6. Convert each of the following equations from polar form to rectangular form. a) r² = 9 b) r = 7 sin 0.
Answers
The rectangular form of the equation r = 7 sin θ is: x² + y² = (7 sin θ)², x = 7 sin θ cos θ. Conversion of polar form equation r² = 9 to rectangular form: In polar coordinates, a point (r, θ) in the polar plane is given by r = the distance from the origin to the point, and θ = the angle measured counterclockwise from the positive x-axis to the point.
a) Conversion of polar form equation r² = 9 to rectangular form: In polar coordinates, a point (r, θ) in the polar plane is given by r = the distance from the origin to the point, and θ = the angle measured counterclockwise from the positive x-axis to the point. To convert the polar form equation r² = 9 to rectangular form, we use the conversion formulae:
r = √(x² + y²), θ = tan⁻¹(y/x)
where x and y are rectangular coordinates. Hence, we obtain: r² = 9 ⇒ r = ±3
We take the positive value because the radius cannot be negative. Substituting this value of r in the above conversion formulae, we get: x² + y² = 3², y/x = tan θ ⇒ y = x tan θ
Putting the value of y in the equation x² + y² = 3², we get: x² + x² tan² θ = 3² ⇒ x²(1 + tan² θ) = 3²⇒ x² sec² θ = 3²⇒ x = ±3sec θ
Again, we take the positive value because x cannot be negative. Therefore, the rectangular form of the equation r² = 9 is: x² + y² = 9, y = x tan θ isx² + (x² tan² θ) = 9⇒ x²(1 + tan² θ) = 9⇒ x² sec² θ = 9⇒ x = 3 sec θ.
b) Conversion of polar form equation r = 7 sin θ to rectangular form: In polar coordinates, the conversion formulae from rectangular to polar coordinates are: r = √(x² + y²), θ = tan⁻¹(y/x)
Hence, we obtain: r = 7 sin θ = y ⇒ y² = 49 sin² θ
We substitute this value of y² in the equation x² + y² = r², which gives: x² + 49 sin² θ = (7 sin θ)²⇒ x² = 49 sin² θ - 49 sin² θ⇒ x² = 49 sin² θ (1 - sin² θ)⇒ x² = 49 sin² θ cos² θ⇒ x = ±7 sin θ cos θ
Again, we take the positive value because x cannot be negative. Therefore, the rectangular form of the equation r = 7 sin θ is: x² + y² = (7 sin θ)², x = 7 sin θ cos θ.
Conversion of equations from polar form to rectangular form is an essential process in coordinate geometry. In polar coordinates, a point (r, θ) in the polar plane is given by r = the distance from the origin to the point, and θ = the angle measured counterclockwise from the positive x-axis to the point. On the other hand, in rectangular coordinates, a point (x, y) in the rectangular plane is given by x = the distance from the point to the y-axis, and y = the distance from the point to the x-axis. To convert the polar form equation r² = 9 to rectangular form, we use the conversion formulae:
r = √(x² + y²), θ = tan⁻¹(y/x)
where x and y are rectangular coordinates. Similarly, to convert the polar form equation r = 7 sin θ to rectangular form, we use the conversion formulae: r = √(x² + y²), θ = tan⁻¹(y/x)
Here, we obtain: r = 7 sin θ = y ⇒ y² = 49 sin² θ
We substitute this value of y² in the equation x² + y² = r², which gives: x² + 49 sin² θ = (7 sin θ)²⇒ x² = 49 sin² θ - 49 sin² θ⇒ x² = 49 sin² θ (1 - sin² θ)⇒ x² = 49 sin² θ cos² θ⇒ x = ±7 sin θ cos θ
Again, we take the positive value because x cannot be negative. Therefore, the rectangular form of the equation r = 7 sin θ is: x² + y² = (7 sin θ)², x = 7 sin θ cos θ.
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Related Questions
A 875
B 5550
C3840
D 835
Answers
Write an expression for the nth term of the sequence. (your formula should work for n = 1, 2, .) 1 2 , 1 3 , 1 7 , 1 25 , 1 121 ,
Answers
The nth term of the sequence can be expressed as: \(1 / (n^2)\)
The given sequence is: 1, 2, 1/3, 1/7, 1/25, 1/121, ...
To find an expression for the nth term of this sequence, we can observe that each term is the reciprocal of a specific pattern of numbers: 1, 2, 3, 4, 5, 6, ...
Notice that the numerator of each term follows a pattern of increasing consecutive positive integers: 1, 2, 3, 4, 5, 6, ...
The denominator of each term follows a pattern of perfect squares: \(1^2, 2^2, 3^2, 4^2, 5^2, 6^2, ...\)
Therefore, the nth term of the sequence can be expressed as:
\(1 / (n^2)\)
So, the expression for the nth term of the sequence is \(1 / (n^2)\). This formula will work for n = 1, 2, 3, and so on.
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Van has 45 new baseball cards. He puts them in a binder that holds 9 cards on each page. How many pages dose he fill?
Answers
To find out how many pages Van fills with his new baseball cards, we need to divide the total number of cards by the number of cards that fit on each page.
He has 45 baseball cards and each page holds 9 cards.
So, we can divide 45 by 9.
45 / 9 = 5
Therefore, Van fills 5 pages with his new baseball cards.
It's worth noting that this method gives the number of full pages that are filled and if there are any remaining cards, we have to add one more page for the remaining cards.
show that x^2-y^2= (x+y)(x-y) with venn diagram
Answers
Answer:
\( \sqrt[]{ \frac{0}{0} } \)
Explainion are shown in pictures above
An item is regularly priced at $27. It is on sale for 20% off the regular price.
Answers
Answer:
0.0135
Step-by-step explanation:
. The number which when divided by 19 gives the quotient 4 and remainder
4 is
(a) 76 (b) 80 (c) 72 (d) 152
Answers
Answer:
b option 80 is the correct answer
please mark me as the
have a beautiful day
Answer:
a) 76
Step-by-step explanation:
Divide each number by 19.
76/19=4
80/19=4.21
72/19=3.79
152/19=8
Suppose that an individual has a body fat percentage of 19.3% and weighs 187 pounds. How many pounds of her weight is made up of fat? Round your answer
to the nearest tenth.
Answers
Answer:
The individual has 36.09 pounds of body fat.
Step-by-step explanation:
Given that an individual has a body fat percentage of 19.3% and weighs 187 pounds, to determine how many pounds of her weight is made up of fat the following calculation must be performed:
(187 x 19.3) / 100 = X
3.609.1 / 100 = X
36.091 = X
Therefore, the individual has 36.09 pounds of body fat.
ill mark brainlisr pls help
Answers
Answer:
$12.00
Step-by-step explanation:
6% off $199 =
6% × $199 = $11.94
$11.94 is rounded or estimated to $12.00
Let a, b, p = [0, 27). The following two identities are given as cos(a + B) = cosa cosß-sina sinß, cos²q+sin² = 1, Hint: sin o= (b) Prove that 0=cos (a) Prove the equations in (3.2) ONLY by the identities given in (3.1). cos(a-B) = cosa cosß+sina sinß, sin(a-B)=sina cosß-cosa sinß. I sin (a-B)=cos os (4- (a − p)) = cos((²-a) + p). cos²a= 1+cos 2a 2 (c) Calculate cos(7/12) and sin (7/12) obtained in (3.2). (3.1) sin² a (3.2) (3.3) 1-cos 2a 2 (3.4) respectively based on the results
Answers
Let a, b, p = [0, 27). The following two identities are given as cos(a + B) = cosa cos ß-sina sin ß, cos² q+sin² = 1, Hint: sin o= (b)Prove that 0=cos (a)Prove the equations in (3.2) ONLY by the identities given in (3.1).
cos(a-B) = cosa cos ß+sina sin ßsin(a-B)=sina cos ß-cosa sin ß.sin (a-B)=cos os (4- (a − p)) = cos((²-a) + p).cos²a= 1+cos 2a 2(c) Calculate cos(7/12) and sin (7/12) obtained in (3.2).Given: cos(a + B) = cosa cos ß-sina sin ß, cos² q+sin² = 1, Hint:
sin o= (b)Prove:
cos a= 0Proof:
From the given identity cos² q+sin² = 1we have cos 2a+sin 2a=1 ......(1)
also cos(a + B) = cosa cos ß-sina sin ßOn substituting a = 0, B = 0 in the above identity
we getcos(0) = cos0. cos0 - sin0. sin0which is equal to 1.
Now substituting a = 0, B = a in the given identity cos(a + B) = cosa cos ß-sina sin ß
we getcos(a) = cosa cos0 - sin0.
sin aSubstituting the value of cos a in the above identity we getcos(a) = cos 0. cosa - sin0.
sin a= cosaNow using the above result in (1)
we havecos 0+sin 2a=1
As the value of sin 2a is less than or equal to 1so the value of cos 0 has to be zero, as any value greater than zero would make the above equation false
.Now, to prove cos(a-B) = cosa cos ß+sina sin ßProof:
We have cos (a-B)=cos a cos B +sin a sin BSo,
we can write it ascus (a-B)=cos a cos B +(sin a sin B) × (sin 2÷ sin 2)cos (a-B)=cos a cos B +(sin a sin B) × (1-cos 2a ÷ sin 2)cos (a-B)=cos a cos B +(sin a sin B) × (1-cos 2a) / 2sin a
We have sin (a-B)=sin a cos B -cos a sin B= sin a cos B -cos a sin B×(sin 2/ sin 2) = sin a cos B -(cos a sin B) × (1-cos 2a ÷ sin 2) = sin a cos B -(cos a sin B) × (1-cos 2a) / 2sin a
Now we need to prove that sin (a-B)=cos o(s4-(a-7))=cos((2-a)+7)
We havecos o(s4-(a-7))=cos ((27-4) -a)=-cos a=-cosa
Which is the required result. :
Here, given that a, b, p = [0, 27),
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Which statements about geometric sequences are true?
There is more than one correct answer. Select all that apply.
Geometric sequences have a common ratio between terms.
Geometric sequences are restricted to the domain of natural numbers.
Geometric sequences can have a first term of 0.
Geometric sequences have a common difference between terms.
Geometric sequences only increase, never decrease.
Answers
Answer:
Step-by-step explanation:
Geometric sequences have a common ratio between terms. TRUE
Geometric sequences are restricted to the domain of natural numbers. FALSE
Geometric sequences can have a first term of 0. FALSE. If this were true, then every member the sequence would also be 0.
The correct statements are,
Geometric sequences have a common ratio between terms.
Geometric sequences :A geometric series is one which has a constant ratio between successive terms.
A geometric series is the sum of the terms of a geometric sequence.Geometric sequences may be increase or decrease.Geometric sequences can not have a first term of 0, if the first term is zero, it will not result in a Geometric Progression.Learn more about the Geometric sequences here:
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determine whether the integral is convergent or divergent. if it is convergent, evaluate it. (if the quantity diverges, enter diverges.) [infinity] 0 1 6 1 x dx
a. convergent
b. divergent
Answers
To determine whether the integral is convergent or divergent, we need to analyze the given integral is b. divergent
The integral you provided seems to have some typos, but I will assume you meant the following: ∫(1/x) dx from 1 to ∞
Now, we will determine if this integral is convergent or divergent.
Step 1: Set up the improper integral
∫(1/x) dx from 1 to ∞
Step 2: Rewrite the integral with a limit
lim (b→∞) ∫(1/x) dx from 1 to b
Step 3: Find the antiderivative of the integrand
The antiderivative of 1/x is ln|x|
Step 4: Evaluate the antiderivative at the limits and subtract
lim (b→∞) [ln|b| - ln|1|]
Step 5: Simplify the expression
lim (b→∞) [ln(b) - 0]
Step 6: Determine the limit
As b approaches ∞, ln(b) approaches ∞.
Since the limit is ∞, the integral is divergent. Therefore, the answer is: b. divergent
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There are 3 x as many blue counters as yellow. 45 blue counters are removed. There are now still 63 more blue counters than yellow. How many blue counters were there to start with
Answers
After 45 blue counters are removed, the remaining number of blue counters is 63 more than yellow counters. Setting up an equation, we find that there were 162 blue counters to start with.
Let's start by assigning variables to the unknowns:
Let x be the number of yellow counters.
Then, the number of blue counters is 3x.
After 45 blue counters are removed, the number of blue counters is 3x - 45.
Finally, there are 63 more blue counters than yellow, so we can set up the equation:
3x - 45 = x + 63
Solving for x, we get:
2x = 108
x = 54
Therefore, there were 3x = 3(54) = 162 blue counters to start with.
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Use the given conditions to write an equation for the line in point-slope form and general form. Passing through \( (-5,5) \) and parallel to the line whose equation is \( 5 x-3 y-2=0 \)
Answers
The equation in point-slope form is y - 5 = (5/3)(x + 5). The general form of the equation is 5x - 3y + 40 = 0.
To find the equation of a line parallel to the given line and passing through the point (-5,5), we can use the point-slope form of a linear equation. The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line. First, we need to find the slope of the given line, and then we can substitute the values into the point-slope form to obtain the equation. Additionally, we can convert the equation to the general form, which is Ax + By + C = 0, by rearranging the terms.
The given line has the equation 5x - 3y - 2 = 0. To find the slope of this line, we can rearrange the equation in slope-intercept form (y = mx + b), where m represents the slope. Rearranging the equation gives us:
-3y = -5x + 2
y = (5/3)x - 2/3
From the equation, we can see that the slope of the given line is 5/3. Since the line we want to find is parallel to this line, it will have the same slope.
Now, using the point-slope form with the point (-5,5) and the slope 5/3, we can write the equation:
y - 5 = (5/3)(x - (-5))
y - 5 = (5/3)(x + 5)
This is the equation of the line in point-slope form.
To convert it to general form, we can multiply through by 3 to eliminate the fraction:
3y - 15 = 5x + 25
5x - 3y + 40 = 0
This is the equation of the line in general form.
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"Define the difference between experimental data and non
experimental data. Give two examples of experimental data and two
examples of non experimental data and state why each, in detail,.
Answers
Experimental data and non-experimental data differ in their method of collection and the level of control over the variables involved.
Experimental data is obtained through controlled experiments where researchers manipulate independent variables to observe their effects on dependent variables. Non-experimental data, on the other hand, is collected without direct manipulation of variables, often through observation or existing records.
Experimental data involves conducting controlled experiments to gather information. In such cases, researchers actively manipulate variables to study their impact on the outcome. For example, in a study investigating the effects of a new medication on blood pressure, participants may be randomly assigned to receive either the medication or a placebo. The researchers then measure and compare the participants' blood pressure readings to determine the medication's effectiveness. In another example, an experiment could be conducted to examine the impact of different fertilizers on plant growth by treating different groups of plants with specific fertilizers and measuring their growth rates.
Non-experimental data encompasses information collected without direct manipulation of variables. One example is observational data, where researchers observe and record naturally occurring events or behaviors. For instance, a study on animal behavior might involve observing a group of primates in their natural habitat and documenting their social interactions, feeding patterns, or communication. Another example of non-experimental data is survey data, which involves gathering information through questionnaires or interviews. For instance, a market research survey may ask consumers about their preferences for different brands of a product or their likelihood of purchasing certain items.
In experimental data, researchers have control over variables, allowing them to establish cause-and-effect relationships between the independent and dependent variables. Non-experimental data, however, lacks such control, making it more challenging to establish causal relationships. While experimental data provides strong evidence for drawing conclusions, non-experimental data is valuable for exploring relationships, describing phenomena, and generating hypotheses that can be further tested through experiments.
In summary, experimental data is collected through controlled experiments where variables are manipulated, while non-experimental data is obtained through observation or existing records without direct manipulation. Experimental data allows for establishing cause-and-effect relationships, while non-experimental data provides valuable insights for exploration and hypothesis generation. Examples of experimental data include studies on medication effects or fertilizer impact, while examples of non-experimental data include observational studies on animal behavior or surveys on consumer preferences.
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Suzanne is baking cupcakes. Her cupcake recipe calls for 314 cups of flour. Suzanne has already added 235 cups of flour. How much more flour does Suzanne need to add in order to reach the amount called for in the recipe? Enter your answer as a fraction.
Answers
Answer:
179
Step-by-step explanation: First you would wanna add 70, to get to 100.
Then, add 9 to get to 314. add 70+9, that is your final answer.
TTQA: Suzanne needs to add 79 more cups of flour, to get to the anount called for in the recipe.
A three-column table is given. Part 6 B D Part 10 25 35 Whole A C 56 What is the value of C in the table? 15 35 40 46
Answers
The value of C from the column value table is C = 40
Given data ,
Let the table be represented as T
where ,
6 B D
10 25 35
A C 56
Now , the ratio of the table values is r = 35 / 56
r = 5 / 8
So , from the proportion , the value of C is
25 = ( 5/8 ) C
Multiply by 8 on both sides , we get
5C = 200
Divide by 5 on both sides ,we get
C = 40
Hence , the proportion is solved and C = 40
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find the slope of the line through each pair of points
a. (8,-7) and (5,-3)
b. (-5,9) and (5,11)
c. (-8,-4) and (-4,-9)
Answers
Answer:
I think the answers would be :
a . 4/3
b. 1/5
c . - 5/4
hope it helps u ^^
Hey can someone please help quickly thanks!
Answers
Answer:
360,000
Step-by-step explanation:
just multiply by 3
Answer:
3,120 second year 3,244.80 the third year.
Step-by-step explanation:
4% of 3,000 is 120
so 3,120 then 3,244.80 the third year after the 4% is compounded
What is the inverse of the function f(x) = 4x + 8?
Answers
Answer:
y = 1/4x - 2
Step-by-step explanation:
f(x) = 4x + 8
y = 4x + 8
x = 4y + 8
x - 8 = 4y
1/4x - 2 = y
y = 1/4x - 2
Algebra I need help!
Answers
Answer:
≤ ≥ > < is all of them
PLS COME AND LOOK!!! I NEED YOU GENIUSES!!! I WILL GIVE BRAINLIEST!!!!! AT LEAST COME AND LOOK!!!! WILL FOREVER BE GREAT FULL!!! EASY IM JUST DUMB!!
1. Which compound inequalitiy represents the inequality |y – 3| − 4 ≤ −3?
A) −1 ≤ y – 3 AND y – 3 ≤ 1
B) −1 ≤ y – 3 OR y – 3 ≤ 1
C) y – 3 ≤ −1 AND y – 3 ≥ 1
Answers
Answer:
A
Step-by-step explanation:
Answer:
A
Step-by-step explanation:
Which of the following number lines shows the solution to the compound inequality given below?
-2<3r+4<13
Answers
Answer:
We get -2 < r < 3
Corresponding to the fourth choice
The fourth number line is the correct option
Step-by-step explanation:
-2 < 3r+4 < 13
We have to isolate r,
subtracting 4 from each term,
-2-4< 3r + 4 - 4 < 13 - 4
-6 < 3r < 9
divding each term by 3,
-6/3 < r < 9/3
-2 < r < 3
so, the interval is (-2,3)
or, -2 < r < 3
this corresponds to
The fourth choice (since there is no equality sign)
Anyone know this?? Step by step please .
Find the absolute extreme if they exist, as well as the value of x where they occur for the function f(x) = 2x^2+384In x on the domain [1,6].
Answers
1. Find the critical points of the function inside the domain.
2. Evaluate the function at the critical points and at the endpoints of the domain.
3. Compare the values obtained to determine the absolute extreme values.
Let's apply these steps to the function f(x) = 2x^2 + 384 ln(x) on the domain [1,6].
1. Find the critical points of the function inside the domain.
The critical points of a function are the points where the derivative is zero or undefined. We can find the derivative of f(x) as:
f'(x) = 4x + 384/x
To find the critical points, we need to solve the equation f'(x) = 0:
4x + 384/x = 0
Multiplying both sides by x, we get:
4x^2 + 384 = 0
Dividing both sides by 4, we get:
x^2 + 96 = 0
This equation has no real solutions, since x^2 is always positive and 96 is positive too. Therefore, there are no critical points inside the domain [1,6].
2. Evaluate the function at the critical points and at the endpoints of the domain.
Since there are no critical points inside the domain, we only need to evaluate the function at the endpoints:
f(1) = 2(1)^2 + 384 ln(1) = 0
f(6) = 2(6)^2 + 384 ln(6) ≈ 290.7
3. Compare the values obtained to determine the absolute extreme values.
The function f(x) has a minimum value of 0 at x = 1 and a maximum value of approximately 290.7 at x = 6.
Therefore, the absolute extreme values of the function f(x) on the domain [1,6] are:
- Absolute minimum: 0, at x = 1
- Absolute maximum: approximately 290.7, at x = 6
Francis owns a camping supply store and just put in an order for flashlights and sleeping
bags. The number of flashlights ordered was five times the number of sleeping bags ordered.
The flashlights cost $12 each and the sleeping bags cost $45 each. If the total cost for the
flashlights and sleeping bags was $1785, how many sleeping bags did Francis order?
Answers
Answer:
The number of sleeping bags bought is 17.
Step-by-step explanation:
If we let "x" be the number of the sleeping bags that were bought, we can make the following equation....
(5x)(12) + (x)(45) = 1785
By solving the equation we get....
(5x)(12) + (x)(45) = 1785
60x + 45x = 1785
105x = 1785
x = 1785 / 105
x = 17
There for the number of sleeping bags bough is 17.
Use the diagram right above
AB=?
m
Answers
Can someone please tell me what 20% of 100. Thank you soooo much.
Answers
Answer:
20.
Step-by-step explanation:
Answer:
20
Step-by-step explanation:
20% of 100 = 20
Thenks and make me brainliest :))
solve each equation for the indicated variable
g = c + x solve for x
Answers
Answer:
x = g - cStep-by-step explanation:
g = c + x |subtract c from both sides
g - c = c - c + x
g - c = x → x = g - c
Hello there!
Question:-
\(g = c + x\)
We need to find the value of x.
Solution:-
\(\sf \longmapsto \: g = c + x\)
Note that g , c and x equals to 1g,1c and 1x.
\(\sf \longmapsto \: 1g =1 c +1 x\)
Flip the equation:-
\(\sf \longmapsto \: 1c + 1x = g\)
Add -c to both sides:-
\(\sf \longmapsto \: 1c+1x+( - c)=1g+ (- c)\)
On Simplification:-
\(\sf \longmapsto \: 1c - 1c + 1x = 1g - c\)
\(\sf \longmapsto \: 0 + x = 1g - c\)
\(\sf \longmapsto \: x = 1g - c\)
______________________________________
Henceforth the value of x is :-
\(\boxed{\huge\tt x = g - c}\)
_________________________________
Please let me know if you have any questions.
~MisterBrian
Work out
9+13x3
————
15-12+1
Answers
The solution to the work involving the basic mathematics operations is 24
Basic mathematics operationsThe following basic mathematics operations PEDMAS are:
P- ParenthesesE-ExponentsD-DivisionM-MultiplicationA-AdditionS-SubtractionFrom the question, we can notice 4 of the basic mathematics operations,
+ and × at the numerator
- and + at the denominator
So we work as follows:
for the numerator
9 + 13 × 3 = 9 + 39 (multiplication first)
9 + 13 × 3 = 48
for the denominator
15 - 12 + 1 = 16 - 12 (addition first)
15 - 12 + 1 = 2
Therefore,
(9 + 13 × 3)/(15 - 12 + 1 ) = 48/2
(9 + 13 × 3)/(15 - 12 + 1 ) = 24
Hence, from careful application of the basic mathematics operations, we have worked out the solution to be 24.
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Find the location of A' given that A is (4,5). Translate 1 unit left. The location of A is
Answers
Answer:
3,5
Step-by-step explanation:
You just move it to the left
The table below shows a relationship between x and y that is not a function:
PLS HELP
Answers
Answer:
X 11 Y 11
Step-by-step explanation:
the last one need to be removed