You are applying for a \$1000 scholarship and your time is worth \$10 an hour. If the chance of success is $1 -(1/x)$ from $x$ hours of writing, when should you stop?
My way if thinking eventually led to the correct answer which is $\frac{1}{x^2}1000=10$. Solving for $x$ gives you the solution.
I was stuck at this problem and didn't know how to proceed and I tried to find out if I could find the answer just by matching up the units on both sides. Let $p(x)$ be the rate of success as a function of time, $x$. I don't know how they calculated $p(x)$ but I do know that it is dimensionless. Thus, $p'(x)$ will give me the rate of sucess per unit time or just the unit, $1/h$. I know that \$1000 has dollar units and that $10\frac{$}{h}$ has dollar per hour unit, so if I multiply $p'(x)=\frac{1}{x^2}$ with \$1000 I should get the same units as $10\frac{$}{h}$. So I set $\frac{1}{x^2}1000=10$ and solved for $x$. Feeling doubtful, I checked the solution and was surprised how I got it right.
My question is how do you interpret the solution, $\frac{1}{x^2}1000=10$? Like since $p'(x)$ is defined as the success rate per unit time, how come if I multiplied it by \$1000, it got me the solution? I just don't understand it, $$\frac{\text{success rate}}{\text{time}}\cdot \text{currency}$$
How do you interpret this? and yes, I'm aware that the success rate is dimensionless but still, I don't understand the reasoning behind the answer. I just want someone to solve the problem with also providing some commentary on his/her methods of solving it.