Question

alicia is staying in a cottage along a beautiful and straight shoreline. a point q on the shoreline is located 4 kilometers east of the cottage, and an island is located 3 kilometers north of q (see image below). alicia plans to travel from the cottage to the island by some combination of walking and swimming. she can start to swim at any point p between the cottage and the point q. if she walks at a rate of 7 km/hr and swims at a rate of 5 km/hr, what is the minimum possible time it will take alicia to reach the island? enter an exact answer or round to the nearest hundredth of an hour.

Explanation:

Step1: Set up variables

Let the distance she walks along the shore be $x$ km. Then the distance she swims is $\sqrt{(4 - x)^2+3^2}$ km. The time $T$ is the sum of the walking - time and the swimming - time. The walking time $t_1=\frac{x}{7}$ and the swimming time $t_2=\frac{\sqrt{(4 - x)^2 + 9}}{5}$. So, $T(x)=\frac{x}{7}+\frac{\sqrt{(4 - x)^2+9}}{5}$.

Step2: Differentiate the time function

First, expand $(4 - x)^2=16-8x + x^2$. Then $y=\sqrt{16-8x + x^2+9}=\sqrt{x^2-8x + 25}$. Using the chain - rule, the derivative of $y$ with respect to $x$ is $y^\prime=\frac{2x - 8}{2\sqrt{x^2-8x + 25}}=\frac{x - 4}{\sqrt{x^2-8x + 25}}$. The derivative of $T(x)$ is $T^\prime(x)=\frac{1}{7}+\frac{x - 4}{5\sqrt{x^2-8x + 25}}$.

Step3: Set the derivative equal to zero

Set $T^\prime(x)=0$, then $\frac{1}{7}+\frac{x - 4}{5\sqrt{x^2-8x + 25}}=0$. Cross - multiply to get $5\sqrt{x^2-8x + 25}+7(x - 4)=0$. Rearrange to $5\sqrt{x^2-8x + 25}=-7(x - 4)$. Square both sides: $25(x^2-8x + 25)=49(x^2-8x + 16)$. Expand: $25x^2-200x + 625=49x^2-392x + 784$. Combine like terms: $24x^2-192x + 159 = 0$. Divide by 3: $8x^2-64x + 53 = 0$. Using the quadratic formula $x=\frac{64\pm\sqrt{64^2-4\times8\times53}}{2\times8}=\frac{64\pm\sqrt{4096 - 1696}}{16}=\frac{64\pm\sqrt{2400}}{16}=\frac{64\pm20\sqrt{6}}{16}=4\pm\frac{5\sqrt{6}}{4}$. We take the value of $x$ in the interval $[0,4]$, so $x = 4-\frac{5\sqrt{6}}{4}$.

Step4: Calculate the minimum time

Substitute $x = 4-\frac{5\sqrt{6}}{4}$ into the time function $T(x)$.
$T=\frac{4-\frac{5\sqrt{6}}{4}}{7}+\frac{\sqrt{(4-(4 - \frac{5\sqrt{6}}{4}))^2+9}}{5}$
$=\frac{4-\frac{5\sqrt{6}}{4}}{7}+\frac{\sqrt{\frac{150}{16}+9}}{5}=\frac{4-\frac{5\sqrt{6}}{4}}{7}+\frac{\sqrt{\frac{150 + 144}{16}}}{5}=\frac{4-\frac{5\sqrt{6}}{4}}{7}+\frac{\sqrt{\frac{294}{16}}}{5}=\frac{4-\frac{5\sqrt{6}}{4}}{7}+\frac{\frac{7\sqrt{6}}{4}}{5}$
$=\frac{20 - 5\sqrt{6}}{28}+\frac{7\sqrt{6}}{20}=\frac{5(20 - 5\sqrt{6})+7\times7\sqrt{6}}{140}=\frac{100-25\sqrt{6}+49\sqrt{6}}{140}=\frac{100 + 24\sqrt{6}}{140}\approx1.03$ hours.

Answer:

$1.03$ hours