find the area of this semi-circle with radius, r=98cm. give your answer to 1dp

The burger costs $0.625, the souvenir costs $3.75, and the pass costs $5.00.

Let's name the burger price "b," the souvenir price "s," and the pass price "p." Then, using the information provided, we can construct an equation system:

b + s + p = 35.75  (1) (The three things cost a total of $35.75.)

b = (1/6)s               (2) (the burger was one-sixth the price of the memento).

s = (3/4)p     (3) (the memento cost one-quarter the price of the pass)

b + s + p + 2 = 35.75       (4) (After purchasing these products, Dion had                        $2.00 left over.)

To remove s from the equations, we can apply equations (2) and (3):

b + (3/4)p + p = 35.75    (equations (1) and (3))

b = (1/6)(3/4)p                (Applying Equation (2))

b = (1/8)p

This expression can now be substituted by Input b into equation (1):

(1/8)p + (3/4)p + p = 35.75

(7/8)p = 35.75

p = 35.75 / (7/8)

p = $5.00

We may calculate the cost of the keepsake using equation (3):

s = (3/4)p = (3/4)($5.00) = $3.75

Finally, we can calculate the cost of the burger using equation (2):

b = (1/6)s = (1/6)($3.75) = $0.625

As a result, the burger costs $0.625, the souvenir costs $3.75, and the pass costs $5.00.

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The 95% confidence interval for the mean mass of the population is 24.57 ± 1.06. The upper limit of the 95% confidence interval is 25.63.

Given the masses, in grams, of ten ball bearings taken at random from a batch are: 25.9, 24.7, 23.4, 24.5, 25.0, 26.9, 26.4, 25.8, 23.2, 21.9

We have to calculate a 95% confidence interval for the mean mass of the population, supposed normal, from which these masses were drawn.The formula for the Confidence Interval is given by,

CI = X bar± tα/2 * (s/√n)

Where,

CI = Confidence Interval

Xbar = Sample mean

tα/2 = α level of significance/2 (α is the level of significance)

s = sample s

tandard deviationn = sample size

tα/2 = t-distribution value for α/2 and (n-1) degrees of freedomtα/2 for α = 0.05 and (n-1) degrees of freedom

tα/2 = 2.262

Hence,CI = X bar ± tα/2 * (s/√n)

CI = 24.94 ± 2.306 * (1.618/√10)

CI = 24.94 ± 1.36

Upper limit of 95% confidence interval

x2 = 24.94 + 1.36= 26.30

Rounded off to two decimal places, x2 is 26.30.

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The answer is D: w = 85-25/ 5. This equation represents the number of weeks it will take Paul to save enough money to buy the skateboard.

Equations are useful for solving problems, finding unknown values, determining rates of change, and understanding relationships between different variables.

The equation can be written as w = (85 - 25)/5, where 85 is the cost of the skateboard, 25 is the amount he has already saved, and 5 is the Subtracting 85-25, the equation is essentially asking how many 5s are in 60.

The answer is 12, which is the number of weeks it will take Paul to save enough money for the skateboard.

This can be verified by multiplying 12 by 5, which equals 60.

Thus, the correct answer is D: w = 85-25/ 5.

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Question :

Paul gets $5 every week for doing his chores. He is saving his money to buy a skateboard that costs $85. He has already saved $25. Which equation describes the number of weeks (w) that it will take Paul to save enough money to buy the skateboard?

A : w = 85 / 5

B : w = 85 - 25

C : w - 85 / 5 - 25

D: w = 85-25/ 5

If there is sufficient evidence to reject a null hypothesis at the 10% level, then there is sufficient evidence to reject it at the 5% level.

What is hypothesis testing?

A statistical hypothesis test is a technique for determining if the available data are sufficient to support a specific hypothesis. We can make probabilistic claims regarding population parameters through hypothesis testing.

Here,

We have to find the correct statement about hypothesis testing.

We concluded that  If there is sufficient evidence to reject a null hypothesis at the 10% level, then there is sufficient evidence to reject it at the 5% level.

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Answer:

In this case, B is true.

Whether to use a one-sided or a two-sided test is typically decided BEFORE the data are gathered, not after.

Be careful while reading the options on the quiz, because depending on the question you get, the options may be worded slightly differently. Because of the wording, the answer may be different as well (on the FLVS quiz, I got different answer choices for this question). Be careful and read cautiously!

Have a great day.

The values of the variable x are 7 and -1. Options B and D

Functions are simply defined as those equations, expressions or laws that shows the relationship between two variables.

These variables are;

From the information given, we have that;

f(x) = 16

f(x) =  x² - 6x + 9

Now, equate the two functions, we have;

x² - 6x + 9 = 16

collect the like terms

x² - 6x + 9 - 16 = 0

subtract the values

x² - 6x - 7 = 0

solve the quadratic equation

x² - 7x + x - 7 = 0

x(x - 7) + 1( x - 7)

Then, x = 7 and -1

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"type of orange" because it is a bar graph therefore it is supposed to be categories, not numbers :)

The required expected number of employees who will pass the test from given 11 potential employees is equal to 10.

Probability of passing a drug test is 90%,

Then the probability of failing is 10%.

For each potential employee,

Probability of passing is 0.9, and the probability of failing is 0.1.

The number of potential employees who would be expected to pass the test,

Calculated using the binomial distribution with n = 11 and p = 0.9.

P(A = x) = (ⁿCₓ) × pˣ ×  (1-p)ⁿ⁻ˣ

where,

A is the number of potential employees who pass the test

x is the number of potential employees who pass the test

n is the number of potential employees

p is the probability of passing the test

Using this formula, the expected number of potential employees who would pass the test is,

E(A) = n ×p

      = 11 × 0.9

      = 9.9

Rounding to the nearest whole number, we can expect 10 potential employees to pass the test.

Therefore, the expected number of employees from the given  probability who would pass the the test is 10.

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Answer: The median decreases to 6.5.

Step-by-step explanation:

The median before 4 was added was 6.75.

When 4 is added the median is 6.5.

it is important to note that the sample size of 1000 subjects is large enough for statistical inference, but the significance level must be taken into account when interpreting the results.

No. Since the tests were performed at the 1% significance level, as many as 10 subjects may have done significantly better than random guessing.

When conducting hypothesis testing, the significance level (in this case, 0.01 or 1%) is the probability of rejecting the null hypothesis when it is actually true. This means that there is a 1% chance of observing a significant result purely by chance, even if the null hypothesis (in this case, that the subjects do not have ESP) is true.

In this scenario, nine subjects performed significantly better than random guessing, which may suggest that they have ESP. However, due to the possibility of observing significant results by chance, it is not proper to conclude that these nine people have ESP. There is still a chance that these results are purely coincidental.

Additionally, it is important to note that the sample size of 1000 subjects is large enough for statistical inference, but the significance level must be taken into account when interpreting the results.

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Just make a flat line for each time the car stops.

Hope this helps.

To find the area of the shaded polygon, we had to assume that each dotted line was one unit. In which case, the coordinates are:

Hence, the area of the polygon is A [tex]\approx[/tex] 11.15 units.

To get the area of the polygon, we must determine the length of all it's sides.

a) A-B is given by the distance formula:

distance = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Using this formula, we can calculate the distance between points A(1,4) and B(5,6) as follows:

distance = √((5 - 1)² + (6 - 4)²)

= √(16 + 4)

= √(20)
A-B = 4.47213595

A-B [tex]\approx[/tex] 4.47

b)
To find B-C we use the same method:
B-C = 3.16 units.

c)
To find C- D we use the same method:
C- D  [tex]\approx[/tex] 3.16 units.

d) to find D-A  we use the employ the same method:
D - A [tex]\approx[/tex] 2.83



Since this is an irregular quadrilateral, we will require the use of the
length of one of it's diagonals which is also computed using the distance formula to get: 4.47 Units.


Calculate the perimeter of ABCD

Perimeter of ABCD = AB + BC + CD + AD

Perimeter of ABCD = 4.4721359549996 + 3.1622776601684 + 3.1622776601684 + 2.8284271247462

Perimeter of ABCD = 13.625118400083


Calculate the semi-perimeter (s) of ABCD

s  =   Perimeter ÷ 2

s  =   13.625118400083 ÷ 2

s = 6.8125592000413


Calculate the Area (A) using Brahmagupta's Formula

A = √[(s - a)(s - b)(s - c)(s - d)]

A = √(6.8125592000413 - 4.4721359549996)(6.8125592000413 - 3.1622776601684)(6.8125592000413 - 3.1622776601684)(6.8125592000413 - 2.8284271247462)

A = √(2.3404232450417)(3.6502815398729)(3.6502815398729)(3.9841320752951)

A = √(124.24555320337)
A = 11.14654893693

A [tex]\approx[/tex] 11.15 units.

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Answer:

Step-by-step explanation:

Where the above mean exist, the values of x and y  are 10.5 and 3.5 respectively.

The total number of people surveyed is 24. So we can write:

Frequency of 0 sisters + Frequency of 1 sister + Frequency of 2 sisters + Frequency of 3 sisters = Total frequency

5 + x + 5 + y = 24

x + y = 14 --- (1)

Also, we know that the mean number of sisters is 1.25. This means that the total number of sisters divided by 24 should be equal to 1.25. So we can write:

0 × 5 + 1 × x + 2 × 5 + 3 × y = 1.25 × 24

x + 3y = 21 --- (2)

Now we have two equations (1) and (2) with two variables (x and y). We can solve for x and y by simultaneous equations.

Multiplying equation (1) by 3, we get:

3x + 3y = 42 --- (3)

Subtracting equation (2) from equation (3), we get:

2x = 21

x = 10.5

Substituting the value of x in equation (1), we get:

10.5 + y = 14

y = 3.5

Therefore, the values of x and y are 10.5 and 3.5 respectively.

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Answethe two digit is a multiple of 5

Step-by-step explanation:

the two-digit number is 65 so, options that must be true are a and b.

Addition is a basic arithmetic operation that combines two or more numbers to produce a sum or total

Let x stand for the tens digit and y for the units digit.

x + y = 11 (the digits of the two-digit number sum to 11)

x - y = 1 (the positive difference of the digits is 1)

Adding these two equations gives:

2x = 12

So x = 6.

Substituting x = 6 into either of the equations gives y = 5.

Therefore, the two-digit number is 65.

Now we can check which of the options is true:

a. The two-digit number has a product of 30 digits.

6 x 5 = 30, so this is true.

b. The number has two digits and is a multiple of 5.65 is a multiple of 5, so this is true.

c. The two-digit number is even.

65 is not even, so this is false.

d. The two-digit number is 83.

The two-digit number is 65, not 83, so this is false.

Therefore, the options that must be true are a and b.

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The area of the shaded segment of the circle is calculated by substracting the area of triangle from area of sector of circle So, area of shaded segment of the above circle is equals to the (12π - 9√3) cm².

Assume a sector having a radius r and a central angle of α degrees. The sector's area will be A= (α/360∘)πr². Similarly, different shapes have different area formulas. We have been provided a circle of radius 6cm. Assume that its center is O and the segment is AB. Radius of circle , r = 6 cm

Central angle,α = 120°

Area of shaded segment is calculated by the area of sector AOB of circle minus area of triangle AOB formed there. So, Area of sector AOB , A = (α/360∘)πr²

= ( 120°/360°) π6²

= (1/3) 36π = 12π cm²

As we know, OA = OB= 6cm

=> m∠OAB = m∠OBA

and m∠AOB + m∠OAB + m∠OBA = 180∘

=> 120∘ + m∠OAB + m∠OAB= 180∘

=> 2(m∠OAB) = 180∘− 120∘ = 60°

=> m∠OAB = 30° = m∠OBA

So, it is an isosceles triangle. Draw a perpendicular OC on AB then there is formed two right angled triangles. In right angled triangle OAC, 90°, 60°, 30° and OA = 6 cm then OC/6 = Sin 30°

=> OC= 3 cm and

AC = √36 - 9 = √27 = 3√3 cm

Similarly, BC = 3√3 cm so, AB = 6√3 cm. Now, area of triangle AOB = (1/2) × AB × OC = (1/2)× 6√3 ×3 = 9√3 cm²

So, required area = 12π cm² - 9√3 cm². Hence, required value is (12π - 9√3) cm².

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Complete question:

The above figure completes the question. Find the area of the shaded segment of the circle.

Assuming the table models a linear equation, we will get the complete table:

minutes: 10         40         89.3        150        360

cost:       200    225.7      268        320        500

Here we can assume that the table models a linear equation of the form:

y = ax+  b

Where a is the rate of change and b is the y-intercept.

If we know two points on the line, let's say:

(x₁, y₁) and (x₂, y₂), then the rate of change is:

a = (y₂ - y₁)/(x₂ - x₁)

In the table we have two pairs:

(10, 200) and (150, 320)

Then the rate is:

a = (320 - 200)/(150 - 10)

a = 12/14 = 6/7

Then we can write:

y = (6/7)*x + b

To find the value of b, we can replace one of the two points there, using (10, 200) we will get:

200 = (6/7)*10 + b

200 = 60/7 + b

200 - 60/7 = b

1400/7 - 60/7 = b

1340/7 = b

Then the line is:

y = (6/7)*x + 1340/7

Now to complete the table, just replace the correspondent values in the equation.

for the first value we have x = 40

y = (6/7)*40 + 1340/7 = 225.7

if x = 360

y = (6/7)*360 + 1340/7 = 500

if y = 268

268 = (6/7)*x + 1340/7

x = (268 - 1340/7)*(7/6)

x = 89.3

Then the complete table is:

minutes: 10         40         89.3        150        360

cost:       200    225.7      268        320        500

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Answer:

B

Step-by-step explanation:

first you would need to do the inverse of sine to get the angle since normal sine would give you the length and you need the angle.

An easy way to remeber sine, cosine and tangent is

SOH CAH TOA

sin= opposite/ hypotenuse

cosine= adjacent/ hypotenuse

tangent= opposite/adjacent.

first label your triangle with the sides hypotenuse, adjacent and opposite.

hypotenuse is the longest side that is diagonal in this case like ab

opposite is the side that is opposite the angle you are looking for, so in thsi case line cb

adjacent is the like next to the angle you are trying to find so in thsi case like ac

and since we know sine inverse = opposite/hypotenuse = angle

(we are using sine for this question since that is what the question is asking for)

we simply do

sine inverse= 3/5

so the answer is B

i hope this helps! :)

Answer:B

Step-by-step explanation: im sure because i answer that in school im correct

This problem can be solved by the unitary method. If Lee Ann bought 2 cartons of yoghurt, she would take 16 days to finish two cartoons as there were 8 yoghurt in each cartoon.

The unitary method involves first determining the value of a single unit, followed by the value of the necessary number of units.

Always write the things that need to be calculated on the right side and the things that are known on the left side to simplify things. In the aforementioned issue, the number of apples is known, but it is unclear what they are worth. It should be noted that issues with this method are solved using the concepts of ratio and proportion.

Let's say you go to the store to buy six apples. You are informed by the shopkeeper that he is offering 10 apples for Rs 100. In this instance, the value and the units are the price of the apples. Recognizing the units and values is crucial when applying the unitary method to a problem.

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The sοlutiοn tο the  radical equatiοn is x = 5.

In mathematics, a radical is the οppοsite οf an expοnent, which is symbοlised by the symbοl "," alsο referred tο as rοοt. A square rοοt οr a cube rοοt can be used, and the number befοre the symbοl οr radical is regarded as an index number οr degree. The expοnent οf this number cancels the radical and is a whοle number.

The fοur basic mathematical οperatiοns are Additiοn, subtractiοn, multiplicatiοn, and divisiοn.

Starting with:

-5√(2x - 1) = -15

Divide bοth sides by -5:

√(2x - 1) = 3

Square bοth sides:

2x - 1 = 9

Add 1 tο bοth sides:

2x = 10

Divide bοth sides by 2:

x = 5

Hence, the solution to the equation is x = 5.

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the area of the porch is 228 square centimeters. To calculate the area of a porch, we need to use the formula for the area of a rectangle, which is:

Area = length x width

In this case, the length of the porch is 19 cm and the width is 12 cm. So, we can plug these values into the formula:

Area = 19 cm x 12 cm

Area = 228 cm^2

Therefore, the area of the porch is 228 square centimeters.

It's important to note that when we use the term "area," we are referring to the amount of space that is enclosed within the boundaries of the porch. In other words, we are calculating the size of the surface area of the porch, which is measured in square units.

Knowing the area of a porch can be useful in a variety of contexts. For example, if you were planning to install new flooring on the porch, you would need to know the area in order to calculate how much material you would need to purchase. Similarly, if you were planning to paint the porch, you would need to know the area in order to estimate how much paint you would need to cover the surface.

In summary, the area of a porch can be calculated by multiplying its length and width, and is measured in square units.

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In the given problem,  the distance from point A to point B is approximately 658 feet (rounded to the nearest foot).

Let's define some variables:

Using trigonometry, we can write the following equations:

For triangle ACT, we have:

tan(15°) = h/d

For triangle BCT, we have:

tan(50°) = h/(d+x)

We want to find the value of x, so let's rearrange the second equation:

tan(50°) = h/(d+x)

tan(50°)*(d+x) = h

d+x = h/tan(50°)

x = h/tan(50°) - d

Substituting the given values:

x = 115/tan(50°) - d

Now we need to find the value of d. For that, we can use the first equation:

tan(15°) = h/d

d = h/tan(15°)

Substituting the given value:

d = 115/tan(15°)

Now we can substitute both values into the equation for x:

x = 115/tan(50°) - 115/tan(15°)

Using a calculator, we get:

x ≈ 658.4 ft

Therefore, the distance from point A to point B is approximately 658 feet (rounded to the nearest foot).

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The points A', B', and C' are  (3, 1), (0, 3) and (-2, 4) respectively. The solution has been obtained by using the concept of transformation.

What is transformation?

Transformations allow for the translation, rotation, and reflection of shapes on a coordinate plane. Any line, point, or vector can be rotated, translated, or reflected.

We are given coordinates of triangle ABC as A (1, -3), B (3, 0) and C (4, -2).

Now, we need to rotate the triangle ABC 90 degrees about the origin.

We know that the rotation rule for rotating 90 degrees about the origin is that (x, y) becomes (y, -x).

So, from this we get the coordinates as

A' (3, -1)

B' (0, -3)

C' (-2, -4)

Now, on reflecting it over the x - axis, we get

A' (3, 1)

B' (0, 3)

C' (-2, 4)

Hence, the points A', B', and C' are  (3, 1), (0, 3) and (-2, 4) respectively.

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Answer:

Well,

[tex]0.5(5-7x)=8-(4x+6)\\-3.5x+2.5=-4x+2\\0.5x+2.5=2\\0.5x=-0.5\\x=-1[/tex]

Step-by-step explanation:

Hope this helps!^^

Answer: -1

Step-by-step explanation:

first you want to simplify 0.5 (5-7x) which will get you 2.5 - 3.5x

then you will do the same for the other side move the xs to one side then treat it like any other problem

The polynomial to represent the volume of the rectangular prism is:

V = (0.5x - 3) × (x - 4) × (x + 5)

Simplifying this expression gives V = 0.5x³ - 6.5x² - 23x + 60

How to calculate the volume (V) of a rectangular prism?
The volume (V) of a rectangular prism can be calculated using the formula V = l × w × h, where l, w, and h are the length, width, and height of the prism respectively.

Here, the length (l) can be represented by (0.5x - 3), the width (w) can be represented by (x - 4), and the height (h) can be represented by (x + 5).

Therefore, the polynomial to represent the volume of the rectangular prism is:

V = (0.5x - 3) × (x - 4) × (x + 5)

Simplifying this expression gives V = 0.5x³ - 6.5x² - 23x + 60

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The area of circle with radius 4 centimetres and the special math constant, pi to be 3. 142 is equals to the 50.3 cm².

Area of a shape is defined as the total spece covered by shape in two-dimensional. In geometry, the area enclosed circle is calculated by π times the square of radius of circle. Formula of Area = π × r², where 'r' is the radius. The units of area are square of units of radius that if radius in feet then unit of area will be square feet. We have a circle with radius say r = 4 cm.

The value of specific math constant, pi ( i.e., π) is equals to be 3.142. We have to determine the area of circle. The area of circle, A = π× 4²

=> A = 16π

=> A = 16× 3.142

=> A = 50.272 ~ 50.3 cm²

Hence, the required area value is 50.3 cm².

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Complete question:

The above figure completes the question.

Work out he area of this circle. Take pi to be 3. 142 and give your answer to 1 decimal place

Answer:

-9 and -2

Step-by-step explanation:

-9 and -2

two negatives multiply to make a positive (18)

and -9 add -2 is -11

Answer:

24 sin(3x) cos(3x)

Step-by-step explanation:

To find:-

Answer:-

The given function to us is ,

[tex]:\sf\implies y = 4sin^2(3x) \\[/tex]

To find it's derivative we would have to use Chain rule of differentiation , which is ;

[tex]:\sf\implies \pink{ \dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}} \\[/tex]

Taking the given function,

[tex]:\sf\implies y = 4sin^2(3x) \\[/tex]

Differentiate both sides with respect to x,

[tex]:\sf\implies \dfrac{dy}{dx}=\dfrac{d}{dx}4sin^2(3x)\\[/tex]

We can take out the constant as ,

[tex]:\sf\implies \dfrac{dy}{dx}= 4 \dfrac{d}{dx}sin^2(3x) \\[/tex]

Multiply and divide by sin(3x) as ,

[tex]:\sf\implies \dfrac{dy}{dx}= 4 \bigg( \dfrac{d(sin^23x)}{d(sin3x)}\times \dfrac{d\ sin3x}{dx}\bigg)\\[/tex]

Differentiation of sin(nx) is n cos(nx) and

[tex]:\sf\implies \pink{ \dfrac{d(x^n)}{dx} = nx^{n-1} }[/tex]

So that ,

[tex]:\sf\implies \dfrac{dy}{dx}= 4 ( 2sin(3x) \cdot 3cos(3x))\\[/tex]

Simplify,

[tex]:\sf\implies\pink{ \dfrac{dy}{dx}= 24\ sin(3x) \ cos (3x)}\\[/tex]

Hence the derivative of the given function is 24 sin(3x) cos(3x) .

[tex]\rule{200}2[/tex]

Related formulae :-

[tex]\boxed{\boxed{\begin{minipage}{5cm}\displaystyle\circ\sf\;\dfrac{d}{dx}(sin\;x)=cosx \\\\ \circ \;\dfrac{d}{dx}(cos\;x) = -sinx \\\\ \circ \; \dfrac{d}{dx}(tan\;x) = sec^{2}x \\\\ \circ\; \dfrac{d}{dx}(cot\;x) = -csc^{2}x \\\\ \circ \; \dfrac{d}{dx}(sec\;x) = secx \cdot tanx \\\\ \circ \; \dfrac{d}{dx}(csc\;x) = -cscx \cdot cotx \\\\ \circ\; \dfrac{d}{dx}(sinh\;x)=coshx \\\\ \circ\; \dfrac{d}{dx}(cosh\;x)= sinhx \\\\ \circ\;\dfrac{d}{dx}(tanh\;x)=sech^{2}h \\\\ \circ\;\dfrac{d}{dx}(coth\;x)=-csch^{2}x \\\\ \circ\;\dfrac{d}{dx}(sech\;x) =-sechx \cdot tanhx \\\\ \circ\;\dfrac{d}{dx}(csch\;x) = -cschx \cdot cothx\end{minipage}}}[/tex]

[tex]\rule{200}2[/tex]

Answer:

[tex]\dfrac{\text{d}y}{\text{d}x}=12 \sin (6x)[/tex]

Step-by-step explanation:

To find the derivative of y = 4sin²(3x), use the chain rule.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If $y=f(u)$ and $u=g(x)$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

The given function can be written as:

[tex]y=4(\sin 3x)^2[/tex]

Let y = 4u², where u = sin(3x).

Differentiate the two parts separately.

[tex]\dfrac{\text{d}y}{\text{d}u}=8u \quad \text{and} \quad \dfrac{\text{d}u}{\text{d}x}=3 \cos (3x)[/tex]

Put everything into the chain rule formula:

[tex]\begin{aligned} \implies \dfrac{\text{d}y}{\text{d}x}& =\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}\\\\&=8u \times 3 \cos (3x)\\\\&=24u \cos (3x)\\\\&=24 \sin (3x) \cos (3x) \end{aligned}[/tex]

Simplify using the sine double angle identity:

[tex]\boxed{\sin (2 \theta)= 2 \sin \theta \cos \theta}[/tex]

[tex]\begin{aligned} \implies \dfrac{\text{d}y}{\text{d}x}&=24 \sin (3x) \cos (3x) \\\\&=12(2 \sin (3x) \cos (3x))\\\\&=12 \sin (2 \cdot 3x)\\\\&=12 \sin (6x)\end{aligned}[/tex]

Therefore:

[tex]\dfrac{\text{d}y}{\text{d}x}=12 \sin (6x)[/tex]

[tex]\hrulefill[/tex]

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Differentiating $\sin (ax)$}\\\\If $y=\sin(ax)$, then $\dfrac{\text{d}y}{\text{d}x}=a \cos (ax)$\\\end{minipage}}[/tex]