In Indiana, polls are open from 6am to 6pm.

Question

Candidate A receives votes at a rate of f(t)=−1t2+12t thousand votes per hour where t is the number of hours since 6am. Candidate B receives votes at a constant rate of 30 thousand votes per hour.

Assume candidate A had 3 thousand votes at and candidate B had 8 thousand votes from early voting.

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Which candidate wins?

By how many votes? 

Answer 1

When you have a function describing an amount, it's instantaneous rate of change is the derivative of the function. When given a function that describes a rate of change, the amount is the ant-derivative, or integral of the function. There are an infinite number of "anti-derivatives/integrals called indefinite integrals, that come from the same derivative function, so you

EITHER need some specific point on the anti-derivative/integral to pin it down, called the definite integral,

OR you need an interval to integrate over, resulting in what is called a definite integral.

In this case your problem identifies a rate and asks for an amount over a specific time interval, so we will need to integrate the rate over this interval in the process of finding our answer.

Given:

Let t be time where t=0 at 6AM so that t=12 at 6PM

Let V_EA and V_EB be the total votes from early voting respectively for candidates A and B

V_EA=3 thousand

V_EB=8 thousand

Let r_A and r_B be the thousand of votes per hour candidate A and B respectively receive

between 6am and 6pm, or between t=0 and t=12

Let T_A and T_B be the total votes respectively for candidate A and candidate B.

r_A= −t²+12t thousand of votes per hour candidate A receives

between 6am and 6pm, or t=0 and 12

r_B=30 thousand of votes per hour candidate A receives

between 6am and 6pm, or t=0 and 12

T_A=V_EA + Integral[(r_A)dt, 0 to 12]

T_B=V_EB + Integral[(r_B)dt, 0 to 12]

T_A=3 + Integral[(−t²+12t)dt, 0 to 12]dt

T_A=3+[-t³/3+12*t²/2] from t=0 to 12

T_B=8 + Integral[(30)dt, 0 to 12]

T_B=8 + 30t from t=0 to 12

Calculate TA=3+[-t³/3+12*t²/2]

and TB=8 + 30t

both at at t=12 and at t=0.

Then, subtract the results for TA and TB at t=0 from the results for TA and TB at t=12.to get the total votes for candidate A and B.