Question

a 50 foot rope on a dock is connected to a boat at a 35° angle with the boat. how far away is the boat from the dock? ? feet round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle situation where the rope is the hypotenuse ($c = 50$ feet) and we want to find the side opposite the given angle $\theta=35^{\circ}$. We use the sine function, $\sin\theta=\frac{opposite}{hypotenuse}$.

Step2: Set up the equation

Let $x$ be the distance between the boat and the dock (opposite side). So, $\sin(35^{\circ})=\frac{x}{50}$.

Step3: Solve for $x$

Multiply both sides of the equation by 50: $x = 50\times\sin(35^{\circ})$.
We know that $\sin(35^{\circ})\approx0.5736$. Then $x = 50\times0.5736 = 28.68$ feet.

Answer:

$28.68$ feet