Algebra Word Problem Solutions

Finding the roots of $(x^2+7x+6{)}^2+7(x^2+7x+6)+6=x$
I have been trying to solve this one problem from the Duke Math Meet, which does not provide a solution:
Find all solutions of $(x^2+7x+6{)}^2+7(x^2+7x+6)+6=x$
At first I tried to factorize the polynomial, but always had the right-hand side x remain, which was inconvenient. I then tried using the fact that one can write the left-hand side as a composition of functions, and equated that with the inverse of the quadratic plugged into the function, but that was a very nasty equation with square roots, still ending up with a quartic.
What other solution paths are viable for this problem, and is there a way to factor the quartic?

Now building a new tunnel that has a shape of parabolic curve. The tunnel is 10 m wide and at 4 m from either side, the height of the tunnel is 6 m. Find the quadratic equation in standard form that models the ceiling of the new tunnel.
The question is very simple, but I just can't understand what does at 4 m from either side mean in this question while mentioning the width in 10m?
I am not a native English speaker so it is hard for me understand what the information this question is asking.
Any comments and answers will be appreicated!

Modeling differential equation
I am trying to model some differential equation which can explain buying some assets that can increase itself like a virus.
So, let's assume my money is y(t), and number of assets is n(t).
- y(t) to a certain level c, it will increase the n(t) by +1
- n(t) generate profit r %
so I can see something like
$$\begin{gathered} dy(t)/dt=y_0+r\cdot <!--\; \cdot \; -->n(t) \\ dn(t)/dt=If(y(t)\ge <!--\; \ge \; -->c,y(t)-<!--\; -\; -->20andn+1,n) \end{gathered}$$
However, I have no clue how I can model this discrete-continuous variable differential equation. Is there any way to model the system and get the solution?

How do you find the simple interest paid after 3 months on a loan of $9700 borrowed at an annual interest rate of 11%?

How do you find the first three terms if the common ratio in a geometric sequence is 3/2, and the fifth term is 1

Squares of side 3 inches are cut from each corner of a square sheet of paper. The paper is folded to form an open box with a volume of 675 cu in. How long were the sides of the sheet of paper?

Given the first term and the common dimerence er an arithmetic sequence how do you find the 52nd term and the explicit formula: ${\displaystyle a_1=25,d=6}$?

Translate the following phrase into an algebraic expression. Do not simplify. Use the variable names "x" or "y" to describe the unknowns.

Calculate Ln$(i^i)$
My attempt:
Ln$(z)$=$\ln <!--\; \; -->|z|+i\arg <!--\; \; -->z$
$z=0+i^i=0+i\cdot <!--\; \cdot \; -->i$
$$\begin{gathered} |z|=\sqrt{0^2+i^2}=i \\ \arg <!--\; \; -->z=\arctan <!--\; \; -->(i/0) \end{gathered}$$
$1.$ how it can be that the modulus equal to $i$?
$2.$ how can I find the argument?

Fractional/rational form of $\mathrm{0.999...}$
Is it possible to express $\mathrm{0.999...}$, a repeating number, as a fraction? Or as a ratio of two numbers?
Basically all (my) attempts at the problem cancels all the terms and returns $1$. Is it even possible? Or has it been proven to be an exercise of futility?

Your alarm clock is broken and always rings with a delay. For example, last night, you set your alarm for 7:30 AM but your alarm rang at 7:42 AM. However, the delay is not constant. After much data collection, you realize that the alarm clock’s delay time (in minutes) can be reasonably modeled as a continuous random variable T with $f(t)=c{e}^{-<!--\; -\; -->t/15}\text{ for }t\ge <!--\; \ge \; -->0\text{ and }f(t)=0$ otherwise.
Answer the following questions:
(a) What is the value of c?
(b) What is the mean delay?
(c) What is the variance in the delay?
(d) If you wanted your alarm to ring at 7:30 AM, approximately what time should you set your alarm so that 95% of the time, the alarm rings before 7:30 AM?

Find two consecutive integers whose sum is 73

Halima is now three years older than Nekesa If Nekesa's age is represented by x, what will be the sum of their ages in 10 years time?

Write an equation in point-slope form for the given (-4, 2), (-3, -5)

How to prove this logarithm equation?
Given :
${\log }_{12}<!--\; \; -->18=a\text{ and }{\log }_{24}<!--\; \; -->54=b$
prove that:
$ab+5(a-<!--\; -\; -->b)=1$
My attempt: I couldn't solve it in any way, as base were not common. I could solve it if base of second equation was $12$ raised to something, but here it is $12\times <!--\; \times \; -->2$ :( Any help will be greatly appreciated.

Find the integers if the sum of two integers is 2, and their difference is 6

Why must we distinguish between rational and irrational numbers?

A company manufactures and sells x transistor radios per week. The weekly cost and price demand equations are $C(x)=5000+2x$ and $p=10-<!--\; -\; -->0.001x$ where x can be no more than 10000 units. The government has decided to tax the company $2 for each radio produced. Taking into account this additional cost, how many radio should the company manufacture each week in order to maximize its weekly profit? What is the maximum weekly profit? How much should the company charge for the radios to realize the maximum weekly profit?

Find the range of values of k such that f(x) is onto
I am trying to solve the below problem in pre-calculus, I give below the steps I followed , I am stuck as it gets very complicated , need help in solution.
A function $f:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ is defined by
$f(x)=\frac{k{x}^2+6x-<!--\; -\; -->8}{k+6x-<!--\; -\; -->8x^2}.$
Find the intervals of values of k such that f is onto. The answer given in the book is $2\le <!--\; \le \; -->k\le <!--\; \le \; -->14$.
I started by trying to find the range of the function and find the values of k such that the range is = R ( that is range = co domain , in this case co domain is R and hence has to prove that range = R)
Let
$y=\frac{k{x}^2+6x-<!--\; -\; -->8}{k+6x-<!--\; -\; -->8x^2}$
Rearranging and grouping gives
$(k+8y)x^2+6(1-<!--\; -\; -->y)x-<!--\; -\; -->(8+ky)=0.$
This is quadratic in x and for x to be real $det(x)\ge <!--\; \ge \; -->0$, which requires
$36(1-<!--\; -\; -->y{)}^2+4(k+8y)(8+ky)\ge <!--\; \ge \; -->0.$
Rearranging and grouping this equation gives
$(36+32k)y^2+(4k^2+184)y+(36+32k)\ge <!--\; \ge \; -->0.$
Now I have to find the range of values of k such that the above expression is $\ge <!--\; \ge \; -->0$. Totally stuck here. Please help. Also, is there any other easier way of arriving at the range of k ?

Could somebody validate my proof regarding the limit of $\ln <!--\; \; -->(x_n)$ when, $x_n$ tends to $a$?
So, let me cearly state the problem:
Let $(x_n)$ be a convergent sequence, with: $x_n>0$, $\mathrm{\forall <!--\; \forall \; -->}n$, n natural number, and $x_n\to <!--\; \to \; -->a$, with $a>0$. Then $\ln <!--\; \; -->x_n\to <!--\; \to \; -->\ln <!--\; \; -->a$
Here is my idea for a proof:
Our goal is to proof that there $\mathrm{\forall <!--\; \forall \; -->}\mathrm{ϵ<!--\; ϵ\; -->}>0$ there is some $n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, such that $\mathrm{\forall <!--\; \forall \; -->}n\ge <!--\; \ge \; -->n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, we have that $|\ln <!--\; \; -->x_n-<!--\; -\; -->\ln <!--\; \; -->a|<\mathrm{ϵ<!--\; ϵ\; -->}$
So here is what I did. First:
$|\ln <!--\; \; -->x_n-<!--\; -\; -->\ln <!--\; \; -->a|=|\ln <!--\; \; -->\frac{x_n}{a}|$
Then, beacause $\ln <!--\; \; -->x<x$, $\mathrm{\forall <!--\; \forall \; -->}x>0$, it easily follows that $|\ln <!--\; \; -->x|<x$, $\mathrm{\forall <!--\; \forall \; -->}x>0$. Applying this, we have that:
$|\ln <!--\; \; -->x_n-<!--\; -\; -->\ln <!--\; \; -->a|=|\ln <!--\; \; -->\frac{x_n}{a}|<\frac{x_n}{a}=\frac{x_n-<!--\; -\; -->a+a}{a}=\frac{x_n-<!--\; -\; -->a}{a}+1$
Now, we use the basic property of the absolute value: $x_n-<!--\; -\; -->a\le <!--\; \le \; -->|x_n-<!--\; -\; -->a|$, that gives us:
$|\ln <!--\; \; -->x_n-<!--\; -\; -->\ln <!--\; \; -->a|<\frac{x_n-<!--\; -\; -->a}{a}+1\le <!--\; \le \; -->\frac{|x_n-<!--\; -\; -->a|}{a}+1$
Now, we use the fact that $x_n\to <!--\; \to \; -->a$. So, $\mathrm{\forall <!--\; \forall \; -->}\mathrm{ϵ<!--\; ϵ\; -->}>0$ there is some $n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, such that $\mathrm{\forall <!--\; \forall \; -->}n\ge <!--\; \ge \; -->n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, we have that $|x_n-<!--\; -\; -->a|<\mathrm{ϵ<!--\; ϵ\; -->}$. We choose an epsilon that takes the form $a({\mathrm{ϵ<!--\; ϵ\; -->}}_0-<!--\; -\; -->1)$. This choice is possible for any ${\mathrm{ϵ<!--\; ϵ\; -->}}_0$
Now, we have managed to obtain that:
$|\ln <!--\; \; -->x_n-<!--\; -\; -->\ln <!--\; \; -->a|<\frac{a({\mathrm{ϵ<!--\; ϵ\; -->}}_0-<!--\; -\; -->1)}{a}+1={\mathrm{ϵ<!--\; ϵ\; -->}}_0,\mathrm{\forall <!--\; \forall \; -->}n\ge <!--\; \ge \; -->n_{\mathrm{ϵ<!--\; ϵ\; -->}}$
Since $\mathrm{\forall <!--\; \forall \; -->}{\mathrm{ϵ<!--\; ϵ\; -->}}_0>0$ we can find an $n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, such that $\mathrm{\forall <!--\; \forall \; -->}n\ge <!--\; \ge \; -->n_{\mathrm{ϵ<!--\; ϵ\; -->}}$, the above inequality is staisfied, our claim is proved.
So, can you please tell me if my proof si correct? I've tried to find a proof, only for the limit of sequences! This problem has been on my nerves for s while. Also, probably there is some simpler way to do it, but I couldn't find it.