Harvesting Fish. A fish farmer has 5000 catfish in a pond. The number of catfish increases by 8% per month and the farmer harvests 420 catfish per month.

a) (4 points) Find a recursive equation for the catfish population Pn for each month.

b) (4 points) Solve the recursive equation to find an explicit equation for the catfish population.

c) (3 points) How many catfish are in the pond after six months?

d) (3 points) Is harvesting 420 catfish per month a sustainable strategy for the fish farmer?

e) (4 points) What is the maximum number of catfish the fish farmer can harvest to sustain his business?

A mass of 8 kg stretches a spring 16 cm. The mass is acted on by an external force of 7sin⁡(t/4)N and moves in a medium that imparts a viscous force of 3 N when the speed of the mass is 6 cm/s.If the mass is set in motion from its equilibrium position with an initial velocity of 4 cm/s, determine the position u of the mass at any time t. Use 9.8 m/s^2 as the acceleration due to gravity. Pay close attention to the units.

Show that |N| = |Z|, where N is the set of natural numbers and Z is the set of all integers.

A house can be purchased for

​$155 comma 000155,000​,

and you have

​$25 comma 00025,000

cash for a down payment. You are considering the following two financing​ options:

bullet•

Option​ 1. Getting a new standard mortgage with a

7.57.5​%

​(APR) interest and a

3030​-year

term.

bullet•

Option​ 2. Assuming the​ seller's old​ mortgage, which has an interest rate of

5.55.5​%

​(APR), a remaining term of

2525

years​ (the original term was

3030

​years), a remaining balance of

​$97 comma 21797,217​,

and payments of

​$597597

per month. You can obtain a second mortgage for the remaining balance

​($32 comma 78332,783​)

from your credit union at

99​%

​(APR) with a

1010​-year

repayment period.

​(a) What is the effective interest rate of the combined​ mortgage?

The effective interest rate of the combined mortgage is

nothing​%.

1.Prove that{2k+1:k∈N}∩{2k2 :k∈N}=∅.

2.Give two examples of ordered sets where the meaning of ” ≤ ” is not the same as the one used with the set of real numbers R.

True or False. If true, quote a relevant theorem or reason, or give a proof. If false, give a counterexample or other justification.

The set of irrationals in the interval (0, 1) is not countable. (Assume the fact that the set of points in the interval (0, 1) is uncountable.)

Find a path that traces the circle in the plane y=0 with radius r=2 and center (2,0,−2) with constant speed 12.

r1(s)=〈 , , 〉

Find a vector function that represents the curve of intersection of the paraboloid z=7x^2+5y^2 and the cylinder y=6x^2. Use the variable t for the parameter.

r(t)=〈t, , 〉

Need help with Calculus III

Find thesidue of :

1- (Z²) / (Z²+i)³(Z²-i)²

^2-(Z³-2Z⁴) / (Z+4)²(Z²⁺⁴)²

An oil company operates two refineries in a certain city. Refinery I has an output of 200, 100, and 100 barrels of low-, medium-, and high-grade oil per day, respectively. Refinery II has an output of 100, 200, and 600 barrels of low-, medium-, and high-grade oil per day, respectively. The company wishes to produce at least 1000, 1400, and 3000 barrels of low-, medium-, and high-grade oil to fill an order. If it costs $400/day to operate Refinery I and $600/day to operate Refinery II, determine how many days each refinery should be operated to meet the production requirements at minimum cost to the company.

What is the minimum cost?
$

2. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. Develop the linear programming problem and solve the linear programming model to maximize the number of customers served while generating at most $1800 in revenue and honoring all reservations. (Hint: You need 2 decision variables to develop this LP problem.)

Give a vector parametric equation for the line that passes through the point (2,0,5), parallel to the line parametrized by 〈t−3,t−3,−3−3t〉:

Find the simplest vector parametric expression r⃗ (t) for the line that passes through the points P=(0,0,−1) at time t = 1 and Q=(0,−4,−1) at time t = 5

Need help with Calculus III

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if    T is One-to-one then T^-1 exist on R(T) and T^-1 : R(T) à X is also a linear map.

2. Let X, Y and Z be linear spaces over the scalar field F, and let T₁ ϵ B (X, Y) and T₂ ϵ B (Y, Z). let T₁T₂(x) = T₂(T₁x) ∀ x ϵ X.

(i) Prove that T₁T₂ ϵ B (X,Y) is also a bounded linear mapping.

(ii) Prove that ǀǀT₂T₁ǀǀ ≤ ǀǀT₂ǀǀ ǀǀT₁ǀǀ

3. If X is an inner product space, then for arbitrary x, y ϵ X, ǀ< x, y>ǀ ≤ ǀǀxǀǀ ǀǀyǀǀ, prove that the inner product < ∙ > is a continuous function on X by X (Cartesian product) domain.

A construction firm is considering buying a backhoe, since it pays $50/h to rent one, and it needs to use a backhoe for 150 hours per month. Assuming a backhoe lasts 5 years, and has monthly maintenance of $2,000 and a salvage value of $15,000, what’s the most the company should pay for this backhoe? (Use a 6% discount rate)

Bob​ Carlton's golf camp estimates the following workforce requirements for its services over the next two​ years: Quarter 1 2 3 4 5 6 7 8 Demand​ (hrs) 4,200 6,400 3,100 5,000 4,400 6,240 3,800 5,000

Each certified instructor puts in 480 hours per quarter regular time and can work an additional 120 hours overtime.​ Regular-time wages and benefits cost Carlton $7,200 per employee per quarter for regular time worked up to 480 hours, with an overtime cost of $20 per hour.

Unused regular time for certified instructors is paid at $15 per hour

. There is no cost for unused overtime capacity. The cost of​ hiring, training, and certifying a new employee is $10,000. Layoff costs are $4,000 per employee.

Currently 8 employees work in this capacity.

(a) Find a workforce plan using the level strategy that allows for no delay in service. It should rely only on overtime and the minimum amount of undertime necessary. What is the total cost of the​ plan? 701000

(b) Use a chase strategy that varies the workforce level with minimal undertime and without using overtime. What is the total cost of this plan? 809600

(c) Propose a low-cost, mixed strategy and calculate its total cost. (Any strategy that improves on both the chase and level strategies is acceptable; no need to find the optimal schedule.)