Eric wants to stream a 41 minute show over the internet. However, he is having some connection issues, and the show is loading slowly. Specifically, he sees that after 5 minutes of waiting, only 1 minute and 10 seconds of the show has loaded. He is able to play the show while later parts of the show are loading, but he does not want to pause the show once he has started playing. He is willing to wait a certain amount of time before he starts watching the show; what is the MINIMUM WHOLE TOTAL NUMBER OF MINUTES he has to wait before he can play the show without any breaks?
I was given this problem in a math competition, but I couldn't figure out how to solve it.
My reasoning:
Using the slope formula, I found that for every minute he waits, the computer loads 7/30 minutes of the show. I tried visualizing this as a piecewise function with the functions, y=7/30x and y= -23/30x (because at some time he is going to play the video and I combined 7/30x with -x). Somehow I arrived at the answer 54 minutes, which is incorrect.
I'm not sure if this is the best way of solving this problem, however. Can someone show me how should I go about doing this? Thank you.
Answer
If you've waited $t_{wait}$ minutes, you've downloaded $\frac{7}{30}t_{wait}$ of the show. You've got that.
So at the optimal point, $\frac{7}{30}(t_{wait} + 41) = 41.$ In other words, the time you wait, plus the time of the show (during which you're still downloading) must be long enough to download the entire show.
Then $t_{wait} \approx 134.71$, so your answer is $135$ minutes.
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