The letter O has the experimental probability that is closest to its theoretical probability.
Define the term probability?Probability is an area of statistics that deals with the study of random events and their likelihood of occurrence.
The theoretical probability of drawing a particular letter is the number of balls of that letter divided by the total number of balls in the game. For each letter, the theoretical probability is:
B: 15/75 = 0.2
I: 15/75 = 0.2
N: 15/75 = 0.2
G: 15/75 = 0.2
O: 15/75 = 0.2
The experimental probability of drawing a particular letter is the number of times that letter was actually drawn divided by the total number of draws. For each letter, the experimental probability is:
B: 247/1250 = 0.1976
I: 272/1250 = 0.2176
N: 238/1250 = 0.1904
G: 241/1250 = 0.1928
O: 252/1250 = 0.2016
Compare the differences between the theoretical and experimental probabilities for each letter. The letter with the smallest difference is the one whose experimental probability is closest to its theoretical probability.
Here, the differences between the theoretical and experimental probabilities for each letter are:
B: 0.2 - 0.1976 = 0.0024
I: 0.2 - 0.2176 = 0.0176
N: 0.2 - 0.1904 = 0.0096
G: 0.2 - 0.1928 = 0.0072
O: 0.2 - 0.2016 = 0.0016
Based on these calculations, we can see that the letter O has the smallest difference between its theoretical and experimental probabilities, with a difference of only 0.0016.
Therefore, the letter O has the experimental probability that is closest to its theoretical probability.
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1. A car's speed increases as it merges onto a highway. The car travels at 65 mph on the highway until it slows to exit. The car then stops at three traffic lights before reaching its destination. Draw a sketch of a graph that shows the car's speed over time.
Just make a flat line for each time the car stops.
Hope this helps.
write a polynomial to represent the volume of the rectangular prism if 0.5x-3 and x-4 and x+5
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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Simplify the expression. Enter the answer in the box. 3 2/5 + (-7 1/5) = 3 + (-7) + 2/5 + (-1/5)
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. calculate a 95% confidence interval for the mean mass of the population, supposed normal, from which these masses were drawn? what is x2, the upper limit of the 95% confidence interval? enter your answer rounded to two decimal places. for example, if your answer is 12.345 then enter as 12.35 in the answer box.
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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Dion brought $35.75 to the state fair. He bought a burger, a souvenir, and a pass. The burger was
1/6 as much as the souvenir, and the souvenir cost 3/4 the cost of the pass. Dion had $2.00 left over after buying these items. What was the cost of each item?
The burger costs $0.625, the souvenir costs $3.75, and the pass costs $5.00.
What was the cost of each item?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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Given triangle ABC with points:
A - (1, -3)
B - (3, 0)
C - (4, -2)
Rotate ABC 90 degrees counterclockwise about the origin then reflect it over the x-axis. Determine the ordered pairs for A', B', and C'.
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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If a porch is 12 cm by 19 cm what is the area of the point?
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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Please help with this question.
Where the above mean exist, the values of x and y are 10.5 and 3.5 respectively.
What is the explanation for the above response?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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If a shoe size of 4 is added to the data, how does the median change?
The median stays 6.5.
The median decreases to 6.5.
The median stays 6.75.
The median decreases to 6.75.
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.
Amari gathered a random sample of oranges in her town. She calculated data on different variables. For one of the variables that she collected, she constructed a bar graph.
Which of the following variables did she use?
Type of orange
Diameter of the oranges
Number of navel oranges
Price per pound of oranges
"type of orange" because it is a bar graph therefore it is supposed to be categories, not numbers :)
Paul gets $5 every week for doing his chores. He is saving his money to buy a skatboard that costs $85. He has already saved $25. A : w = 85 / 5
B : w = 85 - 25
C : w - 85 / 5 - 25
D: w = 85-25/ 5
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.
What is equation?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
When driving in normal conditions on a rural roadway your following distance should bea) no more than one secondb) three secondsc) twelve secondsd) four seconds or more
When driving in normal conditions on a rural roadway, your following distance should be four seconds or more because it gives you more time to respond to any unexpected occurrences that might occur. therefore option d) four seconds or more is correct.
What is a following distance?Following distance is the amount of space between a car and the car in front of it when driving on the road. It's essential to keep a safe following distance because it gives you more time to react if the car in front of you makes a sudden stop or starts to slow down. It's critical to keep a safe following distance because it's not always possible to avoid sudden stops or slow-downs.
What is a rural roadway?A rural roadway is a road that runs through the countryside. In comparison to city roads, rural roads are less congested, with less traffic and less likelihood of accidents. As a result, it's essential to maintain a safe following distance when driving on rural roads because they're often long and wide, with higher speed limits and less congestion than city roads.
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Find the area of the shaded segment of the circle
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.
Solve the system of linear equations by elimination. -x+y=1 and 3x+y=7
We then substitute -1 for x in the equation x + y = 4 to get -1 + y = 4, which we can then subtract -1 from both sides to get y = 5. The solution is x = -1 and y = 5.
3x + y = 7
-x + y = 1
Add the equations together:
2x + 2y = 8
Divide both sides by 2:
x + y = 4
Subtract the second equation from the first:
-x + y = 1
x - y = -3
Add the equations together:
2x = -2
Divide both sides by 2:
x = -1
Substitute -1 for x in the equation x + y = 4:
-1 + y = 4
Subtract -1 from both sides:
y = 5
The solution is x = -1 and y = 5.
We begin by solving the system of linear equations by elimination. We need to get the variables to one side of the equation and the constants to the other side. We start by adding the two equations together: 3x + y = 7 and -x + y = 1. This gives us 2x + 2y = 8, which we can then divide both sides by 2 to get x + y = 4. We then subtract the second equation from the first to get -x + y = 1 and x - y = -3. We then add these two equations together to get 2x = -2 and divide both sides by 2 to get x = -1. We then substitute -1 for x in the equation x + y = 4 to get -1 + y = 4, which we can then subtract -1 from both sides to get y = 5. The solution is x = -1 and y = 5.
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Find the area of the shaded polygon.
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:
A(1,4)B (5,6)C (6, 3)D (3, 2)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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The distance from Sunderland to Wigan is 150 miles.
Mollie leaves Sunderland in her car at 07:50.
Her average speed on the journey is 60mph.
What time does she arrive in Wigan?
Addison wanted to know if there was a connection between her coffee consumption and how
well she slept that night. For weeks, Addison recorded how many cups of coffee she drank in
the morning and how many hours she slept that night.
0 cups of coffee
1 cup of coffee
6 hours 7 hours
1
5
5
4
What is the probability that a randomly selected day is one when she drank exactly 1 cup of
coffee and is one when she slept about 6 hours?
Simplify any fractions.
Answer:
Step-by-step explanation:
0.5 ( 5 − 7 x ) = 8 − ( 4 x + 6 )
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
Can someone please help me? I’ll give brianliest!
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
7 divided by 70.85 show work
f(x) = 16, x = *
f(x) = x² - 6x + 9
Multiple choice: Finding: x values
-7
7
1
-1
The values of the variable x are 7 and -1. Options B and D
What are functions?
Functions are simply defined as those equations, expressions or laws that shows the relationship between two variables.
These variables are;
The independent variablesThe dependent variablesFrom 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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Lee Ann bought 2 cartons of yogurt. She ate 1 over 8 of a carton of yogurt each dat. How many days did it take Lee Ann to eat all of the yogurt in the 2 cartons?
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 probability of a potential employee passing a drug test is 90%. if you selected 11 potential employees and gave them a drug test, how many would you expect to pass the test?
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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Which of the following is a TRUE statement about hypothesis testing?
A: 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.
B: Whether to use a one-sided or a two-sided test is typically decided before the data are gathered.
C: If a hypothesis test is conducted at the 1% level, there is a 1% chance of rejecting the null hypothesis.
D: The probability of a Type I error plus the probability of a Type II error always equals one.
E: The power of a test concerns its ability to detect a null hypothesis.
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.
what two numbers do you multiply to get 18 and add to get -11
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
Find dy/dx for y=4sin²(3x)
Answer:
24 sin(3x) cos(3x)
Step-by-step explanation:
To find:-
The derivative of y = 4sin²(3x) .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]
Differentiation Rules[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]
A researcher looking for evidence of extrasensory perception (ESP) tests 1000 1000 subjects. Nine of these subjects do significantly better ( P < 0.01 ) (P<0.01) than random guessing.
(a) Nine subjects may seem like a lot of people, but can you conclude that these nine people have ESP? Select the appropriate statement that explains whether or not it is proper to conclude that these nine people have ESP.
Yes. This follows directly from the 1 % 1% significance level.
Yes. This follows directly from the statistical significance.
No. Since the tests were performed at the 1% significance level, as many as 10 subjects may have done significantly better than random guessing.
No. The sample size was not large enough for any statistical inference.
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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The digits of a two-digit number sum to 11 and have a positive difference of 1. Which of these MUST be true?
a. The product of the digits of the two-digit number is 30.
b. The two-digit number is a multiple of 5.
c. The two-digit number is even.
d. The two-digit number is 83.
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.
Define additionAddition 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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Work out he area of this circle. Take pi to be 3. 142 and give your answer to 1 decimal place
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
please does anyone know how to solve this table
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
How to complete the table?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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