The curvature k(t) of the Eequation r(t) = (-2sint) i + (-2sint) j + (-cost) k will be [tex]\frac{2\sqrt{2} }{(8cos^2t + sin^2t)^{3/2}}[/tex]
Given r(t) = (-2sint) i + (-2sint) j + (-cost) k
We can clearly see that r(t) is a vector equation of t.
If r(t): R→R³ is a vector-valued function of a real variable with independent scalar output variables x, y & z
where, r(t) = {x, y, z}
Where x = (-2sint) , y = -2sint and z = -cost
We know that the curvature to be
|| r'(t) X r"(t) || / ||r'(t)||³
This implies we need to find the determinant of the cross product of the derivative and double derivative of the equation r(t) for the numerator
r'(t) = (-2cost) i + (-2cost) j + (sint) k
r"(t) = (2sint) i + (2sint) j + (cost) k
hence we get || r'(t) X r"(t) || to be
[tex]\left|\begin{array}{ccc}i&j&k\\-2cost&-2cost&sint\\2sint&2sint&cost\end{array}\right|[/tex]
[tex]= \left|\begin{array}{ccc}-2cost&sint\\2sint&cost\end{array}\right| i - \left|\begin{array}{ccc}-2cost&sint\\2sint&cost\end{array}\right|j + \left|\begin{array}{ccc}-2cost&-2cost\\2sint&2sint\end{array}\right|k[/tex]
= -2i + 2j
Hence the magnitude will be
√(2² + 2²)
= 2√2 units
magnitude of r'(t) is
[tex]\sqrt{(2sint)^2 + (2sint)^2 + (cost)^2}[/tex]
= [tex]\sqrt{8cos^2t + sin^2t}[/tex]
Hence we get the curvature to be
[tex]\frac{2\sqrt{2} }{(\sqrt{8cos^2t + sin^2t})^3}[/tex]
simplifying this will give us
[tex]\frac{2\sqrt{2} }{(8cos^2t + sin^2t)^{3/2}}[/tex]
Hence the curvature k(t) will be [tex]\frac{2\sqrt{2} }{(8cos^2t + sin^2t)^{3/2}}[/tex]
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Correct Question
Find the curvature k(t) of the curve r(t) = (-2 sin t)i + (-2 sin t)j + (-1 cost)k
In the standard (x, y) coordinate plane which equation represents a line through the point (6, 1) and perpendicular to the line with the equation =3/2 + 1?
A.) = −3/2 − 8
B. ) = −2/3 − 3
C.) = −2/3 + 1
D.) = −2/3 + 5
E.) = −3/2 + 10
Answer:
d. -2/3x+5
Step-by-step explanation:
Because the line is perpendicular, the slope must be the inverse of the original slope. The inverse of 3/2 is -2/3. To find the b value, you plug (6,1) into the equation y=-2/3x+b
1=-2/3(6)+b
1=-4+b
5=b
The final equation is y=-2/3x+5
The highway mileage (mpg) for a sample of 9 different models of a car company can be found below. 41 50 41 25 28 27 36 47 46 Find the mode: Find the midrange: Find the range: Estimate the standard deviation using the range rule of thumb: Now use technology, find the standard deviation: decimal places.) (Please round your answer to 2
For highway-mileages of 9 different models of car, the mode is 41, mid-range is 37.5, range is 25, and standard-deviation is 6.25.
⇒ To find the mode, we identify the value(s) that occur most frequently in the data set. In this case, the mode is 41, as it appears twice, more than any other value.
So, the mode is 41,
⇒ The midrange is calculated by finding the average of the maximum and minimum values in the data set. In this case, the maximum value is 50 and the minimum value is 25. So, the midrange is (50 + 25)/2 = 37.5,
⇒ The range is determined by subtracting the minimum value from the maximum value. In this case, the range is 50 - 25 = 25,
⇒ To calculate the standard-deviation using the range rule of thumb, we divide the range by 4. In this case, the range is 25, so the standard deviation would be 25/4 = 6.25.
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The given question is incomplete, the complete question is
The highway mileage (mpg) for a sample of 9 different models of a car company can be found below. 41, 50, 41, 25, 28, 27, 36, 47, 46.
Find the mode, midrange, range, and standard deviation.
Given the following data set, calculate the values for the five-number summary and fill in the table below: -7, -5, -2, 0, 4, 6, 8, 8, 10, 22, 24 Name Number Minimum First Quartile Median Third Quartile Maximum
The five-number summary for the given data set is: Minimum = -7, First Quartile = 2, Median = 6, Third Quartile = 9, Maximum = 24.
To calculate the five-number summary for the given data set, we need to arrange the data in ascending order and then determine the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values.
The given data set: -7, -5, -2, 0, 4, 6, 8, 8, 10, 22, 24
Arranged in ascending order: -7, -5, -2, 0, 4, 6, 8, 8, 10, 22, 24
Now, let's calculate the values for the five-number summary:
Minimum: The smallest value in the data set is -7.
First Quartile (Q1): This represents the median of the lower half of the data set. Since we have 11 data points, Q1 is the median of the first 5 data points. Q1 = (0 + 4) / 2 = 2.
Median (Q2): The median is the middle value of the data set. Since we have an odd number of data points, the median is the 6th value, which is 6.
Third Quartile (Q3): This represents the median of the upper half of the data set. Q3 is the median of the last 5 data points. Q3 = (8 + 10) / 2 = 9.
Maximum: The largest value in the data set is 24.
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y=-2x-10
2x+5y=6
i need it solved with the substitution method.
a team of 3 employees is preparing 20 reports. it takes mary 30 minutes to complete a report, and it takes matt 45 minutes to complete a report. all reports are completed in 4 1/2 hours. how long does it take the third team member to complete a report?
Given: A team of 3 employees is preparing 20 reports. Mary takes 30 minutes to complete a report. Matt takes 45 minutes to complete a report.
All reports are completed in 4 1/2 hours. To Find: How long does it take the third team member to complete a report?Solution: Let the third employee takes x minutes to complete a report work done by Mary in 1 minute = 1/30Work done by Matt in 1 minute = 1/45Work done by the third employee in 1 minute = 1/x Total work done by all three in 1 minute = 1/30 + 1/45 + 1/x (As all are working together) a Total number of reports to be prepared = 20Therefore, total work = 20Now,
we know that all reports are completed in 4 1/2 hours = 9/2 hours∴ Total time = 9/2 x 60 = 270 minutes according to the problem statement, Total work = Total time x Total work done by all three in 1 minute20 = 270 (1/30 + 1/45 + 1/x)Solving the above equation for x, we get :x = 90 minutes therefore, it takes the third team member 90 minutes to complete a report.
Answer: 90 minutes.
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Let's represent the third team member as t. Mary takes 30 minutes to complete a report, while Matt takes 45 minutes to complete a report.
Thus, it takes the third team member 3 hours to complete a report.
Therefore, we can use the information given to form an equation. We are given that the team is preparing 20 reports, so:
30 minutes/report × M reports + 45 minutes/report × N reports + T minutes/report × O reports = 4.5 hours
To make the equation simpler, let the unit conversion 4.5 hours to minutes:
4.5 hours × 60 minutes/hour = 270 minutes
Thus:
30M + 45N + TO = 270
O= 20 - M - N
From the third team member: TO = T × 20
Therefore:
30M + 45N + T × 20 = 270
Solving for T:
30M + 45N + 20T = 270
T = (270 - 30M - 45N)/20
We know that there are only three members in the team, and that M and N have already been defined, so we can substitute these values:
T = (270 - 30(20) - 45(0))/20
T = 3
Thus, the time taken by the third team member is 3 hours to complete a report.
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I and my friends can't find the answer to this and we need help pls.
plss help meee
i need help on this and it's dueeeee
i will put brainliest for first answer plss help
Step-by-step explanation:
Storage Space of Shed = Volume of Shed.
Volume of Shed = Volume of Rectangular level
+ Volume of Triangular roof
Volume of Rectangular Level =
Length x Width x Height
=
[tex]12 \times 6 \times 8 \\ = 576 {ft}^{3} [/tex]
Volume of Triangular Roof =
Area of Triangular side x Length
=
[tex] \frac{1}{2} \times base \times height \times length \\ = \frac{1}{2} \times 6 \times 4 \times 12 \\ = \frac{1}{2} \times 288 \\ = 144 {ft}^{3} [/tex]
Volume of Shed = 576 + 144
=
[tex]720 {ft}^{3} [/tex]
Write an equation for the linear function graphed above;
Answer:
y = -1/4x + 16
Step-by-step explanation:
the slope is -1/4 and the y-intercept is 16
A glass bead has the shape of a rectangular prism with a smaller rectangular prism removed. What is the volume of the glass that forms the bead?
Thanks in advance!
Answer:
216 cm³
Step-by-step explanation:
large prism volume = 6 x 6 x 8 = 288 cm³
small cutout volume = 3 x 3 x 8 = 72 cm³
288- 72 = 216 cm³
Please consider the following linear congruence, and solve for x, using the steps outlined below. 57x + 13 = 5 (mod 17) (a) (4 points) Use the Euclidean algorithm to find the correct GCD of numbers 57 and 17.
The correct GCD of 57 and 17 is 1, obtained through the Euclidean algorithm.
To find the correct GCD (Greatest Common Divisor) of 57 and 17 using the Euclidean algorithm, we follow these steps:
1.) Divide the larger number (57) by the smaller number (17) and find the remainder:
57 ÷ 17 = 3 remainder 6
2.) Replace the larger number with the smaller number and the smaller number with the remainder:
17 ÷ 6 = 2 remainder 5
3.) Repeat step 2 until the remainder is 0:
6 ÷ 5 = 1 remainder 1
5 ÷ 1 = 5 remainder 0
4.) The GCD is the last nonzero remainder, which is 1.
Therefore, the correct GCD of 57 and 17 is 1.
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Let g be a twice-differentiable function with g'(x) > 0 andg''(x) > 0 for all real numbers x, such that
g(4) = 12 and g(5) = 18. Of the following, which is apossible value for g(6)?
a. 15
b. 18
c. 21
d. 24
e. 27
A possible value for g(6) is 27. The only option greater than 18 is:
e. 27
To determine a possible value for g(6), we can make use of the given information and the properties of the function g(x).
Since g'(x) > 0 for all real numbers x, we know that g(x) is strictly increasing. This means that as x increases, g(x) will also increase.
Furthermore, since g''(x) > 0 for all real numbers x, we know that g(x) is a concave up function. This implies that the rate at which g(x) increases is increasing as well.
Given that g(4) = 12 and g(5) = 18, we can conclude that between x = 4 and x = 5, the function g(x) increased from 12 to 18.
Considering the properties of g(x), we can deduce that g(6) must be greater than 18. Since the function is strictly increasing and concave up, the increase from g(5) to g(6) will be even greater than the increase from g(4) to g(5).
Among the given answer choices, the only option greater than 18 is:
e. 27
Therefore, a possible value for g(6) is 27.
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Write a [tex]y=\frac{4}{5}x-2[/tex] in standard form using integers.
Answer:
4x-5y=10
Step-by-step explanation:
What is the function
The function for this problem is given as follows:
y = 0.25(x + 5)²(x - 4)²
How to define the function?We are given the roots for each function, hence the factor theorem is used to define the functions.
The function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.
The roots of the function in this problem are given as follows:
x = -5 with a multiplicity of 2, as the graph touches the y-axis.x = 4 with a multiplicity of 2, as the graph touches the y-axis.Hence the linear factors are given as follows:
(x + 5)².(x - 4)².The function is:
y = a(x + 5)²(x - 4)²
In which a is the leading coefficient.
When x = 0, y = 100, hence the leading coefficient a is given as follows:
100 = a(5²)(-4)²
400a = 100
a = 0.25.
Hence the function is:
y = 0.25(x + 5)²(x - 4)²
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Let X and Y be two random variables. Suppose that σ2 of X=4, and σ2 of Y=9.
If we know that the two random variables Z=2X−Y and W=X+Y are independent, find Cov(X,Y) and rho(X,Y)
Cov(X, Y) = -1/3 and ρ(X, Y) = -1/18.
Given data:X and Y are two random variables,
σ² of X=4,σ² of Y=9.Z=2X − Y and W = X + Y are independent
To find:
Cov(X, Y) and ρ(X, Y)
Solution:
We know that:
Cov(X, Y) = E(XY) - E(X)E(Y)ρ(X, Y) = Cov(X, Y) / σX σY
Let's find E(X), E(Y), E(XY)E(X) = E(W - Y) = E(W) - E(Y)E(W) = E(X + Y) = E(X) + E(Y)
From this equation, E(X) = E(W)/2 ------- (1)
Similarly, E(Y) = E(W)/2 ------- (2)
To find E(XY), we will use the following equation:
E(XY) = Cov(X, Y) + E(X)E(Y)Using equations (1) and (2) in the above equation:
E(XY) = Cov(X, Y) + E(W)²/4
Now, we will use the independence of Z and W to find Cov(X, Y).Cov(X, Y) = Cov((W - Z)/2, (W + Z)/3)= 1/6[Cov(W, W) - Cov(W, Z) + Cov(Z, W) - Cov(Z, Z)]= 1/6[Var(W) - Var(Z)]
Here,Var(W) = Var(X + Y) = Var(X) + Var(Y) [using independence]= 4 + 9 = 13Var(Z) = Var(2X - Y) = 4Var(X) + Var(Y) - 2 Cov(X, Y)= 4 + 9 - 2 Cov(X, Y)
Now, putting these values in Cov(X, Y),Cov(X, Y) = -1/3
Also,σX = 2 and σY = 3ρ(X, Y) = Cov(X, Y) / σX σY= -1/18
Hence, Cov(X, Y) = -1/3 and ρ(X, Y) = -1/18.
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Leo drew a line that is perpendicular to the line shown on the grid and passes through the point (f, g).
a. True
b. False
Any line perpendicular to it would have an undefined slope and would not pass through point (f, g) where f and g could be any values. In this case, the statement is false.
The statement "Leo drew a line that is perpendicular to the line shown on the grid and passes through the point (f, g)" can be true or false.
It depends on the line shown on the grid and the coordinates of point (f, g).
Two lines are perpendicular if their slopes are opposite reciprocals of each other.
To find the slope of the line perpendicular to the line shown on the grid,
we can use the negative reciprocal of the slope of the given line.
If the line drawn by Leo has this slope and passes through point (f, g), then the statement is true.
If not, then the statement is false.
For example, if the line shown on the grid has a slope of 2/3 and passes through the point (0,0),
then the perpendicular line drawn by Leo would have a slope of -3/2 and pass through point (f, g) where f and g could be any values.
In this case, the statement is true.
However, if the line shown on the grid has a slope of 0 and passes through point (1,1), then any line perpendicular to it would have an undefined slope and would not pass through point (f, g) where f and g could be any values. In this case, the statement is false.
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To write a polynomial in standard form, write the _____ of the terms in descending order.
Answer:
Step-by-step explanation:
Write the sum of the terms in descending order.
If you need 4 eggs to make 12 Yorkshire puddings. How many do you need to make 18
Yorkshire puddings?
Answer:
54
Step-by-step explanation:
First of all,
Cross multiplication
4 = 12
18 = x
Let the value of the number of eggs be represented by x
4x = 12×18
4x = 216
4. 4
(Same as 216÷4)
x = 54
Please help, thank you
y = x + 4 y = −2x − 2 Explain how you will solve the pair of equations by substitution. Show all the steps and write the solution in (x, y) form. Source StylesNormal
Answer:
6x
Step-by-step explanation:
The Parks Department collected data on 20 fishermen at a local lake. Each person caught two fish. The scatterplot below shows the relationship between the weight of the first fish a person caught and the weight of the second fish a person caught.
Study the graph below...
Weight of Fish...
Weight of second fish... ounces..
weight of first fish... ounces...
Which of the following best describes the association of the data?
A. Positive association
B. Negative association
C. No association
D. Non-linear association
Answer:
I think it's A
Step-by-step explanation:
Can someone help with this question? I’m stuck.
Answer:
2
Step-by-step explanation:
2+2 is 4 so therefore the answer is 2.00.
Do all pls 20 points Combine Rationa
Find all the solutions to [x^3 - 1] = 0 in the ring Z/13Z. Make sure you explain why you have found all the solutions, and why there are no other solutions.
The solution to the equation [x³ - 1] = 0 is x = 1
How to determine the solutions to the equationFrom the question, we have the following parameters that can be used in our computation:
[x³ - 1] = 0
Remove the square bracket in the equation
So, we have
x³ - 1 = 0
Add 1 to both sides
This gives
x³ = 1
Take the cube root of both sides
x = 1
Hence, the solution to the equation [x³ - 1] = 0 is x = 1
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solve the given differential equation by undetermined coefficients. y'' − 12y' 36y = 36x 4
The differential equation y'' - 12y' + 36y = 36[tex]x^4[/tex] is solved using the method of undetermined coefficients. The particular solution is found to be y_p = (1/72)[tex]x^6[/tex] - (1/12)[tex]x^4[/tex]+ (1/6)[tex]x^{2}[/tex].
To solve the given differential equation using the method of undetermined coefficients, we assume a particular solution of the form y_p = A[tex]x^6[/tex] + B[tex]x^4[/tex] + C[tex]x^{2}[/tex], where A, B, and C are constants to be determined. We differentiate y_p twice to find its derivatives: y_p' = 6A[tex]x^5[/tex] + 4B[tex]x^3[/tex]+ 2Cx and y_p'' = 30A[tex]x^4[/tex] + 12B[tex]x^{2}[/tex] + 2C.
Substituting these derivatives into the original differential equation, we have:
30A[tex]x^4[/tex] + 12B[tex]x^{2}[/tex] + 2C - 12(6A[tex]x^5[/tex] + 4B[tex]x^3[/tex] + 2Cx) + 36(A[tex]x^6[/tex] + B[tex]x^4[/tex] + C[tex]x^{2}[/tex]) = 36[tex]x^4[/tex].
Simplifying and equating the coefficients of like powers of x, we obtain the following equations:
36A = 0 (coefficient of x^6 term),
-72A + 36B = 0 (coefficient of x^4 term),
-36B + 36C = 36 (coefficient of x^2 term).
Solving these equations, we find A = 0, B = -1/12, and C = 1/6. Therefore, the particular solution is y_p = (1/72)[tex]x^6[/tex]- (1/12)[tex]x^4[/tex]+ (1/6)[tex]x^{2}[/tex].
The general solution of the given differential equation is the sum of the particular solution and the homogeneous solution. However, since the equation does not specify any initial conditions, we only provide the particular solution in this case.
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I'M GIVING BRAINLIEST TO WHOEVER ANSWERS FIRST! GOOD LUCK!
In the following problem, define the variable and then write an expression to represent the number of students at the elementary school. Finally, find the number of students at the middle school if the elementary school has 380 students: The middle school has 24 students less than 3 times the number of students at one of the elementary schools.
Answer:
3x - 24
Step-by-step explanation:
this is probably wrong
Answer:
1116
Step-by-step explanation:
Hey!
We can use the algebraic expression, 3x - 24, to solve.
Just substitute 380 in for x.
⇒3(380) - 24
⇒1140 - 24
⇒ 1116
--------------------------------------------------------------------------------------------------------------
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-Aadi x
Given the Cauchy problem (utt - c²uxx = F(x, t), t> 0, x € (-[infinity]0,00) xe (-00,00) u(x,0) = f(x) (u₂(x,0) = g(x) x € (-00,00) (A) Prove that if f, g are even functions and for every t > 0 the function F(-, t) is even, then for every t > 0 the solution u(,t) is even (i.e. even w.r.t x). (B) Prove that if f, g are periodic functions and for every t≥ 0 the function F(.,t) is periodic, then for every t≥0 the solution u(.,t) is periodic. For part (A) - you can use the lecture notes for Lecture 5 (available in the course website). Write everything in your own words of course.
In part (A) of the problem, it is required to prove that if the initial conditions f(x) and g(x) are even functions and the forcing function F(x, t) is even for every t > 0, then the solution u(x, t) is also even with respect to x for every t > 0. In part (B), the task is to prove that if f(x) and g(x) are periodic functions and the forcing function F(x, t) is periodic for every t ≥ 0, then the solution u(x, t) is also periodic for every t ≥ 0.
To prove part (A), we can use the principle of superposition, which states that if the initial conditions and forcing function are even, then the solution will also possess the property of evenness.
To prove part (B), we can use the fact that if the initial conditions and forcing function are periodic, the solution will be a linear combination of periodic functions. The sum of periodic functions is also periodic, thus making the solution u(x, t) periodic for every t ≥ 0.
By leveraging these principles and the given assumptions about the initial conditions and forcing function, it can be shown that the solutions u(x, t) will also possess the specified properties of evenness or periodicity, depending on the case.
Note: The explanation provided is a general overview of the approach without delving into the mathematical details and formal proofs.
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Mrs.sorestam bought one ruler for 0.49$ one compass for 1.49$ and one mechanical pencil 0.49 at the price shown in the table for each of her 12 students
Answer:
12(x−2.57)=0.36
Step-by-step explanation:
Let x represent the initial amount of money Mrs. Sorenstam had to spend on each student.
The cost of the 3 items is:
1.49+0.59+0.49=2.57
The change left for each student will be:
x−2.57
For 12 students, the change left will be
12(x−2.57) which equals 36 cents, according to the problem
So, the equation to represent this situation will be:
12(x−2.57)=0.36
9 x 10^7 is how many times as large as 3 x 10^3
Answer:
30000 times
Step-by-step explanation:
3 x 10^3 x 30000 = 90000000
Answer:
30000
Step-by-step explanation:
9×10^7 =90000000
also
3×10^3 =3000
divide 90000000 by 3000
=30000
simplify nine to the power of -53×9 to the power of 37
Answer:
Is this for your homework
Step-by-step explanation:
A ship sails 20 km due East, then 12 km due South.
Find the bearing of the ship from its initial position.
Give your answer correct to 2 decimal places.
Answer:
Step-by-step explanation:
20km east - 12km south= 8km east
Answer:
its 120.96
Step-by-step explanation:
i dont have any but i know that is the answer
Find the Minimum & Maximum values of the given function over the interval (-1,
4].
y = e^x+1
Answer:
answer in photo