How do I correctly simplify the expression and where did the student go wrong?

Check the picture below.

The length of the hypotenuse (x) in the given right triangle is approximately 10 units.

Hence the correct option is B.

Given is right triangle with perpendicular side having measurement of 5 units,

The acute angle between the base and the hypotenuse = 30°

The acute angle between the perpendicular and the hypotenuse = 60°

We need to find the length of the hypotenuse (x).

We can use the trigonometric ratio for sine (sin) to relate the lengths of the sides.

In this case, we'll use the sine of the 30° angle:

sin(30°) = opposite/hypotenuse

sin(30°) = 5/x

To solve for x, we can rearrange the equation:

xsin(30°) = 5

x = 5 / sin(30°)

Now, we can calculate the value of x using this equation:

x = 5 / sin(30°)

x ≈ 10 units

Therefore, the length of the hypotenuse (x) in the given right triangle is approximately 10 units.

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Answer:

What are your goals for taking a gap year?

Are you looking to gain new experiences, skills, or perspectives?

Do you want to travel, learn a new language, or volunteer?

Are you trying to improve your mental health or overall well-being?

What are your current academic or career plans?

Do you have a clear idea of what you want to study or what career you want to pursue?

Have you already been accepted into a program or job?

What are your financial circumstances?

Can you afford to take a gap year without negatively impacting your future plans?

Do you have savings or a plan to work during your gap year?

How will taking a gap year affect your relationships with friends and family?

Will you be able to stay in touch with loved ones while you're away?

Will you miss any important events or milestones?

What are the potential risks and rewards of taking a gap year?

Could it delay your academic or career goals?

Could it help you gain valuable experience or clarity about your future plans?

How do you feel about the idea of taking a gap year?

Are you excited about the possibilities, or do you feel uncertain or anxious?

By answering these questions honestly, you should have a better idea of whether taking a gap year is the right decision for you. Keep in mind that there is no one-size-fits-all answer, and what works for one person may not work for another. Ultimately, the decision should be based on your own unique goals, circumstances, and desires.

Step-by-step explanation:

Answer:

Step-by-step explanation:

For a right angle triangle we use Pythagoras theorem

hypotenuse²=x²+(x+18)²

                    =x²+x²+36x+324

                    =2x²+36x+324

Answer: X = -3

Step-by-step explanation:

a. The sketch is attached

b. The vertical height is 18.13 feet

c. The given angle 65° is the angle of elevation.

d. The foot of the ladder is 8.45 feet away from the wall.

7. The angle of elevation is congruent to angle of depression because they are on the alternate sides of two parallel lines.

8.

a. Diagram of triangle CDE is attached

b. The formula that can be used to solve triangle CDE is the Law of Sines

c.

∠ e = 11

c = 75.47

d = 8.77

b. To find the vertical height of the ladder that is touching the wall, we use the trigonometric function

sin 65 = vertical height / height of ladder

sin 65 = vertical height / 20

vertical height = sin 65 * 20

vertical height = 18.13 feet

c. The given angle 65° is the angle of elevation.

d. To find how far the foot of the ladder is from the wall, we can use the trigonometric function cosine.

cos 65° = x / 20

x = 20 cos 65°

x ≈ 8.45 feet

7. The statement is true. The angle of elevation is the angle between the horizontal line and the line of sight from the observer to the object above the horizontal line.

The angle of depression is the angle between the horizontal line and the line of sight from the observer to the object below the horizontal line.

If the two objects are on parallel lines (e.g., the ground and a bridge above it), then the angle of elevation from one object is congruent to the angle of depression from the other object.

8. a. Diagram of triangle CDE is attached

b. The formula that can be used to solve triangle CDE is the Law of Sines

∠ e = 180 - 52 - 17 = 111

sin E / 28 = sin C / c = sin D / d

c = 28 sin 52 / sin 111 = 23.63

d = 28 sin 17 / sin 111 = 8.77

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The answer is A. The game is strictly determined and the value of the game is -7.

A rectangular array of characters, numbers, or phrases arranged in rows and columns is known as a matrix.

Given matrix, where we have to find out, is the game strictly determined for the following matrix or not?

[tex]\left[\begin{array}{ccc}-7&7&-3\\3&8&6\\1&3&-3\end{array}\right][/tex]

Using the minimax theorem, we know that if the game is strictly determined, the value of the game is equal to the value of the saddle point. If the game is not strictly determined, the value of the game may not exist or may be a range of values.

We can check each row and column for minimum and maximum values:

The minimum value in the first row is -7.

The maximum value in the second column is 7.

The minimum value in the third row is -3.

We can see that the element in the first row and second column is both the minimum in its row (-7) and the maximum in its column (7), so it is a saddle point. Therefore, the game is strictly determined and the value of the game is -7.

The answer is A. The game is strictly determined and the value of the game is -7.

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Answer: The top 1% of the world's population holds more than 45% of global wealth.

Step-by-step explanation:

To find the percentage of global wealth concentrated in the hands of the top 1%, we need to divide the wealth of the top 1% by the total amount of global wealth, and then multiply by 100 to get the percentage.

Percentage of global wealth held by top 1% = (Wealth of top 1% / Total global wealth) x 100

We are given that the total amount of global wealth is $418.3 trillion, and the wealth of the top 1% is more than $190 trillion.

So, the percentage of global wealth held by the top 1% is:

(190 trillion / 418.3 trillion) x 100 = 45.38%

Therefore, the top 1% of the world's population holds more than 45% of global wealth.

Answer:

45.42%

Step-by-step explanation:

To find:-

Answer:-

We are here given that, the total global wealth is 418.3T dollars and more than 190T dollars is in hands of top 1% of the people. We are interested in finding out the percentage of global wealth in the hands of top 1% . We can find the percentage as ,

[tex]:\sf\implies \pink{ \% =\dfrac{Wealth\ owned\ by \ 1\% }{Total\ wealth}\times 100}\\[/tex]

On substituting the respective values, we have;

[tex]:\sf\implies \% =\dfrac{\$ 190T}{\$ 418.3T}\times 100\\[/tex]

[tex]:\sf\implies \pink{\% = 45.42\% } \\[/tex]

Hence the required percentage is 45.42% .

The graph οf the equatiοn −7y + 5x = 5(−7 + x) is a straight line passing thrοugh the pοint (0, 5) with a negative slοpe.

A linear equatiοn in twο variables is an equatiοn that can be written in the fοrm ax + by = c, where a, b, and c are cοnstants and x and y are variables.

Tο graph the equatiοn −7y + 5x = 5(−7 + x), we can first find the x- and y-intercepts by setting each variable tο zerο in turn.

When y = 0, we get:

-7y + 5x = 5(-7 + x)

0 + 5x = 5(-7 + x)

5x = -35 + 5x

0 = -35

This is a cοntradictiοn, which means that there is nο x-intercept.

When x = 0, we get:

-7y + 5x = 5(-7 + x)

-7y + 0 = 5(-7 + 0)

-7y = -35

y = 5

Sο the y-intercept is (0, 5).

Therefοre, the graph οf the equatiοn −7y + 5x = 5(−7 + x) is a straight line passing thrοugh the pοint (0, 5) with a negative slοpe.

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The savings plan balance after 6 months with an APR of 2% and monthly payments of $150 is approximately $913.35.

The stated interest rate on a loan or investment is known as the nominal interest rate; it does not account for compounding or any other elements that may have an impact on the real return or expense. Often, it is stated as an annual percentage rate (APR).

The effective interest rate, in contrast, takes into account compounding and other elements that may have an impact on the true cost or return of a loan or investment. It is often represented as an annual percentage yield and represents the actual rate of return or cost that is earned or paid on the principal amount over a certain time (APY).

Given that,

APR = 2% and monthly payments = $150.

For future value of annuity we use the following formula:

FV = Pmt × [(1 + r)⁽ⁿ⁻¹⁾] /r

Substituting the values:

r = 0.02/12 = 0.0016667, n = 6, and Pmt = $150:

FV = $150 × [(1 + 0.0016667)⁽⁶⁻¹⁾]/0.0016667

FV ≈ $913.35

Hence, the savings plan balance after 6 months with an APR of 2% and monthly payments of $150 is approximately $913.35.

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Answer:

To write the time taken to fill a tank to the nearest 5 minutes, we first need to convert 2 hours and 43 minutes to minutes. 2 hours = 2 x 60 = 120 minutes So, the total time taken in minutes is: 120 + 43 = 163 minutes To round this to the nearest 5 minutes, we need to consider the last digit of the number of minutes, which is 3. Since 3 is less than 5, we round down to the nearest multiple of 5, which is 160. Therefore, the time taken to fill the tank to the nearest 5 minutes is: 2 hours and 43 minutes = 160 minutes = 2 hours and 40 minutes

To write the given time to the nearest 5 minutes, round down the minutes value if the last digit is less than 5, and keep the hours value as it is.

To write the given time to the nearest 5 minutes, we need to round it to the closest multiple of 5. The given time is 2 hours and 43 minutes. To round to the nearest 5 minutes, we look at the last digit of the number of minutes, which is 3. Since 3 is less than 5, we round down the minutes to 0 and keep the hours as it is. Therefore, the time to the nearest 5 minutes is 2 hours and 40 minutes.

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Answer:

-12x⁴-10x³+3x²+3x+2

Step-by-step explanation:

(3x²+x−2)(−4x²−2x−1)=3x²(−4x²−2x−1)+x(−4x²−2x−1)−2(−4x²−2x−1)

                                 { ( a+b ) ( a+b )=a ( a+b )+b ( a+b ) by distributive  property }

                                =  -12x⁴-6x³-3x²-4x³-2x²-x+8x²+4x+2

                                =   -12x⁴-10x³+3x²+3x+2  

Answer:

x = 26 degrees

Step-by-step explanation:

Supplementary angles mean that they sum up to be 180 degrees therefore

C + D = 180 degrees

128 degrees + D = 180 degrees

D = 180 degrees - 128 degrees

D = 52 degrees

If the measure of the angle D is two times the value of x then x is...

x = D/2

x = 52 degrees / 2

x = 26 degrees

The estimated salary fοr an emplοyee with 18 years οf experience is $77,201.24.

BODMAS stands fοr Brackets, Order, Divisiοn/Multiplicatiοn, Additiοn/Subtractiοn, and it represents the οrder οf mathematical οperatiοns tο be perfοrmed when evaluating an expressiοn. It ensures that expressiοns are evaluated in a cοnsistent and unambiguοus manner.

Tο calculate the estimated salary fοr an emplοyee with 18 years οf experience using the prοvided fοrmula, we can substitute 18 fοr "Years οf Experience" and sοlve fοr the salary:

Estimated Salary = 29,608.40 + 2632.98(18)

Estimated Salary = 29,608.40 + 47,592.84

Estimated Salary = 77,201.24

Therefοre, the estimated salary fοr an emplοyee with 18 years οf experience is $77,201.24.

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Answer: 1

Step-by-step explanation: From the above data, we observe that 1 is the greatest factor of itself and the smallest factor of any other number. Therefore, 1 is the factor of every number.

The solution of each equation are;

1) x = 6, x = 0

2) x = 3, x = 2

3) x= 4 ±√6

4) x = 6 ± √24/2

An extraneous solution is a solution obtained in a mathematical equation that does not satisfy the original equation, even though it appears to be a valid solution when plugged into the equation.

1) (√6x)^2 = x^2

6x = x^2

x^2 - 6x = 0

x = 6, x = 0

2) √5x - 6 = x

5x - 6 = x^2

x^2 - 5x + 6 = 0

x = 3, x = 2

3) √2x - 1 = x - 3

2x - 1 = (x - 3)^2

2x - 1 = x^2 - 6x + 9

x^2 - 6x - 2x + 9 + 1 = 0

x^2 -8x + 10 = 0

x= 4 ±√6

4) x = 2 + √2x - 11

x -2 = √2x - 11

x^2 - 4x + 4 = 2x - 11

x^2 -4x -2x + 4 + 11 = 0

x^2 - 6x + 15 = 0

x = 6 ± √24/2

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On solving the question we have that the answer of the equation will be [tex]Answer: "f(x) = 1 + x +\frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

A math equation is a mechanism for connecting two statements and indicating equivalence with the equals sign (=). To explain the connection between the two sentences put on each side of a letter, a statistical method can be employed. The software and the logo are usually interchangeable. 2x - 4 equals 2, for example. An equation is a logical expression that asserts the equality of some mathematical expressions in algebra. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14.

[tex]"f(x) = {e^x} \Rightarrow f'(x) = {e^x} \Rightarrow f''(x) = {e^x} \Rightarrow f'''(x) = {e^x}"[/tex]

Then

[tex]"f(x) = f(0) + \frac{{f'(0)}}{{1!}}x + \frac{{f''(0)}}{{2!}}{x^2} + \frac{{f'''(0)}}{{3!}}{x^3} + ... = {e^0} + \frac{{{e^0}}}{1}x + \frac{{{e^0}}}{2}{x^2} + \frac{{{e^0}}}{6}{x^3} + ... = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

[tex]Answer: "f(x) = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

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Answer:

Step-by-step explanation:

The area of the figure is 1120

a) The constant of proportionality is solved to be 9

b) The equation of the relationship is y = 9x

a. To find the constant of proportionality, we can use the formula:

K = y/x

where

k is the constant of proportionality,

y is the dependent variable (output), and

x is the independent variable (input).

In this case, The ordered pair (3, 27) will be used them to solve for k.

where

x = 3

y = 27

k = y/x = 27/3

k = 9

b. The equation for this relationship is written using the formula

K = y/x

rewriting in  terms of y

y = kx

y = 9x

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Answer:

Option D

Step-by-step explanation:

Total number of posters Rhett puts up = 9

                Printing cost of each poster = $2

                                    ∴Total cost on posters = ([tex]\frac{2 dollars}{1 poster}[/tex]) × (9 posters)

                                                                           = $18

Note: The unit multiplier used above is arranged in a way that the current unit of “posters” is cancelled out and the answer is assigned with the new unit of “dollars".

Number of fliers Rhett gives out = x

Printing cost of each flier = $0.20

                                    ∴Total cost on fliers = [tex]\frac{0.20 dollars}{1flier}[/tex] × (x fliers)

                                                                      = $0.20x

Note: The unit multiplier used above is arranged in a way that the current unit of “fliers” is cancelled out and the answer is assigned with the new unit of “dollars”

∴How much it cost Rhett to search for his lost puppy:

y = Total cost on fliers + Total cost on posters

∴[tex]y = 0.20x + 18[/tex]

∴Option D

The recursive formula for the given sequence is:

f(n) = f(n - 1) + 3

Here we have the graph of a sequence, we can see that the first terms are:

f(0) = 4

f(1) = 7

f(2) = 10

f(3) = 13

f(4) = 16

A recursive formula tells us how to find the value of the n-th term of the sequence by using the previous terms.

By looking at the values above, you can see that all the terms are 3 units apart, so the recursive sequence that we can use here is:

f(n) = f(n - 1) + 3

It says that "the n-th term is equal to the previous term plus 3"

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Answer:

Step-by-step explanation:

3.1:

Using the compound angle expansion, we have:

sin 2x = sin(x + x) = sin x cos x + cos x sin x

sin 2x = 2 sin x cos x

Therefore, sin 2x = 2 sin x cos x.

3.2:

Using the result from 3.1:

3.2.1: sin 100° = 2 sin 50° cos 50° = sin 100° = 0.766

3.2.2: sin 46° = 2 sin 23° cos 23° = sin 46° = 0.719

3.2.3: sin 40° = 2 sin 20° cos 20° = sin 40° = 0.642

3.3:

Using the identity sin(a ± b) = sin a cos b ± cos a sin b:

3.3.1: 2 sin 19° cos 19° = sin 38° = sin (45° - 7°) = sin 45° cos 7° - cos 45° sin 7° = (1/√2)cos 7° - (1/√2)sin 7° = -sin (7° - 45°) = -sin 38°

Therefore, 2 sin 19° cos 19° = -sin 38°.

3.3.2: 2 cos 40° sin 40° = sin 80° = sin (45° + 35°) = sin 45° cos 35° + cos 45° sin 35° = (1/√2)cos 35° + (1/√2)sin 35° = sin (35° + 45°) = sin 80°

Therefore, 2 cos 40° sin 40° = sin 80°.

3.3.3: sin 25° cos 155° = (1/2)(sin 180°) = 0

Therefore, sin 25° cos 155° = 0.

The conditional probability is given as follows:

P(B|A) = 1/2, option C.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The events are given as follows:

P(B|A) is the probability that the second digit is even, considering that the first digit was odd. As the digits cannot repeat, and there will be 8 remaining digits, out of which 4 will be even, the probability is given as follows:

P(B|A) = 4/8 = 1/2.

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Answer:

therefore, the expression for John's age in terms of y is 4y - 16.

Step-by-step explanation:

Answer:

John's age = 4y - 16

Step-by-step explanation:

We can start by using the information given to write expressions for the ages of Libby and John in terms of Alfred's age y.

We know that Libby is 4 years younger than Alfred, so her age can be written as:

Libby's age = Alfred's age - 4

We also know that John is 4 times as old as Libby, so his age can be written as:

John's age = 4 × Libby's age

Substituting the expression for Libby's age from the first equation into the second equation, we get:

John's age = 4 × (Alfred's age - 4)

Simplifying this expression, we get:

John's age = 4 × Alfred's age - 16

Therefore, an expression in terms of y for John's age is:

John's age = 4y - 16

Answer:

Step-by-step explanation:

domain is negative infinity, positive infinity

range is -3, positive infinity

Answer:

F(x) is all values that x can be. so the answer is the domain is (-∞,4)

Step-by-step explanation:

The cost price of one ball is equal to Rs. 75

We shall represent the cost price of one bat with the letter 'x', so that the selling price of one bat is 125% of 'x', which is 1.25x.

The cost price of 5 bats is equal to the cost price of 4 balls. So,

5x = 4y, where 'y' is the cost price of one ball.

y = (5/4)x

Also, we know that the profit percentage on one ball is 40%. So, the selling price of one ball is 140% of its cost price, which is 1.4y.

Profit earned on selling one bat and one ball is calculated as:

Profit = Selling price - Cost price

Profit = (1.25x + 1.4y) - (x + y)

Profit = 0.25x + 0.4y

Profit = 0.25x + 0.4(5/4)x {Substituting the value of 'y'}

Profit = 0.75x

given that the above profit is equal to Rs. 45. them,

0.75x = 45

x = 60

The cost price of one bat is Rs. 60 and the cost price of one ball:

y = (5/4)x

y = (5/4) × 60

y = 75

Therefore, the cost price of one ball is Rs. 75.

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The effective interest rate on the loan is also 32%.

What is effective interest rate ?

Effective interest rate is the actual interest rate that is paid or earned on an investment, loan, or other financial product. It takes into account not only the nominal interest rate, which is the stated rate of interest, but also the frequency of compounding or other factors that affect the overall return.

According to the question:
To find the effective interest rate, we first need to calculate the amount of interest Kremena paid on the loan. We can do this by subtracting the amount she received from the amount of the loan:

Interest = Loan amount - Amount received

Interest = $500 - $460

Interest = $40

Next, we can use the formula for simple interest to calculate the interest rate:

I = P * r * t

where I is the interest, P is the principal (loan amount), r is the interest rate, and t is the time in years.

Since Kremena took out the loan for 3 months, which is a quarter of a year, we have:

I = P * r * t

$40 = $500 * r * (3/12)

$40 = $500 * r * 0.25

r = $40 / ($500 * 0.25)

r = 0.32 or 32%

So the interest rate on the loan is 32%. However, this is the nominal interest rate, which does not take into account the effects of compounding. To find the effective interest rate, we need to take into account the fact that Kremena only received $460, which means that she effectively paid more interest than she would have if she had received the full $500.

To calculate the effective interest rate, we can use the formula:

[tex]re = (1 + r/n)^n - 1[/tex]

where re is the effective interest rate, r is the nominal interest rate, and n is the number of compounding periods per year. Since the problem specifies simple interest, there is no compounding, so we can take n = 1. Plugging in the values, we get:

[tex]re = (1 + 0.32/1)^1 - 1[/tex]

re = 0.32 or 32%

So the effective interest rate on the loan is also 32%.

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Answer:

The function f(x) = -8t(t - 8) represents the height of the soccer ball at any given time t, where t is the time in seconds after the ball was kicked.

To find out how long the ball spends in the air, we need to determine the time at which the ball hits the ground. The ball hits the ground when its height is zero, so we can set f(x) = 0 and solve for t:

-8t(t - 8) = 0

This equation has two solutions: t = 0 and t = 8. The first solution corresponds to the time when the ball is kicked and the second solution corresponds to the time when the ball hits the ground. Since we're interested in the time the ball spends in the air, we need to subtract the time the ball was kicked from the time the ball hits the ground:

t = 8 - 0 = 8 seconds

Therefore, the ball spends 8 seconds in the air.