Question

The current in an electric circuit is given by
i=t sin t
where t is the time in seconds. Using the bisection method, estimate the time
required for the current to reach 1 amp (correct up to 2 decimal places)

Answer 1

Answer on Question #51400 – Math – Algorithms | Quantitative Methods

The current in an electric circuit is given by $i = t \sin t$ where $t$ is the time in seconds. Using the bisection method, estimate the time required for the current to reach 1 amp (correct up to 2 decimal places)

Solution

We must solve the equation $t \sin t - 1 = 0$ using the bisection method.

The input for the method is a continuous function $f$, an interval $[a, b]$, and the function values $f(a)$ and $f(b)$. The function values are of opposite sign (there is at least one zero crossing within the interval). Each iteration performs these steps:

1. Calculate $c$, the midpoint of the interval, $c = 0.5 * (a + b)$.

2. Calculate the function value at the midpoint, $f(c)$.

3. If convergence is satisfactory (that is, $a - c$ is sufficiently small, or $f(c)$ is sufficiently small), return $c$ and stop iterating.

4. Examine the sign of $f(c)$ and replace either $(a, f(a))$ or $(b, f(b))$ with $(c, f(c))$ so that there is a zero crossing within the new interval.

Let's start with values of $a = 0$ and $b = 2$.

$$\begin{array}{l} f(t) = t \sin t - 1 \\ f(0) = -1 \\ f(2) = 0.818595 \\ \end{array}$$

So, for example, on the first iteration we get $f(c_n) = -0.158529$ and so must replace the value of the left endpoint of interval $a$ with $c_1 = 1$ hence narrowing the interval.

Iteration process is represented by the following table:

We see that root of the equation $t \sin t - 1 = 0$ is 1.11524 with accuracy up to two decimal places.

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