some mathematics problems

[mathguy]

Hi, I have following questions, solutions of which have not been found, and hope to get some ideas or better the whole solution paths

1. which letter is not homomorphic to letter C? A,J,N,S,O,U,D.U.E

2. if mean of a<b<c<d is 100, what is the minimum value of a+d?

3. write formula to caculate the volume of intersection of x*x+y*y+z*z=4 and r=2*cos(theta)?

4.function of f is a linear function composed of two linear pieces and continuous on [0,2]. f(0)=f(2)=0 and f_max=1, what is the length of the graph of f ?

[lime]

1. Apparently, you did imply homeomorphism, not homomorphism. The letter "C" is not homeomorphic to letters "O", "E" and "A".
It is not homeomorphic to "O" since eliminating any point in "O" would remain it connected, while eliminating any "internal" point in "C" would make it disconnected.
It is not homeomorphic to "E" since eliminating one specific point in "E" would divide it into 3 connected subsets. There is no such point for letter "C".
It is not homeomorphic to "A" since eliminating two specific points in "A" would divide it into 4 conncected subsets. There is no such pair of points for letter "C".


2. There is no solution if the numbers a,b,c,d can be real since it is always possible to make them infinetely close. Therefore, I suppose the right formulation of the problem would imply that a,b,c,d are integers. Also this problem looks like the problem of the linear programming. Please formulate it correctly and we would try to solve it.

[mathguy]

First, Thank you for your interest in my topic and anwering to it. As you thought, in second question, the four symbols are indeed integers, but problem description seems only like this, what I appreciated was general thinking of it, based on some experience. Certainly, you can also add some condition if it is needed. For the fourth question, i have 4 answer choices,
a. (2*2^0.5, 1+5^0.5)
b. (2*2^0.5, 1+5^0.5]
c. [2*2^0.5, 1+5^0.5)
d. [2*2^0.5, 1+5^0.5]
I've no ideas how to work out the both bound values and to decide the type(open or close) of the interval.

[lime]

4. Thanks for answer choices. Now it is clear, that the question was about the possible range of length for described family of curves. Try to be precise next time describing the problem and avoid vague formulations.
Let's consider that "c" is the point that f(c)=1. Here is important to notice that 0<c<2.
The using the properties that f(0)=0 and f(2)=0 we can determine equations for both pieces of our curve. After that, we calculate the length of curve by integrating and looking for its extremums. Apparently, on the segment [0,2] there would be one maximum that appears when c=1 and minimum for c=0 and c=2. But as we mentioned before, c cannot be equal to those border values.
Therefore, the answer is C.

[mathguy]

thank you, lime, I have another problems, which I put in my created topic forward:
1. There are total 90 students, average passing students is 84 and average fail students is 60, What percentage of arithmetic mean of passing students?
A)12%
B)15%
C)40%
D)70%
E)75%
I didn't quit understand the question. Could you tell me how to post a sketch as you did?I am not familiar with the use of button provided under 'subject' title, e.g. I couldn't find proper http://image_url for my image.

[mathguy]

lime, one problem from #59 of practice book:
A cyclic group of order 15 has an element x such that the set {x^3,x^5,x^9} has exactly two elements. the number of elements in the set {x^(13n): n is a positive integer} is 3.
I saw the two elements x^3,x^5(right? because x^9=(x^3)^3), and then 13=5*2+3,--> one element doubled, the other single. I was confused about the elements questioned, distinct elements or repeated element even allowed