Suppose you are deciding whether to buy an electric or a natural gas hot water heater. Assume you live in California.

Note: 1 therm of natural gas equals ~30 kWh of electricity

Natural Gas Price in CA = $13.35 per Thousand Cubic Feet

Electric price in CA = .16 cents per kilowatt-hour (kWh)

Gas Water Heater Uses 258 therms per year

Water heater numbers are based on 40-gallon models with a 0.63 EF rating using 40,000 Btu/hr and based on the U.S. Department of Energy’s (DOE) national average of 64 gallons of hot water use daily.

Electric water heater Uses 4,881.25 kWh per year

An average water heater runs three hours daily. A 50-gallon, 5,500-watt water heater with a .90 EF and an electricity rate of $.16 per kilowatt hour.

For each sub question, show your assumptions and calculations.

1a) Which would you pick to minimize your operating costs?

1b) If the electric water heater cost $1250 and the natural gas water heater cost $1600, how much would this change your calculation for 1a?

1c) Which would you pick to minimize your total energy footprint, and why?

1d) Which would you pick to minimize your environmental footprint, and why?

Keep the Old Car or Buy a Used Car
Manny is an online student who currently owns an older car that is fully paid for. He drives, on average, 160 miles per week to commute to work. With gas prices currently at $2.02 per gallon, he is considering buying a used, fuel-efficient car, and wants to know if it would be a good financial decision.

The old car Manny owns currently gets 15 miles per gallon for average fuel efficiency. It has been a great vehicle, but with its age, it needs repairs and maintenance that average $800 per year (as long as nothing serious goes wrong).

He is considering buying a newer used car that will cost a total of $7,500 over a three-year loan process. The used car gets 27 miles per gallon and would only require an average of $10 per month for general maintenance. To help make a decision Manny wants to calculate the total costs for each scenario over three years. He decides to use the Quantitative Reasoning Process to do this.

Find the total costs (gas, maintenance/repairs, purchase price) for each scenario over the three years.

Round your answers to the nearest dollar.

True/False: If A is an invertible m x m matrix and B is an m x n matrix, then AB and B have the same null space.

(Hint: AB and B have the same null space means that ABX = 0 if and only if BX = 0, where X is in Rⁿ.)

Course: Differential Geometry (Vector Calculus & Linear Algebra)

Provide all proofs

(a) Find the formula for the distance from p to the line y=mx

(b) prove that the set U={(x,y): y<mx} is an open set

Prove the following theorem:

In a Pasch geometry, a quadrilateral is a convex quadrilateral if and only if the vertex of each angle is contained in the interior of the opposite angle.

Problem 2-25 (Algorithmic)

George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 8% for the bond fund and 20% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 40% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 5.5%.

1. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. If required, round your answers to three decimal places.
   

   Let B	=	percentage of funds invested in the bond fund
   S	=	percentage of funds invested in the stock fund

   	B	+	S
   s.t.
   	B					Bond fund minimum
   	B	+	S			Minimum return
   	B	+	S			Percentage requirement

2. Solve the problem. If required, round the answers to one decimal place.
   
   Optimal solution: B = , S =
   
   Value of optimal solution is  % ????

A rectangular box with no top is to have a surface area of

64 m².  Find the dimensions (in m) that maximize its volume.

I got width as X= 8sqrt3/3, length y=8sqrt3/3, and height as z=4sqr3/3 but it is wrong and I don't know why

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

Compute φ(21). Find all the units modulo 21 and write down their inverses.

Find all RING homomorphisms from Z18 to Z18

we are going to focus on basic logic and how you can use logic outside of the classroom. Unfortunately,

many of our daily interactions include logical errors.

Respond to the following prompts in a minimum of 175 words:

• Consider a recent interaction you have had with a co-worker or item that you saw in the media that might have included logical errors.
• Share the example as well as the logical errors that are present in the example. Be sure to use terms and concepts from the content covered this week to support your discussion.

Solve the differential equation using the Laplace transform.   y''' + 3y''+2y' = 100e^-t , y(0) = 0, y'(0) = 0, y''(0) = 0

What is the number of permutations of the six digits 123456 in which the strings 123, 234, 345, 456, 654, 543, 432, 321 don't appear?