a small island is 4 miles from the nearest point p on the straight shoreline of a large lake. if a woman on…

a small island is 4 miles from the nearest point p on the straight shoreline of a large lake. if a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 11 miles down the shore from p in the least time? the boat should be landed miles down the shore from p. (type an exact answer.)

a small island is 4 miles from the nearest point p on the straight shoreline of a large lake. if a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 11 miles down the shore from p in the least time? the boat should be landed miles down the shore from p. (type an exact answer.)

Answer

Explanation:

Step1: Define variables

Let $x$ be the distance down - shore from $P$ where the boat is landed. Then the distance rowed by boat $d_1=\sqrt{4^{2}+x^{2}}$ and the distance walked $d_2 = 11 - x$.

Step2: Calculate time functions

The time taken to row the boat $t_1=\frac{\sqrt{16 + x^{2}}}{3}$ and the time taken to walk $t_2=\frac{11 - x}{4}$. The total time function $T(x)=\frac{\sqrt{16 + x^{2}}}{3}+\frac{11 - x}{4}$.

Step3: Differentiate the time function

First, find the derivative of $y = \sqrt{16 + x^{2}}=(16 + x^{2})^{\frac{1}{2}}$. Using the chain - rule, $y^\prime=\frac{x}{\sqrt{16 + x^{2}}}$. Then $T^\prime(x)=\frac{x}{3\sqrt{16 + x^{2}}}-\frac{1}{4}$.

Step4: Set the derivative equal to zero

Set $T^\prime(x) = 0$, so $\frac{x}{3\sqrt{16 + x^{2}}}-\frac{1}{4}=0$. Cross - multiply to get $4x=3\sqrt{16 + x^{2}}$.

Step5: Square both sides

Square both sides: $(4x)^{2}=9(16 + x^{2})$. Expand to $16x^{2}=144 + 9x^{2}$.

Step6: Solve for x

Subtract $9x^{2}$ from both sides: $16x^{2}-9x^{2}=144$, so $7x^{2}=144$. Then $x^{2}=\frac{144}{7}$, and $x=\frac{12}{\sqrt{7}}=\frac{12\sqrt{7}}{7}$.

Answer:

$\frac{12\sqrt{7}}{7}$