Ms. Garcia wants to estimate how much time students spend on homework, on average, during a typical week. She wants to estimate at the 90% confidence level with a margin of error of at most 30 minutes. A pilot study indicated that the standard deviation of time spent on homework per week is about 154 minutes. What’s the minimum number of students you would need to survey?

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You're asking in the ap section, so i'll give the ap calculus-style answer. alternatively, you can find the vertex of the parabola and use the "y"-coordinate as the answer.

the maximum height can occur at a critical number, that is, a number  such that  .

the derivative of  can be calculated as follows:



find a value of  that is a zero of the derivative:


this is parabola that is opening downward; we can assume there is a maximum at  .

to find that maximum, calculate  .