a rope is attached to the bottom of a hot - air balloon that is floating above a flat field. if the angle of…

Explanation:

Step1: Define variables

Let $y$ be the elevation of the balloon and $x$ be the length of the rope. We know $\sin\theta=\frac{y}{x}$, where $\theta = 55^{\circ}$.

Step2: Differentiate with respect to time $t$

Differentiating $\sin(55^{\circ})=\frac{y}{x}$ (since $\sin(55^{\circ})$ is a constant) with respect to $t$ using the quotient - rule. We get $0=\frac{x\frac{dy}{dt}-y\frac{dx}{dt}}{x^{2}}$. Cross - multiplying gives $x\frac{dy}{dt}-y\frac{dx}{dt}=0$, so $\frac{dy}{dt}=\frac{y}{x}\frac{dx}{dt}$. But $\frac{y}{x}=\sin(55^{\circ})$, and $\frac{dx}{dt}=- 5$ ft/s (negative because the length of the rope $x$ is decreasing).

Step3: Calculate $\frac{dy}{dt}$

Substitute $\sin(55^{\circ})\approx0.8192$ and $\frac{dx}{dt}=-5$ into $\frac{dy}{dt}=\sin(55^{\circ})\frac{dx}{dt}$. Then $\frac{dy}{dt}=0.8192\times(-5)=-4.096$.

Answer:

$-4.10$ ft/s