2020 NJC P1 Q8

A blogger reviews a data plan with unlimited data. The download speed is \(x\) megabytes per second, \(t\) seconds after the download has started. This download speed decreases at a rate proportional to the difference between \(x\) and the eventual limit of \(40\) megabytes per second. The download speed falls from a value of \(60\) megabytes per second to \(50\) megabytes per second in the first \(30\) minutes.

1. Write down a differential equation for this situation. Solve this differential equation to get \(x\) as an exact function of \(t\).[5]

The total amount of data downloaded daily by time \(t\) seconds is \(y\) gigabytes. The fair usage policy enforces that the download speed will be reduced to a speed which cannot be used to stream videos if \(y>500\). You may assume that \(1\) gigabyte is \(1000\) megabytes.

2. The blogger starts using the data plan on the morning of a particular day to stream videos. How long would he be able to stream videos continuously? Give your answer in hours to \(1\) decimal place.[3]