If you already know |aN| and |v|, then the formula aN=k|v|2 gives a convenient way to find the curvature. Use it to find the curvature and radius of curvature of the curve jr(t) = (cos t + t sin t) i + (sin t - t cos t) j?, t > 0.
Answers
The curvature of the curve is k(t) = 1/(1+t2)^(3/2) and the radius of curvature is R(t) = (1+t^2)^(3/2).
To find the curvature and radius of curvature of the curve, we need to find the components of the unit tangent vector T(t) and the normal vector N(t) at a given point. Then we can find the curvature as the magnitude of the rate of change of the unit tangent vector with respect to arc length, which is given by |dT/ds|. The radius of curvature is the reciprocal of the curvature.
First, we find the unit tangent vector T(t) by differentiating r(t) with respect to t and dividing by its magnitude:
r(t) = (cos t + t sin t) i + (sin t - t cos t) j
v(t) = dr/dt = (-sin t + t cos t) i + (cos t + t sin t) j
|v(t)| = sqrt[(-sin t + t cos t)^2 + (cos t + t sin t)^2]
= sqrt[1 + t^2]
Therefore, the unit tangent vector is:
T(t) = v(t) / |v(t)| = (-sin t + t cos t) / sqrt[1 + t^2] i
+ (cos t + t sin t) / sqrt[1 + t^2] j
Next, we find the normal vector N(t) by differentiating T(t) with respect to arc length s and dividing by its magnitude:
dT/ds = |v(t)| dT/dt
N(t) = (1 / |dT/ds|) dT/ds
We have:
dT/dt = (-cos t - t sin t) / (1 + t^2)^(3/2) i
+ (-sin t + t cos t) / (1 + t^2)^(3/2) j
|dT/ds| = |v(t)| / |r'(t)| = sqrt[1 + t^2] / sqrt[(cos t + t sin t)^2 + (sin t - t cos t)^2]
= sqrt[1 + t^2] / sqrt[1 + t^2]
= 1
Therefore, the normal vector is:
N(t) = (-cos t - t sin t) / sqrt[1 + t^2] i
+ (-sin t + t cos t) / sqrt[1 + t^2] j
The curvature is given by:
k = |dT/ds|
We have:
dT/ds = |v(t)| dT/dt = (1 + t^2) (-cos t - t sin t) / (1 + t^2)^(3/2) i
+ (1 + t^2) (-sin t + t cos t) / (1 + t^2)^(3/2) j
= (-cos t - t sin t) / (1 + t^2)^(1/2) i
- (sin t - t cos t) / (1 + t^2)^(1/2) j
Therefore, the curvature is:
k = |dT/ds| = sqrt[(-cos t - t sin t)^2 + (sin t - t cos t)^2] / (1 + t^2)
= sqrt[2] / (1 + t^2)
The radius of curvature is the reciprocal of the curvature:
R = 1 / k = (1 + t^2) / sqrt[2]
To find the curvature and radius of curvature at t = 1, we substitute t = 1 into the expressions for k and R:
k = sqrt[2] / 2
R = (1+t^2)^(3/2)
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Related Questions
what is the volume of a right circular cylinder with a radius of 4 m and a height of 4 m? responses 8π m³ 8 pi, m³ 16π m³ 16 pi, , m³ 64π m³ , 64 pi, , m³ 256π m³ 256 pi, m³
Answers
The volume of a right circular cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
Substituting the values r = 4 m and h = 4 m into the formula, we have:
V = π(4^2)(4)
V = π(16)(4)
V = 64π m³
Therefore, the volume of the right circular cylinder is 64π m³.
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f(c)=3c-8
g(c)=√14-c
Evaluate.
(fog)(-11) =
Answers
Answer: f(g(-11)) = out -23
Step-by-step explanation:
The value of the composite function (fog)(-11) is 3√14 - 41.
What is a composite function?When the output of one function is given to the input of another function they are called composite functions.
In (fog)x which is [f{g(x)}] here the out of the function g(x) is the input of the function f(x).
Given, Are two functions f(c) = 3c - 8 and g(c) = √14 - c.
Now, We know that (fog)(c) is the output of g(c) is in the input of f(c).
So, g(-11) = √14 - 11.
Therefore,
(fog)(-11) = 3(√14 - 11) - 8.
(fog)(-11) = 3√14 - 33 - 8.
(fog)(-11) = 3√14 - 41.
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What is an equation of the line (-1,0) and (1,8)
Answers
Answer:
use the formula
Step-by-step explanation:
(y-y1)=M(x-x1) use this formula after finding the slope
you will get the answer
is a fraction a term? If it's not a term, why is it that we can apply the distributive property to it? the distributive property only works for either terms, or addition and subtraction. a fraction is technically division, so why does it work? Please help!!!!!
Answers
No, a fraction is not a term. The distributive property can be applied to fractions because it is a general mathematical principle.
A fraction is not considered a term in the traditional sense. It is a mathematical expression that represents division. However, the distributive property can still be applied to fractions because the property itself is a fundamental rule of arithmetic that extends beyond specific types of expressions.
The distributive property states that for any real numbers a, b, and c:
a × (b + c) = (a × b) + (a × c).
When working with fractions, we can apply the distributive property as follows:
Let's consider the expression: a × (b/c).
We can rewrite this as: (a × b)/c.
Now, let's distribute the 'a' to 'b' and 'c':
(a × b)/c = (a/c) × b.
In this step, we applied the distributive property to the fraction (a/c) by treating it as a whole.
Although fractions represent division, we can still use the distributive property because it is a general mathematical principle that allows for manipulating expressions involving addition, subtraction, multiplication, and division.
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11. The scale of a dollhouse is l in: 2 ft. Which of the following would most likely be the measurement of the heightof the dollhouse's front doorA 21in2No - - -B. 3-Ft2O c. 14 inO D. 14H12. A flagpole casts a shadow 5 ft. long. At the same time, a 3 ft. yardstick casts a shadow 1.5 ft. long. How tall isthe flagpole?O A. 5 ft.B. 10 ft.O C. 20 ft.O D. 15 ft
Answers
We want to find the height of the dollhouse's front door.
Since the dollhouse's measures is given in inches, we have just two possible right choices:
because the answer should be given in inches.
If the correct option were the third one, 14 in,
then the real door measure would be twice in feet: 28 feet.
If the correct option were the first one, 3 1/2 in,
then the real door measure would be twice in feet: 7 in.
28 feet is a really big door. It is more likely for a house to have a 7 in door.
Answer: A. 3 1/2 in
Step right up for a brainliest
Answers
calculate the tax amount they pay in a form of tax (tax =R14 850)
Answers
Answer:
246373737
Step-by-step explanation:
good luck
10. Use the vertical line test to determine if the graph below
represents a function. If it is not a function, tell where the
vertical line test fails.
Answers
The graph is a function. When drawing a vertical line, there will be two circumstances:
The line intersects only one point on a graph. The line intersects more than one point on a graph ( POI > 1 )For the first circumstance, the graph that has only one intersect with a line - it is a function. For the second circumstance, the graph cannot be a function as it gives more than one y-value for a single x-value.
The conclusion is that if a line only has one intersect on a graph then it is a function. Otherwise, not a function but a relation instead. All functions are relations, but not all relations can be functions.
Answer
The graph passes the vertical line test and is a function.Find the slope and y-intercept of each graph
Answers
Answer:1. m=1 3. m=1/3
Step-by-step explanation: your slope on number one was wrong it is 1. the slope on question 3 is 1/3. subtract two of the x-values. then subtract two y-values. they do not have to correspond with the x-values. the x will be your numerator the y your denominator. simplify the fraction
A swimming pool has to be drained for maintenance. The pool is shaped like a cylinder with a diameter of 9 m and a depth of 1.3 m. Suppose water is pumped out of the pool at a rate of 14 m'³ per hour. If the pool starts completely full, how many hours will it take to empty the pool? Use the value 3.14 for a, and round your answer to the nearest hour. Do not round any intermediate computations.
Answers
First of all, calculate the volume of the swimming pool by using the following formula:
V = π r·h
h: height = 1.3 m
r: radius of the base = 9 m/2 = 4.5 m
replace the previous values into the formula for V:
V = (3.14)(4.5 m)(1.3 m)
V = 18.37 m³
Next, calculate the quotient:
18.37 m³/(14 m³/h) = 1.31 h
Hence, 1.31 hours are required to empty the swimming pool
In statistical experiments, each time the experiment is repeateda. the same outcome must occurb. the same outcome can not occur againc. a different outcome may occurd. None of the other answers is correct.
Answers
In statistical experiments, each time the experiment is repeated, a different outcome may occur. The correct answer is C.
Statistical experiments involve random processes, such as the random selection of individuals from a population, the random assignment of subjects to groups, or the random measurement of a variable.
The outcome of a statistical experiment is influenced by chance and is inherently uncertain, so it is not possible to guarantee that the same outcome will occur every time the experiment is repeated. Different outcomes may occur, even if the experimental conditions are kept constant, due to the random nature of the processes involved.
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What is the slope of y=2+3.5x pls answer
Answers
Answer:
slope = 3.5
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 2 + 3.5x OR y = 3.5x + 2 ← is in slope- intercept form
with slope m = 3.5
3.
Find the rate of change for the situation. A chef cooks 9 lbs of chicken for 36 people and 17 lbs of chicken for 68 people.
A. 4 lb per person
B. \(\frac{1}{4}\) lb per person
C. \(\frac{9}{17}\) lb per person
D. 36 people
Answers
Answer: 1/4
Answer:
Step-by-step explanation:
Here is the complete question: Find the rate of change for the situation:
A chef cooks 9 lbs of chicken for 36 people and 17 lbs of chicken for 68 people.
Given: 1st situation, A chef cook 9 lbs chicken for 36 people.
2nd situation, chef cook 17 lbs chicken for 68 people.
∴ 1st situation, weight of chicken per person=
Weight of chicken per person=
2nd situation, weight of chicken per person=
Weight of chicken per person=
In both the situation chef cook same amount of chicken per person, which is
∴ Rate of change is
Step-by-step explanation:
There are several important information's already given in the question. Based on those information's the answer can be easily deduced.
In the first case,
The amount of chicken the chef cooks for 36 people = 9 pounds
Then
The amount of chicken the chef cooks for 1 person = 9/36
= 1/4 pounds
Now let us take the second case
The amount of chicken the chef cooks for 68 people = 17 pounds
Then
The amount of chicken the chef cooks for 1 person = 17/68
= 1/4 pounds
From the above deductions, it can be concluded that the correct option among all the options given in the question is the third option or option "B".
Answer the following:
a.Find the uniform continuous probability for P(X < 10) for U(0, 50). (Round your answer to 2 decimal places.)
b.Find the uniform continuous probability for P(X > 595) for U(0, 1,000). (Round your answer to 3 decimal places.)
c.Find the uniform continuous probability for P(21 < X < 49) for U(19, 68). (Round your answer to 4 decimal places.)
Answers
a. The probability P(X < 10) is U(0, 50) is 0.20.
b. The probability P(X > 595) is U(0, 1,000) is 0.405.
c. The probability P(21 < X < 49) is U(19, 68) is 0.4762.
a. For a uniform continuous distribution U(0, 50), the probability of an event X < 10 can be calculated by dividing the length of the interval [0, 10] by the length of the entire interval [0, 50]. Since the lengths of both intervals are equal, the probability is 10/50 = 0.20.
b. Similarly, for a uniform continuous distribution U(0, 1,000), the probability of an event X > 595 can be calculated by dividing the length of the interval [595, 1,000] by the length of the entire interval [0, 1,000]. The length of the interval [595, 1,000] is 1,000 - 595 = 405, and the length of the entire interval is 1,000 - 0 = 1,000. Thus, the probability is 405/1,000 = 0.405.
c. For a uniform continuous distribution U(19, 68), the probability of an event 21 < X < 49 can be calculated by dividing the length of the interval [21, 49] by the length of the entire interval [19, 68]. The length of the interval [21, 49] is 49 - 21 = 28, and the length of the entire interval is 68 - 19 = 49. Therefore, the probability is 28/49 = 0.5714.
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What is the value of x in the equation 3x+9/6=x-3
Answers
\( \frac{3x + 9}{6} = x - 3 \\ \)
\(3x + 9 = 6(x - 3)\)
\(3x + 9 = 6x - 18\)
\(6x - 3x = 9 + 18\)
\(3x = 27\)
\(x = \frac{27}{3} \\ \)
answer: \(x = 9\)
1. Determine the missing value in this table of values for the function y = 2
y = 24
0.5
-1
0
1
2
Answers
Answer:
-1
Step-by-step explanation:
Ghana is a great place to work for and it is the most versatile and affordable way to do business with you in the future of the following statements is of a hole in your heart to be added
An electrician leans an extension ladder against the outside wall of a
house so that it reaches an electric box 28 feet up. The ladder makes an
angle of 71° with the ground. Find the length of the ladder. Round your
answer to the nearest hundredth of a foot if necessary.
Answers
Answer:
30 feet
Step-by-step explanation:
i used a ruler, mm side. 1 mm = 1 foot
draw a horizontal line.
draw a vertical line 28mm high
use a protractor to find a line that connects to the base and top of vertical line.
measure said line
Answer:
Step-by-step explanation:
\text{sin }\color{purple}{71}=\frac{\color{green}{\text{opposite}}}{\color{#EB0000}{\text{hypotenuse}}}=\frac{\color{green}{28}}{\color{#EB0000}{x}}
sin 71=
hypotenuse
opposite
=
x
28
\text{sin }71=
sin 71=
\,\,\frac{28}{x}
x
28
\frac{\text{sin }71}{1}=
1
sin 71
=
\,\,\frac{28}{x}
x
28
x\text{ sin }71=
x sin 71=
\,\,28
28
Cross multiply.
\frac{x\text{ sin }71}{\text{ sin }71}=
sin 71
x sin 71
=
\,\,\frac{28}{\text{ sin }71}
sin 71
28
Divide both sides by sin 71
x=\frac{28}{\text{ sin }71}=29.613379...\approx29.61
x=
sin 71
28
=29.613379...≈29.61
Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor and 8% visit neither. Say that the probability of visiting a physical ther apist exceeds the probability of visiting a chiropractor by 16%. What is the probability of a randomly selected person from this group visiting a physical therapist?
Answers
The probability of a randomly selected person from this group visiting a physical therapist is 0.68.
The probability is the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of cases possible. Let A denote the event that the person visits a physical therapist, and let B denote the event that the person visits a chiropractor. Now, we need to use the laws of probability to arrive at each individual probability i.e. P(A) and P(B).
Given: P(A∩B)=0.28
(ii) P(AC∩BC)=0.08
(iii) P(A)−P(B)=0.16
Required: P(A)
By the laws of inclusion/exclusion,
P(A∪B)=P(A)+P(B)−P(A∩B)
Therefore,
P(A)+P(B)=P(A∩B)+P(A∪B)
By DeMorgan's Law,
P(AC∩BC)=1−P(A∪B)
Therefore,
P(A)+P(B)=0.28+1−0.08=1.2
By adding the equation,
P(A)−P(B)=0.16
we get
2P(A)=1.36
Therefore, P(A)=0.68
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Convert the following decimal numbers to the binary number system. a. 8 b. 35 c. 108 d. 176
Answers
The binary representations of the given decimal numbers are: (a) 8 = 1000, (b) 35 = 100011, (c) 108 = 1101100, and (d) 176 = 10110000.
(a) To convert 8 to binary, we repeatedly divide the number by 2 and keep track of the remainders. The remainders, read in reverse order, give the binary representation.
Starting with 8, the division process yields: 8/2 = 4 with a remainder of 0, 4/2 = 2 with a remainder of 0, and 2/2 = 1 with a remainder of 0. The binary representation of 8 is 1000.
(b) To convert 35 to binary, we follow the same process. The division steps are as follows: 35/2 = 17 with a remainder of 1, 17/2 = 8 with a remainder of 1, 8/2 = 4 with a remainder of 0, 4/2 = 2 with a remainder of 0, and 2/2 = 1 with a remainder of 0. The binary representation of 35 is 100011.
(c) For 108, the division steps are: 108/2 = 54 with a remainder of 0, 54/2 = 27 with a remainder of 0, 27/2 = 13 with a remainder of 1, 13/2 = 6 with a remainder of 1, 6/2 = 3 with a remainder of 0, 3/2 = 1 with a remainder of 1. The binary representation of 108 is 1101100.
(d) Finally, for 176, the division steps are: 176/2 = 88 with a remainder of 0, 88/2 = 44 with a remainder of 0, 44/2 = 22 with a remainder of 0, 22/2 = 11 with a remainder of 0, 11/2 = 5 with a remainder of 1, 5/2 = 2 with a remainder of 1, and 2/2 = 1 with a remainder of 0. The binary representation of 176 is 10110000.
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Probability part 2 is shunya is going to spin the spinner 200 times. (B) work out an estimate for the number of times the spinner will land on 3 (please show working out ) :)
Answers
if the same instrument gives consistent results every time it is used for a measurement, then it indicates:
Answers
Every time the same instrument is used for a measurement, it produces consistent results, which denotes: -reliability.
The degree to which a measurement tool produces consistent, repeatable estimates of what is believed to be a true score at the core.
The same set of people receives the same instrument again. The correlation between the results on the two instruments is what defines dependability. The scores ought to be comparable if the outcomes remain constant over time. How long to wait between the two administrations is the key to test-retest reliability.
Reliability is defined as the probability that a given item will perform its intended function with no failures for a given period of time under a given set of conditions.
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If someone can help with this tysm, in class rn tryna get an A
Answers
Answer:
1. 4/9
2. 80,100,6.5
3. 10 and 11
4. 56cm
5. 3 an4
6. Real, Rational, Integer
7. All whole numbers are natural numbers
8. -1 1/2
9. √72
10. √9
Good luck :)
Step-by-step explanation:
Indicate the level of measurement for the data set described. Number of classes taken in a semester by each student at a college Answer :a. Nominal b. Ordinal c. Interval d. Ratio
Answers
The correct answer is option c. Interval. The level of measurement for the data set described is Interval.
This is so because the data set calculates each student's enrollment in classes during the course of a semester at a particular college.
Numerical information that is measured on a scale with evenly spaced intervals but lacks an absolute zero point is called interval data.
The numerical values themselves do not reflect quantities or have any absolute value, but the intervals between them all have the same size.
The interval between 3 classes and 4 classes, for instance, would be the same as the interval between 8 classes and 9 classes if the data set were measuring the number of classes each student took in a semester, but 3 classes and 8 classes do not have any such absolute value.
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Pls help with this algebra ( it’s about equations with variables on both sides) unit practice 1-4).
Answers
Answer:
x = -5
2.1x+45.2 = -7.3-8.4x
10.5x+45.2 = -7.3
10.5x = -52.5
x = -5
Please help me !! would appreciate
Answers
The answers that describe the quadrilateral DEFG area rectangle and parallelogram.
The correct answer choice is option A and B.
What is a quadrilateral?A quadrilateral is a parallelogram, which has opposite sides that are congruent and parallel.
Quadrilateral DEFG
if line DE || FG,
line EF // GD,
DF = EG and
diagonals DF and EG are perpendicular,
then, the quadrilateral is a parallelogram
Hence, the quadrilateral DEFG is a rectangle and parallelogram.
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Please ANSWER NOW, PLEASE HELP , for math 100 points!!!
Answers
Caution:-
Calculated approximate values as we can't use ruler in screen !
As
1/2in=4ft1in=8ft#1
Sides are
6in3in7in4inPerimeter=20in=20(8)=160ft
#2
Perimeter=2(2+1.5)=2(3.5)=7in=7(8)=56ft
#3
Width:-
2in2(8)16ft:
4. A metal sphere of radius a carries a charge Q. It is surrounded, out to radius b, by linear dielectric material of permittivity &. Find the potential at the center (relative to infinity)
Answers
The potential at the center of the metal sphere, relative to infinity, surrounded by linear dielectric material is:
V = (1 / 4πε) * (Q / a)
To find the potential at the center of the metal sphere surrounded by a linear dielectric material, we can use the concept of the electric potential due to a uniformly charged sphere.
The electric potential at a point inside a uniformly charged sphere is given by the formula:
V = (1 / 4πε₀) * (Q / R)
Where:
V is the electric potential at the center,
ε₀ is the permittivity of free space (vacuum),
Q is the charge of the metal sphere,
R is the radius of the metal sphere.
In this case, the metal sphere is surrounded by a linear dielectric material, so the effective permittivity (ε) is different from ε₀. Therefore, we modify the formula by replacing ε₀ with ε:
V = (1 / 4πε) * (Q / R)
The potential at the center is considered relative to infinity, so the potential at infinity is taken as zero.
Therefore, the potential at the center of the metal sphere, relative to infinity, surrounded by linear dielectric material is:
V = (1 / 4πε) * (Q / a)
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If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible? probability
Answers
There is only one way to divide the 12 people into committees of sizes 3, 4, and 5, and the probability of this division occurring is 1.
To find the number of divisions possible and the probability, we need to consider the number of ways to divide 12 people into committees of sizes 3, 4, and 5.
First, we determine the number of ways to select the committee members:
For the committee of size 3, we can select 3 people from 12, which is represented by the combination "12 choose 3" or C(12, 3).
For the committee of size 4, we can select 4 people from the remaining 9 (after selecting the first committee), which is represented by C(9, 4).
Finally, for the committee of size 5, we can select 5 people from the remaining 5 (after selecting the first two committees), which is represented by C(5, 5).
To find the total number of divisions, we multiply these combinations together: Total divisions = C(12, 3) * C(9, 4) * C(5, 5)
To calculate the probability, we divide the total number of divisions by the total number of possible outcomes. Since each person can only be in one committee, the total number of possible outcomes is the total number of divisions.
Therefore, the probability is: Probability = Total divisions / Total divisions
Simplifying, we get: Probability = 1
This means that there is only one way to divide the 12 people into committees of sizes 3, 4, and 5, and the probability of this division occurring is 1.
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draining 15 gallons of water from a fish tank
(Write your answer as an integer)
Answers
Answer:
-15
Step-by-step explanation:
if its just asking for what number would represent a loss of 15 it would be negative 15
i dont know if there's more to the problem or not
Answer:
-15
Step-by-step explanation:
This is because whatever much water was in the tank is now decreasing by 15, therefore, you can write it as -15,
hope this helps!
need help~giving brainliest!!! and 36pts!
In △△KLM, MK (lines over MK)≅ LM.m∠L=43∘ . Find M.m∠M.
Answers
Answer:
Based on the definition of an isosceles triangle, the measure of angle L is: 43°
An isosceles triangle has two side lengths that are equal in size, and also the angles opposite the equal sides are congruent.
Given that KL ≅ MK in △KLM, it means △KLM is an isosceles triangle.
If m∠M = 43°, therefore, m∠M = m∠L (equal angles directly opposite KL and MK).
This implies that: m∠M = 43°
In summary, based on the definition of an isosceles triangle, the measure of angle M is: 43°