use a double integral to find the area of the region. one loop of the rose r = 8 cos(3θ)
Answers
Using a double integral, the area of the region is [tex]\frac{\pi}{12}[/tex].
Consider the following polar equation:
r = 8cos3[tex]\theta[/tex]
What is a polar curve and how does it work?This is the process for measuring the intensity of light. Such intensity varies according to how close or far you are from LED light bulbs. Engineers use polar curves to plot the level of the intensity at any given point. In other words, a polar curve tells an engineer how LED light bulbs spread light across a room or surface.
The objective is to find the area of the region using double integral.
The sketch of the curve attached at the end of solution.
The intersection points of the curve with horizontal axis are given by,
8cos3[tex]\theta[/tex] = 0
cos3[tex]\theta[/tex] = 0
[tex]3 \theta & =\pm \frac{\pi}{2}[/tex]
[tex]\theta & =\pm \frac{\pi}{6}[/tex]
Therefore, the loop is given by the region,
[tex]D=\left\{(r, \theta) \mid-\frac{\pi}{6} \leq \theta \leq \frac{\pi}{6}, 0 \leq r \leq \cos 3 \theta\right\}[/tex]
Let A be the area of the curve.
The area of the loop is given by,
[tex]& A(D)=\iint_D d A \\[/tex]
[tex]& =\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \int_0^{\cos 3 \theta} r d r d \theta \\[/tex]
[tex]& =\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left[\frac{r^2}{2}\right]_0^{\cos 3 \theta} d \theta \\[/tex]
[tex]& =\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{6}} \cos ^2 3 \theta d \theta \\[/tex]
[tex]& =\frac{1}{2} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(\frac{1+\cos 6 \theta}{2}\right) d \theta \\[/tex]
[tex]& =\frac{1}{4}\left[\theta+\frac{1}{6} \sin 6 \theta\right]_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \\[/tex]
[tex]& =\frac{1}{4}\left[\left(\frac{\pi}{6}+\frac{1}{6} \sin \pi\right)-\left(-\frac{\pi}{6}+\frac{1}{6} \sin (-\pi)\right)\right] \\[/tex]
[tex]& =\frac{1}{4}\left(\frac{\pi}{3}\right) \\[/tex]
[tex]& =\frac{\pi}{12} \\[/tex]
Hence, the area of the one loop of the rose [tex]r = 8 \cos(3\theta)[/tex] is [tex]\frac{\pi}{12}[/tex].
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Related Questions
NEED IT ASAP!!! math
Answers
The variables x, y, z associated to the parallelogram are equal to 60, respectively.
How to find the measures of the variables associated to the angles of a parallelogram
A parallelogram is a quadrillateral formed by two pairs of parallel sides. The figure shows a parallelogram since the following relationships are met: BC ║ AD, AB ║ CD. We need three equations to determine the values of x, y and z and this can be obtained by means of parallelogram properties:
y = x
y + (x + y) = 180°
z + (x + y) = 180°
First, substitute x on the second and third equations:
3 · y = 180°
z + 2 · y = 180°
Second, find y:
y = 180° / 3
y = 60
Third, find z:
z = 180° - 2 · y
z = 180° - 2 · 60
z = 180° - 120°
z = 60
Fourth, find x:
x = 60
The values of variables x, y, z are 60, respectively.
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is it increasing or decreasing y=1/2x+2
Answers
Increasing, as the slope is positive (1/2)
Answer:
Increasing.
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}[/tex]
Given linear equation:
[tex]y=\dfrac{1}{2}x+2[/tex]
Comparing the given linear equation with the slope-intercept formula, the slope of the given equation is ¹/₂.
The graph of a linear equation is increasing if its slope is positive.
(y-values increase as x-values increase).
The graph of a linear equation is decreasing if its slope is negative.
(y-values decrease as x-values increase).
Therefore, as the slope of the given linear equation is positive, its graph is increasing.
Tarriq begins to solve the equation 50 + 2x = –190. 50 + 2x = –190 50 – 50 + 2x = –190 – 50 + 2x = –240 to finish solving the equation using the multiplication property of equality, which factor must tarriq use? a. negative 2 b. negative one-half c. one-halfd. 2
Answers
The factor which tarriq musr use if solving the equation using the multiplication property of equality to finish solving is 2.
The correct answer option is option D
How to solve equation?Tarriq solution:
50 + 2x = –190
50 + 2x = –190
50 – 50 + 2x = –190 – 50
2x = –240
Correct solution:
50 + 2x = –190
Subtract 50 from both sides
50 - 50 + 2x = -190 - 50
2x = - 240
divide both sides by 2
x = -240/2
x = -120
In conclusion, x = -120 is the solution to tarriq equation if using multiplication property of equality.
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Sarah took the advertising department from her company on a round trip to meet with a potential client. Including Sarah a total of 11 people took the trip. She was able to purchase coach tickets for $180
and first class tickets for $1200. She used her total budget for airfare for the trip, which was $10,140. How many first class tickets did she buy? How many coach tickets did she buy?
number of first class tickets bought=
Answers
The number of coach tickets is; 3 coach tickets
The number of first class tickets is; 8 first class tickets
How to solve algebra word problems?We are told that she took 11 people on the trip.
Let the number of first class tickets be f
Let the number of coach tickets be c
Thus;
c + f = 11 -----(1)
She was able to purchase coach tickets for $180 and first class tickets for $1200. She used her total budget for airfare for the trip, which was $10,140. Thus;
180c + 1200f = 10140 -----(2)
Make c the subject in eq 1 to get;
c = 11 - f
Put 11 - f for c in eq 2 to get;
180(11 - f) + 1200f = 10140
1980 - 180f + 1200f = 10140
1020f = 8160
f = 8
Thus;
c = 11 - 8
c = 3
So, 3 coach tickets and 8 first class tickets were bought
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A study of child care enrolled 1364 infants and followed them through their sixth year in school. Later, the researchers published an article in which they stated that "the more time children spent in child care from birth to age four-and-a-half, the more adults tended to rate them, both at age four-and-a-half and at kindergarten, as less likely to get along with others, as more assertive, as disobedient, and as aggressive." $^{31}$ (a) Is this an observational study or an experiment? Justify your answer. (b) What are the explanatory and response variables? (c) Does this study show that child care causes children to be more aggressive? Explain.
Answers
(a) Is this an observational study or an experiment?
ans=Observational study – no treatment was imposed
(b) What are the explanatory and response variables?
ans=explanatory – time spent in child care from birth to age 4 response – adult rating of children’s behavior
d) Does this study show that child care causes children to be more aggressive? Explain.
No. Observational studies cannot show cause-and-effect
Researchers=someone whose job is to study a subject carefully, especially in order to discover new information or understand the subject better: She is a leading researcher in the field.
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Identify the measure of angle x please show work Bc I have too !!
Answers
Answer:
x = 145°
Step-by-step explanation:
x and 35° are a linear pair and sum to 180° , then
x + 35° = 180° ( subtract 35° from both sides )
x = 145°
Help me please thank you show your work if you want
Answers
Answer:
8) y = 4
9) y = x
Step-by-step explanation:
8) Parallel = Same slope
y = mx + b
m = 0
4 = - 4(0) + b
4 = b
y = 4
9) y = x + b
7 = 7 + b
b = 0
Joe make tomato auce to ell at a craft fair. He make it in 5 gallon batche. How many pint jar doe he need to hold 5 gallon of auce?
Answers
The number of pint jar needed to hold 5 gallon of sauce is 40 jars.
Pint and gallons are both units of volume. The number of pint jars needed to hold 5 gallon of sauces can be determined by changing the unit of volume. Conversion of units refers to the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors which change the measured quantity value without changing its effects. According to the conversion of units, 1 gallon is equal to 8 pint. Hence, applying the conversion rate to the total volume of tomato sauce made:
5 gallons = 5*8 = 40 pint
If a jar holds 1 pint of sauce, the number of jars to hold the total volume of sauce is 40 pint jar.
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A science museum just opened near Robert's home, and Robert bought a membership for $25. Then, he can pay just $5.75 per visit. What does it cost Robert to become a member and visit the museum 4 times?
Answers
Answer: $48.00
Step-by-step explanation:
His membership was $25.00
And EACH time he comes he pays $5.75.
so your equation is 25.00 + (5.75 x 4) = $48.00
a simple random sample of 64 observations was taken from a population of size 500. the population standard deviation is 16. what is the standard deviation of sample mean?
Answers
64 observations were randomly selected at random from a population of 500. the population standard deviation is 16.The standard deviation of the sample mean is 4 (16/√64).
The standard deviation of the sample mean is calculated by dividing the population standard deviation (16) by the square root of the sample size (64).
Consequently, the sample mean's standard deviation is
16/√64 = 4.
This means that the expected spread of the sample means is 4, which is less than the population standard deviation of 16. This is because the sample is smaller than the population, so the sample standard deviation will be smaller than the population standard deviation.
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NO0000000000000000000000
Answers
To determine whether the system of equations is consistent and independent, consistent and dependent, or inconsistent, we need to first solve the system of equations to find the value of x and y that satisfy both equations.
To solve the system of equations, we can first solve the first equation for y:
y = −3x + 1
Next, we can substitute this expression for y into the second equation and solve for x:
2y = −6x + 2
2 * (−3x + 1) = −6x + 2
−6x + 2 = −6x + 2
0 = 0
Since the two equations are equivalent, the system of equations is consistent and dependent, meaning that there is only one possible solution (x = 0, y = 1) that satisfies both equations.
Please help. The question and answers are in the screenshot
Answers
Answer:
D
Step-by-step explanation:
To solve for a you would need to subtract k from both sides. That would leave you with
y - k = a ([tex]x-h)^{2}[/tex] then, divide both sides by [tex](x-h)^{2}[/tex] to get letter d
The science club is taking a trip to a nature preserve. Before the trip, the divide 3 pounds of trail mix equally into 10 bags to have as a snack. How much trail mix went in each bag?
Answers
Answer: 0.03 pounds
Step-by-step explanation:
To determine how much trail mix went in each bag, you need to divide the total amount of trail mix by the number of bags.
In this case, you would divide 3 pounds of trail mix by 10 bags to find that each bag contained 0.3 pounds of trail mix.
You can also express this amount in ounces. Since there are 16 ounces in a pound, each bag contained 4.8 ounces of trail mix (0.3 pounds * 16 ounces/pound = 4.8 ounces).
mateo is playing a word game where he's trying to make 444-letter words by rearranging the letters in the word iron. how many unique ways are there to arrange the letters in the word iron?
Answers
The word IRON can be arranged is 24 unique ways.
Given:
The word = IRON
The number of unique ways in which the letters in the word IRON can be arranged = ?
The number of letters in IRON = 4
The number of positions = 4
In the first position, any one of the 4 letters can be placed
In the second position any one of the remaining 3 letters can be placed
In the third position any one of the remaining 2 letters can be placed
The fourth position can be filled with the left over letter
So the total number arrangement in unique way
= (Number of ways to fill in 1st position)* (Number of ways to fill in 2nd position)* (Number of ways to fill in 3rd position)* (Number of ways to fill in 4th position)
=> 4 *3*2*1
= 24 ways
Hence, The word IRON can be arranged is 24 unique ways.
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Which ordered pair does NOT lie on the graph of y = x/4 + 5?
Answers
For example you were given the ordered pair (5,3), plug in 3 instead of the y and 5 instead of the x, so the function would be 3 = 5/4 + 5, which means 3 = 1 1/4 + 5, which simplifies to 3 = 6 1/4. Since that’s not true, the ordered pair doesn’t exist in the graph of that function.
function g is a transformation of the parent exponential function which statements are true about function g if the domain of function g is x greater than zero
Answers
The correct statements are; function g is positive over (-∞, ∞), function g has a y-intercept of (0,4) and function g decreasing over the interval (-∞, 0)
What is a function?A function is defined as a relation between a set of inputs having one output each.
Given that, function g is a transformation of the parent exponential function and the domain of function g is x > 0
Studying the graph, we can conclude that,
function g is positive over (-∞, ∞)
function g has a y-intercept of (0,4)
function g decreasing over the interval (-∞, 0)
Hence, The correct statements are; function g is positive over (-∞, ∞), function g has a y-intercept of (0,4) and function g decreasing over the interval (-∞, 0)
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What is 2 to the power of three halves equal to? square root of 8 cube root of 8 cube root of 16 square root of 16
Answers
The answer to 2 to the power of three halves equal to? is square root of 8 cube root of 8 cube root of 16 square root of 16 is [tex]\sqrt{8}[/tex]
How to find the square root of a number?In mathematics, a square root of a number x is a number y such that y² = x; in other words, a number y whose square is x. For example, 4 and −4 are square roots of 16, because 4² = ² = 16.
From the question, we can simplify the question as
[tex]2^{\frac{3}{2} }[/tex]
simplifying further by simplifying the power
[tex]2^{\frac{3}{2} }[/tex] = [tex](2^{3})\frac{1}{2}[/tex]
= [tex]\sqrt{2^{3} }[/tex]
= [tex]\sqrt{8}[/tex]
hence, the solution to the problem is [tex]\sqrt{8}[/tex]
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The 130 juniors at Smithers High School are taking a class trip. Three buses will be taking the students and
chaperones to their destination, and each bus can hold up to 45 people. The first bus will be filled to capacity,
followed by the second and third buses. The function f(x) describes the total number of people on the first
bus, where x
is the number of minutes spent loading the bus.
If the first bus is boarded at a rate of 15 people per minute, what is the domain of the function?
Answers
If the first bus is boarded at a rate of 15 people per minute, the domain of the function is; (15 ≤ x ≤ 45)
How to solve Inequality word problems?We are told that;
Total number of Juniors at Smithers High School that are taking a class trip = 130
Number of buses taking each student to their destination = 3 buses
Capacity of each bus = 45 people
Now, if the function f(x) describes the total number of people on the first
bus, where x is the number of minutes spent loading the bus.
Then since the maximum number of people on the first bus is 45, then it means that the domain of the function is;
(15 ≤ x ≤ 45)
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Here are the first six terms of an arithmetic sequence 3,8,13,18,19,23,28 find the an expression for the nth term
Answers
Answer:
nth term = a+(n-1)d
Step-by-step explanation:
where a=first term
n=nth term
d=common difference=2nd term - 1st term=5
a menorah holds one candle for each night of hanukkah and and extra candle called the shamash that is used to light the other candles. how many candles does a hanukkah menorah hold?
Answers
The primary eight candles or oil lamps must be on a plane that is higher or lower than the shamash.
A nine-branched candelabrum known as a Hanukkah menorah, also known as a hanukkiah, is lighted during the eight-day Jewish festival of Hanukkah. Eight of the nine branches are held up by lights (candles or oil lamps), which stand throughout the eight nights of the festival. One additional light is lighted each night until all eight branches are illuminated on the last night. The candle on the ninth branch, known as the shamash (Arabic for "helper" or "servant"), is used to light the other eight branches.
The seven-branched menorah used in the first Temple in Jerusalem is commemorated by the Hanukkah menorah, which is separate from it. It's one of the most commonly manufactured pieces of Jewish ritual art, along with the Star of David and the menorah with seven branches.
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What i the um of the fraction? Ue the number 18 equivalent fraction to help find the anwer -3/41/2
Answers
fractions with the same denominator added together
You must add the numerators and keep the same denominator when adding fractions with the same denominator. Since the denominators of the two fractions are the same, we must add the numerators while maintaining the same denominator, which is 4.
What are the parts of fraction?
A fraction consists of two components. The numerator is the figure at the top of the line. It details the number of equal portions that were taken from the total or collection. The denominator is the figure that appears below the line.
The number below the bar is called the denominator . It tells the number of equal parts into which the whole has been divided. The number above the bar is called the numerator. It tells how many of the equal parts are being considered.
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a survey of 600 non-fatal accidents showed that 214 involved the use of a cell phone. find a point estimate for p, the population proportion of non-fatal accidents that involved the use of a cell phone.
Answers
The population proportion of non-fatal accidents that involved the use of a cell phone is 0.03567
A population proportion is known as the share of a population that belongs to a particular group. The Confidence intervals are used to estimate population proportions.
as per given in the question,
The total number of accidents (n) are 600
The accidents involving cell phones (x) are 214
the sample proportion ([tex]\hat p[/tex]) = x/n
where
x is the count of the successes
n is the total size of the population
P is the population proportion
=> 214/600
=> 0.3567
the estimated point for population proportion is
[tex]\hat p[/tex] = 0.03567
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Multiply: (√2x³ +√12x)(2√10x³ + √6x²
2√10x +2√3x³ +4√15x³ +6√2x
2x²√5 +2x√3x + 2x³√30 +3x√2x
4x √√5+2x²√√3x + 4x³√30+ 6x√√2x
x√20+x²√√6x + x³√120 + x√12x
Answers
The result of the multiplication of (√2x³ +√12x)(2√10x³ + √6x²) is
[tex]2\sqrt{20}x^6 + \sqrt{12}x^5 + 2\sqrt{120}x^4 + \sqrt{72}x^3[/tex]
What is multiplication?
Multiplication is a mathematical operation in which one number is multiplied by another. It is usually used to calculate the product of two or more numbers. It is an important operation in mathematics and is used in a variety of ways, including calculating the area of a rectangle, the volume of a cube, and the total cost of items purchased at a store. Multiplication is often represented by the multiplication symbol, which is usually an 'x' or an asterisk (*). When two numbers are multiplied, the result is called the product. For example, if two numbers are multiplied together, the product is the answer to the question "what is the result of multiplying these two numbers together?" The answer is the product of the two numbers.
Here the multiplication given is
(√2x³ +√12x)(2√10x³ + √6x²)
[tex]2\sqrt{20}x^6 + \sqrt{12}x^5 + 2\sqrt{120}x^4 + \sqrt{72}x^3[/tex]
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What value of x is in the solution set of the inequality –5x – 15 > 10 20x? –2 –1 0 1
Answers
X is an algebraic variable which has no defined value. X is kind of universe variable taken in most of the equations.
The value of x in the solution set of - 5x - 15 > 10 + 20x is x < - 1 and the required value from the solution set is -2.
Solution:
Given: - 5x - 15 > 10 + 20x
Let us solve the inequality by isolating the variable x.
Add 5x on both sides
- 5x - 15 + 5x > 10 + 20x + 5x
- 15 > 10 + 25x
Subtract 10 from both sides
- 15 - 10 > 10 - 10 + 25x
- 25 > 25x
Divide both sides by - 25
-1 > x
x < - 1
The value of x in the solution set of - 5x - 15 > 10 + 20x is x < - 1 and the required value from the solution set is -2.
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Answer:
-2
Step-by-step explanation:
i did it
You just won the lottery! You now have $25,000! You make a smart decision and invest all $25,000 to earn interest on it over the next 10 years.
You need to find 3 local banks and find their savings interest rates.
Then you need to decide which bank will earn you the most money after investing $25,000 for 10 years. Use the SIMPLE INTEREST FORMULA.
1) Which bank yielded the highest amount of interest earned? Which bank yielded the lowest amount of interest earned?
2) What is the difference in the principal invested for banks that yielded the highest and lowest interests from Question 1? What is the difference in the interest earned for the banks from Question 1?
3) Which bank would you invest your $25,000? WHY?
Answers
The difference between the interest compounded daily and the interest compounded annually is $67.13.
To calculate the interest earned by your money annually, you calculate as follows;
To determine the interest, use FV - amount invested
FV = P (1 + r)^n
the above represents,
FV = Future value
P = Present value
R = interest rate
N = number of years
$25,000 (1.0725) = 26,812.50
26,812.50 - $25,000 = $1,812.50
What is the interest earned in your money is compounded on a daily basis?
$25,000x (1 + 0.0725 /365)^365 = 26,879.63
26,879.63 - $25,000 = $1879.63
What is the difference in interest?
$1879.63 - $1,812.50 = $67.13
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Jan used 22.5 gallons of water in the shower. This amount is 7.5 gallons less than the amount she used for washing clothes. Write and solve an equation to find the amount of water Jan used to wash clothes, where g is the number of gallons.
Pls show the equation
Answers
Answer: 22.5g+(22.5g-7.5g)
Step-by-step explanation: Its because its all the information combined.
I need help with A help please
Answers
Answer:
x = 12
y = 4
Step-by-step explanation:
Part B
Gradient
Gradient = (y2 - y1) / (x2 - x1)
Gradient = (6 - 3) / (10 - 4)
Gradient = 0.5
Line of equation
y - y1 = m (x - x1)
y - 3 = 0.5 (x - 4)
y - 3 = 0.5x - 2
y = 0.5x - 2 + 3
y = 0.5x + 1
f(x) = mx + b ---> f(x) = 0.5x + 1
m = 0.5
b = 1
Part A
x = 12
y = 4
(Put the values of the graph in the line of equation)
I hope my answer helps you.
You have a piece of fabric with an area of 32 1/3 ft. ² if there are five people in your group what i the area of fabric that each peron can ue
Answers
With the mentioned area of fabric, the five person in group will get 6.47 ft² of fabric per person.
Firstly converting the area of fabric from mixed fraction to fraction.
The area of fabric = (32×3 + 1)/3
Performing multiplication and addition on Right Hand Side of the equation
The area of fabric = 97/3 ft²
Amount of fabric each person gets = (97/3)/5
Performing division on Right Hand Side of the equation
Amount of fabric each person gets = 97/(3×5)
Amount of fabric each person gets = 97/15
Amount of fabric each person gets = 6.47 ft²
Thus, each person will get the fabric of area 6.47 ft².
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find constant rate of change
Answers
Answer: B
Step-by-step explanation:
The constant rate of change or slope is rise/run
Let's pick 2 points ( 4, 12) ( 10, 0)
We see that the y decreases by 12 and the x increase by 6, so our slope is
-12/6 = -2 hours per person
So the answer is B
the surface of an ellipsoid is intersection with a plane to form a space curve. find the vector function that represents the space curve of intersection of the ellipsoid and the plane
Answers
The vector function that represents the space curve of intersection of the ellipsoid and the plane is given by[tex]$\mathbf{r}(t) = (x(t), y(t), z(t))$[/tex] and [tex]x(t) = a\cos(t)$ y(t) = b\sin(t)$ $z(t) = \frac{-d - ax(t) - by(t)}{c}$$[/tex]
Let [tex]$\mathbf{x} = (x, y, z)$[/tex] be the position vector of a point on the ellipsoid, and [tex]$a, b, c$[/tex] be the corresponding semi-major axes. Then the equation is
[tex]$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$$[/tex]
The equation of the plane is
[tex]$$ax + by + cz + d = 0.$$[/tex]
The vector function that represents the space curve of intersection of the ellipsoid and the plane is given by
[tex]$$\mathbf{r}(t) = (x(t), y(t), z(t)),$$[/tex]
where
[tex]$$x(t) = a\cos(t)$$$$y(t) = b\sin(t)$$$$z(t) = \frac{-d - ax(t) - by(t)}{c}$$[/tex]
The vector function that represents the space curve of intersection of the ellipsoid and the plane is given by [tex]$\mathbf{r}(t) = (x(t), y(t), z(t))$[/tex]where [tex]$x(t)$[/tex]and [tex]$y(t)$[/tex] are the coordinates of points on the ellipsoid, and [tex]$z(t)$[/tex] is determined by the equation of the plane. The vector function allows us to represent the intersection of the ellipsoid and the plane as a space curve.
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