Dr Pi

My question in this: Every time I take a shower my eyes hurt the entire day until I sleep for a long time! It's not dry eyes, it doesn't turn red, this happens even when I have no soap around or keep my eyes very tightly shut (water or steam still manages to make them sting). It's made me absolutely hate showers and sometimes I can only take 1 shower a weak or less if I want to get work done, etc.. What may it be? My roommates do not get it. I've tried Goggles but the goggles work very poorly at keeping water from leaking through (or steam). It's not very painful but it's painful enough so that it becomes very hard to want to keep my eyes open at all. And I don't believe it's the chlorine, since otherwise swimming in swimming pools would have the same effect on my eyes, which it doesn't

Under some circumstances when two parallel springs, with constants k1 and k2, support a single mass, the effective spring constant of the system is given by

k = 4k1k2/(k1 + k2).

A mass weighing 20 pounds stretches one spring 6 inches and another spring 2 inches. The springs are attached to a common rigid support and then to a metal plate. As shown in the figure, the mass is attached to the center of the plate in the double-spring arrangement.

q1: Determine the effective spring constant of this system.

q2: Find the equation of motion x(t) if the mass is initially released from the equilibrium position with a downward velocity of 3 ft/s. (Use g = 32 ft/s^2 for the acceleration due to gravity.)

attempt:

Δx1 = 6 in. = 1/2 ft.

Δx2 = 2 in. = 1/6 ft.

F = 20 Ibs

g = 32 ft/s^2

m = F/g = 5/8 slugs

k1 = F/Δx1 = 40

k2 = F/Δx2 = 120

k = 120

x''(t) + k/m*x = 0

x''(t) + 192*x = 0

x'(0) = 3

x(0) = 0

x(t) = 1/8*sqrt(3)*sin(8*sqrt(3)*t)

edit: I got the correct answer (first time my units were incorrect) while typing this out but I decided to post the question anyways for future reference for others. Admins can decide what to do with the question. First person to give a response (maybe physics explanation as to why the sprint constant k = 4k1k2/(k1+k2) can get 10 pts.) that is useful can get 10 pts.

The general formula for a spheroid is the following:

(x^2 + y^2)/a^2 + z^2/c^2 = 1

where this spheroid has semi-axis at x,y = a and z = c.

How would I compute the surface area of this?

Suppose n people are queued, where n is a positive integer less than 366. Each person is asked in turn to write down their birthday (month and day). We assume no one in the queue was born in a leap year, and that every possible birthday is equally likely.

(e) Suppose n = 30. What is the probability that two of the 30 people in the queue have the same birthday? (This counter intuitive result is called the Birthday Paradox).

(f) Suppose n = 30 and the first person has already recorded their birthday. What is the probability that one of the 29 people still in the queue has the same birthday as the first? Why do your answers to parts e and f compare the way they do?

Attempt:

For part (e) I tried 1 - (1-1/365)(1-2/365)*...*(1-30/365) = 0.706

For part (f) I have no clue....

f(z) is a complex valued function.

The picture size on ordinary 35-mm camera film is 24mm x 36mm. Focal lengths of lenses available for 35-mm cameras typically include 28, 35, 50 (the "normal" lens), 85, 100, 135, 200, and 300 mm, among others. Which of these lenses should be used to photograph the following objects, assuming that the object is to fill most of the picture area? (a) a building 240m tall and 160m wide at a distance of 600m

Let's say I wanted to Calculate 454 Mod 8 is a calculator. Then I can do remainder(454,8) in the calculator and it'd output 6. Now let's say I want to do -454 Mod 8 in the Calculator and I try remainder(-454,8). I get a domain error. So my question is, how can I find -454 Mod 8 with a single function in my Ti-84 calculator?

If there is no single function that does this, is there a Ti-84 program that I can create myself (if so what's the code) or find online and download?

You can now sell 40 cars per month at $20,000 per car, and demand is increasing at a rate of 2 cars per month each month. What is the fastest you could drop your price before your monthly revenue starts to drop? HINT [Revenue = Price × Quantity.]

Given a perfectly competitive market structure, a firm's total fixed costs are $195, average variable costs are $3, marginal revenue is $5, and the quantity demanded is 60. Assuming the firm is a profit maximizer, what are the firm's total profits?

So far I have:

Profit = R - C

Max. Profit ==> MR = MC ==> MC = MR = $5

AVC = $3

TVC = AVC * Q = $180

TC = TFC + TVC = $375

Q = 60

Assume that one laborer produces 6 units of output, two laborers product 14 units, three produce 20 units, and four product 24 units. If the cost is $20 per laborer and fixed costs are $100, the average total cost at 20 units of output is:

a: $60

b: $3.33

c: $8.00

d: $100

e: 160

My attempt:

Total cost = 20x + 100 where x = # of Laborers

Average cost = Total cost/x = 20 + 100/x

At 20 units of output, we have 3 laborers working, thus Average cost = 20 + 100/3 = $53.33 which is none of the answer choices but is close to $60.

∫∫F•n̂ dS where F = <xz,0,-y> and S is the part of the sphere x²+y²+z² = 4 lying above the plane z = 1 (n̂ points radially outward).

Prove the following: (xy)^(1/2) = (x)^(1/2) * (y)^(1/2) <==> x,y > 0

I believe this is true but proving it looking quite difficult.

I know φ(m)φ(n) = (m-1)(n-1).

So all I would have to show is that φ(mn) = (m-1)(n-1) as well. So how?

GCD(a,m) = 1 and GCD(x,m) = 1

==>

GCD(ax,m) = 1

f(x) = 6(4-x)^(-1/3) on interval [0,4)

Finding the volume of f(x) revolved about the y-axis.

Answer should end up being:

(432/5)*Pi*(2)^(1/3)

If you know that a Hill Cipher was used with a 3 x 3 key, how would you find the key given:

"friday" encrypted is RLWWVA.

(with 3 x 3 case I'm getting 6 equations & 9 unknowns?)

How about for a 2 x 2 case?

How are the solutions to the sing congruence ax = b mod(mn) related to the system of congruences below? Does GCD(m,n) play a role? Prove your answer.

ax = b (mod m)

ax = b (mod n)

An example where this is applied:

x = 5 (mod 21)

x = 8 (mod 15)

x = 12 (mod 14)

Cannot directly use Chinese Remainder Theorem but after some modification one can.

Using the above property we can write this system as the following new system:

x = 2 (mod 3)

x = 5 (mod 7)

x = 3 (mod 5)

x = 0 (mod 2)

Which can now be solved using the Chinese Remainder Theorem.

Create a Mathematicar function that converts a sequence of numbers from the set {0,1, . . . ,25}

to a character string consisting of the letters represented by those numbers.