A kite glides horizontally at an altitude of 30 m while we...

A kite glides horizontally at an altitude of 30 m while we unspool the string. Consequently, the angle made between the string and the horizon diminishes. We would like to determine the rate at which this angle decreases once 40 m of string has been unspooled, given that, at that instant, the kite's horizontal velocity is 1 m/s. To solve this problem, let ? be the angle in radians made between the string and the horizontal, x the kite's horizontal position in meters since being attached to the ground, and t the time in seconds. We further suppose that the string is straight and taut. (a) Sketch a diagram of this question and use it to express ? as a function of x . ? = rad (b) What is the value of x at the moment in question? Give the exact value. x = m (c) What is the value of d?/dt at the same moment? Give the exact value, paying attention to the sign. d?/dt = rad/s

Transcript

00:01 This question, it says that a kite glides horizontally at an altitude of 30 meters while we unspool the string.

00:09 And consequently, the angle made between the string and the horizon diminishes.

00:14 Okay? we would like to remind the rate at which the angle decreases once 40 meter of string has been unspooled, given that at that instant the kite is moving horizontally with one meter per second.

00:26 So if we draw a rough sketch over here, this is where.

00:30 We are standing and this is where the kite is and this kite moves horizontally like this and as the kite moves horizontally definitely the angle which it makes with the horizontal keeps on decreasing because the kite goes on goes towards this and the angle keeps decreasing all right so at one point in time this is already given as 30 meters let's say this at one particular point is x and this at one particular point is y so we have to uh okay theta is given in radiance and the angle between uh and the x is the right's horizontal position uh in the meters being attached to the ground uh okay and t's in second so sketch a diagram for this equation and use theta to express as a function of x all right so definitely we can uh use theta as a function of x what is theta theta is the angle made between the string and the horizontal which is definitely what we have taken so this is also theta this is also 30 and this is also x so theta can definitely be written in terms of x using tan function because tan theta is nothing but opposite or adjacent so the value of theta is nothing but the inverse tangent or tan inverse of 30 over x this is part a part b is what is the value of x at the moment in this question so it is asking that what is the value of x when uh this the 40 meters of string has been unspooled it means that this y is 40 meters this y is 40 meters because that's that's what is being unspooled and now we can use the pythagorean pythagoras theorem because we know that y square is equal to x square plus 30 square and y is given us 40 so we have 40 square is equal to x square plus 30 square which means that the value of x square is 40 square minus 30 square.

02:34 So this is 1600 minus 900, which is nothing but 700 as the value of x is square, but we are interested in the value of x.

02:46 So that's root of 700.

02:48 If you want to simplify this perhaps this will become 700 can be written as seven times 100 because why we are writing it like this? because the hundred square root we know that is 10.

02:57 So 10 comes out and we have 10 root 7 as the value of x...

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