an airplane is flying at an elevation of 3200 ft. a man is 2200 ft from the base of a radio tower. the…

an airplane is flying at an elevation of 3200 ft. a man is 2200 ft from the base of a radio tower. the airplane flew over the man heading in the direction of the radio tower. if the plane is flying at a rate of 400 ft/sec, how fast is the distance between the man and the plane increasing when the plane passes directly over the radio tower? round your answer to three decimal places.

an airplane is flying at an elevation of 3200 ft. a man is 2200 ft from the base of a radio tower. the airplane flew over the man heading in the direction of the radio tower. if the plane is flying at a rate of 400 ft/sec, how fast is the distance between the man and the plane increasing when the plane passes directly over the radio tower? round your answer to three decimal places.

Answer

Explanation:

Step1: Establish the distance - relationship

Let $x$ be the horizontal distance of the plane from the man, $y = 3200$ ft be the elevation of the plane (constant), and $z$ be the distance between the man and the plane. By the Pythagorean theorem, $z^{2}=x^{2}+y^{2}=x^{2}+3200^{2}$.

Step2: Differentiate with respect to time $t$

Differentiating both sides of the equation $z^{2}=x^{2}+3200^{2}$ with respect to $t$, we get $2z\frac{dz}{dt}=2x\frac{dx}{dt}$. Then $\frac{dz}{dt}=\frac{x}{z}\cdot\frac{dx}{dt}$.

Step3: Find the values of $x$, $z$ when the plane is over the tower

When the plane passes directly over the radio - tower, $x = 2200$ ft. And $z=\sqrt{x^{2}+y^{2}}=\sqrt{2200^{2}+3200^{2}}=\sqrt{4840000 + 10240000}=\sqrt{15080000}=20\sqrt{37700}\approx3908.964$ ft. We know that $\frac{dx}{dt}=- 400$ ft/sec (negative because $x$ is decreasing as the plane moves towards the tower).

Step4: Calculate $\frac{dz}{dt}$

Substitute $x = 2200$, $z\approx3908.964$, and $\frac{dx}{dt}=-400$ into $\frac{dz}{dt}=\frac{x}{z}\cdot\frac{dx}{dt}$. So $\frac{dz}{dt}=\frac{2200}{3908.964}\times(-400)\approx - 225.129$ ft/sec. The negative sign just indicates the direction of change. The speed (magnitude) is approximately $225.129$ ft/sec.

Answer:

$225.129$