The population of a city increases at a rate proportional to the num of inhabitants present at any time t.If the population of city was 2 lakhs in 1990 and 2.5lakhs in 2000.What will be the population in 2010?

Asked by aksharaprachi | 14th May, 2015, 09:07: PM

Expert Answer:

L e t space P space d e n o t e s space t h e space p o p u l a t i o n space a t space a n y space t i m e space t. T h u s comma space f r o m space t h e space g i v e n space s t a t e m e n t space o f space t h e space p r o b l e m comma space w e space h a v e comma fraction numerator d P over denominator d t end fraction proportional to P rightwards double arrow fraction numerator d P over denominator d t end fraction equals lambda P comma space w h e r e space lambda space i s space a space c o n s tan t. rightwards double arrow fraction numerator d P over denominator P end fraction equals lambda d t rightwards double arrow integral fraction numerator d P over denominator P end fraction equals integral lambda d t rightwards double arrow integral fraction numerator d P over denominator P end fraction equals lambda integral d t rightwards double arrow log P equals lambda t plus C W h e n space t equals 0 left parenthesis L e t space u s space c o n s i d e r space t equals 0 space i n space t h e space y e a r space 1990 right parenthesis comma space t h e space i n i t i a l space p o p u l a t i o n space o f space c i t y space w a s space 2 space l a k h s rightwards double arrow log 200000 minus lambda cross times 0 equals C rightwards double arrow log 200000 equals C T h u s comma space P o p u l a t i o n space a t space a n y space t i m e space t space i s space g i v e n space b y log P equals lambda t plus log 200000... left parenthesis 1 right parenthesis I t space i s space g i v e n space t h a t comma space a f t e r space 10 y e a r s comma space t h e space p o p u l a t i o n space i s space 2.5 space l a k h s. T h u s comma space w h e n comma space t equals 10 comma space P equals 250000 S u b s t i t u t i n g space t h e space v a l u e s space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma log 250000 equals 10 lambda plus log 200000 rightwards double arrow 10 lambda equals log 250000 minus log 200000 rightwards double arrow 10 lambda equals log open parentheses 250000 over 200000 close parentheses rightwards double arrow 10 lambda equals log open parentheses 250000 over 200000 close parentheses rightwards double arrow 10 lambda equals log open parentheses 1.25 close parentheses rightwards double arrow 10 lambda equals log open parentheses 1.25 close parentheses rightwards double arrow 10 lambda equals 0.2231 rightwards double arrow lambda equals 0.02231 T h u s space t h e space p o p u l a t i o n space o f space t h e space c i t y space d u r i n g space 2010 comma space t h a t space i s space w h e n space t equals 20 space y e a r s comma log P equals 20 cross times 0.02231 plus log 200000 rightwards double arrow log P equals 0.4462 plus 12.2061 rightwards double arrow log P equals 12.6523 rightwards double arrow P equals a n t i log open parentheses 12.6523 close parentheses rightwards double arrow P almost equal to 3 comma 12 comma 481 T h u s comma space t h e space p o p u l a t i o n space i n space 2010 space w i l l space b e space 3 comma 13 comma 000.

Answered by Vimala Ramamurthy | 15th May, 2015, 11:09: AM