1. **Problem statement:** A hard drive initially contains 1500 megabytes of data and loses 5 megabytes every hour due to corruption. 2. **Formula:** The amount of data left after $t$ hours is given by the linear equation: $$D(t) = D_0 - rt$$ where $D_0 = 1500$ MB is the initial data, $r = 5$ MB/hour is the rate of data loss, and $t$ is time in hours. 3. **Part (a):** Find the data left after 20 hours. $$D(20) = 1500 - 5 \times 20 = 1500 - 100 = 1400$$ So, 1400 megabytes remain after 20 hours. 4. **Part (b):** Find the time when half the data is lost. Half the data is $\frac{1500}{2} = 750$ MB. Set $D(t) = 750$: $$750 = 1500 - 5t$$ Solve for $t$: $$5t = 1500 - 750 = 750$$ $$t = \frac{750}{5} = 150$$ So, half the data is lost after 150 hours. 5. **Part (c):** Find the time when all data is lost. Set $D(t) = 0$: $$0 = 1500 - 5t$$ Solve for $t$: $$5t = 1500$$ $$t = \frac{1500}{5} = 300$$ All data is lost after 300 hours. **Final answers:** - (a) 1400 megabytes - (b) 150 hours - (c) 300 hours