Answer
(a) $t= -4.904$
(b) $t= -12.951$
Work Step by Step
(a)
Expression given:
$ \frac{3}{8e^{0.2t}}+5 = 6$
Multiply both sides by $8e^{0.2t}$:
$3+5\times(8e^{0.2t}) = 6\times(8e^{0.2t})$
Expand:
$ 3+40e^{0.2t} = 48e^{0.2t}$
Rearrange:
$ 3 = 8e^{0.2t}$
Simplify:
$ \frac{3}{8} = e^{0.2t}$
Take the natural logarithm of both sides:
$\ln \frac{3}{8} = 0.2t$
Solve using calculator or quadratic formula to find:
$ t = \frac{\ln \frac{3}{8}}{0.2} \approx -4.904$
(b)
Expression given:
$ \frac{3}{8e^{0.2t}}+5 = 10$
Multiply both sides by $8e^{0.2t}$:
$3+5\times(8e^{0.2t}) = 10\times(8e^{0.2t})$
Expand:
$ 3+40e^{0.2t} = 80e^{0.2t}$
Rearrange:
$ 3 = 40e^{0.2t}$
Simplify:
$ \frac{3}{40} = e^{0.2t}$
Take the natural logarithm of both sides:
$\ln \frac{3}{40} = 0.2t$
Solve using calculator or quadratic formula to find:
$ t = \frac{\ln \frac{3}{40}}{0.2} \approx -12.951$
