Area of the region enclosed by one loop of the curve is pi/288.
To find the area of the region enclosed by one loop of the curve, we can use the formula for the area of a polar region:
A = 1/2 * ∫(r^2)dθ
We can use this formula with the equation for r given:
A = 1/2 * ∫(sin^2(6θ))dθ
This integral can be simplified by noting that sin^2(x) = (1/2)(1 - cos(2x))
Thus,
A = 1/2 * ∫(1/2)(1 - cos(12θ))dθ
This integral can be solved by noting that the antiderivative of (1/2)(1 - cos(12θ)) is (1/24)(sin(12θ) + θ).
Therefore, the area enclosed by one loop of the curve is
A = 1/2 * ((1/24)(sin(12θ) + θ)) evaluated from 0 to pi/6
A = (1/48)(sin(2pi) + pi/6)
= (1/48)(0 + pi/6)
= pi/288
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How do you find the y-intercept without a calculator?
In the equation y = mx + c if we substitute the value of x as zero we will get a point on the y axis given by y =(m × 0) + c ⇒ y = c. Therefore the point of the line on the y axis is (0, c) and so c is also called the y -intercept of the line.
Now, According to the question:
i) from the general equation of line we know that any line can be represented in the form y = mx + c , where m is a constant and is called the slope of the line and c is also a constant and is the y-intercept.
ii) In the equation y = mx + c if we substitute the value of x as zero we will get a point on the y axis given by y =(m × 0) + c ⇒ y = c. Therefore the point of the line on the y axis is (0, c) and so c is also called the y intercept of the line.
iii) If for example we have the line 3y = 6x + 12 then we first get this equation of the line to match the general form of the equation of the line, that is y = mx + c. Therefore dividing both the left and the right sides of the given equation of the line by 3 we get y = 2x + 4 which matches with the general form of the equation for line and we see that the slope of the given line is m = 2 and the y intercept is given by c = 4.
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PLS HELP
using factorisation solve
E=(a+1)(a-1)-3(a+1)²
Answer:
Step-by-step explanation:
factoring
e =
(a+1)(a-1-3(a+1))
= (a+1)(a-1-3a-3)
= (a+1)(-2a-4)
= -2(a+1)(a+2)
The mean and median of the five Numbers are the same 4 6 x y 10
The value of x is 7 and value of y is 8
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
Given data set is {4, 6, x, y, 10}
The mean and the median are the same value.
The median is the middle value, which is x, and the mean is:
M = (4 + 6 + x + y + 10)/5
Then we have that:
(4 + 6 + x + y + 10)/5 = x
Now let's solve this:
(4 + 6 + x + y + 10) = 5x
20 + x + y = 5x
20 + y = 4x
And x and y are whole numbers. Then we can take x = 7 to get:
20 + y = 4*7
y = 28 - 20 = 8
Then we have x = 7 and y = 8.
Hence, the value of x is 7 and value of y is 8 in the data set 4 6 x y 10
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f(n) = 2n-1 find the arithmetic sequence ?
An arithmetic sequence is a sequence of numbers in which the difference between any two successive members of the sequence is a constant value. The general form of an arithmetic sequence is:
a, a + d, a + 2d, a + 3d, ...
where "a" is the first term of the sequence, and "d" is the common difference between successive terms.
Given the function f(n) = 2n - 1, we can see that the function outputs a sequence of integers, with a common difference of 2. So f(n) is not an arithmetic sequence.
Arithmetic sequence is a list of numbers in which each number is the sum of the previous number and a constant number.
Example: 1,3,5,7,9 is an arithmetic sequence because the constant difference is 2.
If you have any other question feel free to ask.
(I NEED THIS RIGHT AWAY!) (PLS HELP) Lynette earns $5 by delivering newspapers. She saves $3 and she spends the rest. If she saved $27 one month, how much did she spend?
Answer:
$18
Step-by-step explanation:
To find how many times she has saved money, you would do 27/3=9. This means that she worked 9 times. Since she worked 9 times, she earned $45 in total. To find how much she saved you would do 45-27=$18
Suppose that ƒ(−1)=10 and ƒ'(x)≥−3 for all values of x. What is the least possible value of ƒ(3)?
Using the mean value theorem, the least possible value of f(3) is - 2
What is the mean value theorem?The mean value theorem states that for a function f(x) on the interval (a, b), there exists a value c such that
f'(c) = f(b) - f(a)/(b - a)
How to find the least possible value for f(3)?Since ƒ(−1) = 10 and ƒ'(x) ≥−3 for all values of x, we require the least possible value of f(3).
Using the mean value theorem where
b = 3, a = -1 and f'(c) ≥ 3.We choose f'(c) = 3 since f'(x) = 3 is the minimum value of f'(x).
So, substituting the values of the variables into the equation, we have that
f'(c) = f(b) - f(a)/(b - a)
f'(x) = f(3) - f(-1)/(3 - (-1))
f'(x) = f(3) - f(-1)/(3 + 1)
f'(x) = f(3) - f(-1)/4
So, making f(3) subject of the formula, we have that
f(3) = f(-1) + 4f'(x)
Since
f'(x) = 3 and f(-1) = 10,Substituting the values of the variables into the equation, we have that
f(3) = f(-1) + 4f'(x)
f(3) = 10 + 4(-3)
f(3) = 10 - 12
f(3) = - 2
So, the least possible value is - 2
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Tamika is ordering a taxi from an online taxi service. The taxi charges $3. 50 just for the pickup and then an additional $0. 75 per mile driven. How much would a taxi ride cost if Tamika is riding for 5 miles? How much would a taxi ride cost that is m miles long?
Cost for 5 miles:
Cost for m miles:
If the pickup charge for taxi is $3.50 and additional charge of $0.75 per mile , then the cost for riding 5 miles is $7.25 and cost for riding m miles is $[tex]3.50+0.75m[/tex] .
The pickup charge that taxi service charges is = $3.50 ;
the additional charge for per miles driven is = $0.75 ;
we have to find the cost for riding 5 miles ;
So , the total cost for riding 5 miles is = [tex]Pickup Charge + (Additional Charge)\times(Number Of Miles Driven)[/tex] ;
substituting the values , the total cost equation becomes ;
[tex]Total Cost = 3.50 + 0.75\times5[/tex]
= [tex]3.50+3.75[/tex]
= 7.25
So , the cost for 5 miles is = $7.25 ;
to find cost for driving "m" miles ,we substitute the number of miles = m;
[tex]Total Cost = 3.50+0.75\times m[/tex]
= [tex]3.50+0.75m[/tex]
Therefore , the cost for riding m miles is $ [tex]3.50+0.75m[/tex] .
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The rate of growth dP / dt of a population of bacteria is proportional to the square root of t, where P is the population size and t is the time in days (0 ? t ? 10) as shown below.
dP/dt=k \sqrt{t}
The initial size of the population is 300. After 1 day the population has grown to 600. Estimate the population after 8 days. (Round your answer to the nearest whole number.)
The population of bacteria after 8 days is 7088
Here, we have been given an equation for the rate of growth of a population of bacteria
dP/dt = k√t
where P is the population size
and t is the time in days (0 ≤ t ≤ 10)
We rewrite above equation as,
dP = k√t dt
Integrating both sides, we get,
P = (2/3) × kt^(3/2) + C
We find the value of C.
For t = 0, P = 300 as the initial size of the population is 300.
Substitute t = 0 in above equation.
300 = 0 + C
C = 300
So, the above equation becomes,
P = 300 + (2/3) × kt^(3/2) .......(1)
Now we find the value of k.
After 1 day the population has grown to 600.
substitute t = 1, P = 600 in equation (1)
600 = 300 + (2/3) × k
(2/3)k = 300
k = 900/2
k = 450
So, the equation of population of bacteria after time t is:
P = 300 + [(2/3) × 450 × t^(3/2)]
P = 300 + 300 × t^(3/2)
P = 300 (1 + t^(3/2))
To find the population after 8 days, substitute t = 8
P = 300 (1 + 8^(3/2))
P = 7088.22
P ≈ 7088
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The null and alternative hypotheses divide all possibilities in to:
a. two sets that may or may not overlap .
b. as many sets as necessary to cover all possibilities c. two non-overlapping sets d. two sets that overlap reset selection
Answer:
Step-by-step explanation:
b
NEED HELP ASAP
PLEASE REFER TO PICTURE
Determine the prices for which quantity demanded is greater than quantity supplied.
The prices where the quantity demanded is greater than quantity supplied are :
$ 2$ 1$ 0When is quantity demanded greater than quantity supplied ?Quantity demanded refers to the amount of goods and services that people and businesses in an economy demand.
This amount of goods and services demanded will be more than the quantity supplied when the price of the good or service is less than the equilibrium price.
The equilibrium price here is $ 3 and so the prices less than $ 3 are $2, $1, and $ 0.
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What is the value of a if 2 is a zero of the polynomial 4x² 2x 5a?
The value of a is [tex]-\frac{12}{5}[/tex] if 2 is a zero of the polynomial [tex]$4 x^2-2 x+5 a$[/tex].
As per the given data, the polynomial is p(x) = [tex]$4 x^2-2 x+5 a$[/tex]
The polynomial is zero polynomial if x is 2
The places where a polynomial becomes zero overall are known as the zeros of the polynomial. The term "zero polynomial" refers to a polynomial with a value of zero (0).
Replace x with 2 we get
Therefore [tex]4(2)^2-2(2)+5 \mathrm{a}=0[/tex]
[tex]\Rightarrow[/tex] 4 (4) - 2 (2)+5 (a) = 0
[tex]& \Rightarrow[/tex] 16 - 4 + 5 (a) = 0
[tex]& \Rightarrow[/tex] 12 + 5 (a) = 0
Add -12 on both sides we get
12 + 5 (a) - 12 = 0 - 12
5 (a) = -12
Divided by 5 into both sides we get
a = [tex]-\frac{12}{5}[/tex]
Therefore the value of a is [tex]-\frac{12}{5}[/tex]
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How do you graph the transformation of an absolute value function?
By translating, stretching and contracting, or reflecting, two functions from the same function family can become one another. Any function from a function family can be obtained by applying one or more transformations to a parent function.
By altering the graph of the parent absolute value function, many other absolute value functions can be created. The resulting absolute value functions take the following form:
[tex]y = |x-h| + k[/tex], where h and k are real numbers.
Absolute value functions can be created using the same kinds of operations that produce new linear functions. They also have a similar impact on absolute value functions. The transformations may, however, appear differently graphically due to some important discrepancies between linear functions and absolute value functions.
Here are some of the transformations:
Vertical Transformations
Translation up k units, k > 0; [tex]y = |x|+k[/tex]Translation down k units, k < 0; [tex]y = |x| + k[/tex]Horizontal Transformations
Translation to the right h units, h > 0; [tex]y = |x - h|[/tex]Translation to the left h units, h < 0; [tex]y = |x - h|[/tex]This is how we graph the transformation of an absolute value function.
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What type of triangle is 70 70 40?
70-70-40 is an isosceles triangle.
The isosceles triangle has three acute angles, which means that the angles are less than 90°.The summation of three angles of an isosceles triangle is always 180°. In an isosceles triangle, if two sides are equal, then the angles opposite to the two sides correspond to each other and are also always equal.The two equal sides are defined as the legs and the third side is defined as the base of the triangle.The two angles opposite the legs are equal and are always acute, so the categorization of the triangle as acute, right, or obtuse depends only on the angles between its two legs.Read more about the isosceles triangle:
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Solve 3x^2 - 6x - 9?
Answer:
3(x+1)(x-3)
Step-by-step explanation:
Well, this isn't an equation, but we can factor it like this:
[tex]3x^2-6x-9=3(x^2-2x-3)=3(x+1)(x-3)[/tex]
Need the answer for these
Answer:
Step-by-step explanation:
2A. 3x=6 ⇒ x=2
2B. 3n=2(n-3) ⇒ 3n=2n-6 ⇒ n= -6
1. cross multiplication method
F(x) = 3x -4 find each value of F(0)
Answer:
Step-by-step explanation:
Rewrite the function as an equation.
y
=
3
x
−
4
Use the slope-intercept form to find the slope and y-intercept.
Tap for fewer steps...
The slope-intercept form is
y
=
m
x
+
b
, where
m
is the slope and
b
is the y-intercept.
y
=
m
x
+
b
Find the values of
m
and
b
using the form
y
=
m
x
+
b
.
m
=
3
b
=
−
4
The slope of the line is the value of
m
, and the y-intercept is the value of
b
.
Slope:
3
y-intercept:
(
0
,
−
4
)
What is the product of - 2 1/2 and - 3 1/3
In the figure, JKLM-EFGH. Complete the following table,
type your answer
Value of x
Value of y
Scale factor of JKLM to EFGH
type your answer.
type your answer
Value of z
type your answer.
The values of x , y and z are 80,2,110 respectively from the given figure where JKLM ~ EFGH
What is Quadrilateral ?
Quadrilateral can be defined as the type of the polygon in which it contains four sides.
Given ,
JKLM ~ EFGH
Since , JKLM ~ EFGH
3/30 = 11/z
z = 30*11/3
z =110
3 / 30 = 8/ x
3 * x = 8 * 30
x = 80
3 / 30 = y/ 20
y = 20 * 3 / 30
y = 2
Hence the values of x , y and z are 80,2,110 respectively.
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What is the difference between an undecidable problem and an intractable problem?
Therefore , the solution of the given problem of tractable problem comes out to be Undecidable troubles are those for which there will never be a computer solution.
What does a tractable problem actually mean?a technique that completes a manageable task in a polynomial period of time. Utilizing the polynomial upper bound is advantageous. A problem is said to be intractable if it cannot be solved in a polynomial amount of time. The bound has a minimum that is exponential.
Here,
Intractable problems would be those in which there is significant proof that, while they can be solved by a desktop, they cannot be solved quickly enough to be actually useful in practice.
Undecidable troubles are those for which there will never be a computer solution, whereas undecidable difficulties are those for which there will never be a computer solution.
Therefore , the solution of the given problem of tractable problem comes out to be Undecidable troubles are those for which there will never be a computer solution.
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Find the radius of convergence R of the series. [infinity] n bn (x − a)n, b > 0 n = 1
R= ____
Find the interval of convergence of the series.
The radius of convergence R = 1/b
The radius of convergence for a power series is given by the formula:
1/R = lim sup |bn|^(1/n)
In this case, the radius of convergence is given by:
1/R = lim sup |bn|^(1/n) = lim sup b^(1/n) = b^(1/n) as b > 0.
So R = 1/b
As for the interval of convergence, it is the set of values of x for which the series converges. The series converges when |x-a| < R, and diverges when |x-a| > R. So the interval of convergence is (a-R, a+R) which is (a-1/b, a+1/b)
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a company spent 762500 to make a movie that will be shown in theaters for 8 weeks. If the movie makes 1250000 in its first week how much money does the movie need to earn in the next 7 weeks to make profit of atleast 2,000,000
The movie need to earn $ 1,512,500 in the next 7 weeks.
What is Unitary Method?The unitary technique involves first determining the value of a single unit,
followed by the value of the necessary number of units.
For Example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
A company spent 762500 to make a movie.
The total movie will be earned
= 762500 + 2000000
= 2,762,500
First week the movie earned 1, 250,000.
So, in 7 weeks the movie needs to earned
= 2,762,500 - 1, 250,000
= 1,512,500
Hence, the movie earn 1,512,500 more.
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Shay and Nadine solve this problem in two different ways.
Mark has $25 in his bank account.
He makes $7 per hour babysitting.
How many hours must he babysit to have a total of $165 in his account?.
Use the drop-down menus to complete the sentences below.
Shay's
Assignment
25+7h-165
25-25 7h-165-25
7h-140
7h-140
7h 7h
A-20
Nadine's
Assignment
166-25-140
You need to make
$140 babysitting
140
-20
You need to babysit
for 20 hours
Please Help!
On solving the provided proportionality question we got that, U = 7.72, is rounded off nearest to tenth digit.
What is proportionality?When two quantities or variables are linearly related in mathematics, the term "proportional" is used. Both quantities increase by two times when one increases by two. Each variable decreases in proportion to the other when it falls to 1/100th of its previous value.We must determine the ratio of the two quantities for each value supplied in order to determine if two quantities are proportionate. The proportionate link between them is evident if their proportions are equal. Their connection is not proportionate if all the ratios are out of balance.
Here,
U/5 = 17/11
U = 17/11 X 5
U = 7.72
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Solve each system by graphing
I’m looking for the answer but also a explanation of how you solved it so I can solve the other 1s without help
Answer: its hard to give an answer but read explanation
Step-by-step explanation: so the first number (1/3x) is your slope and the second number is the y-intercept (5). so for example, you're going to put positive 5 on the y-axis and because you slope is a fraction, rise/run will make you go 3 to the right and 1 up. Then you do the same with the second equation except, the slop is negative and you'll have to go down.
help please (photo attached)
Answer:
slope = -1
Step-by-step explanation:
any 2 points:
(2,2) (0,4)
to find the slope we use the formula:
[tex] \frac{y2 - y1}{x2 - x1} [/tex]
slope is:
2/-2 = -1
Answer:
The slope is =
2/-2 = -1
Step-by-step explanation:
Also i hope you have a wonderful day !
what is the total cost with sales tax? price 120.00 tax 5%
Answer:
5% of 120 is 6, so you would add that 6 to 120.
120 + 6 = 126.
The total cost will sales tax would be $126.00.
Step-by-step explanation:
Hope it helps! =D
Why is it called revolution?
The term "revolution" is often used to refer to a dramatic and sudden change in a situation, especially number of a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another.
Revolution is a term used to refer to a dramatic and sudden change in a situation, especially a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another, thus making it a revolution.
The term "revolution" is often used to refer to a dramatic and sudden change in a situation, especially number of a political situation. It is derived from the Latin word revolutio, which means "a turn around". The term was first used in the 1500s to describe a political movement of overthrowing a government or social system and replacing it with another.
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The tables show the prices for ordering team jerseys from two different companies. You can use the information to write ratios showing the relationship between the cost and the corresponding number of jerseys. 1/5=5 25/5=5 50/10=5 75/15=5. 5/1=5 20/5=4 40/10=4 69/15=4. These ratios are all equivalent. These ratios are not all equivalent. This relationship is proportional. This relationship is not proportional. Explain how you can tell whether a group of ratios represents a proportional relationship.
Even though this relationship is proportional, not all of these ratios are equal.
What is the solution to the 3 to 5 ratio?Verify that the ratios provided are equal. The ratios of 3:5 and 15:25 are both equal. Because the first ratio 3: 5 can be derived by multiplying the ratio 15: 25 by 5 on both the numerator and denominator. The initial ratio of 3: 5 can be multiplied by 5 to get the ratio 15: 25, and vice versa.
What is an illustration of a 4:1 ratio?This indicates that you would require an 8 fl oz can (a quarter of a quart) of Part B if you were ordering a quart of Part A. You will have 1.25 after combining Parts A and B.
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On a coordinate plane, a line is drawn from point L to point M. Point L is at (negative 4, 8) and point M is at (4, 8).
The rule DO,0.25 (x, y) → (0.25x, 0.25y) is applied to the segment LM to make an image of segment L'M', not shown.
The coordinates of L' in the image are
.
The coordinates of M' in the image are
.
The length, L'M', is
.
The slope of the original segment and dilated segment are
.
Answer:
L' = (-1,2)
M' = (1,2)
L'M' = 2
The slope of the original segment and dilated segment are both zero
Step-by-step explanation: just did it on edge!
Given that K is the midpoint of segments RM and JL, and prove Δ RKJ ≅ Δ MKL.
Statements
Reasons
1. K is the midpoint of segments RM and JL
1. Given
2.
2.
3. and
3.
4. RK = MK and JK = LK
4. Definition of Congruent Segments
5. Δ EGF ≅ Δ EGH
5.
What is Reason #3?
a
Given
b
Definition of Midpoint
c
CPCTC
d
SSS Congruence Postulate
The complate table is,
Statement Reasons
1) K is the midpoint of segments Given
RM and JL.
2) RJ ≅ ML SSS Congruence Postulate
3) RK ≅ MK , JK ≅ LK Definition of Midpoint
4) RK = MK , JK = KL Definition of Congruence
Segments
5) Δ EGF ≅ Δ EGH CPCTC
What is Line segment?A line segment is a part of line having two endpoints and it is bounded by two distinct end points and contain every point on the line that is between its endpoint.
Given that;
K is the midpoint of segments RM and JL.
Now,
Since, K is the midpoint of segments RM and JL.
Hence, We get;
Statement Reasons
1) K is the midpoint of segments Given
RM and JL.
2) RJ ≅ ML SSS Congruence Postulate
3) RK ≅ MK , JK ≅ LK Definition of Midpoint
4) RK = MK , JK = KL Definition of Congruence
Segments
5) Δ EGF ≅ Δ EGH CPCTC
Thus, The complete table is shown above.
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