a hunter is at a point on a riverbank. he wants to get to his cabin, located 3 miles north and 8 miles west…

Explanation:

Step1: Define variables and time - function

Let the distance the hunter travels along the riverbank be $x$ miles. Then the distance he travels on the rough - rocky ground is $\sqrt{3^{2}+(8 - x)^{2}}$ miles. The time taken on the riverbank $t_1=\frac{x}{5}$ and the time taken on the rough - rocky ground $t_2=\frac{\sqrt{9+(8 - x)^{2}}}{3}$. The total time function $T(x)=\frac{x}{5}+\frac{\sqrt{9+(8 - x)^{2}}}{3}$.

Step2: Differentiate the time - function

First, rewrite the function as $T(x)=\frac{1}{5}x+\frac{1}{3}(9+(8 - x)^{2})^{\frac{1}{2}}$. Using the sum rule and chain rule of differentiation, $T^\prime(x)=\frac{1}{5}+\frac{1}{3}\times\frac{1}{2}(9+(8 - x)^{2})^{-\frac{1}{2}}\times2(8 - x)\times(- 1)=\frac{1}{5}-\frac{8 - x}{3\sqrt{9+(8 - x)^{2}}}$.

Step3: Set the derivative equal to zero

Set $T^\prime(x) = 0$, then $\frac{1}{5}-\frac{8 - x}{3\sqrt{9+(8 - x)^{2}}}=0$. Cross - multiply to get $3\sqrt{9+(8 - x)^{2}}=5(8 - x)$. Square both sides: $9(9+(8 - x)^{2}) = 25(8 - x)^{2}$. Expand: $81+9(64-16x+x^{2})=25(64 - 16x+x^{2})$. $81 + 576-144x+9x^{2}=1600-400x + 25x^{2}$. Rearrange to get $16x^{2}-256x + 943 = 0$.

Step4: Solve the quadratic equation

Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $16x^{2}-256x + 943 = 0$, where $a = 16$, $b=-256$, and $c = 943$. $x=\frac{256\pm\sqrt{(-256)^{2}-4\times16\times943}}{2\times16}=\frac{256\pm\sqrt{65536-60352}}{32}=\frac{256\pm\sqrt{5184}}{32}=\frac{256\pm72}{32}$. We get two solutions: $x_1=\frac{256 + 72}{32}=\frac{328}{32}=10.25$ (rejected since $x\leqslant8$) and $x_2=\frac{256 - 72}{32}=\frac{184}{32}=5.75$.

Answer:

B. 5.75