# Tagged Questions

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### Question on simple quadratic word problem regarding weekly revenue when price of merchandise is lowered

A store owner sells headphones at 24 dollars a piece with roughly 1000 sold per week. The store owner finds that for every 1 dollar decrease on the price per headphone he sells 100 more headphones per ...
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### Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
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### Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
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### Alternative to Hungarian Algorithm to determine minimum cost?

Is there a graphic calculator (CAS technology) method to solve minimum cost problems/allocations that are normally completed with the Hungarian Algorithm... Hungarian Algorithm is time consuming, ...
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### Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
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### Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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### find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
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### Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
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### Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
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### Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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### Not so easy optimization of variables?

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example ...
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### How to solve this type of equation with posynomial form?

I have an equation with the following form where the goal is to find $x$: $$\sum_k c_k x^{\gamma_k} = 1$$ where $c_k, \gamma_k \in \Re^+$ and $\gamma_k > 1$ Alternatively using $y = \log(x)$ I can ...
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### Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
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### Minimize sum of smallest and largest among integers on the real line.

Suppose there are 3 non-negative integers $x$, $y$ and $z$ on the real line. We are told that $x + y + z = 300$. Without loss of generality, assume $x$ to be the smallest integer, and $z$ to be the ...
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### The minimum value of $(\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)$ if $x+y+z=1$

$x, y, z$ are three distinct positive reals such that $x+y+z=1$, then the minimum possible value of $(\frac{1}{x}-1) (\frac{1}{y}-1) (\frac{1}{z}-1)$ is ? The options are: $1,4,8$ or $16$ ...
### How to find the minimum value of $px+qy$ when $xy=r^2$?
The question says: "Find the minimum value of $px+qy$ when $xy=r^2$." No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...