Boost Your Math Modeling Skills

How many seconds are there in one day?
A)86400 seconds
B)36000 seconds
C)86000 seconds
D)90000 seconds

Modeling phenomena using random variables
A load balancer in a cloud computing system is composed of N servers. The balancer, When assigning the connections to the servers, decide at random which one to use. But nevertheless, Due to some prioritization of the servers, 10% of them have a higher utilization level than the rest. Model the problem of access to the servers of the cloud computing system, for that purpose define clearly the sample space ($\mathrm{\Omega <!--\; \Omega \; -->}$), the observations in the sample space ($\mathrm{\omega <!--\; \omega \; -->}$) and the probability function of the events of the problem. Also define an impossible event. Finally, define the random variables X and Y, which respectively represent the events of accessing a system server and accessing any server of the most used. Which of these random variables gives more information about the problem?

What are the equations modelling a vertical spring system with two masses?
Modeling a vertical spring system with one mass is a pretty common problem. I looked around online and found some horizontal spring systems with two masses, but no examples of a vertical one.
I'm curious, how would you set up equations modeling a vertical spring system like this:
$$\begin{gathered} -<!--\; -\; -->-<!--\; -\; -->-<!--\; -\; -->-<!--\; -\; --> \\ \wedge <!--\; \wedge \; --> \\ \vee <!--\; \vee \; --> \\ \wedge <!--\; \wedge \; --> \\ \vee <!--\; \vee \; --> \\ (m_1) \\ \wedge <!--\; \wedge \; --> \\ \vee <!--\; \vee \; --> \\ \wedge <!--\; \wedge \; --> \\ \vee <!--\; \vee \; --> \\ (m_2) \end{gathered}$$
Where the first spring has constant $k_1$, and the second $k_2$. I'll let $y_1(t)$ be the position of the top mass away from its equilibrium, and $y_2(t)$ the position of the bottom mass away from its equilibrium. I choose down to be the positive direction.
For the bottom mass, there is an upward force of $k_2{y}_2$, and a downward gravitational force of $m_{2}g$. So one equation should be
$m_2{y}_2^{″}=-<!--\; -\; -->k_2{y}_2+m_{2}g$
For the top mass, the first spring pulls up with force $-<!--\; -\; -->k_1{y}_1$ and a downward gravitational force of $m_{1}g$. I'm not sure how to account for the forces of the second spring and second mass acting of the first mass. Is the equation something like
$m_1{y}_1^{″}=-<!--\; -\; -->k_1{y}_1+m_{1}g+\text{ something?}$
I'm just curious how you would correctly set up the equations for this system.

Method of Modeling Problem Complexity
Is there a standard way to model the complexity of a mathematical problem? If so, what are the best online resources to learn more about it?
Honestly don't have an even a basic way to model the complexity of a problem other than "depth" of concepts required to solve the problem and the smallest number of steps require to reach an answer. For example, "$1+1=2$" I would venture to guess is less complex than "$1\times <!--\; \times \; -->2=2$".

Understanding compactness theorem on modeling a sentence
T is a theory and $\mathrm{\varphi <!--\; \varphi \; -->}$ is a sentence with $T\models <!--\; \models \; -->\mathrm{\varphi <!--\; \varphi \; -->}$. I read notes with a quote like this:
By Compactness Theorem, a finite subset $T_0\subseteq <!--\; \subseteq \; -->T$ has $T_0\models <!--\; \models \; -->\mathrm{\varphi <!--\; \varphi \; -->}$.
I thought the Compactness Theorem was something like "a theory has a model iff every subset of the theory has a model". That is $M\models <!--\; \models \; -->T⟹<!--\; ⟹\; -->M\models <!--\; \models \; -->T_0$. (I believe it follows from Completeness of FOL and proofs being finite). So how do we show the claim with compactness? I think it has something to do with $\mathrm{\varphi <!--\; \varphi \; -->}$ being a sentence. If we replaced ϕ with an infinite theory T′ then we cannot claim $T_0\models <!--\; \models \; -->T^{\prime }$.

Tornado Damage Modeling
A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is
$f(x,y)=6[1-<!--\; -\; -->(x+y)]\text{, }x>0,y>0,x+y<1\text{.}$
Determine the probability that the portion of a claim representing damage to the house is less than 0.2.
What I did was establish $f_x(X)=3-<!--\; -\; -->6x$ since X represents damage to the house. Then I integrated the function from 0 to 0.2 and I got an answer of 0.48, however, the actual answer is 0.488.

Modeling some constraints
We have two decision variables $x\in <!--\; \in \; -->{\mathbb{Z}}^{0+}$ that is the main decision variable and $0⩽<!--\; ⩽\; -->y⩽<!--\; ⩽\; -->1$ that is an auxiliary decision variable.
Now based on the nature of the problem we are studying, we know that
$$\begin{gathered} y=\frac{1}{x}, \\ x=0\iff <!--\; \iff \; -->y=\mathrm{\varnothing <!--\; \varnothing \; -->}, \\ x>0\iff <!--\; \iff \; -->y⩾<!--\; ⩾\; -->0. \end{gathered}$$
Now the challenge is how to formulate the dynamics of these two decision variables in the constraints of a mathematical program. We have x in the denominator and since at optimality x can actually be zero, this would create a problem. Obviously, we cannot write $xy=1$ because if the optimal solution is that $x=0$, the equality would be violated.
Any suggestions would be much appreciated.

I'm reading through Shape by Jordan Ellenberg and came across this claim, modeling the movement of a mosquito as a binomial distribution.
The mosquito is fixed to a straight line. Each day, it can choose to whether to fly a kilometer to the northeast or a kilometer to the southwest. It is unbiased, so each path is equally likely.
The claim is that
The chance that a mosquito on its two-hundredth day of life is at least 40km from home is just under 3 in 1000.
A footnote adds that the exact computation is
"What is the probability that a binomial random variable with $p=0.5$ and $n=200$ takes value at least 120?"
To cover a distance of, say 40 km northeast, the mosquito would need to have moved 120 km northeast and 80 km southwest. Representing the movement northeast as a "success", this could be the binomial distribution described above.
$X\sim <!--\; \sim \; -->Bin(0.5,200)$
And that we are looking to calculate $P(X\ge <!--\; \ge \; -->120)$
However, it strikes me that the chance that the mosquito is at least 40km from home would be twice of this computed value, since it could be a net 40km northeast or a net 40km southwest. Am I misunderstanding the claim above or modeling the probability incorrectly?

Modeling salt and water with differential equations
A 5-gallon bucket is full of pure water. Suppose we begin dumping salt into the bucket at a rate of 1/4 pounds per minute. Also, we open the spigot so that 1/2 gallons per minute leaves the bucket, and we add pure water to keep the bucket full. If the salt water solution is always well mixed, what is the amount of salt in the bucket after<br?(a) 1 minute?
(b) 10 minutes?
(c) 10 minutes?
(d) 1000 minutes?
(e) a very, very long time?
My question is about the model. I let x be the amount of salt and t the minutes passed. Then
$\frac{dx}{dt}=\frac{1}{4}-<!--\; -\; -->\frac{1}{10}x$
Some algebra gives me
$\frac{dx}{dt}=\frac{5-<!--\; -\; -->2x}{20}$
and I integrate
$\int <!--\; \int \; -->\frac{20}{5-<!--\; -\; -->2x}dx=\int <!--\; \int \; -->dt$
to obtain $-<!--\; -\; -->10\ln <!--\; \; -->|5-<!--\; -\; -->2x|=t+C$. After some algebra, and keeping in mind that $x(0)=0$, I get
$|5-<!--\; -\; -->2x|=5e^{-<!--\; -\; -->\frac{t}{10}}$
I have two sets of solutions: $x=\frac{5}{2}(1-<!--\; -\; -->e^{-<!--\; -\; -->\frac{t}{10}})$ and $x=\frac{5}{2}(1+e^{-<!--\; -\; -->\frac{t}{10}})$
At this point I'm confused about what to do. Either model provides "sensible" answers, in that the amount of salt I have is positive, as we might expect. So which one do I use?

Modeling the relationship between perimeter and area
Is there any equation that models the relationship between the area and perimeter of a rectangle?

Spliced probability distribution modeling
I have a simulation model for insurance claims that works as follows:
1. Assume random values come from two distributions defined as
$\begin{array}{rcl}{f}_1(x)& ,& 0<x\le <!--\; \le \; -->c\\ f_2(x)& ,& x>c\end{array}$
A random value will fall in $f_1(x)$ with probability p and $f_2(x)$ with probability $1-<!--\; -\; -->p$.
So the model (in Excel) uses rand() to determine which distribution to sample from. $f_2(x)$ is shifted at c so the minimum value from this distribution is c while $f_1(x)$ is set to be capped at c. A determined number of values is generated in the model and then summed to give an aggregate value.
I'm trying express the limited expected value of this distribution and am wondering is the distribution simply defined as:
$f_X(x)=\{\begin{array}{ll}p\cdot <!--\; \cdot \; -->f_1(x),& 0<x\le <!--\; \le \; -->c\\ (1-<!--\; -\; -->p)\cdot <!--\; \cdot \; -->f_2(x),& x>c\end{array}$
or am I missing something? This distribution will not be continuous at c unless I force it to through solving for p but assume that's not a requirement.

Modeling Exponential Equation with 2 given points?
So let's say that the population of rabbits increase according to the law exponential growth.
If a certain population of rabbits has 100 rabbits after the second day and 300 after the fourth day, can we figure out how many rabbits that there were to begin with, or do we not have enough information?

Modeling a greatest integer function
I'm trying to model a function that resembles a greatest integer function. The domain is from [0, $\mathrm{\infty <!--\; \infty \; -->}$). The inputs from 0 to 1.5 (non-inclusive) need to be mapped to an output of 0, and 1.5 to $\mathrm{\infty <!--\; \infty \; -->}$ mapped to 1. But, I'm trying to not use a piecewise function. Is it possible to accomplish this?
Here's what I've tried:
$f(x)=\lfloor \frac{x}{1.5}\rfloor$

Modeling drunkeness over time
So you have the function of drunkenness,D(t), the amount of alcohol in your blood, for $t>0$, over time,t in hours:
$D(t)=T-<!--\; -\; -->ct$
c being the arbitrary amount of alcohol your liver will process in 1 hour. T being the total amount of alcohol you have drunk at time t.
So T does depend on t, but it still a totally arbitrary amount.
As a piece wise function based on $T=c_{n}t$. $c_n=nc$ with n being some non negative number, representing the number of drinks of strength c had in 1 hours time:
$T=\begin{array}{cc}\{& \begin{array}{cc}t=1& n_{1}c\\ t=2& n_{2}c\\ ...\\ t=n& n_{n}c\end{array}\end{array}$
So to find T for a certain time, t,
$T(t=n)=n_{1}c+n_{2}c...+...n_{n}c=(n_1+n_2+n_n)c$
The number of drinks had each hour is totally arbitrary. Now what I want to do is to compare this between different values of c. Like comparing the same amount of drinking between a light beer (call it 1c) and a craft beer (2c). Is there a way to generalize this.
For example:
$T=\begin{array}{cc}\{& \begin{array}{cc}t=1& 2c\\ t=2& 1c\\ t=3& 3c\end{array}\end{array}$
$T(t=3)=6c$
So
$D(t=3)=3c$
but if $c=2c$
$T(t=3)=12c$
and
$D(t=3)=9c$
So drunkenness is not a linear relationship related to the strength of alcohol you are drinking.

Basic calculus question about epidemic modeling
I am a medical student attempting to build mathematical models of disease, but am struggling with the calculus. In particular, I am very confused by this equation shown in attached photo, and feel worse because it is called "elementary". How is the second equation the integral of the first?
Feel free to ignore the first sentence, before "we consider."
The assumption (i) requires a fuller mathematical explanation, since the assumption of a recovery rate proportional to the number of infectives has no clearepidemiological meaning. We consider the "cohort" of members who were all infected at one time and let u(s) denote the number of these who are still infectivea time units after having been infected. If a fraction a of these leave the infective class in unit time then
$u^{\prime }=\mathrm{\alpha <!--\; \alpha \; -->}u,$
and the solution of this elementary differential equation is
$u(s)=u(0)e^{-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}s}$

I need help modeling the velocity profile of a river's current
I am trying to solve Zermelo's Navigation Problem.
One of the cases I'm looking at is when the river's current is a function of the x-position only.
From what I learned in Fluid Mechanics courses, I know that at the two ends when (i.e. the river banks) the velocity should be zero. Then in the center the velocity is at its maximum value.
In other words: $v(x=0)=v(x=L)=0$, and $v(x=0.5L)=V_{\text{max}}$
Everything I learned in the past was these velocities as function of radius, which makes sense for pipes and tubes, but since this can be thought of a 2D rectangular flow, I can't figure this out.
I know it should be a quadratic expression.

Modeling a center of mass of a thin wire
I am asked to find the moment about the x axis for a thin wire of constant density. This thin wire lies along the curve $y=\sqrt{x}$ and the limits for integration are $x=0$ and $x=2$.
I know from my textbook that the moment about the x axis is: $M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm$
Because this is a thin wire, I know that I need to subdivide the wire into small segments for integration. I have the following for relevant data for each segment:
Length: $dl=\sqrt{x}dx$
mass: $dm=\mathrm{\delta <!--\; \delta \; -->}dl=\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}dx$
It's the part about the distance of the center of mass to the x axis that I think I'm missing. I have the following:
$\stackrel{~<!--\; ~\; -->}{y}=\sqrt{x}$
Therefore, my final integral is:
$M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm={\int <!--\; \int \; -->}_0^2\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}\sqrt{x}dx=\mathrm{\delta <!--\; \delta \; -->}{\int <!--\; \int \; -->}_0^{2}xdx=\mathrm{\delta <!--\; \delta \; -->}{\frac{1}{2}{x}^2|}_0^2=\mathrm{\delta <!--\; \delta \; -->}2$
This particular problem is an odd numbered problem and so I know that I've got it incorrect. Please help me to see where I'm going wrong.

Use of moore-penrose inverse when modeling PCA
In the derivation for principal component analysis we model our observed data points y as the result of a linear transformation restricted to being an axis change, W, applied to a set of uncorrelated variables x, where x lives in a lower dimensional space than y.
Thus, we can represent our observations as $y=Wx$, and what we are trying to find as $x=W^{\prime }y$, where W' is the pseudo-inverse of W.
Is the reason we choose to model the mapping of y onto the pca axes with use of a pseudo-inverse due to the fact that we want to be able to use PCA on systems where y doesn't completely follow our assumptions? Where not all points in y are contained in the column space of W?

Set of differential equations modeling biological system
Im trying to describe a moleculatr biological system using some differntial equations, However, differntial equations is not my strong side.
I'm thinking my equation set is quite trivial but i just cant seem to manage it. for 3 distinct groups im trying to calculate $V_1(t)$, expressed by:
$V_1(t)=V_2^{\prime }(t)k1$
$V_2(t)=V_1^{\prime }(t)k_2+V_3^{\prime }(t)k_3$
$V_3(t)=V_2^{\prime }(t)k_4$
for $k_1,k_2,k_3,k_4$ some constants.

Simple physics problem using differential equation modeling
There are 50 litres of dilution, 90% water and 10% of alcohol in a tank. Then, 50% of water and 50% of alcohol is poured in a tank, at the speed of 5 litres per minute, and the dilution is outflowing from a tank at the speed of 5 litres per minute. How much alcohol is in a tank after 10 minutes?
I have a solution that use differential equations, but I can't understand how to model the problem by differential equations. Obtained differential equation is:
$2-<!--\; -\; -->\frac{5}{50-<!--\; -\; -->t}y(t)=\frac{dy}{dt}\Rightarrow <!--\; \Rightarrow \; -->y^{\prime }+\frac{5}{50-<!--\; -\; -->t}y=-<!--\; -\; -->2$
which is a linear differential equation.
General solution:
$y(t)=C(50-<!--\; -\; -->t{)}^5+\frac{1}{2}(50-<!--\; -\; -->t)$
Now, setting $y_0=5$, which is the amount of alcohol at the beginning, we evaluate constant $C\Rightarrow <!--\; \Rightarrow \; -->$
$C=-<!--\; -\; -->\frac{20}{{50}^5}$
Going back to the general solution:
$C=-<!--\; -\; -->\frac{20}{{50}^5}$ litres.
Also, what would be the approach without using differential equations?