# A plane flying with a constant speed of 14 km/min passes over a ground radar station at an altitude of 13 km and climbs at an angle of 20 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 5 minutes later?

the plane's velocity is

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To find the rate at which the distance from the plane to the radar station is increasing 5 minutes later, we use the concept of related rates. We'll need to use trigonometry and the Pythagorean theorem to solve this problem.

Let x be the horizontal distance from the plane to the radar station. Then, the distance from the plane to the radar station at any time t is given by ( \sqrt{x^2 + 13^2} ).

Since the plane is flying with a constant speed of 14 km/min, after 5 minutes, the horizontal distance x covered by the plane is ( 14 \times 5 = 70 ) km.

Now, we can differentiate the equation ( \sqrt{x^2 + 13^2} ) with respect to time t to find the rate at which the distance is increasing:

[ \frac{d}{dt} \sqrt{x^2 + 13^2} = \frac{1}{2} (x^2 + 13^2)^{-1/2} \cdot 2x \cdot \frac{dx}{dt} ]

Substituting the values we have:

[ \frac{dx}{dt} = \frac{1}{\sqrt{70^2 + 13^2}} \cdot 70 \cdot 14 ]

[ \frac{dx}{dt} ≈ \frac{70 \times 14}{\sqrt{70^2 + 13^2}} ]

[ \frac{dx}{dt} ≈ \frac{980}{\sqrt{4900 + 169}} ]

[ \frac{dx}{dt} ≈ \frac{980}{\sqrt{5079}} ]

[ \frac{dx}{dt} ≈ \frac{980}{71.30} ]

[ \frac{dx}{dt} ≈ 13.75 \text{ km/min} ]

So, the distance from the plane to the radar station is increasing at a rate of approximately 13.75 km/min 5 minutes later.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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