Algebra Word Problem Solutions

For positive functions f,g, if $g(x)=x\cdot <!--\; \cdot \; -->f(x)$, and $g^{\prime }(x)\ge <!--\; \ge \; -->0$, $g^{″}(x)\le <!--\; \le \; -->0$, is $f^{\prime }(x)\le <!--\; \le \; -->0$?

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?

How do I equate coefficients?
I got the problem into this form:
$A+B=0$
$C-<!--\; -\; -->B=1$
$8A+4B-<!--\; -\; -->C+D=10$
$-<!--\; -\; -->4B+4C-<!--\; -\; -->D+E=3$
$16A-<!--\; -\; -->4C-<!--\; -\; -->E=36$
I really don't know what to do after that. I suck at equating coefficients. It never makes sense to me. Please help and provide steps and a good explanation. Thank you.

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?

Solve the equations
$x+y=35$
$x-<!--\; -\; -->y=13$

Are the graph y=x and y=3 perpendicular?

Evaluate using long division first to write f(x) as the sum of a polynomial and a proper rational function.
$\int <!--\; \int \; -->\frac{xdx}{3x-<!--\; -\; -->4}$

Inverting the exponential decay equation?
How would I algebraically isolate $x$ in the following exponential decay equation, so that $x$ is most easily derived?
$y=A{e}^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}$

How do you write the expression for the nth term of the sequence given ${\displaystyle 1,2,\frac{2^2}{2},\frac{2^3}{6},\frac{2^4}{24},\frac{2^5}{120},...}$?

Solving ${\log }_2<!--\; \; -->(x^2)=x$ explicitly?
I'm having problems getting a proper step-by-step solution to this equation.
${\log }_2<!--\; \; -->(x^2)=x$
I know the results are 2 and 4, but so far I can get only solutions like these:
$2^x=x^2\text{or}\frac{\log <!--\; \; -->(x)}{x}=\frac{\log <!--\; \; -->(2)}{2}.$
I'd like something along the lines of $x=\cdots <!--\; \cdots \; -->$
Is it possible with some basic knowledge?

Do all monic polynomials factor into monic irreducibles?
Do all monic polynomials in $\mathbb{Z}/n\mathbb{Z}[x]$ where n is prime factor into monic irreducibles?
Just by thinking about these monic polynomials and how they only act on a finite number of elements makes me think that the assertion is true, because if d is a root of a monic polynomial f(x) then $f(x)=(x-<!--\; -\; -->d)g(x)$ where g(x) is monic, and you could apply the same rule iteratively until you exhaust all roots. But I am not sure how to really prove this, it sounds really similar to one part of the fundamental theorem of arithmetic, but I can't find how to quantify a polynomial the same way you would an integer for an inductive proof. How would I go about starting a proof of this?

Write the equation in point slope form given (4,5) and (-3,8)

Graph the line that passes through (4,-2) perpendicular to the graph 3x−2y=6

Derivative of log and exponential function
How would you find the derivative of:
$y=e^{\sqrt{x}}+\ln <!--\; \; -->\sqrt{x}$
Would I simply use the product rule?

Algebra: logarithms word question
Initially there are $2000$ bacteria in a given culture. The number of bacteria $N$ is tripling every hour so $N=2000\cdot <!--\; \cdot \; -->3^t$, where $t$ is the measurable in hours.
a) How many bacteria are present after 4 hours?
b) How long is it until there are 1000000 bacteria?

Find the equation of the line that passes through the origin and is perpendicular to the line that passes through the following points: (9,2),(−2,8)

Find a standard form equation for the line with point (2, 5) and is perpendicular to the line 4x + 9y − 7

What $n^{\frac{1}{{\log }_2<!--\; \; -->n}}$ means?
I was confused with about the $n^{\frac{1}{{\log }_2<!--\; \; -->n}}$ expression. I am not sure how to make mathematical sense of it - i.e. express it in another way for easier understanding. I tried to plug in some numbers like 4 and 8. I got the answer 2 in both cases. Is it what this expression means? 2?

Jim wants to build a rectangular parking lot along a busy street but only has 1900 feet of fencing available. If no fencing is required along the street, find the maximum area of the parking lot.

Is 10, 5, 2.5, 1.25, 0.625, 0.3125 a geometric sequence?