Question

Solve the problem in Example 4 if the river is \(5{\rm{ km}}\) wide and point \(B\) is only \(5{\rm{ km}}\) downstream from \(A\).

Short answer

The girl should row the boat directly to \(B\).

Step-by-step solution

Optimization of a Function`

TheOptimization of a Function\(f\left( x \right)\)for any real value of\(c\)can be given as:

\(\begin{aligned}{l}f\left( x \right)\left| {_{x = c} \to \max \to \left\{ \begin{aligned}{l}{\rm{if }}f'\left( x \right) > 0\,\,\,\,\,\,\forall x < c\\{\rm{if }}f'\left( x \right) < 0\,\,\,\,\,\,\forall x > c\end{aligned} \right.} \right.\\{\rm{and}}\\f\left( x \right)\left| {_{x = c} \to \min \to \left\{ \begin{aligned}{l}{\rm{if }}f'\left( x \right) < 0\,\,\,\,\,\,\forall x < c\\{\rm{if }}f'\left( x \right) > 0\,\,\,\,\,\,\forall x > c\end{aligned} \right.} \right.\end{aligned}\)

Optimizing the Time

The obtained function from example 4 is:

\(T\left( x \right) = \frac{{\sqrt {{x^2} + 25} }}{6} + \frac{{5 - x}}{8},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \le x \le 5\)

For optimization, we have:

\(\begin{aligned}{c}\frac{{dT}}{{dx}} = 0\\\frac{d}{{dx}}\left( {\frac{{\sqrt {{x^2} + 25} }}{6} + \frac{{5 - x}}{8}} \right) = 0\\\frac{x}{{6\sqrt {{x^2} + 25} }} - \frac{1}{8} = 0\\8x = 6\sqrt {{x^2} + 25} \\64{x^2} = 36\left( {{x^2} + 25} \right)\\x = \frac{{15}}{{\sqrt 7 }}\end{aligned}\)

Now, we have:

Therefore, the function has no critical value.

Hence, the girl should row the boat directly to \(B\).