Suppose a survey shows that a mayoral candidate is gaining votes at a rate of $2000 t+1000$ votes per day, where $t$ is the number of days since she announced her candidacy. How many supporters will the candidate have after 50 days, assuming that she had no supporters at $t=0$ ? (Give your answer as an exact number.) supporters:

Step 1 ：Suppose a survey shows that a mayoral candidate is gaining votes at a rate of \(2000 t+1000\) votes per day, where \(t\) is the number of days since she announced her candidacy.

Step 2 ：We need to find out how many supporters the candidate will have after 50 days, assuming that she had no supporters at \(t=0\).

Step 3 ：The number of supporters the candidate gains each day is given by the function \(2000t + 1000\).

Step 4 ：To find the total number of supporters after 50 days, we need to integrate this function from 0 to 50. The integral of a function gives the area under the curve, which in this case represents the total number of supporters.

Step 5 ：The integral of the function from 0 to 50 gives the total number of supporters the candidate will have after 50 days.

Step 6 ：Final Answer: The candidate will have \(\boxed{2550000}\) supporters after 50 days.