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**1. Homework Statement**

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?

**2. Homework Equations**

Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##.

**3. The Attempt at a Solution**

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. 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 success 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 as I felt that my reasoning is inadequate.