Minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
To express the length L(p) of curve C in terms of p, we can use the formula for the length of a curve in cylindrical coordinates. In cylindrical coordinates, the arc length element ds can be given by:
ds² = dr² + r² dt² + dz²
Since dr = 0 (as r = a is constant along the curve C), and dt = -a sin(t) dt (from the parametrization), we have:
ds² = a² sin²(t) dt² + dz²
Integrating ds over the curve C from t = 0 to t = tmax, we get:
L(p) = ∫[0,tmax] √(a² sin²(t) + p'(t)²) dt
where p'(t) denotes the derivative of p(t) with respect to t.
To find the differential equation for the function p(t) that minimizes L(p), we can apply the Euler-Lagrange equation of the calculus of variations. The Euler-Lagrange equation is given by:
d/dt (dL/dp') - dL/dp = 0
Differentiating L(p) with respect to p' and p, we have:
dL/dp' = 0 (since p does not appear explicitly in L(p))
dL/dp = d/dt (dL/dp') = d/dt (a² sin²(t) p'(t) / √(a² sin²(t) + p'(t)²))
Using the chain rule, we can simplify the expression:
dL/dp = (a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2)
Setting the Euler-Lagrange equation equal to zero, we get:
(a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2) = 0
Simplifying further, we have:
p''(t) - (sin(t) cos(t) / sin²(t)) p'(t)² = 0
This is the differential equation that the function p(t) must satisfy to minimize L(p).
To solve this differential equation, we can make the substitution u = p'(t). Then the equation becomes:
du/dt - (sin(t) cos(t) / sin²(t)) u² = 0
This is a separable first-order ordinary differential equation. By solving it, we can obtain the solution for u = p'(t). Integrating both sides and solving for p(t), we get:
p(t) = C exp(-cot(t)) + h
where C is a constant determined by the initial condition p(0) = 0, and h is the value of p at t = tmax.
Therefore, the minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
where C is a constant determined by the initial condition, and h is the value of p at t = tmax.
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Use synthetic division to evaluate (2x2 − 7x + 5) ÷ (x − 1).
1. 2x - 9
2. 2x - 5
3. 2x^2 - 5x
4. 2x^2 - 10
A recipe calls for 1/3 of a cup of butter. If Matthew is making 1/4 of the recipe, how much butter should he use?
Answer:
1/12 a cup
Step-by-step explanation:
1/3*1/4=1/12
Answer:
1/12
Step-by-step explanation:
1/3*1/4=1/12
Jada makes sparkling juice by mixing 3 cups sparkling water with 5 cups apple juice. a. How much sparkling water does Jada need if she uses 30 cups of apple juice? b. How much apple juice does Jada need if she uses 9 cups sparkling water?
Answer:
(A) 22
(B) 12
Step-by-step explanation:
Jada makes sparkling juice by mixing 3 cups of sparkling water and 5 cups of Apple juice
(A) Jada uses 30 cups of Apple juice,the amount of sparkling water needed is
3+5
= 8
30-8
= 22
(B) 9+3
= 12
can anyone help me with this question?
Answer:
45 and 13…......
Step-by-step explanation:
devide
Which is the solution to the equation solved for p? (q + p)q = (s + r)s
Answer:
Option B
Step-by-step explanation:
[tex](q + p)q = (s + r)s[/tex]
Lets open the brackets of L.H.S.
[tex]=> q^2 + pq = (s+ r)s[/tex]
Lets substract q² from both the sides.
[tex]=> q^2 + pq - q^2 = (s + r)s - q^2[/tex]
[tex]=> pq = (s + r)s - q^2[/tex]
Lets divide both the sides by q.
[tex]=> \frac{pq}{q} = \frac{(s + r)s - q^2}{q}[/tex]
[tex]=> p = \frac{(s + r)s }{q} - \frac{q^2}{q} = \frac{(s + r)s}{q} - q[/tex]
The integral [, sin(x - 2) dx is transformed into S1, g(t)dt by applying an appropriate change of variable, then g() is: g(t) = sin g(t) = cos (5) O This option O This option g(t) = -cos (3) g(t) = sin O This option
By introducing the appropriate change of variable x = t + 2, we transformed the given integral [, sin(x - 2) dx] into S1, g(t)dt = [, sin(t) dt], where g(t) = sin(t). (option a)
To transform the given integral [, sin(x - 2) dx] into the form S1, g(t)dt, we need to apply a suitable change of variable. In this case, we'll introduce a new variable, t, and express x in terms of t. Let's denote this change of variable as x = h(t), where h(t) represents the transformation of t to x.
To determine the appropriate change of variable, we can start by looking at the expression inside the sine function: x - 2. We need to find a suitable expression for x that can help us simplify the integral. Let's set x - 2 equal to t:
x - 2 = t.
Now, we solve this equation for x to express it in terms of t:
x = t + 2.
Next, we differentiate both sides of the equation x = t + 2 with respect to t to find dx/dt:
dx/dt = d(t + 2)/dt.
Differentiating t + 2 with respect to t gives us:
dx/dt = 1.
Now, we have dx/dt = 1, which means dx = dt. We can substitute this value into the original integral:
[, sin(x - 2) dx] = [, sin(t) dt].
As you can see, the original integral [, sin(x - 2) dx] has been transformed into the form [, sin(t) dt] using the appropriate change of variable x = t + 2. Therefore, the integral S1, g(t)dt is now:
S1 = [, sin(t) dt],
and g(t) = sin(t).
Hence the correct option is (a).
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Martin recorded the low temperatures at his house for one week. The temperatures are shown below.
-7, -3, 4, 1, 2, 8, 7
Approximately what was the average low temperature for the week?
Α. 7
B. "1
C. 1
D "8
rectangle has its vertices at M(1,1), N(6,1), O(6,5), P(1,5). The rectangle is translated to the left 4 units and down 3 units. What are the new coordinates? (Answer Format: M(#,#) N(#,#) O(#,#) P(#,#))
Answer:
7aabbbjdkdbdjdkxbsks n sbs
The table below shows the relationship between the number of teaspoons of baking powder in a mix and the height of fudge brownies in centimeters. Which equation represents the height of fudge brownies with x teaspoons of baking powder?
Making Fudge Brownies
Baking Powder (tsp) 5 6 7 8
Height of Brownies (cm) 2.15 2.43 2.71 2.99
A.
y = 0.28x + 0.75
B.
y = 0.75x +0.56
C.
y = 0.75x + 0.28
D.
y = 0.56x + 0.75
Answer:
A.
y = 0.28x + 0.75
Step-by-step explanation:
Check all that applies using teaspoons of baking powder, x = 5
A.
y = 0.28x + 0.75
When x = 5
y = 0.28x + 0.75
= 0.28(5) + 0.75
= 1.4 + 0.75
y = 2.15
B.
y = 0.75x +0.56
When x = 5
y = 0.75x +0.56
= 0.75(5) + 0.56
= 3.75 + 0.56
= 4.31
C.
y = 0.75x + 0.28
When x = 5
y = 0.75x + 0.28
= 0.75(5) + 0.28
= 3.75 + 0.28
= 4.03
D.
y = 0.56x + 0.75
When x = 5
y = 0.56x + 0.75
= 0.56(5) + 0.75
= 2.8 + 0.75
= 3.55
The equation represents the height of fudge brownies with x teaspoons of baking powder is
y = 0.28x + 0.75
a dice was rolled 60 times what is the probability of rolling a 3
Answer: 2/15
Step-by-step explanation:
Experimental probability is the actual results of the experiment (not what should have happened, which is theoretical probability.
A 3 was rolled 8 times, so the results are 8/60=4/30=2/15
Express the confidence interval (0.069.0.159) in the form of p-E
the confidence interval (0.069, 0.159) can be expressed as 0.114 ± 0.045
In a confidence interval, the lower bound represents the point estimate minus the margin of error, and the upper bound represents the point estimate plus the margin of error. In this case, the lower bound of the confidence interval is 0.069 and the upper bound is 0.159. To express it in the form of p ± E, we need to find the point estimate and the margin of error.
The point estimate, p, is the average or central value of the interval. In this case, it would be the midpoint between the lower and upper bounds:
p = (0.069 + 0.159) / 2 = 0.114
The margin of error, E, can be calculated by taking half the difference between the upper and lower bounds:
E = (0.159 - 0.069) / 2 = 0.045
Therefore, the confidence interval (0.069, 0.159) can be expressed as 0.114 ± 0.045.
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The table shows the costs
for 2, 3, 4, and 5 vegetable
seed packets. What is the
unit rate?
240g in the ratio 5:3:2
Answer:
120:72:48
Step-by-step explanation:
Remark
Whatever the answer is (call it x to begin with), the equation starts as
5x + 3x + 2x = 240
What this means is that when you solve for x the results when multiplied by the ratio, should add to 240
Equation
5x + 3x + 2x = 240
Solution
Combine all the terms on the left.
10x = 240 Divide by 10
x = 240/10
x = 24
Check
5*24 = 1203*24 = 722*24 = 48Total = 240mrs chambers is 62 years old and plans to retire. over her life she earned an average of $2,500 per month after adjusting for inflation the formula for calculating the full monthly social security benefits is 90% of the first $680 earned the 32% of the earnings over $680 what will her monthly benefit be if her full retirement benefit is reduced by 30% for retiring at 62 years old
Answer: 60
Step-by-step explanation:
Working together, Daniel and Donnie can clean the llama pen in 45 minutes. On his own, Daniel can clean the pen in an hour. How long does it take Donnie to clean the llama pen on his own?
Answer:
It takes him 15 minutes.
Step-by-step explanation:
Donnie clean the llama pen in 180 minutes on his own.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
Daniel and Donnie can clean the llama pen in 45 minutes.
And, On his own, Daniel can clean the pen in an hour.
Now,
Let time to clean the llama pen in on his own = x minutes.
Since, Daniel and Donnie can clean the llama pen in 45 minutes.
So, In 1 minute, Daniel and Donnie clean the pen = 1/45
And, In 1 minute Daniel to work = 1/60
Thus, We can formulate;
1/x + 1/60 = 1/45
Solve for x as;
(60 + x) / 60x = 1/45
60 + x = 60x / 45
60 + x = 4x/3
180 + 3x = 4x
180 = 4x - 3x
180 = x
Thus, Donnie clean the llama pen in 180 minutes on his own.
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At midnight, the temperature in Escondido was 68 degrees. The temperature was dropping at a steady rate of 2 degrees per hour. After how many hours was the temperature colder than -4 degrees?
Answer:
72 hours
Step-by-step explanation:
Answer: 34 hours
Step-by-step explanation: 68 divided by 34 is 2 and thats the rate the temperature is dropping.
Find the surface area of the square pyramid shown below.
(15 points)
Answer:
340
Step-by-step explanation
Base=10x10=100
Sides= {(12x10)/2}x4=240
Add sides plus base, 240+100=340
1. A cooler is in the shape of a rectangular
prism. What is the volume of the cooler?
Round to the nearest tenth if necessary.
I
Answer:
what is the answerrrrrrrrrrrrrrrrr
Step-by-step explanation:
The volume of the cooler is 2772 in³
What is a volume?Volume is the amount of space taken up by an object
What is the volume of a rectangle?The formula for calculating the volume of a rectangular prism is volume = length × width × height.
According to the question
Length= 22in.
breadth= 12 in.
width= 10 1/2 in. or 21/2 in.
So, the volume= 22× 12× (21/2)
=2772 in³
Thus the volume of a rectangular prism is 2772 in³
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The measures of the angles of a triangle are in the extended ratio 7:14:15.
Find the measures of the angles of the triangle.
Please i need help
Can you post a picture?
It would make it so much easier.
In your answers below, for the variable λ type the word lambda; for the derivative d/dx X(x) type X' ; for the double derivative d^2/dx^2 X(x) type X''; etc. Separate variables in the following partial differential equation for u(x,t): t^2uxx+x^2uxt−x^2ut=0
The final answer of the question which is the specific form of the solution u(x, t) = X(x)T(t)
What is the quadratic equation?
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.
To separate variables in the partial differential equation t²uxx + x²ux²t - x²zzut = 0, we assume that the solution can be written as a product of two functions, u(x, t) = X(x)T(t).
Let's begin by substituting this into the given equation:
t²(X''(x)T(t)) + x²(X(x)T'(t)) - x^2(X(x)T(t)) = 0
Now, we divide the equation by t²X(x)T(t) to separate the variables:
(X''(x) / X(x)) + (x^2 / t^2)(T'(t) / T(t)) - (x^2 / t^2) = 0
Next, we rearrange the terms:
(X''(x) / X(x)) + ((x² / t²)(T'(t) / T(t))) - (x² / t²) = 0
To satisfy this equation for all x and t, the left-hand side and the right-hand side must be equal to a constant value. Let's call this constant -λ^2:
(X''(x) / X(x)) + ((x² / t²)(T'(t) / T(t))) - (x² / t²) = -λ²
Now, we have two separate ordinary differential equations:
1. X''(x) / X(x) - x² / t² = -λ²
2. T'(t) / T(t) = λ²
Solving each equation separately will give us the general solutions for X(x) and T(t).
The general solution for u(x, t) can then be obtained by taking the product of X(x) and T(t):
u(x, t) = X(x)T(t)
The specific form of the solutions will depend on the boundary conditions or initial conditions given for the problem.
Hence, the final answer of the question which is the specific form of the solution u(x, t) = X(x)T(t)
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50% of x is 19. *reamber is means divide*
Answer:
9.5
Step-by-step explanation:
can y'all help me pls
Answer:
The change in delta x would be 0 in the second option because the change is exactly the same.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided The top five test scores in Mr. Rhodes's class were 98, 92, 96, 97, and 97. What is the mean absolute deviation?
Step-by-step explanation:
mad = 98 + 92 + 96 + 97 + 97 / 5
= 480/5
= 96
mad = (98 -96) + (92 - 96) + (96 -96) + (97- 96) + (97 -96) / 5
= 2 + (-4) + 0 + 1 + 1 / 5
=0/5
mad = 0
brainliest is appreciated
Answer:
1.6
Step-by-step explanation:
To find the mean absolute deviation of the data, start by finding the mean of the data set. Find the sum of the data values, and divide the sum by the number of data values. Find the absolute value of the difference between each data value and the mean: |data value – mean|.
98+92+96+97+97=480
480/5 values=96 is the mean or average of the set of numbers
so we just find the absolute value of the data value minus the mean.
For the first one it would be |98-96|=2
then continue on and get a set of values
{2,4,0,1,1}
now find the average of that to get the mean absolute deviation
2+4+0+1+1=8
8/5 values
1.6
Hope that helps :)
"In a randomized controlled trial, insecticide-treated bednets were tested as a way to reduce malaria. Among 320 infants using bednets, 10 developed malaria. Among 279 infants not using bednets, 24 developed malaria. Use a 0.01 significance level to test the claim that the incidence of malaria is lower for infants using bednets. Do the bednets appear to be effective? Conduct the hypothesis test by using the results from the given display." solve for the test statistic, p, and what is the conclusion result?
To test the claim that the incidence of malaria is lower for infants using bednets, a hypothesis test is conducted using the given data. The test statistic is calculated and compared to the critical value at a significance level of 0.01.
The null hypothesis (H0) assumes that the incidence of malaria is the same for infants using bednets and those not using bednets. The alternative hypothesis (Ha) suggests that the incidence of malaria is lower for infants using bednets.
To conduct the hypothesis test, the test statistic can be calculated using the formula:
test statistic (p) = (p1 - p2) / sqrt[(p(1 - p) / n1) + (p(1 - p) / n2)]
Where:
p1 = proportion of infants using bednets who developed malaria
p2 = proportion of infants not using bednets who developed malaria
n1 = number of infants using bednets
n2 = number of infants not using bednets
Using the given data, the test statistic is computed as:
p = (10/320) - (24/279) / sqrt[((10+24)/(320+279)) * (1 - ((10+24)/(320+279))) * ((1/320) + (1/279))]
By comparing the calculated test statistic with the critical value at a significance level of 0.01, a conclusion can be made regarding the effectiveness of bednets in reducing malaria incidence.
If the test statistic falls within the critical region (beyond the critical value), the null hypothesis is rejected in favor of the alternative hypothesis, indicating that the bednets appear to be effective in reducing malaria incidence. Conversely, if the test statistic does not fall within the critical region, there is not enough evidence to reject the null hypothesis, suggesting that the bednets may not be effective in reducing malaria incidence.
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2.
A customer went to a garden shop and bought some potting soil for $11.50 and 9 shrubs. The total bill was $94.75. Write and solve an equation to find the price of each shrub.
A. 9p + $11.50 = $94.75; p = $11.75
B. 9p + 11.5p = $94.75; p = $4.62
C. 9(p + $11.50) = $94.75; p = $5.00
D. 9p + $11.50 = $94.75; p = $9.25
Answer:
D. 9p + $11.50 = $94.75; p = $9.25
Step-by-step explanation:
To answer this problem, let p be the price of each scrub. The total bill (t) is the sum of the price of the potting soil (s) and of the 9 scrubs (n). This is depicted in the equation,
$11.50 + 9p = $94.75
The value of p from the equation is 9.25. Thus, each scrub costs $9.25. The answer is letter D.
...................................................................................................................................................
Let
p---------> the price of each scrub
s----------> the price of the potting soil
T--------> total bill
we know that
in this problem
substitute the values in the formula above
solve for p
therefore
the answer is the option D
calculista
Genius
19K answers
437.3M people helped
help please i will give brainliest to first answer
Answer:
m<ADC= 117
Step-by-step explanation:
a little help pls this old i don't remember any of it
Answer:
Bowling ball
Step-by-step explanation:
A bowling ball has much more mass than all three of the other choices and the amount of kinetic energy is determined based on a object's mass and velocity, but in this case, all 4 balls are traveling at the same velocity. Therefore, we know that the bowling ball has much more mass so therefore it has the most kinetic energy.
hurry!!!! I need help
Estimate the answers of these:
1. 489+1210
2. 3694-2016
3. 5082×312
4. 761÷34
Write the equation of the circle.
Circle C that passes through (-5, 1) with center C(1,-4).
Answer:
answer is on this page
Step-by-step explanation:
https://www.mathpapa.com/algebra-calculator.html