If the 0's are indeed zeroes, then it would appear that you're using degrees, so that the equation
r(θ) = 2 sin(0°) + 3 sin(90°)
describes a circle r = 3 (since sin(0°) = 0 and sin(90°) = 1). In this case, the area would simply be π r ² = 9π.
But I suspect you meant to use θ, so the curve has the far more interesting equation,
r(θ) = 2 sin(θ) + 3 sin(9θ)
Since sin(9θ) has a period of 2π/9 and sin(θ) has a period of 2π, their sum has a period of 2π. (That is, 2π times the LCM of 1 and 1/9, which is 1.) But we observe r(θ) is odd, since
r(-θ) = 2 sin(-θ) + 3 sin(-9θ)
… = -2 sin(θ) - 3 sin(9θ)
… = - r(θ)
which tells us that the curve closes itself over a half-period. So the area bounded by the curve is given by the integral,
[tex]\displaystyle\int_0^\pi\int_0^{r(\theta)}r\,\mathrm dr\,\mathrm d\theta[/tex]
If you're not familiar with double integrals yet, all you need to know is that this reduces to the area formula you may/should be familiar with,
[tex]\displaystyle\frac12\int_0^\pi r(\theta)^2\,\mathrm d\theta[/tex]
Compute the integral:
[tex]=\displaystyle\frac12\int_0^\pi(2\sin(\theta)+3\sin(9\theta))^2\,\mathrm d\theta[/tex]
[tex]=\displaystyle\frac12\int_0^\pi(4\sin^2(\theta)+12\sin(\theta)\sin(9\theta)+9\sin^2(9\theta))\,\mathrm d\theta[/tex]
Apply some identities:
sin²(θ) = (1 - cos(2θ)) / 2
sin(θ) sin(9θ) = (cos(9θ - θ) - cos(9θ + θ)) / 2 = (cos(8θ) - cos(10θ)) / 2
sin²(9θ) = (1 - cos(18θ)) / 2
[tex]=\displaystyle\frac14\int_0^\pi(4(1-\cos(2\theta))+12(\cos(8\theta)-\cos(10\theta))+9(1-\cos(18\theta))\,\mathrm d\theta[/tex]
[tex]=\displaystyle\frac14\int_0^\pi(13 - 4\cos(2\theta) + 12\cos(8\theta) - 12\cos(10\theta) - 9\cos(18\theta))\,\mathrm d\theta[/tex]
[tex]=\dfrac14 \left(13\theta-2\sin(2\theta)+\dfrac32\sin(8\theta)-\dfrac65\sin(10\theta)-\dfrac12\sin(18\theta)\right)\bigg|_0^\pi[/tex]
[tex]=\dfrac1{40} \left(130\theta-20\sin(2\theta)+15\sin(8\theta)-12\sin(10\theta)-5\sin(18\theta)\right)\bigg|_0^\pi[/tex]
[tex]=\dfrac{130\pi-20\sin(2\pi)+15\sin(8\pi)-12\sin(10\pi)-5\sin(18\pi)}{40}[/tex]
= 13π/4
please help on this question i give brainliest
Answer:
1. 9 bounds of carbon dioxide
2. 27 bounds of carbon dioxide
Step-by-step explanation:
So let's find a unit rate: 1 to 9
"Why is it 1:9?"
If I can find divide 72 pounds into 8 hours? What would I find? You would find the pounds for each other
Example: 72/8=9. So If 9 carbon dioxide per hour. I can do 3 hours is 27 pounds of carbon dioxide.
So to sum it up, 72/8=9 so 1:9 can be used to figure out any amount of pounds used in a certain number of hours. So If 3 hours, then 27 pounds.
Answer:
1. 9 pounds of carbon dioxide
2. 27 pounds of carbon dioxide
Step-by-step explanation:
Well, I just know it.
Answer part C. I want step by step on this problem.
Answer:
[tex]Rate = 58007956.5217[/tex]
Step-by-step explanation:
Solving (c): The rate of change:
In this concept, the formula to use is:
[tex]Rate = \frac{V2 - V1}{t2 - t1}[/tex]
Where
[tex](t1,v1) = (0,62817000)[/tex] ----- In 2004
[tex](t2,v2) = (23,1397000000)[/tex] ------ In 2026
So, we have:
[tex]Rate = \frac{1397000000 - 62817000}{23 - 0}[/tex]
[tex]Rate = \frac{1334183000}{23}[/tex]
[tex]Rate = 58007956.5217[/tex]
Pleaseeee help I'll give Brainliest urgent!!!!!!!!!!!!!!!!!
Answer:
The answer is
'The range of the data for girls is less than the range of the data for boys'
I'm don't know dfhgh fhfghfg hgf hgfh
‼️DIMENSIONAL ANALYSIS‼️
‼️ASAP!!! BRAINLIEST!!‼️
PLS HELP!!! SHOW ALL WORK +STEPS!! Thx!
Look at point w,x,y,and z on the number line
Answer:
u need toADD THE PITRUE
Step-by-step explanation:
Find the volume of the cone.
Answer:
[tex]14.50[/tex]
...........
WILL MARK BRAINLIEST! Solve for x and each angles measure. Only asking for #5. Thank you!
Answer:
Step-by-step explanation:
2x + 3x + 135 = 180 Every triangle has 180 degrees
5x + 135 = 180 Subtract 135 from both sides.
5x = 45 Divide by 5
x = 9
2x = 18
3x = 27
Other angle 135
Total 280
Please help me I wanna go to sleep :(
Answer:
a. -2
b. 48, -96, 192
Step-by-step explanation:
Consider a graph of the equation y = x + 6. What is the y-intercept? (5 points) 6 −1 1 −6
D I am not 100% sure, If I am wrong sorry
Which example models a negative integer A.Students in a classroom B. The age of a parent C. Money lost from a pocket D. the height of a tree
Answer:
C- Money lost from a pocket
Step-by-step explanation:
It would be C because something is decreasing ( now a negative) from what they did have.
ill mark brainlist plss help
it is oxygen no doubt about it
How do I solve 1 1/3 × 2 2/4 in steps?
Answer:
3 1/3
Step-by-Step
first step is too make a an improper fraction, 1 1/3 equals 4/3 and 2 2/4 equals 10/4. next your gonna multiply 4/3 and 10/4, 4x10=40,3x4 equals 12, so it would be 40/12. but its improper, we need to reduce it. the lowest number that can go in both numbers is 4. 40 divided by 4 =10. 12 divided by 4=3. 12/3= 3 1/3
Answer:
10/3
Step-by-step explanation:
That's 1 1/3 times 2 1/2.
Converting these into improper fractions, we get 4/3 times 5/2.
Multiplying (4/3) by (5/2), we get 20/6, or, after reduction, 10/3
Help
Which product are greater than 2 and 5\6? Select all that apply
Answer:
B, C, & E
Step-by-step explanation:
(9x10^10) - (4x10^10)
Answer: 50000000000 is your answer.
Step-by-step explanation:
-6y+11x=-34
-4y+7x=-24
Answer:
no idea but mabye someone whose NOT in 3rd grade will know?
Change the following into m square 30,000cm²
Answer: 3sqm
Step-by-step explanation:
Select the correct comparison.
A. The typical value and the spread are both greater in set B.
B. The typical value is greater in set B. The spread is greater in set A.
C. The typical value and the spread are both greater in set A.
D. The typical value is greater in set A. The spread is greater in set B.
Answer:
The typical value is greater in set A. The spread is greater in set B.
Step-by-step explanation:
A P E X, hope it helps.
Answer:
it's B
Step-by-step explanation:
took the quiz
4 to the power of3 -(2+2to the power of 3) x 5
Answer:
14
Step-by-step explanation:
Given the following equation;
4³ - (2 + 2³) * 5
4*4*4 - (2 + 2*2*2) * 5
64 - (2 + 8) * 5
64 - (10) * 5
64 - 50 = 14
Note: the rule of BODMAS was applied.
5/8 62.5/100 as a ratio (simplified)
Answer:
5:8, 25:1
Step-by-step explanation:
i guess this is ans
PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA POINTS*..
IM GIVING 40 POINTS !! DONT SKIP :((.
Answer: C
Step-by-step explanation: The layers of rocks implies superposition and changes in species implies evolution
Select all expressions that are equal to 3⋅3⋅3⋅3⋅3.
3^5
3^4 ⋅ 3
5 ⋅ 3
5^3
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{3\times3\times3\times3\times3}\\\\\mathsf{3\times3=\bf9}\\\\\mathsf{9\times9\times3}\\\\\mathsf{9\times9=\bf81}\\\\\mathsf{81\times3=\bf result}\\\\\mathsf{= \bf \underline{243}}\large\checkmark[/tex]
[tex]\text{Option A. }\downarrow\\\mathsf{3^5}\\\\\mathsf{3^5= 3\times3\times3\times3\times3}\\\\\mathsf{3\times3\times3\times3\times3}\\\\\mathsf{3\times3 = 9}\\\\\mathsf{9\times9\times3}\\\\\mathsf{9\times9 = 81}\\\\\mathsf{81\times3= 243}\\\\\large\text{\bold{Option A.} could possibly be your \bf ANSWER}\huge\checkmark[/tex]
[tex]\text{Option B. }\downarrow\\\mathsf{3^4\times5}\\\mathsf{3^4 = 3\times3\times3\times3}\\\\\mathsf{3\times3\times3\times3}\\\\\mathsf{3\times3=9}\\\\\mathsf{9\times9= 81}\\\\\mathsf{81\times3=243}\\\\\large\text{Option B. could possibly be your \bf ANSWER}\huge\checkmark[/tex]
[tex]\text{Option C. }\downarrow\\\mathsf{3\times5}\\\\\maths{3+3+3+3+3}\\\\\mathsf{3+3=6}\\\\\mathsf{6+6+3}\\\\\mathsf{6+6=12}\\\\\mathsf{12+3=15}\\\\\mathsf{15\neq243}\\\\\large\text{Option C is NOT your ANSWER}[/tex]
[tex]\text{Option D.}\downarrow\\\mathsf{5^3}\\\\\mathsf{5^3=5\times5\times5}\\\\\mathsf{5\times5\times5}\\\\\mathsf{5\times5=25}\\\\\mathsf{25\times5 = 125}\\\\\mathsf{125\neq243}\\\\\large\text{Option D. is NOT your ANSWER}[/tex]
[tex]\boxed{\boxed{\large\text{Answers: \boxed{\boxed{\mathsf{\bf{Option\ A.)\ 3^5 \ \& \ Option\ B.)\ 3^4 \times 3}}}}}}}}\huge\checkmark[/tex]
[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
Answer:
A) 3^5 B) 3^4*3
Step-by-step explanation:
Select the correct answer.
Does the equation below represent a relation, a function, both a relation and a function, or neither a relation nor a function?
y = 3:2 - 95 + 20
OA neither a relation nor a function
OB. both a relation and a function
OC function only
OD. relation only
Answer:
B. both function and relation
Step-by-step explanation:
please help me please help me please help
Answer:
a) September; b)October
Step-by-step explanation:
Just do the calculations, for example: 71+(-293). You will see September has a drop less 210 dollars and October has the biggest change of sales.
A bear loves visiting his friend, the piglet. The distance between their houses is 300m , and the bear always walks from his house to the piglet's house in exactly the same amount of time. But today, the bear stops at a beehive after walking —1 of the 4 distance. He eats honey for 2.5 minutes and then continues to the piglet's house at a speed 15 m/min faster than usual. The total journey to the piglet's house takes the exact same amount of time as it usually does, when the bear does not stop for honey. How long does the journey take?
Answer:
10mins
Step-by-step explanation:
you can draw a rtd chart for this problem
rate (m/min)| time (mins) | distance (m )
regular: (300/x) | x | 300m
fist 1/4 : (300/x) | 75: (300/x) | 75m
last 3/4: | (300/x)+15 | 225:(300/x + 15) | 225m
then, make an equation for the times:
x= (75: (300/x)) + (225: ( (300/x)+15) + 2.5 = x
// get the 2.5 mins from the stop for honey, and we know that regular speed is equal to the one with a stop
solve equation to get x=-20/3 , x=10, because speed is positive, x=10 is the most reasonable answer
which of the following is like a radical to 3 sqrt 6x2 ??? PLEASE HURRY
Answer:
[tex]3\sqrt{6x^2} = 3x\sqrt{6}[/tex]
Step-by-step explanation:
Given
[tex]3\sqrt{6x^2[/tex]
Required
Express as a radical
Split
[tex]3\sqrt{6x^2} = 3\sqrt{6} * \sqrt{x^2[/tex]
[tex]\sqrt{x^2} = x[/tex]
So:
[tex]3\sqrt{6x^2} = 3\sqrt{6} * x[/tex]
This gives:
[tex]3\sqrt{6x^2} = 3x\sqrt{6}[/tex]
Solve using the quadratic formula.
q2 + 69 + 9 = 0
Answer:
the value of q = -3
Step-by-step explanation:
The computation of the equation is shown below:
Given that
q^2 + 6q + 9 = 0
q^2 + 3q+ 3q + 9
q(q + 3) + 3(q + 3)
(q+3) (q+3)
q = -3
Hence, after solving it the value of q = -3
Find the unknown number 10% of what number is 300
Answer:3,000
Step-by-step explanation:
Answer:
3000
Step-by-step explanation:
300 x
10 100
300x100
30,000
30,000/10
3000
In a standard 52-card deck, half of the cards are red, and half are black. The 52 cards are evenly divided into 4 suits: spades, hearts, diamonds, and clubs. Each suit has three face cards (jack, queen, king), and an ace. Each suit also has 9 cards numbered 2 to 10.
Dawn removes two cards, without replacement.
What is the probability that Dawn removed two black cards?
What is the probability that Dawn removed a diamond and a spade?
What is the probability that Dawn removed two face cards?
Answer:
Byhrnegngrnhumhrnim jmmumtjmmtjmtj
4. The teacher recommends that students read for 20
minutes each night. How many pages will the
student complete?
proportional relationships
Find three consecutive numbers whose sum shall equal 45.
Answer:
the three consecutive numbers are 14, 15, 16
Answer:
Since the consecutive numbers, so the first one is x, the next one is x+1, and so on....I hope this will help
Step-by-step explanation:
x + (x + 1) + (x + 2) = 45
x + x + 1 + x + 2 = 45
3x + 3 = 45
3x = 42
x = 14
Now that we know x = 14, let's add one to it.
x + 1 = 14 + 1 = 15.
Now we have our middle number, which is 15. You can always check your answer by adding 14 + 15 + 16 = 45. Whoo!