find the area of the following region. the region inside limaçon r=4-3cos0
Answers
The area of the region inside the limaçon r = 4 - 3cos(θ) is 7π square units.
The polar equation for a limaçon is given by r = a ± b*cos(θ), where "a" is the distance from the pole to the loop of the limaçon, and "b" is the distance between the pole and the midpoint of the loop.
Here, we have r = 4 - 3cos(θ), which is a limaçon with a = 1 and b = 3. To find the area of the region inside this limaçon, we can use the formula:
A = (1/2)∫(θ1 to θ2) [r(θ)]^2 dθ
where θ1 and θ2 are the polar angles where the limaçon intersects the x-axis.
To find θ1 and θ2, we can set r = 0 and solve for θ:
4 - 3cos(θ) = 0
cos(θ) = 4/3
θ1 = arccos(4/3)
θ2 = -arccos(4/3) + 2π
(Note that we take θ2 as a negative angle because cos(θ) is an even function.)
Now, we can plug in r = 4 - 3cos(θ) and integrate:
A = (1/2)∫(θ1 to θ2) [4 - 3cos(θ)]^2 dθ
= (1/2)∫(θ1 to θ2) [16 - 24cos(θ) + 9cos^2(θ)] dθ
This integral can be evaluated using trigonometric identities and integration by substitution. The final result is:
A = (25π + 27√3)/6 ≈ 19.63
Therefore, the area of the region inside the limaçon r = 4 - 3cos(θ) is approximately 19.63 square units.
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Related Questions
Which function has a range of all real numbers greater than or equal to -4?
A. f(x) = x-4
B. f(x) = -4x
C. f(x) = (x+1)^2 - 4
D. f(x) = -|x-3|-4
Answers
Answer:
C. \(f(x)=(x+1)^2-4\)
Step-by-step explanation:
A. the graph is a straight line with slope 1. It goes up/down infinitely far so the range is all real numbers. Not A!
B. The graph is a straight line with slope -4, so, like the function in A, the range is all real numbers. Not B!
D. The graph is an absolute value function y = |x|, reflected over the x-axis, shifted right 3 units, then shifted down 4 units. So the graph starts witha "vee" shape opening up, becomes a vee opening down, then ultimately gets shifted down 4 units. The range is all real numbers less than or equal to -4.
See the attached graphs. image2 is the function in answer choice D.
[A→(B→C)]→[B→(A→C)] (A∧B)→(A→B
′
)
′
(A→C)∧(C→B
′
)∧B→A
′
[A→(B∨C)]∧C
′
→(A→B)
Answers
(A→C)∧(C→B′)∧B→A′ is the given proposition. Now, we have to prove that [A→(B→C)]→[B→(A→C)].Proof:We have to prove that [A→(B→C)]→[B→(A→C)] is a tautology.By using the conditional proof method, we have to assume that [A→(B→C)] is true and then show that [B→(A→C)] is also true.
For this, we have to use the rules of inference. Let's begin:1. Assume A → (B → C) is true.2. By Simplification, A is true because we have (A ∧ B) given in the premises.3. By Simplification, B is also true because we have (A ∧ B) given in the premises.4. By Modus Ponens, (B → C) is true.5. By Modus Ponens, (A → C) is true.6. By Simplification, we get C is true.7. By Modus Ponens, (B → (A → C)) is true.8. By Modus Ponens, [(A → (B → C)) → (B → (A → C))] is true. Thus, the proof is completed.Note: Please provide the complete question to receive the correct answer.
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Find the H.C.F of these expressions.(2a²+6ac,4a²c+12ac²)
Answers
Answer:
Hello,
Answer 8a²c(a+3c) =8a³c+24a²c
Step-by-step explanation:
2a²+6ac=2a(a+3c)
4a²c+12ac²=4ac(a+3c)
H.C.F=2a(a+3c)*4ac=8a²c(a+3c)
Answer:
1st expression:2a×a+2×3×a×c
2nd expression:2×2×a×a+2×2×3×a×c×c
here,
2×a×a+2×3×a×c
2a^a+6ac
a right triangle has two legs measuring 9 and 40 respectively, what is the length of the hypotenuse?
Answers
Answer:
The length of the hypotenuse is 41.
Step-by-step explanation:
The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
In this case, the lengths of the legs are 9 and 40. Using the Pythagorean theorem, we can find the length of the hypotenuse:
\(\sf:\implies h^2 = 9^2 + 40^2\)
\(\sf:\implies h^2 = 81 + 1600\)
\(\sf:\implies h^2 = 1681\)
Taking the square root of both sides, we get:
\(\sf:\implies h = \sqrt{1681}\)
Simplifying the square root, we get:
\(\sf:\implies \boxed{\bold{\:\:h = 41\:\:}}\:\:\:\green{\checkmark}\)
Therefore, the length of the hypotenuse is 41.
Greetings! ZenZebra at your service, hope it helps! <33
62c + 86 = 32
A. -1.90
B. -.87
C. .35
D. 4.63
Answers
Answer:B
62ç=32-86
62c=-54
C=0.87
what time does a 12-hour clock read a) 80 hours after it reads 11:00? b) 40 hours before it reads 12:00? c) 100 hours after it reads 6:00?
Answers
a) 80 hours after 11:00 on a 12-hour clock would be 7:00.
b) 40 hours before 12:00 on a 12-hour clock would be 4:00.
c) 100 hours after 6:00 on a 12-hour clock would be 10:00.
a, To find this, we need to divide 80 by 12 (the number of hours on the clock), which gives us a quotient of 6 and a remainder of 8. We then add the remainder to the starting time of 11:00, giving us 7:00.
b, To find this, we need to subtract 40 from 12 (the number of hours on the clock), which gives us 8. We then subtract 8 from the starting time of 12:00, giving us 4:00.
c, To find this, we need to divide 100 by 12 (the number of hours on the clock), which gives us a quotient of 8 and a remainder of 4. We then add the quotient (which represents a full cycle of 12 hours) to the starting time of 6:00, giving us 6+8=14:00, which is equivalent to 2:00 on a 12-hour clock. Finally, we add the remainder of 4 to 2:00, giving us 10:00.
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suppose that 25% of adults exercise regularly. if 11 adults randomly selected, what is the probability that four or less exercise regularly? assume the binomial distribution applies.
Answers
The probability that four or fewer adults exercise regularly is 0.0781.
The following are the step-by-step procedures for getting the probability of less than or equal to 4 adults who exercise regularly:
Determine the binomial distribution parameters n = 11, the number of trials (n) in the experiment p = 0.25,The probability of success on each trial (p)q = 1 - p = 0.75, the probability of failure on each trial (q)
Get the probability that 4 or fewer adults exercise regularly by using the binomial probability formula:P(X ≤ 4) = Σ (¹¹Cx) (0.25)x (0.75)11 - x where X = 0, 1, 2, 3, 4
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)P(X ≤ 4)
= [(¹¹C₀) (0.25)⁰ (0.75)¹¹] + [(¹¹C₁) (0.25)¹ (0.75)¹⁰] + [(¹¹C₂) (0.25)² (0.75)⁹] + [(¹¹C₃) (0.25)³ (0.75)⁸] + [(¹¹C₄) (0.25)⁴ (0.75)⁷]
P(X ≤ 4) = [(1) (1) (0.0824)] + [(11) (0.25) (0.1074)] + [(55) (0.0625) (0.1394)] + [(165) (0.0156) (0.1715)] + [(330) (0.0039) (0.2036)]
P(X ≤ 4) = 0.0824 + 0.2948 + 0.2143 + 0.0847 + 0.0400
P(X ≤ 4) = 0.7162 or 71.62%
Now get the probability that more than 4 adults exercise regularly by subtracting P(X ≤ 4) from 1:P(X > 4) = 1 - P(X ≤ 4)P(X > 4) ⇒ 1 - 0.7162P(X > 4) ⇒ 0.2838 or 28.38%
Therefore, the probability that four or fewer adults exercise regularly is 0.0781.
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Help please ): pleaseeee
Answers
Answer:
answer it yourself
Step-by-step explanation:
The sum of 54 anid six times a number is 216. Find the number.
Answers
Answer:
x = 27
Step-by-step explanation:
Step 1: Write out equation
54 + 6x = 216
Step 2: Solve for x
6x = 162
x = 27
Answer:
x=27
Step-by-step explanation:
54+6x=216
Collect like terms:
6x=216-54
6x=162
Divide both sides by the coefficient of x
6x\6=162\6
x=27
Can someone please help with this ?
Answers
Answer: -9n+20
This is the same as 20-9n
================================================
Explanation:
The jump from 11 to 2 is "minus 9"
The jump from 2 to -7 is also "minus 9".
Assuming this pattern continues on, we have an arithmetic sequence with
a = 11 = first termd = -9 = common differenceThe nth term can be found like so
\(a_n = a + d(n-1)\\\\a_n = 11 + (-9)(n-1)\\\\a_n = 11 -9n + 9\\\\a_n = -9n+20\\\\\)
Let's check the answer by trying n = 3
\(a_n = -9n+20\\\\a_3 = -9*3+20\\\\a_3 = -27+20\\\\a_3 = -7\\\\\)
This shows the third term is -7, which matches what the original sequence shows. The answer is partially confirmed. I'll let you check the other values of n. You should get 11 when trying n = 1, and you should get 2 when trying n = 2.
Divide. Write your answer in simplest terms. 2/5 / 7/8
Answers
Answer:
The answer to your problem is, \(\frac{16}{35}\)
Step-by-step explanation:
= \(\frac{2}{5} / \frac{8}{7}\)
= \(\frac{2*8}{5*7}\)
= \(\frac{16}{35}\)
Thus the answer to your problem is, \(\frac{16}{35}\)
Hope this is as simple you can get.
7-20. The two triangles below are congruent. Write a congruence statement relating the triangles. Describe a sequence of transformations that maps one onto the other. N E 10 10 8 .
Answers
Step-by-step explanation:
m ( < E ) = m ( < N )
m ( < H ) = m ( < J )
m ( < A ) = m ( < O )
NH = EJ
NA = EO
AH = JO
work out m and c for the line: 5x-3y+4=0
Answers
Answer:
\(m = \frac{5}{3}\)
\(b = \frac{4}{3}\)
Step-by-step explanation:
The question requires you to rearrange the equation into the form:
y = mx + c
Which is the straight line equation, meaning rearrange for y:
Add 3y to both sides of the equation:
5x - 3y + 4 = 0
5x - 3y + 4 + 3y = 0 + 3y
5x + 4 = 3y
Divide both sides of the equation to isolate y:
\(\frac{5x}{3} + \frac{4}{3} = \frac{3y}{3}\)
\(y = \frac{5}{3}x + \frac{4}{3}\)
This means that \(m = \frac{5}{3}\) and \(b = \frac{4}{3}\).
Hope this helps!
Answer:
m = 5/3 and c = 4/3.
Step-by-step explanation:
5x - 3y + 4 = 0
-3y = -5x - 4
y = (-5/-3)x - 4/-3
y = 5/3 x + 4/3
y = mx + c
So m = 5/3 and c = 4/3
For any given flight, an airline tries to sell as many tickets as possible. Suppose that on average, 20% of ticket holders fail to show up, all independent of one another. Knowing this, an airline will sell more tickets than there are seats available (i.e., overbook the flight) and hope that there is a sufficient number of ticket holders who do not show up, to compensate for its overbooking. Using the Central Limit Theorem, determine n, the maximum number of tickets an airline can sell on a flight with 400 seats so that it can be approximately 99% confident that all ticket holders who do show up will be able to board the plane. Use the de Moivre-Laplace 1/2-correction in your calculations. Hint: You may have to solve numerically a quadratic equation
Answers
Let p = 0.2 be the probability of a ticket holder failing to show up. Then, the probability of a ticket holder showing up is q = 1 - 0.2 = 0.8. Given that the airline wants to be 99% confident that all ticket holders who do show up will be able to board the plane, we want to find the maximum number of tickets that the airline can sell, such that the probability of more than 400 passengers showing up is no more than 1%.
Solution:
n = the maximum number of tickets that the airline can sell on a flight with 400 seats.
First, we need to check whether the conditions for applying the Central Limit Theorem are satisfied or not. Here, we are dealing with a binomial distribution, which satisfies the following conditions for applying the Central Limit Theorem:
np = 400(0.8)
= 320 ≥ 10nq
= 400(0.2)
= 80 ≥ 10
Since both of the above conditions are satisfied, we can approximate the binomial distribution with a normal distribution with mean
μ = np = 320 and variance σ2 = npq = 64.
The probability of more than 400 passengers showing up is P(X > 400), where X is the number of passengers showing up, which can be approximated by a normal distribution with the same mean and variance. Hence, we need to find the value of n, such that
P(X > 400) = P(Z > (400 - 320 + 0.5)/√64)
= P(Z > 2.37) ≤ 0.01
where Z is the standard normal variable with mean 0 and variance
1. Using the standard normal distribution table, we find that
P(Z > 2.37) = 0.009 is the closest value to 0.01. Therefore, we can set
P(Z > 2.37) = 0.009
in the above equation and solve for n as follows:
2.37 = (n - 320 + 0.5)/8n - 320 + 0.5
= 2.37 × 8n - 319.5 = 18.96n
n ≈ 338.55
≈ 339 (rounded up to the nearest integer)
Therefore, the maximum number of tickets that the airline can sell on a flight with 400 seats is n = 339,
such that it can be approximately 99% confident that all ticket holders who do show up will be able to board the plane. Given that an airline wants to sell as many tickets as possible for a flight, there is always a risk that some ticket holders might not show up on the day of the flight. The airline usually overbooks the flight to compensate for this risk. However, overbooking can lead to inconvenience for passengers, as there might be more passengers than the available seats on the flight. In order to avoid such inconvenience, it is necessary to determine the maximum number of tickets that the airline can sell, such that the probability of more than 400 passengers showing up is no more than 1%.To solve this problem, we used the Central Limit Theorem to approximate the binomial distribution with a normal distribution, since np and nq are both greater than or equal to 10. Using the de Moivre-Laplace 1/2-correction, we found that n = 339 is the maximum number of tickets that the airline can sell on a flight with 400 seats, such that it can be approximately 99% confident that all ticket holders who do show up will be able to board the plane.
Thus the airline can sell a maximum of 339 tickets on a flight with 400 seats, such that it can be approximately 99% confident that all ticket holders who do show up will be able to board the plane.
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A skater is on an ice rink. His right skate has an area of 7.8 cm2 in contact with the ice, but his left skate only has 3 cm2 in contact with the ice.
With his right foot he exerts a pressure on the ice of 50 N/cm2.
He applies an equal force with his left foot.
What is the pressure exerted by his left foot in N/cm2?
Answers
Answer:
130 N/cm2
Step-by-step explanation:
7.8 cm2 -> 50N/cm2
3 cm2 -> x N/cm2
x = (50×7.8) / 3
x = 130N/cm2
use <, >, or = to compare 3 gallons_11.37 litters
Answers
Answer: 3 gallons < 11.37 liters
Step-by-step explanation:
First you want to convert gallons to liters (or liters to gallons).
3 gallons is equivalent to 11.356 liters.
As you can see, all digits up to the hundredths digit are equal, while the hundredths digit of 11.37 is greater than the hundredths digit in 11.356(3 gallons), meaning that 11.37 liters is greater than 3 gallons.
I hope this helps.
please help me
thanks
Answers
Step-by-step explanation:
what is your question
wha
Deena had 30 dollars to spend on 3 gifts. She spent 10 1 4 dollars on gift A and 5 4 5 dollars on gift B. How much money did she have left for gift C?
Answers
Answer:
$14.41
Step-by-step explanation:
5.45+10.14=15.59 30-15.59=14.41
A large cubical block of wood is floating upright in a lake. The density of water is 1000 kg/m You can assume the block has constant density and is the shape of a perfect cube with side length 2 meters, There are only two forces acting on the block at any given time: the downward force due to gravity, and a buoyant force acting upward. Recall Archimedes principle, which states "A fully or partially submerged object is acted on by a buoyant force, equal in magnitude to the weight of the water displaced by the object. If the block is slightly depressed and then released, it bobs up and down, reaching its highest point once every 2 seconds. Using this information, determine the density of the block, in kg/m".
Answers
A block of wood has a density of p (kg/m^3). The water density is 1000 kg/m^3. The block of wood is 2 meters long and has a cubic shape. If the block is slightly depressed and then released, it bobs up and down, reaching its highest point once every 2 seconds.
Since the block is a cube with side length 2 meters, its volume is V = L^3 = 2^3 = 8 m^3.The buoyant force acting on the block is Fb = 1000 kg/m^3 * 9.8 m/s^2 * 8 m^3 = 78400 N.
According to Archimedes' principle, the buoyant force acting on the block is equal to the weight of the water displaced by the block. Therefore, the weight of the water displaced by the block is 78400 N.
The mass of the block is given by m = p * V = p * 8 m^3. Therefore, the weight of the block of wood is Fg = p * 8 m^3 * 9.8 m/s^2.The block of wood bobs up and down once every 2 seconds. This means that the time it takes for the block to complete one cycle is T = 2 seconds. The frequency of the block's motion is f = 1/T = 1/2 Hz. The period of the block's motion is the time it takes for the block to complete one cycle, which is T = 2 seconds.
we get f = (1/2π) * √(78400 N/(p * 8 m^3 * 9.8 m/s^2) - 1) = 0.25 Hz. \Solving for the density of the block of wood, we get p = 78400 N/(8 m^3 * 9.8 m/s^2 * (2π * 0.25 Hz)^2 + 1) = 410 kg/m^3.
Therefore, the density of the block of wood is 410 kg/m^3.
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This is a 30-60-90 triangle.
What is the measure of x?
18
X
X
x = 9V [?]
![This is a 30-60-90 triangle.What is the measure of x?18XXx = 9V [?]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/WHwBxWYFBinHDZb8ibXwqHQKOQhDXENs.png)
Answers
Answer:
\( \huge\: x =9 \sqrt{ \boxed{3}} \)
Step-by-step explanation:
Given is a 30°-60°-90° triangle and x is the length of side opposite to 60° angle.
\( \therefore \: x = \frac{ \sqrt{3} }{2} \times 18 \\ \\ \therefore \: x = \sqrt{3} \times 9 \\ \\ \huge\therefore \: x =9 \sqrt{ \boxed{3}} \)
Answer:
3
Step-by-step explanation:
If 5x+2=52, then what does x equal?
Answers
Answer:
x=10
Step-by-step explanation:
If we subtract 2 from 52 we get 50 and ten mutiplys into 50.
how to graph y=6/x-2
Answers
Solve for M: 0.72M = -5.76
Answers
\(m = - 8\)
Step-by-step explanation:
1) Divide both sides by 0.72.
\(m = - \frac{5.76}{0.72} \)
2) Simplify- 5.76/0.72 to 8.
\(m = - 8\)
Therefor the answer is m = -8.
Answer:
m=-8\(u^{2} w\)
Step-by-step explanation:
0.72m=-5.76UwU
m=-5.76/0.72
m=-8\(u^{2} w\)
Calculate the final price. Round all answers to the hundredths place and make sure to write your answer in the form of $12.34. Video Game: $60, Discount: 40%
Answers
The final price will be:
\(p=(60)-(60)(0.4)\Rightarrow p=36\)So, the final price of the game is $36.00.
Area of a square is 196 cm²
. Find the perimeter of the square. (ii) If the square is altered into a rectangle by increasing the length by x
centimeters, and decreasing the breadth by y centimeters, find the area of
the rectangle in terms of x and y.
Answers
Step-by-step explanation:
Area of Square=side²
Side=√(Area of the square)
Given area of the square=196cm²
Side of the square=√196=14cm
Perimeter of square=4*side=4*14=56cm
Length of the rectangle=(14+x)cm
Breadth of the rectangle=(14-y)cm
Area of rectangle=Length*Breadth=
(14+x)(14-y)=14(14)+14(-y)+x(14)+x(-y)=(196-14y+14x-xy)cm²
please help me i need this right now please please
Answers
Answer:
60
Step-by-step explanation:
Evaluate.
{4−[−2−(1+3)]}⋅(−6)
(answers)
−13
−32
−48
−60
Answers
Answer:
-60
Step-by-step explanation:
Evaluate:First do the operation in the inner most brackets.
{4 - [-2 - (1 + 3)] }*(-6) ={ 4 - [-2 - 4] } * (-6)
= { 4- [ -2 - 4] } * (-6)
= {4 - [-6] } * (-6)
= { 4 + 6} * (-6)
= 10 * (-6)
= -60
find the radius and diameter of a circle whose area is 15400 mm²
Answers
Solution:
Area=15400mm^2
Now,
Area=pi×r^2
or,15400=22/7×r^2
or,15400×7=22×r^2
or,107800=22r^2
or,107800/22=r^2
or,4900=r^2
or,r=70
Also,
Diameter (d)=r×2
=70×2
=140
the population proportion of college students that stated, if given a choice, they would prefer to start their own business rather than work for someone else is 72%. assume you collected a random sample of 100 students and 78 stated they would prefer to start their own business rather than work for someone else, what is the sample proportion? (give your answer as a decimal and round to two decimal places.)
Answers
78% of the students in the sample stated they would prefer to start their own business rather than work for someone else.
The sample proportion is a statistic that measures the proportion of individuals in a sample who have a particular characteristic of interest. In this case, we are interested in the proportion of college students who stated they would prefer to start their own business rather than work for someone else.
The sample proportion can be calculated as the number of individuals in the sample who have the characteristic of interest (in this case, preferring to start their own business) divided by the total number of individuals in the sample. Using the information provided in the question, we have:
Number of individuals in the sample who prefer to start their own business: 78
Total number of individuals in the sample: 100
So the sample proportion is:
sample proportion = 78/100 = 0.78
Rounding this to two decimal places, we get:
sample proportion ≈ 0.78
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Which choice is an irrational number?
Answers
\(\sqrt{7}\) and \(\sqrt{15}\) are irrational numbers.
Option B and Option C are the correct answer.
Given,
- 8\(\pi\) / \(\pi\)
- \(\sqrt{7}\)
- \(\sqrt{15}\)
- 4.179
We need to find which are irrational numbers.
What is an irrational number?An irrational number is a number that is non-recurring and non-terminating.
Example: 22/7, \(\sqrt{2}\).
We have,
- 8\(\pi\)/\(\pi\)
= 8 it is not an irrational number
- \(\sqrt{7}\)
= 2.6457513111
It is an irrational number because it is non-terminating and non-recurring.
-\(\sqrt{15}\)
= 3.8729833462
It is an irrational number because it is non-terminating and non-recurring.
- 4.179
It is terminating so it is not an irrational number.
Thus \(\sqrt{7}\) and \(\sqrt{15}\) are irrational numbers.
Option B and Option C are the correct answer.
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