27. balloons and motorcycles a hot - air balloon is 150 ft above the ground when a motorcycle (traveling in…

Explanation:

Step1: Set up variables

Let (y) be the height of the balloon, (x) be the horizontal distance of the motorcycle from the point directly beneath the balloon initially, and (z) be the distance between the balloon and the motorcycle. By the Pythagorean theorem, (z^{2}=x^{2}+y^{2}).

Step2: Find (x) and (y) at (t = 10s)

The motorcycle's speed (v_{x}=58.67\ ft/s), so (x = 58.67\times10=586.7\ ft). The balloon's initial height (y_{0}=150\ ft) and its rising speed (v_{y}=10\ ft/s), so (y=150 + 10\times10=250\ ft).

Step3: Differentiate (z^{2}=x^{2}+y^{2}) with respect to (t)

Differentiating both sides with respect to (t) gives (2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}), or (\frac{dz}{dt}=\frac{x\frac{dx}{dt}+y\frac{dy}{dt}}{z}).

Step4: Find (z) at (t = 10s)

Using (z^{2}=x^{2}+y^{2}), (z=\sqrt{586.7^{2}+250^{2}}=\sqrt{344216.89 + 62500}=\sqrt{406716.89}\approx638\ ft).

Step5: Substitute values into (\frac{dz}{dt}) formula

Substitute (x = 586.7\ ft), (\frac{dx}{dt}=58.67\ ft/s), (y = 250\ ft), (\frac{dy}{dt}=10\ ft/s), and (z\approx638\ ft) into (\frac{dz}{dt}=\frac{x\frac{dx}{dt}+y\frac{dy}{dt}}{z}). (\frac{dz}{dt}=\frac{586.7\times58.67+250\times10}{638}=\frac{34421.689+2500}{638}=\frac{36921.689}{638}\approx57.9\ ft/s)

Answer:

The rate of change of the distance between the motorcycle and the balloon (10) seconds later is approximately (57.9\ ft/s)