A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a s

Oberhangaps5z 2022-05-09 Answered
A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a sector with vertex angle theta and welding the straight edges of the straight edges of the remaining piece together. Find theta so that the resulting cone has the largest possible volume.

Specifically, the question is asked in the context of wanting derivatives, multiple max/min equations, and hopefully more calc rather than trig or geo.

I have gotten as far as using 10m as the hypotenuse for a triangle formed by the height of the cone, radius of the base of the cone, and slant. I'm not sure where to go from there, because I can't determine how to find height and/or radius, without which I'm not sure I can continue.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Haylie Cherry
Answered 2022-05-10 Author has 14 answers
The requested volume will be (why?):
V ( θ ) = π R 2 h 3 = π ( 2 π θ 2 π r ) 2 r 2 ( 2 π θ 2 π r ) 2
In here, the given variable is r = 10, and the unknown variable θ must be found, by taking the derivative d V ( θ ) d θ .

You can change the variable and analyze better for the variable R = 2 π θ 2 π r, which will be much easier:
V ( R ) = π R 2 r 2 R 2
You will find the optimal is given for R = 2 3 r, or θ = 2 π ( 1 2 3 ) = 66.1 o
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2022-05-14
I have a non-linear maximization problem and I want to convert it to be a minimization problem, can I do so by multiplying it by a negative sign, or is that wrong; and if that is wrong what should I do?
asked 2022-05-07
Statement:

A manufacturer has been selling 1000 television sets a week at $ 480 each. A market survey indicates that for each $ 11 rebate offered to a buyer, the number of sets sold will increase by 110 per week.

Questions :

a) Find the function representing the revenue R(x), where x is the number of $ 11 rebates offered.

For this, I got ( 110 x + 1000 ) ( 480 11 x ). Which is marked correct

b) How large rebate should the company offer to a buyer, in order to maximize its revenue?

For this I got 17.27. Which was incorrect. I then tried 840999, the sum total revenue at optimized levels

c) If it costs the manufacturer $ 160 for each television set sold and there is a fixed cost of $ 80000, how should the manufacturer set the size of the rebate to maximize its profit?

For this, I received an answer of 10, which was incorrect.
asked 2022-05-08
The problem is like
m a x x u ( x 1 , x 2 , . . . , x L ) = i = 1 L x i a i ,
s . t . i L x i C
for each i, a i > 0 is a scalar;
C is a constant that is strictly greater than 0;
x = ( x 1 , x 2 , . . . , x L ) R + L . Characterize the optimal x as a function of C or a i .
Hint: to solve the problem we should discuss the cases when C i a i and C i a i .
Thank you!
asked 2022-05-10
We have a function:
f ( x , y , z ) = x cos ( ω t + y + ϕ 1 ) + z cos ( ω t + y + ϕ 2 ) .
I want to solve the following maximization problems:
max x , y , z max t f ( x , y , z ) or max x , y , z < T > f ( x , y , z ) d t .
Surely, x and z are nonnegative, and y is in [ 0 , 2 π ).

Can someone give me hints for solving the problem, or let me know some references to solve this types of problem?
asked 2022-05-07
Some have proposed that for a natural centrality measure, the most central a node can get is the center node in the star network. I've heard this called "star maximization." That is, for a measure M ( ), and a star network g with center c ,
{ ( c , g ) } arg max ( i , g ) N × G ( N ) M ( i , g )
where N is the set of nodes and G considers all unweighted network structures.

I'd like to learn about some centrality measures that don't satisfy this property, but "star maximization" isn't a heavily used term, so I am having trouble in searching for many such measures. What are some such measures of centrality?
asked 2022-04-07
I am trying to solve the following question:
Maximize  f ( x 1 , x 2 , , x n ) = 2 i = 1 n x i t A x i + i = 1 n 1 j > i n ( x i t A x j + x j t A x i )
subject to
x i t x j = { 1   i = j 1 n   i j
where xi's are column vectors ( m × 1 matrices) with real entries and A is an m × m ( ( n < m )) real symmetric matrix.

From some source, I know the answer as
f max = n + 1 n i = 1 n λ i ,
λ i being the eigenvalues of A sorted in non-increasing order (counting multiplicity). But I am unable to prove it. I will appreciate any help (preferably with established matrix inequality, or Lagrange's multiplier method).
asked 2022-04-06
Let w , a R n , and B R + + n × n (the set of n × n positive definite matrices).
We know that the following function (which is a specific form of the Rayleigh quotient) has a unique maximum, and a closed-form solution in w:
f ( w ) = w T a a T w w T B w
It's maximum is achieved at w = B 1 a and its value is f ( w ) = a T B 1 a. (by using the generalized eigenvalue decomposition)

Now here is my question:
If I have w , a 1 , a 2 R n , and B 1 , B 2 R + + n × n , and the following function:
g ( w ) = w T a 1 a 1 T w w T B 1 w + w T a 2 a 2 T w w T B 2 w
Then, what can we say about the maximum of g ( w ), can we still solve for w in closed-form?