Question

The bacteria concentration in a reservoir varies as

c= e^t - (t^3/6 ) (e^0.3t) - t^2/2 - t

where is the time in seconds. Use the Newton-Raphson method to estimate the
time required for the bacteria concentration to reach 1 (correct up to 2 decimal
places)

Answer 1

Answer on Question #52495 – Math – Algorithms | Quantitative Methods

The bacteria concentration in a reservoir varies as

$$c = e ^ {\wedge t} - \left(t ^ {\wedge} 3 / 6\right) \left(e ^ {\wedge} 0. 3 t\right) - t ^ {\wedge} 2 / 2 - t$$

where is the time in seconds. Use the Newton-Raphson method to estimate the time required for the bacteria concentration to reach 1 (correct up to 2 decimal places)

Solution

$$c = e ^ {t} - \frac {t ^ {3}}{6} e ^ {0. 3 t} - \frac {t ^ {2}}{2} - t = 1 \rightarrow e ^ {t} - \frac {t ^ {3}}{6} e ^ {0. 3 t} - \frac {t ^ {2}}{2} - t - 1 = 0;$$

Newton-Raphson method:

$$t _ {n + 1} = t _ {n} - \frac {f (t _ {n})}{f ^ {\prime} (t _ {n})}$$

Here $f(t) = e^{t} - \frac{t^{3}}{6} e^{0.3t} - \frac{t^{2}}{2} - t - 1, f^{\prime}(t) = e^{t} - \frac{t^{2}(t + 10)}{20} e^{0.3t} - t - 1$

Thus, the bacteria concentration will reach 1 in 2.36 sec.

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