a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a…

a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a restaurant. a woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. see parts (a) and (b) below. t = (sqrt(x^2 + 4))/2+(13 - x)/3 (type an expression.) the interval of interest of the objective function is 0,13. (simplify your answer. type your answer in interval notation.) to minimize the total travel time, the boat should land miles from the restaurant. (type an exact answer, using radicals as needed.)

a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a restaurant. a woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. see parts (a) and (b) below. t = (sqrt(x^2 + 4))/2+(13 - x)/3 (type an expression.) the interval of interest of the objective function is 0,13. (simplify your answer. type your answer in interval notation.) to minimize the total travel time, the boat should land miles from the restaurant. (type an exact answer, using radicals as needed.)

Answer

Explanation:

Step1: Differentiate the time - function

We have the time function $T(x)=\frac{\sqrt{x^{2}+4}}{2}+\frac{13 - x}{3}$. First, use the chain - rule to differentiate $\frac{\sqrt{x^{2}+4}}{2}$. Let $u = x^{2}+4$, then $\sqrt{u}=u^{\frac{1}{2}}$. The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2}u^{-\frac{1}{2}}$, and the derivative of $u$ with respect to $x$ is $2x$. So the derivative of $\frac{\sqrt{x^{2}+4}}{2}$ with respect to $x$ is $\frac{x}{2\sqrt{x^{2}+4}}$. The derivative of $\frac{13 - x}{3}$ with respect to $x$ is $-\frac{1}{3}$. Then $T^\prime(x)=\frac{x}{2\sqrt{x^{2}+4}}-\frac{1}{3}$.

Step2: Set the derivative equal to zero

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

Step3: Square both sides

Square both sides of the equation $3x = 2\sqrt{x^{2}+4}$: $(3x)^{2}=4(x^{2}+4)$. Expand to get $9x^{2}=4x^{2}+16$.

Step4: Solve for $x$

Subtract $4x^{2}$ from both sides: $9x^{2}-4x^{2}=16$, which simplifies to $5x^{2}=16$. Then $x^{2}=\frac{16}{5}$, and $x=\frac{4}{\sqrt{5}}=\frac{4\sqrt{5}}{5}$.

Step5: Find the distance from the restaurant

The distance from the restaurant is $13 - x$. Substitute $x=\frac{4\sqrt{5}}{5}$ into $13 - x$, we get $13-\frac{4\sqrt{5}}{5}=\frac{65 - 4\sqrt{5}}{5}$ miles.

Answer:

$\frac{65 - 4\sqrt{5}}{5}$