An aircraft is climbing at a 30 degree angle to the horizontal. how fast is the aircraft gaining altitude if it's speed is 500 miles per hour?

The speed rate at which the aircraft is gaining altitude is;

dx/dt = 250 miles per hour

Since the aircraft is climbing at 30° to the horizontal. it means there will be a vertical component which will be the altitude and also a horizontal component which will be the horizontal distance covered. while the hypotenuse will be how far it has climbed.

Let us call the vertical component x and the hypotenuse component z.

Thus, from trigonometric ratios, we can say that;

sin 30 = x/z

We are looking for how fast the aircraft is gaining altitude. Thus, using Liebniz notation, we can say this rate is; dx/dt. Thus;

sin 30 = (dx/dt)/(dz/dt)

0.5 = (dx/dt)/(dz/dt)

0.5 × (dz/dt) = dx/dt

Its' speed is 500 mph. This means dz/dt = 500

Thus;

dx/dt = 0.5 × 500

dx/dt = 250 mph

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The aircraft is gaining altitude at

Explanation:

Draw a vector diagram. (Right angle triangle speed as the hypotenuse)

Let

x -----> horizontal component of the speed

y ----> vertical component of the speed

s ----> speed in miles per hour

we know that

we have

substitute

therefore

The aircraft is gaining altitude at

yes he does : )