Answer:
(a) 8.15
(b) 12.92
Step-by-step explanation:
Given: P = $3000, r = 0.085
[tex]A = Pe^{rt}[/tex]
Where
A is the Amount
P is the Principal
r is the rate
t is the time
(a) For the amount to double, A = 2 × P
A = 2 × $3000
A = $6000
[tex]6000 = 3000e^{0.085t}[/tex]
[tex]\frac{6000}{3000} = e^{0.085t}[/tex]
[tex]2 = e^{0.085t}[/tex]
Take [tex]log_{e}[/tex] of both sides
[tex]log_{e}2 = log_{e}e^{0.085t}[/tex]
But [tex]log_{e}e = 1[/tex]
∴ [tex]ln2 = 0.085t[/tex]
[tex]t = \frac{ln2}{0.085}[/tex]
[tex]t = \frac{0.693}{0.085}[/tex]
t = 8.15
(b) For the amount to double, A = 3 × P
A = 3 × $3000
A = $9000
[tex]9000 = 3000e^{0.085t}[/tex]
[tex]\frac{9000}{3000} = e^{0.085t}[/tex]
[tex]3 = e^{0.085t}[/tex]
Take [tex]log_{e}[/tex] of both sides
[tex]log_{e}3 = log_{e}e^{0.085t}[/tex]
But [tex]log_{e}e = 1[/tex]
∴ [tex]ln3 = 0.085t[/tex]
[tex]t = \frac{ln3}{0.085}[/tex]
[tex]t = \frac{1.0986}{0.085}[/tex]
t = 12.92
