Answer to Question #265361 in Differential Equations for mary

Question

1. The bureau of census record in 1975 shows that the population in the country doubles compared to that of 1955. In what year will the population quadruple? Ans. 1995
2. The man puts a total amount of Php 30,000.00 to an account. For the first three years, the bank pays 4% compounded continuously and for the succeeding 7 years, the bank pays 5.5%, compounded continuously. How much will he have for the first 10 years? Ans. Php 49,709.57

Answer 1

1)

Rate of increase in population is proportional o the population

Hence, $P'(t) \alpha P(t)$

$P(t)=P_0e^{kt}$ ,$t>0$ and k is a positive constant

Let P₀ be the population in 1955

$2P_0=P_0e^{k(20)}$

$k=\frac{ln\ 2}{20}=0.0347$

When the population quadruple;

$4P_0=P_0e^{0.0347t}$

$t=\frac{ln\ 4}{0.0347}=39.95 \approx40 yrs$

=1955+40

=1995

2)

P=30000

r₁=$\frac{4}{100}$ for first 3 yrs

r₂=$\frac{5.5}{100}$ for the next 7yrs

$A=Pe^{rt}$

Amount after first 3 yrs,

A=30000*e^(0.04*3)

=33824.90555

This is the new principal

Amount accumulated after the next 7yrs,

A$=Pe^{rt}$

$=33824.90555*e^{(\frac{5.5}{100}*7)}$

=49709.56561

$\approx Php \ 49709.57$