Question

an island is 7 km due north of its closest point p along a straight shoreline. a town is on the shoreline 8 km west of the point p. xavier can run at a rate of 8 km/h and ride a boat at a rate of 5 km/h. to get to the island, hes going to first run a certain distance then take a boat to complete the trip. if he wants to minimize the time it will take to get from the town to the island, what distance will he need to cover by boat? round to 3 decimal places.

Explanation:

Step1: Define variable

Let $x$ = distance (km) from point $P$ to the boat launch point along the shoreline. So, the running distance is $(8 - x)$ km, and the boat distance is $\sqrt{x^2 + 7^2} = \sqrt{x^2 + 49}$ km.

Step2: Set up time function

Time = distance / speed. Total time $T(x) = \frac{8 - x}{8} + \frac{\sqrt{x^2 + 49}}{5}$.

Step3: Find derivative of $T(x)$

Compute $T'(x) = -\frac{1}{8} + \frac{x}{5\sqrt{x^2 + 49}}$.

Step4: Set derivative to 0 (minimize time)

$$-\frac{1}{8} + \frac{x}{5\sqrt{x^2 + 49}} = 0$$
Rearrange to: $\frac{x}{5\sqrt{x^2 + 49}} = \frac{1}{8}$
Cross-multiply: $8x = 5\sqrt{x^2 + 49}$
Square both sides: $64x^2 = 25(x^2 + 49)$

Step5: Solve for $x$

Expand: $64x^2 = 25x^2 + 1225$
Subtract $25x^2$: $39x^2 = 1225$
Solve: $x^2 = \frac{1225}{39}$, so $x = \sqrt{\frac{1225}{39}} \approx 5.607$ km (we take positive $x$ as it is a distance)

Step6: Calculate boat distance

Substitute $x$ into boat distance formula: $\sqrt{x^2 + 49} = \sqrt{\frac{1225}{39} + 49}$
Simplify: $\sqrt{\frac{1225 + 1911}{39}} = \sqrt{\frac{3136}{39}} \approx \sqrt{80.4103} \approx 8.967$ km

Answer:

8.967 km