For parametric equations x= a cos t and y= b sin t, describe how the values of a and b determine which conic section will be traced
Answers
In mathematics, a conic section is
a curve obtained as the intersection of the surface of
a cone with a plane. The four types of conic section are
the hyperbola, the parabola, the circumference and the ellipse.
For the problem we have this parametric
equation:
(1) [tex]\left\{{{x=acost}\atop{y=bsint}}\right[/tex]
From geometry, we know that we can express a
circumference in terms of parameters like this:
(2) [tex]\left \{ {{x=rcost} \atop {y=rsint}}\right[/tex]
being r the radius of the circumference.
On the other hands, we know that a ellipse can
be expressed in terms of parameters like this:
(3) [tex]\left \{ {{x=acost} \atop {y=bsint}}\right[/tex]
Therefore, we will have three answers that are
the cases for the values a and b, namely.
Case 1:
Circumference
To the case of a circumference, the more simple
ordinary equation is given by:
(4) [tex]x^{2} + y^{2} = r^{2}[/tex]
Substituting (1) into (4):
[tex]a^{2}cos^{2}t+b^{2}cos^{2}t=r^{2}[/tex]
But because of the equation (2), necessarily:
[tex]a = b = r[/tex]
Case 2: Ellipse
(focal axis matches the x-axis)
In this case, the simple ordinary equation is
given by:
(5) [tex]\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1[/tex]
being a and b semi-major axis and semi-minor
axis respectively.
Given that a an b are variables of the
parametrization, and a and b are variables of the ellipse as well, to avoid
confusion we will modify the equation (5) like this:
(6) [tex]\frac{x^{2}}{a'^{2}}+\frac{y^{2}}{b'^{2}}=1[/tex]
So, substituting (2) into (6):
[tex]\frac{a^{2}cos^{2}t}{a'^{2}}+\frac{b^{2}cos^{2}t}{b'^{2}}=1[/tex]
Necessarily:
[tex]a=a'[/tex] and [tex]b=b'[/tex]
and given that the focal axis matches the
x-axis, then:
[tex]a>b[/tex]
Case 3: Ellipse
(focal axis matches the y-axis)
In this case, applying the same previous
reasoning, the simple ordinary equation is given by:
(7) [tex]\frac{x^{2} }{b'^{2}}+\frac{y^{2}}{a'^{2}}=1[/tex]
being a' and b' semi-major axis and semi-minor
axis respectively.
So, substituting (2) into (7):
[tex]\frac{a^{2}cos^{2}t}{b'^{2}}+\frac{b^{2}cos^{2}t}{a'^{2}}=1[/tex]
Necessarily:
[tex]a = b'[/tex] and [tex]b = a'[/tex]
and given that the focal axis matches the y-axis, then:
[tex]a<b[/tex]
Finally, the conclusions are:
1. If [tex]a = b[/tex] then a circumference will be traced. (See Figure 1)
2. If [tex]a>b[/tex] then a ellipse will be traced with focal axis matching the x-axis. (See Figure 2)
3. If [tex]a<b[/tex] then a ellipse will be traced with focal axis matching the y-axis. (See Figure 3)
How the values of a and b determine which conic section will be traced was discussed thoroughly.
What is a conic section?A conic section is a curve obtained as the intersection of the surface of a cone with a plane.
The given parametric equations are:
[tex]x=acos t[/tex]
[tex]\frac{x}{a} =cost[/tex]....(1)
[tex]y=bsint[/tex]
[tex]\frac{y}{b} =sint[/tex]....(2)
Adding the squares of (1) and (2)
[tex](\frac{x}{a} )^2+(\frac{y}{b} )^2=cos^{2} t + sin^{2} t[/tex]
We know [tex]cos^{2} t + sin^{2} t=1[/tex]
So, [tex]\frac{x^{2} }{a^{2} } +\frac{y^2}{b^2} =1[/tex]........(3)
If [tex]a=b[/tex], (3) will be reduced into:
[tex]x^{2} +y^{2} =a^{2}[/tex] representing a circle.
If [tex]a > b[/tex], (3) will represent an ellipse with the length of the major axis > length of the minor axis.
if [tex]a < b,[/tex] (3) will represent an ellipse with the length of the major axis < length of the minor axis.
Thus, How the values of a and b determine which conic section will be traced was discussed thoroughly.
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Related Questions
Solve for c in the equation 3x^2-18x+5=47
Answers
The general quadrilatic equation is ax²+bx+c=0.
So, arranging the equation in this form we get,
3x²-18x+5-47=0
3x²-18x-42=0
The value of C = -42
Kali invests $3516 in a savings account with a fixed annual interest rate of 7% compounded 3 times per year. What will the account balance be after 7 years
Answers
Which line of symmetry for the parabola (x-1)^2= 4(y-1)^?
Answers
Vertex of the parabola (1,1).
So line symmetry is x=1.
The line of symmetry for the parabola is x = 1.
The line of symmetry for the parabola is x = 1. In the given equation (x-1)^2 = 4(y-1)^2, the x-coordinate of the vertex is 1, which means the line of symmetry is a vertical line passing through x = 1.
Which of the following sequences of numbers are arithmetic sequences? Check all that apply.
A.99, 100, 101, 102, 103, ...
B.-3, -5, -7, -9, -11, ...
C.2, -2, 2, -2, 2, ...
D.1, 6, 11, 16, 21, ...
E.-4, 7, -10, 13, -16, ...
Answers
__________________
If we observe the given options, we can note:
In option A, the difference between two consecutive terms is the same and is equal to 1.
In option B, the difference between two consecutive terms is the same and is equal to -2.
In option C, the difference is not the same.
In option D, the difference between two consecutive terms is the same and is equal to 5.
In option E, the difference is not the same.
So, the correct options are A,B and D
Sandy releases a javelin 1.6 m above the ground with an initial vertical velocity of 25 m/s. How long will it take the javelin to hit the ground?
Answers
A(t)=V'(t)=S''(t)
A(t)=-9.8m/s
⇒V(t)=-9.8t/1+C m/s
Where C is the initial velocity:
⇒V(t)=-9.8t+25
⇒S(t)=-9.8t²/2+25t/1+C
⇒S(t)=-4.9t²+25t+C
but the C is the initial vertical displacement, hence:
S(t)=-4.9t²+25t+1.6m
The time taken to hit the ground will therefore be:
S(t)=0
-4.9t²+25t+1.6=0
Using a quadratic function, it is found that it will take 5.17 seconds for the javelin to hit the ground.
What is the quadratic function for a projectile's height?Considering meters as the measure of height, it is given by:
h(t) = -4.9t² + v(0)t + h(0).
In which:
v(0) is the initial velocity.h(0) is the initial height.In this problem, we have that v(0) = 25, h(0) = 1.6, hence:
h(t) = -4.9t² + 25t + 1.6
It hits the ground when h(t) = 0, hence:
-4.9t² + 25t + 1.6 = 0
4.9t² - 25t - 1.6 = 0.
Which is a quadratic function with coefficients a = 4.9, b = -25, c = -1.6. Hence:
[tex]\Delta = (-25)^2 - 4(4.9)(-1.6) = 656.36[/tex]
[tex]x_1 = \frac{25 + \sqrt{656.36}}{9.8} = 5.17[/tex]
[tex]x_2 = \frac{25 - \sqrt{656.36}}{9.8} = -0.06[/tex]
For time we want the positive value, hence, it will take 5.17 seconds for the javelin to hit the ground.
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How many arrangements are possible using the letters in the word fuzzy if each letter "z" is distinctly different than the other? how many arrangements are possible if the letter "z" is interchangeable with the other? explain your reasoning?
Answers
First answer is 5!, second answer is 5!/2 because swapping two z's does not generate a new permutation.
Case 1: If the letter z is distinctly different
If we have the letter z which is distinctly different than the other "z", then we have five letters in the word F U Z Z Y and the number of arrangements would be = 5*4*3*2*1 =5! = 120 different arrangements
Case 2: If the letter Z is interchangeable
If the letter "Z" is interchangeable with the other "z" then we have 2 letters which cancel out each other that would be represented by 2*1 = 2! = 2
So, the number of arrangements in this case would be 5!/2! = 5*4*3*2*1/2*1 = 60 arrangements
I need Help with this TRIG question
Answers
(15) sin 30.5 = ------ (15)
15
(15) sin 30.5 = x
Plug this part into a calculator.
7.61 = x
Area of a triangle: 1/2bh
1/2 (7.61)(13)
3.805(13)
49.469
49.47 un² is your final answer since you are supposed to round to the tenths place. I hope this helps.
Help on homework! Which of the following can have a negative effect on your credit score?
A. Getting a promotion at work
B. Late payments on a car loan
C. Paying credit card balances in full
D. low debt-to-income ratio
Answers
The answer is - B.
Hope this helps you!
~DL
plz help me!!!!
A cross country coach records the number of miles his athletes on the Junior Varsity and Varsity teams ran today and displays the data in the provided dot plots. Given the shape of each distribution, determine which measures of center and spread are appropriate for him to use to summarize the data from each team.
Answers
b.) For the Varsity Team, the median would be the appropriate measure of the center since the data is skewed left and not evenly distributed so median could be used since it does not account for outliers while we use IQR or interquartile range in measuring the spread of data since IQR does not account for the data that is skewed.
Answer:
Step-by-step explanation:
The quarters, buckled, and dimes totaled 20 and their value was $2.05. How many kind were there if there were 3 times as many dimes as quarters
Answers
Let q, n, d represent the numbers of quarters, nickels, and dimes, respectively. Then you have three equations in the unknowns:
q +n +d = 20
25q +5n +10d = 205
-3q +d = 0
A calculator can give the solution to this matrix equation in short order. It is
q = 3
n = 8
d = 9
There are 3 quarters, 8 nickels, and 9 dimes.
_____
If you use the third equation to write an expression for d, you can substitute that into the other two equations.
q +n +3q = 20
25q +5n +10(3q) = 205
Simplifying, these are
4q +n = 20
11q +n = 41
Subtracting the first from the second, we find
7q = 21
q = 3
From there, it is easy to find d=9, n=8.
A travel website reports that in a particular city, the probability of rain on any day in April is 40%. What is the expected number of rainy days in this city during the month of April?
Answers
0.4(30)
(0.4 represents 40%)
Multiply:
0.4(30) = 12
The expected amount of rainy days is 12 days.
(2n + 3)(n – 4) = 0
need help asap!!!
Answers
2n + 3 = 0
2n = - 3
n = - 3 / 2
n - 4 = 0
n = 4
So n = -3/2, 4
please help no will mark brainlyst
Explain how you would find 90% of 120.
Answers
land is the same as the correspondent in "number" land; so we can write:
100%
90%
= 120x
Where X is the correspondent of 90% in numbers.
Rearranging: x = 120⋅
90% 100% =108
To find out you will have to multiply 90 with 120 and then divide what you get with 100.
90 x 120 = 10800
10800/100 = 108
so 90% of 120 is 108.
Hope this helped :)
Someone please help me?
Answers
CBD = 180 - 98/2
CBD = 180 = 49
CBD = 131
Answer is the third one
Determine the number of real solutions for each of the given equations. Equation Number of Solutions y = -3x2 + x + 12 y = 2x2 - 6x + 5 y = x2 + 7x - 11 y = -x2 - 8x - 16
pppppppllllllleeeeaaaassssseeeee ANSWER FASRT
Answers
if its <0, no real soln
its =0, 1 real soln
its >0, 2 real solns
for
y = -3x2 + x + 12, 1^2-4(-3)(12) > 0, two solns
y = 2x2 - 6x + 5, -6^2-4(2)(5) < 0, no soln
y = x2 + 7x - 11, 7^2-4(1)(-11) > 0, two solns
y = -x2 - 8x - 16, -8^2-4(-1)(-16) = 0, one soln
According to the discriminant:
[tex]y = -3x^2 + x + 12[/tex] has two real solutions.[tex]y = 2x^2 - 6x + 5[/tex] has zero real solutions.[tex]y = x^2 + 7x - 11[/tex] has two real solutions.[tex]y = -x^2 - 8x - 16[/tex] has one real solution.The number of real solutions of quadratic equation [tex]y = ax^2 + bx + c[/tex] depends on the discriminant [tex]\Delta = b^2 - 4ac[/tex].
If [tex]\Delta > 0[/tex], the equation has two real solutions.If [tex]\Delta = 0[/tex], the equation has one real solution.If [tex]\Delta < 0[/tex], the equation has zero real solutions.Equation [tex]y = -3x^2 + x + 12[/tex]
The coefficients are: [tex]a = -3, b = 1, c = 12[/tex]The discriminant is: [tex]\Delta = 1^2 - 4(-3)(12) = 145[/tex].Positive, thus two real solutions.Equation [tex]y = 2x^2 - 6x + 5[/tex]
The coefficients are: [tex]a = -2, b = -6, c = 5[/tex]The discriminant is: [tex]\Delta = (-6)^2 - 4(2)(5) = -4[/tex].Positive, thus zero real solutions.Equation [tex]y = x^2 + 7x - 11[/tex]
The coefficients are: [tex]a = 1, b = 7, c = -11[/tex]The discriminant is: [tex]\Delta = (7)^2 - 4(1)(-11) = 93[/tex].Positive, thus two real solutions.Equation [tex]y = -x^2 - 8x - 16[/tex]
The coefficients are: [tex]a = -1, b = -8, c = -16[/tex]The discriminant is: [tex]\Delta = (-8)^2 - 4(-1)(-16) = 0[/tex].Positive, thus one real solutions.A similar problem is given at https://brainly.com/question/19776811
A-Town wants to know the number of cell phones in each household the town divides the population up by neighborhood and surveys a random number of households and eat what type of survey is this
Answers
A-Town is conducting a survey using a random sample method by dividing the population into neighborhoods and surveying random households in each, which ensures each household has an equal chance of being selected and provides a representative sample.
Explanation:A-Town's approach to determining the number of cell phones in each household through a survey is an example of a random sample. The method involves dividing the population into neighborhoods and surveying a random selection of households within each area. This technique helps to ensure that every household has an equal chance of being selected, which should provide a representative sample of the town's population. Such methods are necessary to obtain reliable data that reflects the true distribution and characteristics of the target population without regional bias.
Using a representative sample is crucial in research and polling to measure accurately public opinion or collect data on phenomena. As in the given example where the market research firm surveys 500 randomly selected residents to estimate the percentage of adults with cell phones, the firm employs statistical measures like confidence intervals to infer about the entire population's cell phone ownership based on the sample data. Moreover, the concept of probability sampling is also important for achieving a sample that can be generalized to the whole population with reasonable accuracy.
Farmer toby's chickens and pigs have a total of 30 heads and 84 legs. a chicken has 2 legs, a pig has 4 legs, and each animal has 1 head. how many of each type of animal does toby have
Answers
C = h + 2l
P = h + 4l
C = 18h + 36l
P = 12h + 48l
What is the value of the rational expression 2x+1/x2 when x = 5?
Answers
[tex] \dfrac{2x + 1}{ {x}^{2} } \\ = \dfrac{2(5) + 1}{ {5}^{2} } \\ = \dfrac{11}{25} [/tex]
Answer is 11/25.
Hope this helps. - M
The solution is A = 11/25
The value of the equation A = ( 2x + 1 ) / x² is 11/25
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
A = ( 2x + 1 ) / x²
when x = 5 , substitute the value of x as 5 in the equation , we get
A = ( 2 ( 5 ) + 1 ) / 5²
The value of A = ( 10 + 1 ) / 25
The value of A = 11/25
Therefore , the value of A = 11/25
Hence , The value of the equation A = ( 2x + 1 ) / x² is 11/25
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The slope of a line in a blueprint must be . A line is drawn that contains the points 5/6 < | m | <7/4. Determine if the line meets the required specifications.
A.No, the rooftop does not meet the specifications, because the absolute value of the roof’s slope is greater than the parameters.
B.No, the rooftop does not meet the specifications, because the absolute value of the roof’s slope is less than the parameters.
C.Yes, the rooftop does meet the specifications, because the absolute value of the roof’s slope is within the parameters.
D.Yes, the rooftop does meet the specifications, because the absolute value of the roof’s slope is less than the parameters.
Answers
The equation of a parabola is 1/32 (y−2)2=x−1 .
What are the coordinates of the focus?
(1, 10)
(1, −6)
(−7, 2)
(9, 2)
Answers
The given equation of parabola is
[tex] \frac{1}{32} (y-2)^2 = x-1 [/tex]
Which can also be written as
[tex] x = \frac{1}{32} (y-2)^2 +1 [/tex]
Here vertex (h,k) is (1,2)
And value of a is
[tex] a = \frac{1}{32} [/tex]
Formula of focus is
[tex] (h+ \frac{1}{4a} , k) [/tex]
Substituting the values of h,k and a, we will get
[tex] (1+ \frac{1}{4*(1/32) } , 2} = (1+ 8,2) = (9,2) [/tex]
Therefore the correct option is the last option .
Answer: The correct option is (D) (9, 2).
Step-by-step explanation: We are given to find the co-ordinates of the focus for the following parabola:
[tex]\dfrac{1}{32}(y-2)^2=x-1~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We know that the standard form equation of a parabola is
[tex](y-k)^2=4p(x-h),[/tex]
where the co-ordinates of the focus are (h+p, k).
From equation (i), we have
[tex]\dfrac{1}{32}(y-2)^2=x-1\\\\\\\Rightarrow (y-2)^2=32(x-1)\\\\\Rightarrow (y-2)^2=4\times 8(x-1).[/tex]
Comparing the above equation with the standard form equation of the parabola, we get
h = 1, k = 2, and p = 8.
Therefore, the co-ordinates of the focus are
[tex](h+p,k)=(1+8,2)=(9,2).[/tex]
Thus, option (D) is CORRECT.
the mean mark for the class was 70%. There are 30 students in the class, 20 of whom are boys. The mean mark for the boys is 62%. What is the mean mark for the girls?
Answers
Total marks for the class = 70 x 30 = 2100
Mean mark for the boys = 62%
Total marks for the boys = 62 x 20 = 1240
Total marks for the girls = 2100 - 1240 = 860
Mean marks for the girls = 860 ÷ 10 = 86%
Answer: 86%
What are the coordinates of the vertex of the table? Is it a minimum or a maximum?
x y
0 1
-1 -2
-2 -3
Question 1 options: A.(-4, 1); minimum B.(-2, -3); maximum C.(-2, -3); minimum D.(1, 0); maximum.
pleeeeeeeeez help me
Answers
A (0,1)
B(-1,-2)
C(-2,-3)
using a graph tool
see the attached figure
the vertex is the point C (-2,-3) is a minimum
the answer is
minimum (-2, -3)
Please help me idk this
Answers
1- You have the following information given in the problem above:
- The jug helds 16 cups of milk.
- Isaac drank 1 7/8 cups and his brother drank 1 1/2 cups.
2- Keeping this on mind, you have:
1 7/8=15/8
1 1/2=3/2
=16 cups-15/8 cups-3/2 cups
=101/8 cups
The answer is: 101/8 cups
In sakura’s garden, for every 5 red flowers, there are 10 yellow flowers. There are a total of 75 yellow and red flowers in her garden. How many ref flowers are in Sakura’s garden?
Answers
We have then:
x: number of red flowers
y: number of yellow flowers.
We write the system of equations:
x + y = 75
y = 2x
Solving the system we have:
x = 25
y = 50
Answer:
there are 25 red flowers in Sakura's garden
Use the graph below to answer the question that follows:
What is the rate of change between the interval x = π and x = three pi over two?
pi over 16
negative 16 over pi
pi over 4
negative 4 over pi
Answers
Jake is younger than Sophie. Sophie is 14 years old. Write an inequality that compares Jake's age in years, (j), to Sophie's age.
Answers
To express that Jake is younger than Sophie who is 14, we use the inequality j < 14, where j symbolizes Jake's age in years.
Explanation:The question is asking for an inequality that represents Jake's age compared to Sophie's age. Since we are told that Jake is younger than Sophie and Sophie is 14 years old, we can use an inequality symbol to express this relationship. The appropriate inequality is j < 14, where j represents Jake's age in years. The symbol < means 'less than,' which correctly compares Jake's age to Sophie's, indicating that Jake's age is less than 14 years.
The total cost (c) in dollars of renting a sailboat for n days is given by the inequality c≥120+60n. what is the maximum number of days for which a sailboat could be rented if the total cost was $360?
Answers
c≥120 + 60n
We substitute the value of c = 360 in the inequality:
360≥120 + 60n
We clear the value of n:
360 - 120 ≥ 60n
240 ≥ 60n
240/60 ≥ n
4 ≥ n
Answer:
The maximum number of days for which a sailboat could be rented if the total cost was $ 360 is:
4 days
Answer:
4 days
Step-by-step explanation:
The function h=−16t2+26t models the height h (in feet) of the dolphin after t seconds. after how many seconds is the dolphin at a height of 5 feet? round your answers to the nearest tenth.
Answers
Answer:
The dolphin is at height 5 feet after 1.40 seconds or 0.228 seconds.
Step-by-step explanation:
Given : The function [tex]h=-16t^2+26t[/tex]models the height h (in feet) of the dolphin after t seconds.
To Find: After how many seconds is the dolphin at a height of 5 feet?
Solution:
Function: [tex]h=-16t^2+26t[/tex]
h is the height
t is the time
Now we are supposed to find After how many seconds is the dolphin at a height of 5 feet
Substitute h = 5
So, [tex]5=-16t^2+26t[/tex]
[tex]16t^2-26t+5=0[/tex]
[tex]16t^2-26t+5=0[/tex] ---1
Using quadratic formula : [tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
General quadratic equation: [tex]ax^2+bx+c=0[/tex] ---A
On Comparing equation 1 with A
a = 16
b = -26
c=5
Substitute the values in the quadratic formula.
[tex]x = \frac{-(-26)\pm\sqrt{(-26)^2-4(16)(5)}}{2(16)}[/tex]
[tex]x = \frac{26\pm 18.86796}{32}[/tex]
[tex]x = \frac{26+18.86796}{32},\frac{26-18.86796}{32}[/tex]
[tex]x = 1.4021,0.2228[/tex]
Thus the dolphin is at height 5 feet after 1.40 seconds or 0.228 seconds.
The dolphin is at a height of 5 feet at 0.2 second and 1.4 seconds.
Polynomial
Polynomial is an expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Let h represent the height of dolphin at t seconds.
Given the function:
h = −16t² + 26tFor a height of 5 ft.:
5 = −16t² + 26t
16t² - 26t + 5 = 0
t = 0.2 or t = 1.4
The dolphin is at a height of 5 feet at 0.2 second and 1.4 seconds.
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Nikolai has a jar filled with 120 marbles. he has 72 red marbles, 17 blue marbles, 13 green marbles, and 18 purple marbles. what is the probability that he will randomly select a blue marble, without replacement, and then a purple marble from the jar
Answers
Probability of choosing a blue marble: 17/120
Probability of choosing a purple marble: 18/119
17/120 x 18/119 = 3/140
You will have a 3/140 chance of both of these occurring.
The probability of selecting a blue marble followed by a purple marble from a jar containing 120 marbles with different colors is approximately 2.2%.
The probability of grabbing a blue marble and then a purple marble is determined by multiplying the probability of grabbing a blue marble first and the probability of grabbing a purple marble second. Here's the calculation:
Probability of selecting a blue marble first: 17/120
Probability of selecting a purple marble second (after one blue marble is already taken out): 18/119
Multiplying the probabilities: (17/120) * (18/119) ≈ 0.022, or 2.2%
Convert 7.2*10^-3 to standard form please help thanks
Answers
=7.2 * 10^−3=7.2 * 1/10*10*10=0.0072
Hope this helps!!