A liquid culture of infectious bacteria was accidentally dumped into a river 900 m upstream of the intake pipe for the municipal water plant; 10^(11) organisms are introduced into the river which has a cross-section of 0.6 m^2 .

(a) If the river velocity is 40 cm/s and the longitudinal dispersion coefficient is 0.1 m^2 /s, what is the maximum concentration expected at the plant?

(b) How spread out (in the direction of flow) is the cloud of organisms at the time the peak concentration occurs at the intake pipe? The “spread” can be described by the standard deviation (σ) on either side of the peak.

(c) How does the answer to (a) change if the organisms have a half-life of 10 hours in the river?

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal basis {v1,v2.....vk-1}. Work out a orthonormal basis of W in terms of that of V and the orthogonal complement of V in W.

Provide detailed reasoning.

Show that the structure (Zn*, ⊗) is a group (It is important to identify the number theoretic background required for this problem.)

Let f₁ = 1 and f₂=1 and for all n>2 Let f_n = f_n-1+f_n-2. Prove that for all n, there is no prime p that divides noth f_n and f_n+1

solve the following LP. Formulate and algebraically solve the problem.

what is the new optimal z value

show that the current basis is optimal

max z=65x1+25x2+20x3

8x1+6x2+x3<=48

4x1+2x2+1.5x3<=20

2x1+1.5x2+0.5x3<=8

x2<=5

x1,x2,x3>=0

Differential Equations

Solve for the IVP

ty³ dt - ( t⁴ + y⁴ ) dy = 0

y (1) = 2

Recall that the absolute value |x| for a real number x is defined as the function

|x| = x if x ≥ 0 or −x if x < 0.
Prove that, for any real numbers x and y, we have
(a) | − x| = |x|.
(b) |xy| = |x||y|.
(c) |x^-1| = 1/|x|. Here we assume that x is not equal to 0.
(d) −|x| ≤ x ≤ |x|.

1. Write a function that will receive two input arguments that is the height in inches (in.) and the weight in pounds (lb) of a person and return two output arguments that is the height in meters (m) and mass in kilograms (kg).

1. Determine in SI units the height and mass of a 5 ft.15 in. person who weight 180 lb.
2. Determine your own height and weight in SI units.

Note: 1 meter = 39.3701 inch, 1 foot = 12 inches, 1 pound = 0.453592 kilogram

Solve this please using MATLAP

QUESTION:

USING MATLAB, Carry out three iterations of the Gauss-Seidel method, starting from the initial vector Use the  ,  and  norm to calculate the residual error after each iteration, until all errors are below 0.0001.

[Use 5 decimal place accuracy in you calculations]

The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs $210. The Bradleys’ have a budget of $52,500 for next year. The government limits the number of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the profit from each acre of tobacco is $520. The Bradleys’ want to know how many acres of each crop to plant in order to maximize their profit.

a. Formulate the linear programming model for the problem and solve.

b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The neighbor is offering the land to them for $110 per acre. Should the Bradleys’ lease the land at that price? What is the maximum price the Bradleys’ should pay their neighbor for the land, and how much land should they lease at that price?

d. The Bradleys’ are considering taking out a loan to increase their budget. For each dollar they borrow, how much additional profit would they make? If they borrowed an additional $1,000, would the number of acres of corn and tobacco they plant change?

1. Let V and W be vector spaces over R.

a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation.

b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) = λ(R(v)) is also a linear transformation.

c) Let E(V) be the set of all linear operators T: V → V. Check that E(V) is a vector space with the addition and scalar multiplication defined above.

d) Suppose dim V = n. What is dim(E(V))? Justify your answer.

On the day of their son’s birth, Mr. and Mrs. Su decided to set aside a sum of

money to provide for his college education. They wish to make a single deposit in a bank that pays 9% compounded annually in order to provide a payment of $12,999 on each of the son’s 18 th, 19th, 20th, 21st birthdays. How much should they deposit?

please show how to slove step by step with formula not excel

This is Discrete Math

Write a small paragraph about Russell-Zermelo? (3 to 5) lines

What is Russell-Zermelo paradox? and what is it used for?  (least 3 lines)

What is the proof of Russell-Zermelo paradox?

Give at least two examples of Russell-Zermelo paradox from life.

Let V -^Φ -> W be linear. Show that ker (Φ) is a subspace of V and Φ (V) is a subspace of W.