Differential Equation: Application of D.E: Exponential Decay

Let x = amount of radium at any time.
$\dfrac{dx}{dt} = kt$

$\dfrac{dx}{x} = k \, dt$

$\displaystyle \int \dfrac{dx}{x} = k \int dt$

$\ln x = kt + c$
 

When t = 15 yrs, x = 81%
$\ln 81 = 15k + c$   ←   (1)
 

When t = 25, x = 70.4%
$\ln 70.4 = 25k + c$   ←   (2)
 

From (1) and (2)
$k = -0.014$

$c = 4.605$
 

Hence,
$\ln x = -0.014t + 4.605$       answer for part (a)

$t = \dfrac{4.605 - \ln x}{0.014}$
 

For x = 50%
$t = \dfrac{4.605 - \ln 50}{0.014}$

$t = 49.5 ~ \text{years}$       answer for part (b)

Another solution (By Calculator - CASIO fx-991ES PLUS):
MODE 3 5

AC
$t = 50\hat{x} = 59.4 ~ \text{years}$

Note:
$\hat{x}$ = SHIFT 1 5 4