The Mechanical Universe- Chapter 3- question 6

Question

Suppose your income $y$ is directly proportional to the number $x$ of hours you work: $y = cx$, where $c$ is a constant. In addition, suppose you're a big spender and the money $z$ you spend varies with income as $z(y) = a + by^2$, where $a$ and $b$ are also constants. For your income to be larger than your expenses, what conditions on $y$, $a$, and $b$ must be met?

I'm not entirely sure how I am supposed to write the answers(There is no solution on the back of the book). I know that $a < 0$ and $|a/y^2|>b$ but is that how I am supposed to express the conditions? Or am I supposed to express the conditions in terms of $c$ and $x$?

Answer 1

The income is $y$, the expenses is $a+by^2$, so the condition income greater than expenses is $$y>a+by^2$$ You can rewrite this as $$by^2-y+a<0$$ Calculate the roots of $by^2-y+a=0$. If $b>0$, then $y$ has to be between the roots. if $b<0$, $y has to be on the outside.

Answer 2

If income is greater than spending.

$y\ge z\\ y\ge a+by^2\\ 0\ge by^2 - y + a$

if $a,b > 0$

$\frac {1 - \sqrt{1-4ab}}{2} \le y \le\frac {1 + \sqrt{1-4ab}}{2}$

and $ 1-4ab \ge 0$