### Instruction: In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay  is the income minus the tax on that income.).### Response: Let's start by writing the amount of tax collected in terms of the income.
The amount of tax collected is $x\%$ of the income, which means it is $\frac{x}{100}$ times the income.
Since the take-home pay is the income minus the amount of tax collected, we can write an expression for the take-home pay.
That expression is $x*1000 - \frac{x}{100}*x*1000$.
Right. Now, let's simplify this expression.
The two terms in the expression are $x*1000$ and $\frac{x}{100}*x*1000$.
Right, let's simplify the first term $x*1000$.
$x*1000 = 1000x$.
Now, let's simplify the second term $\frac{x}{100}*x*1000$.
$\frac{x}{100}*x*1000 = \frac{x^2}{100}*1000 = 10x^2$.
Now, let's put the two terms back together.
The expression is $1000x - 10x^2$.
Right. Now let's write this expression in standard form.
The standard form of a quadratic is $ax^2 + bx + c$. So we can rewrite the expression as $-10x^2 + 1000x$.
Now, to find the maximum take-home pay, we need to find the vertex of this quadratic.
The x-coordinate of the vertex is $-\frac{b}{2a}$.
So, the x-coordinate of the vertex is $-\frac{1000}{-20} = 50$.
That means that an income of $50*1000 = 50000$ dollars will yield the maximum take-home pay.