Answer:
C) (6,2)
Step-by-step explanation:
To solve the system of equations, solve one of the equations for y, then plug the result into the other equation to get the value of x. When you get x, substitute it back into one of the original equations to find the value of y, which is 2.
To find the solution to the system of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution and solve for x and y. The solution is (6, 2).
Explanation:To find the solution to the system of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution:
From the second equation, we can isolate y by subtracting 5x from both sides: y = 32 - 5x
Substitute this expression for y in the first equation: 3x - 2(32 - 5x) = 14
Simplify and solve for x: 3x - 64 + 10x = 14, 13x - 64 = 14, 13x = 78, x = 6
Now substitute the value of x back into one of the original equations and solve for y: 5(6) + y = 32, 30 + y = 32, y = 2
Therefore, the solution to the system of linear equations is x = 6 and y = 2, which can be written as (6, 2).
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Which equation has only one solution?
Ox-51 = -1
1-6 - 2x1 = 8
15x + 10 = 10
O |-6x +31 = 0
Answer:
I would say the third one ( 15x + 10 = 10 )
Step-by-step explanation:
I say this because, if we take 10 from both sides, 15x = 0. Even if you tried to divide by 15, x = 0. x also =0 if you try to do it the other way by finding what times 15 +10 = 10 it would be zero. The third one is your Answerr
Since the only solution to the equation is 0, hence 15x + 10 = 10 is the only equation With a solution.
Equation of functions With only one solution
Equations With just one solution are knoWn to have a leading degree of one and Without a modulus sign
Fro the linear function 15x + 10 = 10
Check:
Subtract 10 from both sides
15x + 10 - 10 = 10 - 10
15x = 10 - 10
15x = 0
Dividde both sides by 15 to have:
15x/15 = 0/15
x = 0
Since the only solution to the equation is 0, hence 15x + 10 = 10 is the only equation With a solution.
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A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same. The cone is tilted at an angle so its peak touches the edge of the cylinder’s base. What is the volume of the space remaining in the cylinder after the cone is placed inside it?
Answer:
[tex]\frac{11}{12}\pi r^{2}h\ units^{3}[/tex]
Step-by-step explanation:
we know that
Te volume of the cone is equal to
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]
we have
[tex]r=(r/2)\ units[/tex]
substitute
[tex]V=\frac{1}{3}\pi (r/2)^{2}h[/tex]
[tex]V=\frac{1}{12}\pi r^{2}h[/tex]
Te volume of the cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
we know that
To find the volume of the space remaining in the cylinder after the cone is placed inside it, subtract the volume of the cone from the volume of cylinder
so
[tex]\pi r^{2}h-\frac{1}{12}\pi r^{2}h=\frac{11}{12}\pi r^{2}h\ units^{3}[/tex]
Columbia,South Carolina, is located at 34degrees north latitude. Use the equation to estimate the average annual snowfall for Columbia.
The question is missing the specific equation needed to estimate average annual snowfall for Columbia, South Carolina at 34 degrees north latitude. Factors like latitude and climate patterns influence snowfall levels, and based on Columbia's location, low snowfall would be expected.
Explanation:The student is asking about an equation to estimate average annual snowfall for a specific location based on its latitude, in this case, Columbia, South Carolina, which is at 34 degrees north latitude. Unfortunately, the explicit equation needed to estimate the snowfall is not provided in the question or the contextual information. Generally, to estimate snowfall or precipitation levels using latitude, additional climatic data such as elevation, proximity to oceans or large bodies of water, and prevailing wind patterns are also necessary.
However, referring to the geographical regions mentioned, it's noted that certain latitudes often correspond with specific climate patterns. The comparison with other regions suggests that areas farther away from the equator, especially those close to either the Arctic Circle or the Antarctic Circle, experience more extreme temperatures and snowfall. Given the subtropical zone where Columbia, South Carolina, is located, the expected average snowfall would likely be relatively low compared to higher latitudes.
-3x + 1 + 10x = x + 4
x = 1/2
x = 5/6
x = 12
x = 18
Answer:
A
Step-by-step explanation:
Given
- 3x + 1 + 10x = x + 4 ( simplify left side )
7x + 1 = x + 4 ( subtract x from both sides )
6x + 1 = 4 ( subtract 1 from both sides )
6x = 3 ( divide both sides by 6 )
x = [tex]\frac{3}{6}[/tex] = [tex]\frac{1}{2}[/tex]
1. Describe the domain and range of this equation.
y = 3(x + 1)^2 – 6
Indicate the method you would use to prove the two A's 2. If no method applies, enter none.
Answer:
HL
Step-by-step explanation:
You have a the two hypotenuses from the two triangles are congruent.
You also have a pair of of legs from each of the triangles that are congruent.
So HL means hypotenuse-leg which is what you have in the pic!
Final answer:
To prove two 'A's are equivalent, methods such as direct proof, disjunctive syllogism, conditional proof, and indirect proof can be applied. The selection of the method depends on the nature of the proof and the preference of the solver.
Explanation:
To establish that both 'A's are equivalent in a given proof, one might utilize several methods including direct proof, disjunctive syllogism, conditional proof, or indirect proof. For instance, a disjunctive syllogism can be employed when an argument has an either/or scenario, which allows for a conclusion to be drawn when one of the options is eliminated. Conversely, if you aim to reach a conclusion in the form of a conditional A → B, you might opt for a conditional proof method where you start a subproof by assuming A and then proceed to derive B within that subproof. In cases where a direct proof is challenging, you might resort to an indirect proof approach by assuming the negation of what you're attempting to prove and then showing this leads to a contradiction.
Direct proof and indirect proof are both formally legitimate, even though one might offer a more straightforward path to the solution or resonate more with the problem-solver's reasoning style. When planning an experiment or mapping out a mathematical proof, it is vital to consider the different methods and choose one that aligns with the nature of the problem and the proof you wish to establish.
Simplify.
(5y + 9) + y
6y + 9
14 + y
14y
15y
The answer you are looking for is 6y + 9.
In the equations (5y + 9) + y, you'd combine like terms to find the answer. You aren't distributing anything into the parenthesis, and 5y and 9 are not like terms (since the 9 doesn't have a "y" after it). That being said, you simply add "y" and 5y together to get 6y, and add the 9 to the end. Thus getting 6y + 9 as an answer.
I hope this helps!
Answer:
A. 6y+9
Step-by-step explanation:
Distributive property:
↓
[tex]A(B+C)=AB+AC[/tex]
First, you remove parenthesis.
5y+9+y
Group like terms:
↓
5y+y+9
Then, you add by similar into elements.
5y+y=6y
6y+9 is the correct answer.
what is the equation of the graph that represents f(x)=x^4 stretched vertically by 2 and shifted up 3 spaces
Answer: [tex]g(x)=2x^4+3[/tex]
Step-by-step explanation:
These are some transformations for a function f(x):
If [tex]f(x)+k[/tex], then the function is shifted up "k" units.
If [tex]mf(x)[/tex], and [tex]k>1[/tex], then the function is stretched vertically by a factor of "m".
Knowing this transformation and knowing that the function [tex]f(x)=x^4[/tex] is stretched vertically by 2 and shifted up 3 spaces, then we can conclude that new function, which we can call g(x), is:
[tex]g(x)=2(x^4)+3[/tex]
[tex]g(x)=2x^4+3[/tex]
what is the factorization of the polynomial below? x^2+6x+8
Answer:
(x+2) (x+4)
Step-by-step explanation:
x^2+6x+8
What 2 numbers multiply together to give us 8 and add together to give us 6
(2*4) =8
(2+4) = 6
(x+2) (x+4)
Answer:
(x + 4)(x+2)
Step-by-step explanation:
We must multiply 8 and 1, and find two numbers which add to 6:
8 * x(suppose x is 1) = 8
Two numbers which add to 6, but also multiply to 8:
4 and 2
4 * 2 = 8
4 + 2 = 6
Hence, the answer would be (x + 4)(x+2)
What is the integration of cos3x
Answer:
1/3 sin(3x)+C
Step-by-step explanation:
int (cos(3x) dx)
Let u=3x then du=3 dx so 1/3 du=dx
rewriting integral
int(1/3 cos(u) du)
now evaluating
1/3 sin(u)+C since (sin(u))'=cos(u)
Replace u with 3x
Answer is 1/3 sin(3x)+C
Answer:
Step-by-step explanation:
Integral of cos3x = ⅓sin3x
So when integrating just simply multiply by reciprocal of the cooeffecient of the angle and the integral of that particular trig ratio, in this case it's the sinx.
The results of a survey of common allergies was organized into a Venn diagram. Answer the questions about the following sets: D = {x | x is a person allergic to dogs}; C = {x | x is a person allergic to cats}; P = {x | x is a person allergic to pollen} How many people are not allergic to any of the three choices? How many people are allergic to all three choices? How many people are allergic to both dogs and cats but not allergic to pollen? How many people are allergic to cats only?
Answer:
How many people are not allergic to any of the three choices? 22
How many people are allergic to all three choices? 1
How many people are allergic to both dogs and cats but not allergic to pollen? 7
How many people are allergic to cats only? 18
A survey of common allergies
The survey of the common allergies was made to see the percentage of people affected by the same sort of allergies and analysis the disease. The Venn diagrams are thus made in order to refer to the types of diseases that elate and people can be identified.
As per the answer, the Venn people allergic to cats are 18, not allergic are 22, and allergic to all three is one.
The Venn diagram is one that is used to depict the relationships between the two or more finite groups of things. The diagram of the people affected by dogs, cats, and pollen is shown.The number of people who are not allergic to any of the three choices is 22. The people who are allergic to all 3 are one. The number of people allergic to both, but not to pollen is 7. Those people who are allergic to cats are 18.Hence the Venn diagram shows us the relationships.Learn more about the survey of common.
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Which of the following equations are equivalent to -2m - 5m - 8 = 3 + (-7) + m?
-15m = -4m
-7m - 8 = m - 4
-3m - 8 = 4 - m
m - 4 = -7m - 8
-8 - 7m = -4 + m
-8 - 3m = 4 - m
Answer:
B) -7m - 8 = m - 4
D) m - 4 = -7m - 8
E) -8 - 7m = -4 + m
Step-by-step explanation:
Solve all of the equations to see if they end up with the same result. Start with the original:
-2m - 5m - 8 = 3 + (-7) + m
-7m - 8 = -4 + m (now we know B and D and E are correct)
-7m - m - 8 = -4 + m - m
-8m - 8 = -4
-8m - 8 + 8= -4 + 8
-8m = 4
Answer:
-7m - 8 = m - 4
m - 4 = -7m - 8
-8 - 7m = -4 + m
Step-by-step explanation:
Given equation,
-2m - 5m - 8 = 3 + (-7) + m,
Combining like terms,
-7m - 8 =-4 + m
Subtract m from both sides,
-8m - 8 = -4
Add 8 on both sides,
-8m = 4
Divide both sides by -8,
m = [tex]-\frac{1}{2}[/tex]
(i) -15m = -4m ⇒ -15m + 4m = 0 ⇒ -11m = 0 ⇒ m = 0
(ii) -7m - 8 = m - 4 ⇒ -7m - m = -4 + 8 ⇒ -8m = 4 ⇒ m = [tex]-\frac{1}{2}[/tex]
(iii) -3m - 8 = 4 - m ⇒ -3m + m = 4 + 8 ⇒ -2m = 12 ⇒ m = -6
(iv) m - 4 = -7m - 8 ⇒ m + 7m = -8 + 4 ⇒ 8m = -4 ⇒ m = [tex]-\frac{1}{2}[/tex]
(v) -8 - 7m = -4 + m ⇒ -7m - m = -4 + 8 ⇒ -8m = 4 ⇒ m = [tex]-\frac{1}{2}[/tex]
(vi) -8 - 3m = 4 - m ⇒ -3m + m = 4 + 8 ⇒ -2m = 12 ⇒ m = -6
The solution to the system of equations shown is (2,0).
3x – 2y = 6
x + 4y = 2
When the first equation is multiplied by 2, the sum of the two
equations is equivalent to 7x = 14
Which system of equations will also have a solution of (2,0)?
Answer:
D
Step-by-step explanation:
The system of equation will also have a solution of (2,0) are,
x + 4y = 2, 7x = 14.
Given that,
The solution to the system of equations shown is (2,0).
3x – 2y = 6 , x + 4y = 2
When the first equation is multiplied by 2, the sum of the two equations is equivalent to 7x = 14.
We have to determine,
Which system of equations will also have a solution of (2,0).
According to the question,
To determine the system of the equation after applying all the given conditions in the steps, follow all the steps given below.
System of equations; 3x – 2y = 6 , x + 4y = 2.
Step1; Multiply the first equation by 2,[tex]2 \times (3x-2y) = 2 \times 6\\\\6x - 4y = 12[/tex]
Step2; Adding equation 1 after multiplying by 2 from equation 2.[tex]6x - 4y + x + 4y = 12+ 2\\\\7x = 14\\\\x = \dfrac{14}{7}\\\\x = 2[/tex]
Step3; When x = 2 the value of y is,
[tex]6(2) - 4y = 6\\\\12-4y = 6\\\\-4y = 6-12\\\\-4y == -6\\\\y = \dfrac{-6}{-4}\\\\y = \dfrac{3}{2}[/tex]
Hence, The required system of equation will also have a solution of (2,0) are, x + 4y = 2, 7x = 14.
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The radius of a circle is 2 centimeters. What is the circle's area?
Answer:
I just know its formula 2 pie r square
Step-by-step explanation:
Answer: A = 12.56 cm
Step-by-step explanation:
A = 3.14 x r^2
A = 3.14 x 2^2
A = 3.14 x 4
A = 12.56 cm
What is the lateral area of a prism which has a lateral edge of 4 inches and has a regular pentagonal base with an edge of 12 inches?
Check the picture below.
so the pentagonal prism in the picture is been seen from the bottom, namely so we can see its base, however the base is at the bottom, what does that mean, it means that the pentagonal bases are the top and bottom of the prism, that matters, because the other sides are the lateral sides.
so, if we notice, the lateral sides are really just 5 rectangles, each one a 12x4, so if we simply get the area of all those rectangles.
5(12 * 4) = 240 in².
The lateral area of the given regular pentagonal prism is 240 square inches, calculated using the formula Lateral Area = Perimeter of Base * Lateral Edge.
Explanation:The lateral area of a prism, particularly a regular pentagonal prism, can be calculated using the formula Lateral Area = Perimeter of Base * Lateral Edge. In this case, the base is a regular pentagon implying that all its sides are equal. Therefore, its perimeter would be 60 inches (12 inches * 5 sides).
The lateral edge is given as 4 inches. Applying the values to the formula, we get Lateral Area = 60 inches * 4 inches = 240 square inches.
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Which of the following lists the angles from smallest to largest?
ORST
STR.
OTRS
Which of the following lists the angles from smallest to largest?
Answer:
T, R, S
Multiply: 4x^3/4x^2(2^3
Answer:
The answer is B
Step-by-step explanation:
happy cheating
Answer:
The answer is B
Step-by-step explanation:
And it's not cheating just trying to make it through high school.
Ryans final exam has true false questions with two points each multiple-choice questions worth five points each . let X be the number of true false questions he gets correct and let Y be the number of multiple choice questions he gets correct... he needs at least 90 points on the exam to get an a in the class . using the values of variables given write an inequality describing this
Answer: 2X+5Y≥ 90
Step-by-step explanation:
since you get two points per/ each (key word of multiplication) true & false question, this will be represented by 2X
since you get five points per/ each multiple choice question, this will be represented by 5Y
he needs at least 90 points in total to get an A in his class
A circle has a radius of 5 in. A central angle that measures 150° cuts off an arc.
Explain how to find the arc length exactly, and then approximate it to one decimal place.
Answer:
Part 1) The exact value of the arc length is [tex]\frac{25}{6}\pi \ in[/tex]
Part 2) The approximate value of the arc length is [tex]13.1\ in[/tex]
Step-by-step explanation:
step 1
Find the circumference of the circle
The circumference of a circle is equal to
[tex]C=2\pi r[/tex]
we have
[tex]r=5\ in[/tex]
substitute
[tex]C=2\pi (5)[/tex]
[tex]C=10\pi\ in[/tex]
step 2
Find the exact value of the arc length by a central angle of 150 degrees
Remember that the circumference of a circle subtends a central angle of 360 degrees
by proportion
[tex]\frac{10\pi}{360} =\frac{x}{150}\\ \\x=10\pi *150/360\\ \\x=\frac{25}{6}\pi \ in[/tex]
step 3
Find the approximate value of the arc length
To find the approximate value, assume
[tex]\pi =3.14[/tex]
substitute
[tex]\frac{25}{6}(3.14)=13.1\ in[/tex]
Answer:
13.1 (rounded to tenths)
Step-by-step explanation:
150 ° into radian is 5/6.
150°/1 (π/180) =5π/6.
Then multiply the radian angle by the radius.
5π/6 (5) = 25π/6
25π/6 = 13.1 (rounded to tenths)
The net of a triangular prism is shown below.
The perimeter of the base of the prism is
units.??
The prism's base perimeter is 28 units, calculated by summing the lengths of its sides or using the rectangle perimeter formula.
Step 1: Identify the Prism and Its Base
Given a triangular prism, focus on the rectangular base formed by sides S1, S2, S3, and S4.
Step 2: Understand Perimeter Calculation
Recall that the perimeter of any shape is the sum of all its sides.
Step 3: Label the Sides of the Base
Define the sides of the rectangular base:
S1 = 4 + 5
S2 = 5
S3 = 4 + 5
S4 = 5
Step 4: Apply Perimeter Formula
Utilize the formula for the perimeter of a rectangle: P = 2 * (length + width).
For the rectangular base, length = S1 + S3 and width = S2.
Step 5: Calculate Perimeter
Substitute the values into the formula: P = 2 * (9 + 5) = 2 * 14 = 28 units.
Step 6: Verify Using Summation
Confirm the result by adding the individual sides: P = S1 + S2 + S3 + S4 = 9 + 5 + 9 + 5 = 28 units.
If f(x) = 2x2 + 3x and g(x) = x – 2, what is (f + g)(2)?
Answer:
14
Step-by-step explanation:
(f + g)(x) = f(x) + g(x) = 2x² + 3x + x - 2 = 2x² + 4x - 2
To evaluate (f + g)(2) substitute x = 2 into (f + g)(x)
(f + g)(2) = 2(2)² + 4(2) - 2 = 8 + 8 - 2 = 14
Answer:
14
Step-by-step explanation:
First find an algebraic formula for (f + g)(x). To do this, combine like terms from f and g: (f + g)(x) = 2x^2 + 4x -2
Next, substitute 2 for x: (f + g)(2) = 2(2)^2 + 4(2) - 2 = 8 + 8 - 2 = 14
Solve using cross multiplication method, ax + by = a^2 ; bx + ay = b^2
Answer:
x=a²+ab+b²/a+b , y=-ab/a+b
Step-by-step explanation:
The system of the given equation may b written as:
ax+by-a²=0
bx+ay-b²=0
Here,
a1=a,b1=b,c1= -a²
a2=b,b2=a and c2= -b²
By cross multiplication we get
x/b*(-b²)-(-a²)*a = -y/a*(-b²)-(-a²)*b = 1/a*a-b*b
x/-b³+a³ = -y/-ab²+a²b = 1/a²-b²
Now
x/-b³+a³ = 1/a²-b²
x=a³-b³/a²-b²
x=(a-b)(a²+ab+b²)/(a-b)(a+b)
x=a²+ab+b²/a+b
And,
-y/-ab²+a²b = 1/a²-b²
-y=a²b -ab²/a²-b²
y=ab²-a²b/a²-b²
y=ab(b-a)/(a-b)(a+b)
y= -ab(a-b)/(a-b)(a+b)
y= -ab/a+b
Hence x=a²+ab+b²/a+b , y=-ab/a+b....
-2xy-x+4y para y=-3 x=9
Answer: 33
if you add them all up,they are 33
Ceilia uses the steps below to solve the equation -3/8 (-8-16d) + 2d= 24
Step 1. Distribute -3/8 over the expression in parentheses
Step 2.Simplify like terms
Step 3. Subtract 3 from both sides
Step 4. Divide both sides by the equation by -14
Which step had an error?
The incorrect step and with error is subtract 3 from both sides. Option 3
We have that the equation is;
-3/8 (-8-16d) + 2d= 24
multiply the values, we get;
-24 + 48d + 2d = 24
add the like terms
-24 + 50d = 24
50d = 48
d = 48/50
The other steps are;
Subtract 3 from both sides (However, the instruction mentions subtracting 3, but there is no number 3 in the equation.
Divide both sides by the equation by -14
Graph the linear equation. Find three points that solve the equation, then plot on the graph. -y=x+1
Answer:
The graph in the attached figure
Step-by-step explanation:
we have
-y=x+1
Find the y-intercept
For x=0
-y=0+1
y=-1
The y-intercept is the point (0,-1)
Find the x-intercept
For y=0
-0=x+1
x=-1
The x-intercept is the point (-1,0)
Find a third point
For x=1
-y=1+1
-y=2
y=-2
The third point is (1,-2)
Plot the three points to graph the linear equation
see the attached figure
Note Remember that to graph the linear equation is sufficient with two points
Answer:
(2,1) (0,-1) (-6,7)
Step-by-step explanation:
Writing Equations of Parallel Lines
What is the equation of the line parallel to the given line
with an x-intercept of 4?
y =
5
4 -3 -2 -1
Answer:
y= 4x + (-16)
Step-by-step explanation:
The slope of the given lime is m=4
x-intercept of the parallel line: (4,0)
plug that into the equation y= mx+b to find that b= -16
since the slope stays the same for parallel lines, and we just found b, we have the new line equation
The equation of parallel line with x intercept of 4 will be : y= 4x + (-16)
Given,
x intercept : 4
Slope = 4
Now,
x-intercept of the parallel line: (4,0)
As the equation of line to be obtained is parallel . Thus both lines will have same slope .
Slope of parallel line = 4
Substitute the value to find y intercept,
y = mx + b
0 = 4(4) + b
b = -16
So,
Equation of parallel line: y = 4x + (-16)
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What is the distance between points D and H?
round to the nearest 10th of a unit.
3.7 units
5.2 units
7.3 units
9.4 units
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ D(\stackrel{x_1}{9}~,~\stackrel{y_1}{8})\qquad H(\stackrel{x_2}{1}~,~\stackrel{y_2}{3})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ DH=\sqrt{(1-9)^2+(3-8)^2}\implies DH=\sqrt{(-8)^2+(-5)^2} \\\\\\ DH=\sqrt{64+25}\implies DH=\sqrt{89}\implies DH\approx 9.4[/tex]
Please please please help me!!!!!!
Answer:
Step-by-step explanation:
Use Pythagorean theorem.
c^2 = a^2+b^2
c^2 = 4^2+6^2
c^2 = 16+36
c^2 = 52
c = √52 = 7.21
Convert the Cartesian equation (x 2 + y 2)2 = 4(x 2 - y 2) to a polar equation.
Choices:
r4 = -4r2
r2 = 4cos2θ
r2 = 4sin2θ
ANSWER
[tex]{r}^{2} = 4 \cos2\theta[/tex]
EXPLANATION
The Cartesian equation is
[tex] {( {x}^{2} + {y}^{2} )}^{2} = 4( {x}^{2} - {y}^{2} )[/tex]
We substitute
[tex]x = r \cos( \theta) [/tex]
[tex]y = r \sin( \theta) [/tex]
and
[tex] {x}^{2} + {y}^{2} = {r}^{2} [/tex]
This implies that
[tex] {( {r}^{2} )}^{2} = 4(( { r \cos\theta) }^{2} - {(r \sin\theta) }^{2} )[/tex]
Let us evaluate the exponents to get:
[tex] {r}^{4} = 4({ {r}^{2} \cos^{2}\theta } - {r}^{2} \sin^{2}\theta)[/tex]
Factor the RHS to get:
[tex] {r}^{4} = 4{r}^{2} ({ \cos^{2}\theta } - \sin^{2}\theta)[/tex]
Divide through by r²
[tex]{r}^{2} = 4 ({ \cos^{2}\theta } - \sin^{2}\theta)[/tex]
Apply the double angle identity
[tex]\cos^{2}\theta -\sin^{2}\theta= \cos(2 \theta) [/tex]
The polar equation then becomes:
[tex]{r}^{2} = 4 \cos2\theta[/tex]
To convert the Cartesian equation to polar form, we substitute x and y with polar coordinates r and theta. Simplifying the equation with trigonometric identities leads to a polar equation, r^2 = 4cos^2\theta, which is the correct option among those given.
Explanation:To convert the Cartesian equation (x^2 + y^2)^2 = 4(x^2 - y^2) to a polar equation, we use the relationships x = r\cos\theta and y = r\sin\theta. Substituting these into the given equation, we get:
(r^2\cos^2\theta + r^2\sin^2\theta)^2 = 4(r^2\cos^2\theta - r^2\sin^2\theta)
This simplifies to:
r^4 = 4r^2(\cos^2\theta - \sin^2\theta)
Using the double angle identity for cosine, \cos(2\theta) = \cos^2\theta - \sin^2\theta, we can further simplify:
r^4 = 4r^2\cos(2\theta)
Dividing both sides by r^2, as long as r \neq 0, gives us:
r^2 = 4\cos(2\theta)
However, this is not one of the provided options, so we must go further and use another trigonometric identity:
\cos(2\theta) = 2\cos^2\theta - 1
The equation r^2 = 4\cos(2\theta) can then be rewritten as:
r^2 = 4(2\cos^2\theta - 1)
Since this is still not aligning with the provided options, it is important to check the original approach. There may have been a simplification error or a misinterpretation of the trigonometric identities. The correct polar equation that corresponds to the given choices should indeed be r^2 = 4\cos^2\theta, which is found by recognizing that \cos(2\theta) can also be written as 2\cos^2\theta - 1.
What is the solution to the system of equations graphed below?
Answer:
B
Step-by-step explanation:
The solution to a system of equations given graphically is at the point of intersection of the 2 lines, that is
solution = (0, - 4) → B
Answer:
B (0,-4)
Step-by-step explanation:
The solution to a system of equations is where the graphs intersect.
The two lines cross at x=0, y= -4