Let two vector functions be defined by r1(t) = (cost, sint,t) r2(t) = (1 + t, t^2,t^3) a. Find the intersection point of the functions. Is it a collision point? b. Find the equation of the line that goes through the intersection point that is also orthogonal to both curves at that point.
Answers
a) The intersection point (2,1,1) which is not a collision point since the two curves do not have the same tangent vector at that point.
b) the equation of the line that goes through the intersection point and is orthogonal to both curves at that point,
-5(x - 2) - 2cos(1)(y - 1) = 0
Now, We can simplify as,
a. The intersection point of the two vector functions, we need to solve the equation:
r₁(t) = r₂(t)
We can write this as three separate equations:
cos t = 1 + t
sin t = t²
t = t³
Hence, From the third equation, we can see that t = 0 or t = 1.
If t = 0, then the first two equations give us:
cos(0) = 1 + t
1 = 1 + t
And, sin(0) = 0
0 = t
This leads to the trivial solution (0,0,0) which is not a collision point since the two curves do not intersect at t=0.
If t = 1, then we have:
cos(1) = 2
sin(1) = 1
1 = 1
This gives us the intersection point (2,1,1) which is not a collision point since the two curves do not have the same tangent vector at that point.
b. To find the equation of the line that goes through the intersection point and is orthogonal to both curves at that point, we need to find the tangent vectors of the two curves at the intersection point.
The tangent vector of r1(t) at t=1 is given by:
r₁'(t) = (-sint, cost, 1)
So, r₁'(1) = (-sin(1), cos(1), 1)
The tangent vector of r₂(t) at t=1 is given by:
r₂'(t) = (1, 2t, 3t)
So, r₂'(1) = (1, 2, 3)
Now, we can find the normal vector of the plane that contains both tangent vectors by taking the cross product:
n = r₁'(1) x r₂'(1) = (cos(1) - 6t, -sin t - 3t, 2t - 2cost)
n = (-5, -2cos(1), 0)
Since, n is orthogonal to both tangent vectors.
Therefore, the line we seek is the intersection of the plane that contains both tangent vectors and the plane that contains the intersection point and is orthogonal to n.
The first plane has equation:
r = (2,1,1) + s(-sin(1), cos(1), 1)
The second plane has equation:
n . (r - (2,1,1)) = 0
Substituting for n and simplifying, we get:
-5(x - 2) - 2cos(1)(y - 1) = 0
Thus, the equation of the line that goes through the intersection point and is orthogonal to both curves at that point,
-5(x - 2) - 2cos(1)(y - 1) = 0
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Related Questions
The temperature of a pot of water was 180.3°F and cools at a rate of -2.5°F per minute. a. What is the temperature after 20 minutes? b. Look for Relationships How many minutes will it take to cool from 180.3°F to 100.3°F?
Answers
Answer:
A. 130.3F B. 32 minuits
Step-by-step explanation:
A: if It cools 2.5F a minuite than we will do 20x2.5=50
180.3-50=130.3
B: 180.3-100-3=80. So that means it cooled 80 degrees. we then divide 80 by 2.5. 80/2.5=32. and we know that it has cooled for 32 minuits
It takes 32 minutes to cool from 180.3°F to 100.3°F.
Given that, the temperature of a pot of water was 180.3°F and cools at a rate of -2.5°F per minute.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Let the temperature after 20 minutes be x.
Now, x = -2.5 × 20
= - 50° F
So, 180.3-50 = 130.3° F
Time taken to cool from 180.3°F to 100.3°F = 180.3-100.3=80°F
= 80/2.5
= 32 minutes
Therefore, it takes 32 minutes to cool from 180.3°F to 100.3°F.
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A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $30 per foot and on the other three sides by a metal fence costing $10 per foot. If the area of the garden is 450 ft2, find the dimensions of the garden minimizing the cost.
Answers
The dimensions that minimize the cost are x ≈ 13.42 ft and y ≈ 33.58 ft.
We are given that;
Brick wall costing= $30
Area= 450ft^2
Now,
To find the critical points, we take the derivative of C and set it equal to zero:
C’ = 50 - 9000/x2 C’ = 0 50 - 9000/x2 = 0 9000/x2 = 50 x2 = 9000/50 x2 = 180 x = ±√180 x ≈ ±13.42
Since x must be positive, we only consider x ≈ 13.42 as a critical point. To check if this is a minimum or maximum, we can use the first derivative test or the second derivative test. Using the second derivative test, we find:
C’’ = 18000/x3
Since x ≈ 13.42 is positive, C’’ > 0 at this point, which means C has a local minimum at x ≈ 13.42.
To find the absolute minimum, we also need to evaluate C at the endpoints of its domain:
C(0) = undefined C(450) = 50(450) + 20(1) = 22520
So, C has an absolute minimum at x ≈ 13.42 with a value of:
C(13.42) ≈ 50(13.42) + 20(33.58) ≈ $1,337
To find the dimensions of the garden that minimize the cost, we use y = 450/x and plug in x ≈ 13.42:
y ≈ 450/13.42 y ≈ 33.58
Therefore, by area the answer will be x ≈ 13.42 ft and y ≈ 33.58 ft.
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If you save 10% off headphone that normally
cost $8.90, how much do you have to pay?
Answers
Answer:
1.1
Step-by-step explanation:
looked it up
Multiple Choice
Which of the following is a solution to the equation y=3x – 1?
A. (4,1)
B. (2.5)
C. (4,3)
D. (0,-3)
Answers
Answer:
B: (2,5)
Step-by-step explanation:
y=3x-1
(5)=3(2)-1
5=6-1
5=5 (correct)
Answer:
hola necesito puntos, solo dejame comentar
Step-by-step explanation:
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Question 2 ""If the Vpp is 10 V, then the Vavg is:"" O 20 V O 3.53 V O 3.18 V O 5 V
"
Answers
The correct answer is option O: 5 V.
To determine the average voltage (Vavg) given a peak-to-peak voltage (Vpp) of 10 V, we need to consider the relationship between Vavg and Vpp in an alternating current (AC) waveform.
The average voltage of an AC waveform is related to its peak-to-peak voltage by the formula: Vavg = 0.5 * Vpp.
Applying this formula to the given Vpp of 10 V, we can calculate the Vavg as follows: Vavg = 0.5 * 10 V = 5 V.
The average voltage is equal to half of the peak-to-peak voltage, resulting in an average voltage of 5 V for a Vpp of 10 V.
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The graph below shows the relationship between the pumpkin weight and growth time
Answers
Where is the graph and what is the question?
Click to add text
Combine Like Terms:
6y + 4z + 9x + 3y
Answers
Answer:
9y + 4z + 9x
Step-by-step explanation:
1/4 of ... is 1/5
What is the ...
Answers
Answer:
4/5
Step-by-step explanation:
Just multiply 1/5 times 4, because that's the reciprocal of 1/4
Consider the following random priority mechanism for the assignment of dorm rooms to college students.
Random Priority
initialize R to the set of all rooms
randomly order the agents
for i=1, 2, ..., n do
assign the ith agent her favorite room r from among those in R
delete r from R
Does this mechanism DSIC, no matter which random ordering is chosen by the mechanism?
Answers
No, this mechanism does not satisfy Dominant Strategy Incentive Compatibility (DSIC) no matter which random ordering is chosen by the mechanism.
DSIC requires that each agent has a dominant strategy, meaning that regardless of what other agents do, it is always in an agent's best interest to report their true preferences.
In this mechanism, the problem lies in the step where the ith agent is assigned her favorite room from the set R.
Since the rooms are assigned based on the agent's preferences, an agent has an incentive to misreport her preferences in order to increase her chances of getting her most preferred room.
For example, if an agent knows that her most preferred room is more likely to be available at a later stage, she may strategically misreport her preferences to increase the likelihood of getting that room.
This introduces the possibility of manipulation and strategic behavior, which violates the DSIC property.
Therefore, the mechanism described does not satisfy DSIC, regardless of the chosen random ordering.
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In ΔQRS, s = 5.2 inches, ∠S=129° and ∠Q=30°. Find the length of r, to the nearest 10th of an inch.
Answers
The measure of the length r (SQ) of the triangle ΔQRS will be 2.40 inches.
What is the sine law?The Law of Sines The law of trigonometric functions, sine law, sine formula, or sinusoidal rule is a trigonometric equation that relates the sizes of any triangle's edges to the sines of its orientations.
The sine law in the triangle ΔQRS is given as,
QR / sin S = SQ / sin R = SR / sin Q
The third angle of the triangle ΔQRS is given as,
∠Q + ∠R + ∠S = 180°
∠R + 30° + 129° = 180°
∠R = 21°
Then the measure of the length r (SQ) will be calculated as,
5.2 / sin 129° = SQ / sin 21°
SQ = 2.40 inches
The measure of the length r (SQ) of the triangle ΔQRS will be 2.40 inches.
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9/11 of a number is 36. What is the number?
Answers
Answer:
44
Step-by-step explanation:
36/9 * 11 will get the answer
9/11 * x = 36
x = 44
(multiplied each side by 11/9 to cancel out fraction)
i have four identical oranges. how many ways are there for me to divide these oranges into at most three groups? (by definition, a group must have at least one orange.)
Answers
Answer:
put 1 orange in each group and then cut the last in thirds and each group will have 1 and 1/3 oranges
Step-by-step explanation:
hElP mE plS anD ty I hAd tO rEpoSt thiS :')
Answers
Answer:
1: below its initial height.
2: if Jimmy has 20 balls and Chris take 15,how many balls does Jimmy have?
3: I would want Mrbeast to be the President because he would help everyone.
a fair coin is tossed n times, and the number of heads, n, is counted. the coin is then tossed n more times. find the expected total number of heads generated by this process.
Answers
The anticipated total number of heads produced by this method is 1/2n=N when fair coin is tossed n times, and the results are tallied by counting the heads.
Given that,
A fair coin is tossed n times, and the results are tallied by counting the heads. The coin is then thrown a further n times.
We have to calculate the anticipated total number of heads produced by this method.
We know that,
Either head or tail will appear when a coin is tossed.
The sample space of a random experiment is the collection of all feasible results. Therefore, if your random experiment involves flipping a coin, the sample space is "Head, Tail," or more tersely, "H, T."
So,
1/2n=N
Therefore, the anticipated total number of heads produced by this method is 1/2n=N when fair coin is tossed n times, and the results are tallied by counting the heads.
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Which of the following number lines represents the solution set of x/-1 > -6?
Will give brainliest to best answer!
:3
Answers
Answer:
Second Line
Step-by-step explanation:
x/-1 > -6
x < 6
Donna bought 5 bags of dog treats for $13.10. What is the cost per bag of dog treats?
A. $18.10
B.$3.28
C.$2.62
D.$0.38
Answers
The current in a certain resistor is 0.50A when it is connected to a potential difference of 110V. If the operating potential is 130V, the current in the same resistor is
Answers
Answer:
0.59 A
Step-by-step explanation:
The current in a certain resistor is 0.50 A
The potential difference is 110v
The operating potential is 130v
The current in the same resistor can be calculated as follows
= 110/0.50
= 220
= 130/220
= 0.59
Hence the current is 0.59 A
Write an expression showing 4 less than the number j.
Answers
Answer:
j-4 is how you show 4 less than a number j
Can someone please explain how I do this problem?
Answers
Step-by-step explanation:
click on the number rates on the graph so basically you want to put the answer in the numbers where they said, so the y would go in the middle on 3 and 4 and the x would just go in the 3What is an inherent zero? describe three examples of data sets that have inherent zeros and three that do not.
Answers
The inherent zero is simply a zero. The three instances for both are-
the average age among college graduates, average monthly body weight, maximum wind velocity during a hurricane with intrinsic zero.Year of birth, body weight in space, and year of automobile accidents are three that do not.What is inherent zero?The absolute zero is another name for the intrinsic zero. This is not included in the interval scale because, while the disparity between the two observations makes sense, the ratio does not because it lacks an inherent zero.
Some key features of inherent zero are-
In statistics, the inherent zero is regarded as the starting point or reference point for the ratio scale. The scale with an inherent zero is called as the ratio scale, since we can then compute the ratio between observations and comment on the number of times one observation is less than or bigger than another.To know more about the inherent zero, here
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Helpmeeeeeeeeeeeeeeeeeeeeeeee
Answers
Answer:
B)
Step-by-step explanation:
distances:
AB = 6
BC = √4+25 = √29
CD = √0+144 = 12
AD = √5²+4² = √41
I WILL GIVE BRAINIEST, PLEASE HELP
Answers
Answer:
A = (1, 10)
B = (4,10)
C = (4, 14)
D = (1, 16)
Step-by-step explanation:
Okay, so first, we move, or translate the quadrilateral PQRS down -8 spaces on the graph, turning old points:
P (1, -2) = (1, -10) Q (4, -2) = (4, - 10) S (1, -8) = (1, -16) and R (4, -6) = (4, -14)
Now, we need to rotate this new box into -x, y. So our new points are:
(1, -10) = (-1, 10) (4, -10) = (-4, 10) (1, -16) = (-1, 16) and (4,-14) = (-4, 14)
Now we're reflecting off the y-axis, making our x change. So for our final points:
A(1, 10) B (4,10) D(1, 16) and C(4, 14)
Hope this helps! Please check my math and comment wether this worked or not! :)
Answer:
hi
Step-by-step explanation:
A jury pool consists of 12 angry people and 8 chill people. How many 12 member juries:Can be made?Consist of 7 chill and 5 angry people?Consist of at least 10 angry people?
Answers
Given that the number of angry people=12 and the number of chill peoples=8.
The total number of peoples=The number of angry people + the number of chill people.
The total number of people in juries=12+8=20
Let A be an event that made 12 members of juries consists of 7 chill and 5 angry people.
The probability of A =P(A)
\(P(A)=\frac{8C_7+12C_5}{20C_{12}}\)\(=(\frac{8!}{7!(8-7)!}+\frac{12!}{5!(12-5)!})\div\frac{20!}{12!(20-12)!}\)\(=(\frac{8\times7!}{7!}+\frac{12\times11\times10\times9\times8\times7!}{5\times4\times3\times2\times(7)!})\div\frac{20\times19\times18\times17\times16\times15\times13\times12!}{12!\times8\times7\times6\times5\times4\times3\times2}\)\(=800\div8997.857=0.0889\)\(P(A)=0.09\)Hence only one 12 member jury pool can be made that consist of 7 chill and 5 angry people.
Hence the answer is 1.
1. y = 7x
2. y = x - 5
3. y = x + 2
4. y = 3x + 6
Answers
Answer:
whats the question?
Step-by-step explanation:
which property is
used below?
A. Distributive Property
B. Multiplication Property of Inequality
C. Subtraction Property of Inequality
D. Associative Property of Multiplication
Answers
Answer:
B
Step-by-step explanation:
.
Multiplication Property of Inequality because you are multiplying each side by 3. I hope this helps! :)
the temperature at midnight is -3 degrees the weatherman predicts the temperature to be 25 degrees warmer by noon the next day.find the temperature at noon
Answers
Answer:the tempature will be 22 degrees (i dunno if you measure in Farenheight or celcius so dunno)
Step-by-step explanation: because if the tempature
would rise it would be 22 whatever you measure with
What is function and not function?
Answers
Answer: A function is a relation between domain and range such that each value in the domain corresponds to only one value in the range. Relations that are not functions violate this definition. They feature at least one value in the domain that corresponds to two or more values in the range.
Step-by-step explanation:
Suppose a survey of 979 business owners found that more than ought . which part of the survey represents the descriptive branch of statistics? make an inference based on the results of the survey.
Answers
Women make up 64% of the sample and are typically the household's main investment.
There is a correlation between American women and being the main provider in their home.
What is Descriptive Statistics?Descriptive statistics, also known as brief informative coefficients, are used to summarize a specific data collection, which may be a sample of a population or a representation of the entire population. Descriptive statistics include measures of variability and central tendency (spread). Measures of central tendency include the mean, median, and mode, whereas measures of variability include the standard deviation, variance, minimum and maximum variables, kurtosis, and skewness.
For instance, the sum of the following data set is 20: (2, 3, 4, 5, 6). A 4 (20/5) is the mean. A data set's mode is the number that appears most frequently, while its median is the value that is in the middle of the range of values. It is the value that separates a data set's higher values from lower values. There are many less common ways to use descriptive statistics, but they are nevertheless essential.
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The correct question is - Suppose a survey of 580 women in the United States found that more than 64% are the primary investor in their household. Which part of the survey represents the descriptive branch of statistics? Make an inference based on the results of the survey.
Suppose in 2008 the average cost of a movie ticket in the United States was $7. If
inflation causes ticket prices to increase by 4.5% a year, in what year will a ticket cost
$25?
a. 2025
b. 2037
c. 2029
d. 2040
Answers
Answer:
B) 2037
Step-by-step explanation:
using the problem an exponential equation can be made
7(1.045)× =25
--solve for x--
x=28.9
it will take 28.9 year for the price to get to $25 so add 2008 to 28.9 to get 2036.9
2037 is closest to 2036.9 so the answer is B
7(1+0.045)^29= 25
there are 13 marbles in each package.65 marbles way 500h what is the weight of the package of marbles
Answers
Answer:
The weight is 100.
Step-by-step explanation:
65 divided by 13 = 5
500 divided by 5 = 100