Question

Let
V denote the volume of a tetrahedron a, b, c the lengths of the
three sides of one of its faces, and assume that each edge of the
tetrahedron is equal in length to the opposite edge. Express V in
terms of a, b, and c

Answer #1

Let f : A → B, and let V ⊆ B.
(a) Prove that V ⊇ f(f−1(V )).
(b) Give an explicit example where the two sides are not
equal.
(c) Prove that if f is onto then the two sides must be
equal.

1. A six-sided box has a square base and a surface area of 54
m^2. Let V denote the volume of the box, and let x denote the
length of one of the sides of the base. Find a formula for V in
terms of x.
2. What is the maximum possible volume of the box in Problem 1?
Note that 0< x≤3√3.

Let A, B and C be three events. With set notation, denote the
following events (a) - (e).
(a) A happens and neither B or C happens.
(b) Both A and B happen but C does not.
(c) At least one of A, B, C happens.
(d) None of A, B, C happens.
(e) Out of events A, B and C, at most one event happens.
(f) In words, what is event (A ∪ C) c ∪ B?.
(g) In...

Hurry pls. Let W denote the set of English words. for u,v are
elements of W (u~v have the same first letter and same last letter
same length)
a) prove ~ is an equivalence relation
b)list all elements of the equivalence class[a]
c)list all elements of [ox]
d) list all elements of[are]
e) list all elements of [five]
find all three letter words x such that [x]has 5 elements

Consider Theorem 3.25:
Theorem 3.25. Let f : A → B, let S, T ⊆ A, and let V , W
⊆ B.
1. f(S ∪T) = f(S)∪f(T)
2. f(S ∩T) ⊆ f(S)∩f(T)
3. f-1(V ∪W) = f-1(V
)∪f−1(W)
4. f-1(V ∩W) = f-1(V
)∩f−1(W)
(a) Prove statement (2).
(b) Give an explicit example where the two sides are not
equal.
(c) Prove that if f is one-to-one then the two sides must be
equal.

Consider the following problem: A farmer with 650 ft of fencing
wants to enclose a rectangular area and then divide it into four
pens with fencing parallel to one side of the rectangle. What is
the largest possible total area of the four pens?
(a) Draw a diagram illustrating the general situation. Let x
denote the length of each of two sides and three dividers. Let y
denote the length of the other two sides.
(b) Write an expression for...

(a) Let (a, b, c) denote the result of throwing three dice of
colours, amber, blue and crimson, respectively., e.g., (1, 5, 3)
represents throwing amber dice =1, blue dice = 5, crimson dice = 3.
What is the probability of throwing these three dice such that the
(a, b, c) satisfy the equation b 2 − 4ac ≥ 0? [7 marks]
(b) From a survey to assess the attitude of students in their
study, 80% of them are highly...

Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose that T is a linear transformation from V to itself and
T(u) = u + v,
T(v) = u, T(w) =
v.
1. Find the matrix of T relative to the ordered basis B.
2. A typical element of V looks like
au + bv +
cw, where a, b and c
are scalars. Find T(au +
bv + cw). Now
that you know...

Let A be the sum of the last three digits, let B be the last
three digits, and let C be the last digit of your 8-digit student
ID. Example: for 20245347, A = 14, B = 347, and C=7.
The speed of a wave in a string is given by v = √(FT /μ),
where FT is the tension in the string and μ = mass / length of the
string.
A 2.00 m long string has a mass...

A cube of silver (density = 10.5 g/cm^3) has a mass of 90.0 g a
resistivity of 1.59x10^-8. The atomic number of silver is 47 and
its molar mass is 107.87 g/mol
a.) find the length of the cubne on each side.
b.)What is the resistance between opposite faces of the
cube?
c.) assume each silver atom contributes one conduction electron.
average drift speed of electrons when potential difference of 10^-5
V is applied to the opposite faces

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