Expert: Sherman D. - 8/17/2008

Question
QUESTION: The number of journalists in a metropolitan region is increasing at a rate of 18t^2 + 60t journalists per year. If t = 0 represents the present, what will be the increase in the total number of journalists in the region over the next three years? A. 168 B. 258 C. 342 D. 432 I would appreciate being shown how to solve this, please.


ANSWER: 18(3)^2 + 60(3)
18(9) + 180
162 + 180
342

ANS : C.

---------- FOLLOW-UP ----------

QUESTION: Thank you for such a quick reply but unfortunately the study guide says that the answer is D.  I went back and forth with someone on another site but couldn't figure out the clues provided.  Here is the exchange, perhaps you could shed some light???

The "rate" tells you the instantaneous increase: in a time increment of dt, the increase is [18t^2 + 60t]dt. You need to integrate the incremental increase from t=0 to t=3.

........TTotal Increase = Int(0,3)[18 t^2 + 60 t]dt

The formula gives the rate of change. When you evaluate it at t=3, the statement is
...."At the end of the 3rd year, the rate of increase is 342 journalists per year."

But that is not what the question asks for. Rather, it wants to know the sum of the increase over the total time span of 3 years. You have to keep track of what the "units" are on the numbers. To find  units of (number), you have to  multiply (number per year) times (years).  The rate (number per year) is changing all the time according to the given formula. Youi know that the total time is 3 years, but you do not  know what the average rate of increase is over the 3 years. The only way to find the sum is to integrate.

........Total Increase = Int(0,3)[18 t^2 + 60 t]dt

Integrate each of the two terms using the rule for powers of t. Evaluate the integral at t=3 and at t=0, and subtract.  

Answer
i didn't think about it, but here it is.

the integration of 18t^2 + 60t is 6t^3 + 30t^2

if you plug in 3 you'll get 432 as your answer.

if you wanna know how i got it, just ask. if your no familiar with integration, it's a reverse derivative. if you find the derivative of the integration, you will end right back up with your previous equation.