A tank contains 60 kg of salt and 1000 L of water. Pure water enters a tank at the rate 12 L/min. The solution is mixed and drains from the tank at the rate 6 L/min.

(a) What is the amount of salt in the tank initially?
amount =  (kg)

(b) Find the amount of salt in the tank after 3 hours.
amount =   (kg)

(c) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.)
concentration =   (kg/L)

Let R be a commutative ring with identity with the property that every ideal in R is principal. Prove that every homomorphic image of R has the same property.

An unknown radioactive element decays into non-radioactive substances. In 340 days the radioactivity of a sample decreases by 37 percent. (a) What is the half-life of the element? (b) How long will it take for a sample of 100 mg to decay to 74 mg? time needed: (days)

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 205 degrees Fahrenheit when freshly poured, and 1 minutes later has cooled to 190 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 150 degrees.
The coffee will reach a temperature of 150 degrees in how many minutes?

A round-robin tournament involving n plays is modeled with digraph D where, for every two distinct vertices (players) u and v, either (u,v) is an edge (player u defeats player v) or (v,u) is an edge (player v defeats play u). Prove that if D is acyclic, i.e., no directed cycles, then there always exists a player who has defeated everyone (out-degree is n – 1) and a player who has lost to everyone (in-degree is n – 1).

Use the dual simplex method to solve the following linear programming problems. Clearly indicate all the steps, the entering and departing rows and columns and rows, the pivot and the row operations used. Use the simplex method to solve the following linear programming problems. Clearly indicate all the steps, the entering and departing rows and columns and rows, the pivot and the row operations used. 2.2.1 An electronics manufacturing company has three production plants, each of which produces three different models of a particular MP3 player. The daily capacities (in thousands of units) of the three plants are shown in the table. Basic model Gold model Platinum model Plant 1 8 4 8 Plant 2 6 6 3 Plant 3 12 4 8 The total demands are 300,000 units of the Basic model, 172,000 units of the Gold model, and 249,500 units of the Platinum model. The daily operating costs are $55,000 for plant 1, $60,000 for plant 2, and $60,000 for plant 3. How many days should each plant be operated in order to fill the total demand while keeping the operating cost at a minimum? What is the minimum cost? Use the method of the Dual.

Jack has two children. What is the probability that both are boys. In addition, what is the probability the oldest is a boy, at least one is a boy, at least oen boy is born on a monday.

Thanks for the help!

5. For each piecewise linear function, graph the function and find:      2 lim x f x      2 lim x f x (a) { ? + 3 ?? ? < 2 2? + 1 ?? ? ≥ 2 (b) ?(?) = { ? − 2 ?? ? < 2 ? + 1 ?? ? ≥ 2 6. For each piecewise linear function, graph the function and find:    x  0 lim f x    x  0 lim f x   x 0 lim f x   (a) ?(?) = −|?| + 4 (b) ?(?) = − 2|?| �

Problem 1 [20 pts]: An experiment consists of tossing a coin 6 times. Let X be the random variable that is the number of heads in the outcome. Find the mean and variance of X.

Thank You

Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine.

Factor the following polynomials as a product of irreducible in the given polynomial ring.

A. p(x)=2x^2+3x-2 in Q[x]

B. p(x)=x^4-9 in Q[x]

C. p(x)=x^4-9in R[x]

D. p(x)=x^4-9in C[x]

E. f(x)=x^2+x+1 in R[x]

F. f(x)=x^2+x+1in C[x]

Let A =

(a) Calculate the matrix exponential e^(At). (Hint: It might help to write down the power series expansions for the hyperbolic functions
cosh(t) =(e^t + e^(−t))/2
and sinh(t) =(e^t −e^(−t))/2
and then try to write eAt in terms of these two functions.)
(b) Use your matrix from part (a) to solve the nonhomogeneous initial value problem
x' =

x +

, x(0) =

. (Hint: You might need the identity cosh^2(t)−sinh^2(t) = 1.)

If a set K that is a subset of the real numbers is closed and bounded, then it is compact.

Show that (0, 1) and (−1, 1) have the same cardinality.

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential?

function, please show E is a vector space by checking all the 10 axioms. (Notice: you may use the properties of vector addition and scalar multiplication in R2)