7 = 66

Of those, 26 play only football, hence the probability is:

p = 26/66 = 13/33

Which means that option c is correct for question 19.

Of those same 66 students, 9 + 7 = 16 play baseball but not football, hence the probability is:

p = 16/66 = 8/33

Which means that option d is correct for question 20.

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Using the permutation formula, there are 157,410 ways for the medals to be distributed.

The number of possible permutations of x elements from a set of n elements is given by:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

In this problem, 3 athletes are chosen from a set of 55, hence the number of ways is given by:

P(55,3) = 55!/52! = 157,410

157,410 ways for the medals to be distributed.

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Answer: [tex]y= x-8[/tex]

Step-by-step explanation:

[tex]y-x=-8[/tex]

[tex]y-x+x=-8+x[/tex]

Final Answer: [tex]y=x-8[/tex]

Answer:

10 + 4i

Step-by-step explanation:

2 sqrt (25) + sqrt (-16) =

2 * 5     + 4i

10 + 4i

Answer:

[tex]10 + 4i[/tex]

Step-by-step explanation:

Original Equation:

[tex]2\sqrt{25} + \sqrt{-16}[/tex]

Simplify sqrt(25)

[tex]2*5+\sqrt{-16}[/tex]

Simplify multiplication

[tex]10+\sqrt{-16}[/tex]

Split the radical -16 into two, using the identity: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{ab}[/tex]

[tex]10 + \sqrt{-1}*\sqrt{16}[/tex]

Simplify the sqrt(16) and sqrt(-1) using definition of an imaginary number

[tex]10 + 4i[/tex]

The value of the probability P(A and D) is 0.30

Using the probabilities on the tree diagram, we have the following parameters

P(A) = 0.5

P(D/A) = 0.6

The probability P(A and D) is then calculated using the following formula

P(A and D) = P(A) * P(D/A)

Substitute the known values in the above equation

P(A and D) = 0.5 * 0.6

Evaluate the product

P(A and D) = 0.30

Hence, the value of the probability P(A and D) is 0.30

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The linear inequality that is graphed in the xy plane is: C. 2x + 3y ≤ 4.

Values in the shaded part are the solution of a a linear inequality. Thus, a dotted or dashed line is used on the graph when the inequality sign is either "<" or ">". On the other hand, when a line that is not dotted or dashed is used when the inequality sign is either "≤" or "≥". These lines, dotted or not are the boundary lines.

Also, when the shaded area is above the boundary line, the sign "≥" or ">" is used. When the shaded part is beneath the boundary line, "≤" or "<" is used in the linear inequality.

The graph given has a boundary line that is not dashed or dotted, and also, the shaded part is beneath the boundary line. Therefore, the inequality sign to use is "≤".

Find the slope:

Slope (m) = rise/run = -4/3 / 2 = -4/6

m = -2/3

y-intercept (b) = 4/3.

Substitute m = -2/3 and b = 4/3 into y ≤ mx + b:

y ≤ -2/3x + 4/3

Rewrite

3y ≤ -2x + 4

2x + 3y ≤ 4

The answer is: C. 2x + 3y ≤ 4.

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Answer:

m = 3/2

Step-by-step explanation:

-7(m-1)= -(3m-1)

-7m + 7 = -3m + 1

-4m = -6

m = 3/2

Check.
-7(m-1)= -(3m-1)

-7(3/2 - 1) = -(3*3/2 - 1)

-7(1/2) = -(4.5 - 1)

-3.5 = -3.5

Use the distributive property to multiply −7 by m−1.

To find the opposite of 3m−1, find the opposite of each term.

The opposite of −1 is 1.

Add 3m to both sides.

Combine −7m and 3m to get −4m.

Subtract 7 from both sides.

Subtract 7 from 1 to get −6.

Divide both sides by −4.

Reduce the fraction -6/-4 to its lowest expression by extracting and canceling −2.

Answer: m=3/4 ✅

Answer:

(4,8)

Step-by-step explanation:

hihihihihihihihohihih

Applying the definition of the median of a triangle: x = 9; CD = 28; DB = 28.

The median of a triangle can be defined as a line segment in a triangle that joins the vertex of a triangle to the midpoint of the side of the triangle that is opposite the vertex.

Given triangle ABC, where AD is the median and D is the midpoint of side AB, therefore:

Side CD equals side DB.

Side CD = 5x - 17

Side DB = 3x + 1

Make both segments equal to each other. Therefore, we would have the equation:

5x - 17 = 3x + 1

Solve for x

Add both sides by 17

5x - 17 + 17 = 3x + 1 + 17

5x = 3x + 18

Subtract 3x from both sides

5x - 3x = 3x + 18 - 3x

2x = 18

Divide both sides by 2

2x/2 = 18/2

x = 9

CD = 5x - 17

Plug in the value of x

CD = 5(9) - 17

CD = 45 - 17

CD = 28 units.

DB = 3x + 1

Plug in the value of x

DB = 3(9) + 1

DB = 27 + 1

DB = 28 units.

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Answer:$2346

Step-by-step explanation: Assuming that the students' numbers start at 1, we have 1+2+3+4.....+65+66+67+68 as the total amount of money raised. We can see that 1+68 = 69 and 2+67 also equals 69. So, we can use this method to figure out how many 69s are in the sum. Since 68 divided by 2 is 34, there are 34 69s in the sum. 34x69 = 2346.

Answer: [tex]50^{\circ}[/tex]

Step-by-step explanation:

[tex]m\angle CAB=m\angle CAD=80^{\circ}[/tex] (angle bisector)

[tex]m\angle ACB=20^{\circ}[/tex] (one-eighth of [tex]\angle BAD[/tex])

[tex]m\angle ABC=80^{\circ}[/tex] (sum of angles in a triangle)

[tex]\overline{AC} \cong \overline{BC}[/tex] (converse of base angles theorem)

[tex]m\angle ADC=50^{\circ}[/tex] (base angles theorem)

Answer:

See below

Step-by-step explanation:

[tex]A = \{2, 4, 6\}[/tex]

[tex]A^c =\{1, 3, 5\}\\B = \{4, 5, 6\}\\B^c = \{ 1,2, 3\}[/tex]

To prove 1 we have

[tex](A \bigcup B) = \{2, 4, 6\} \bigcup \{4, 5, 6\} = \{2, 4, 5, 6\}\\(A \bigcup B) ^c =\bold{\{1,3\}}\\A^c \bigcap B^c = {\{1, 3, 5\} \bigcap \{1, 2, 3\} =\bold{\{1,3\}}\\[/tex]

Hence proved

To prove 2

[tex](A \bigcap B) = \{2, 4, 6\} \bigcap \{4, 5, 6\} = \{4, 6\}\\\\(A \bigcap B)^c = \bold{\{1,2, 3, 5\}}\\A^c \bigcup B^c = \{1, 3, 5\} \bigcup \{1, 2, 3\} = \bold{\{1,2,3,5\}}\[/tex]

Hence proved

Notes

The complement of a set is the set of all elements not in the set

Here the set of all outcomes is {1, 2, 3,4,5, 6}

So if A = {2, 4, 6} then the complement of A s=is the set of outcomes not in A ie the set {1, 3, 5} which is the set of odd numbers on a die throw

The union of two sets is the set of all elements in both sets without duplication

The intersection of two sets is the set of all elements common to both sets

Answer:

  H

Step-by-step explanation:

Plane K is shown with two lines. Points on one of them are A, B, G. Points on the other one are G, M, N.

The point in plane K that is not shown on any line is point H.

Answer:

Step-by-step explanation:

9. D

10.D

11.E

12.D

Answer:

9.

[tex]x^2-4\\(x-2)(x+2)[/tex]

10.

[tex]x^2-25\\(x-5)(x+5)[/tex]

11.

[tex]36x^4-4x^2\\4x^2(9x^2-1)\\4x^2(3x+1)(3x-1)[/tex]

12.

[tex]x^2+100\\[/tex]

This polynomial cannot be factored by using the difference of squares method

Using a linear function, it is found that it will take 13 hours for the candle to burn down to a height of .25 inches.

A linear function is modeled by:

y = mx + b

In which:

Considering the situation described, the y-intercept is of 10 while the slope is of -0.75, hence the height of the candle after t hours is given by:

h(t) = 10 - 0.75t

The height will be of 0.25 inches when h(t) = 0.25, hence:

0.25 = 10 - 0.75t

0.75t = 9.75

t = 9.75/0.75

t = 13 hours.

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Answer: 55

Step-by-step explanation: There is one simple theorem we need to know in order to solve for this.

Triangle Angle-Sum Property: All three angles interior angles of a triangle add up to 180°.

Thus, we could label the points A, B, and C, and set up an algebraic expression.

Let A = 27, B = 98, and C = x.

A + B + C = 180°.

Substituting A, B, and C we get:

27 + 98 + x = 180°.

Adding, we get:

125 + x = 180°

Subtracting 125 by 180, we get:

x = 55°

Thus, the angle X is 55°.

We could have simply solved this by just doing 180 - 98 - 27 = 55 in the first place, but I wanted to show you how I got such results.

the inequality gives the 'y' value as -4

An inequality is a mathematical tool that compares two values, expressing when one is less than, greater than, or not equal to the other value.

For instance,

a ≠ b says that a is not equal to b

a < b says that a is less than b

a > b says that a is greater than b

From the information given, that 3y+1 is less than -11 . It can be expressed as;

3y + 1 ∠ -11

Let's collect like terms

3y ∠ -11 - 1

add like terms

3y ∠ -12

make 'y' the subject of formula

y ∠ -12/3

y ∠ -4

The inequality gives the 'y' value as -4

Thus, the inequality gives the 'y' value as -4

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The single payment six months from now is required to settle the debt will be $4042.

The interest rate monthly will be 8/1200.

The number of time period given is 6 months.

Therefore, the future value will be:

= 2000(1 - 8/1200)^6

= 2081.35

The present value will be:

= 2000(1 - 8/1200)^3

= 1960.53

Therefore, the single payment will be:

= 2081.35 + 1960.53

= 4042

In conclusion, the correct amount is $4042.

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The upper estimate and lower estimate become R5 = 16 and L5 = −64

According to the statement

we have given that the some values of the increasing function f and we have to find the lower estimate and upper estimate values.

So, For this purpose, we know that the

the values of x is 10 14 18 22 26 30

And the values of f(x) is −14 −8 −1  1  4  7

So,

Δx = 4 because

L5 = 4(−12 − 6 − 2 + 1 + 3) = −64,

And

R5 = 4(−6 − 2 + 1 + 3 + 8) = 16.

And from these functions we see that the

The function being increasing, the Riemann sum with left endpoint L5 = −64 is a lower estimate,

And while the sum with right endpoints R5 = 16 is an upper estimate.

So, The upper estimate and lower estimate become R5 = 16 and L5 = −64

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Answer:

x = 31

Step-by-step explanation:

Given:

f(x) =

[tex]2 {x}^{2} - 1[/tex]

g(x) = x + 2

We will first find g(2).

g(2) = 2 + 2 = 4

Next we will find f(g(2)).

f(g(2))= f(4) =

[tex]2( {4}^{2} ) - 1 \\ = 2(16) - 1 \\ = 32 - 1 \\ = 31[/tex]

Answer:

ratio = 4

Step-by-step explanation:

According to the given table:

• the number of days with no fire incidents

  = 16

• the number of days with more than 5 fire incidents

  = 2 + 2

  = 4

Conclusion :

the ratio of the number of days with no fire incidents

to the number of days with more than 5 fire incidents is :

16 to 4 (16 : 4)

Then

The ratio = 4

Answer:

A

Step-by-step explanation:

If digits can be repeated, that means there are 10 options for each place in the pin code. 10*10*10*10 = 10,000

If digits can not be repeated, there are 10 options for the first digit, 9 options for the second digit, 8 options for the third digit, and 7 options for the fourth digit. 10*9*8*7 = 5040