A farmer has a mixing tank of capacity 1200 liters which she half-filled with pure, fresh water. She pumps into the tank a concentrated liquid fertilizer (CLF) at rate 3 liters/minute, containing 1/3 ∼ 0.333 kg/liter of nitrate (a salt of nitric acid). In addition, she pours the dry powder fertilizer (DPF, the same chemical as a soluble powder) at rate 1/12 = 0.08333 kg/min in the tank; her aim is to get a right solution concentration 0.12kg/liter for the type of soil she has in her field. Unfortunately, the tank is leaking: when the farmer checks it after 90 min (that is, 1.5 hr) she finds the tank containing only 330 liters of solution. Assume the leak is at the bottom of the tank. Then:

(a) Determine the leakage w in liters/min.

(b) In what time would the tank become empty? Let N(t) be the amount of nitrate in the tank at time t.

(c) Write the InValProblem for N(t) and

(d) solve it, for t in min, N(t) in kg.

(e) Find the amount N(t) and the concentration c(t) of nitrate in the tank at time t = tcheck when the farmer checks the tank. Find out:

(f) how much of the CLF in liters and

(g) how much of the DPF in kg has been pumped/poured in the tank by the time t = tcheck she checks the tank. In absence of leakage: determine

(h) the time t = t ∗ when c(t) would be at the right l evel 0.12 and

(i) the volume V of the liquid solution in the tank at t = t ∗ .

The United States Postal Service (USPS) uses 11-digit serial numbers on its money orders. The first ten digits identify the document, and the last digit is the check digit. The check digit is obtained by a11 = a1 + · · · + a10 (mod 9). For example, one of the money order has serial number 16094004377. (a) The first ten digits of the serial number on a USPS money order are 7306125986. Find the last digit (the check digit). (b) Will this scheme detect all single-digit errors? Prove your statement. (c) Will this scheme detect all transposition error? Why?

Explain the difference between the actual definition of a Riemann Integral of function f on the interval [a,b] and the conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part 2)

use euclidean algorithm to find integers m,n such that 1693m+2019n=1

Suppose there are two lakes located on a stream. Clean water flows into the first
lake, then the water from the first lake flows into the second lake, and then water from the second
lake flows further downstream. The in and out flow from each lake is 500 liters per hour. The first
lake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water.
A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being
continually mixed perfectly by the stream.

a) Find the concentration of toxic substance as a function of time in both lakes.
b) When will the concentration in the first lake be below 0.001 kg per liter?
c) When will the concentration in the second lake be maximal?

A girl who has two siblings is chosen at random and the number X of her sisters is counted. Describe how to simulate an observation on X based on U ∼ unif[0, 1] in Matlab.

Explain the difference between the actual definition of a Riemann Integral of function f on the interval [a,b] and the conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part 2)

This problem is an example of critically damped harmonic motion. A mass m=6kg is attached to both a spring with spring constant k=96N/m and a dash-pot with damping constant c=48N⋅s/m . The ball is started in motion with initial position x0=5m and initial velocity v0=−24m/s . Determine the position function x(t) in meters. x(t)= Graph the function x(t) . Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c=0 ). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t)=C0cos(ω0t−α0). Determine C0, ω0 and α0. C0= ω0= α0= (assume 0≤α0<2π ) Finally, graph both function x(t) and u(t) in the same window to illustrate the effect of damping.

For each of the following problems show the fully augmented problem and simplex table solution, Also, show which extreme points are feasible and identify the optimal solution.

a) Maximize 12?$ + 18?' subject to 6?$ + 5?' ≤ 60

?$+3?' ≤15 ?$ ≤9

?' ≤4 ?$,?' ≥0

b)

Minimize 3.5?$ − 2.5?' s.t. ?$ − 0.5?' ≥ 2

10?$ + 3?' ≤ 30 0.5?$ + ?' ≥ 5

?$,?' ≥ 0

Check whether the following families of functions of t are linearly independent or not

(a) t^2 + 1, 2t, 4(t + 1)^2

(b) sin(t) cos(t), sin(2t) + cos(2t), cos(2t)

(c) e^2t , e^-2t , 2e^t

(d) 2e^t , 3 cosh(t), 13 sinh(t)

(e) 1/((t^2)-1) , 1/(t + 1), 1/(t-1)

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector.

a.) suppose Ax=0. Show that A^TAx=0.

b.)Suppose A^TAx=0. show Ax=0.

c.) by part a and b, we can conclude that Nul(A) = Nul(A^TA), and thus dim(Nul A) = dim(Nul(A^TA)), and thus nullity(A) = nullity(A^TA). prove the columns of A are linearly independent iff A^TA is invertible.

The following matrix is the augmented matrix for a system of linear equations. A =

(a) Write down the linear system of equations whose augmented matrix is A.

(b) Find the reduced echelon form of A.

(c) In the reduced echelon form of A, mark the pivot positions.

(d) Does the system have no solutions, exactly one solution or infinitely many solutions? Justify your answer

Show that the sequence an = (−1)^n doesn’t converge to 1 nor −1. Can it converge to anything other than 1 and −1?

we are given n chips which may be working or defective. A working chip behaves as follows: if we connect it to another chip, the original chip will correctly output whether the new connected chip is working or is defective. However, if we connect a defective chip to another chip, it may output any arbitrary answer (defective---->might say the other one is working /defective).

In the class, we saw that if strictly more than half the chips are working, then there is an algorithm that finds a working chip using O(n) tests.

1) Prove that even when we only have a single working chip and a single defective chip (i.e.,n = 2), there is no algorithm that can find the working chip in general.