Question

A kite glides horizontally at an altitude of 20 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 30 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

          A kite glides horizontally at an altitude of 20 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 30 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

Show more…

A kite glides horizontally at an altitude of 20 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 30 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

Added by Michael S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jacob Fry

10/17/2022

Step 1

We have a right triangle formed by the ground (horizontal), the kite's string (hypotenuse), and the altitude of the kite (vertical). The angle θ is between the ground and the kite's string. Show more…

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A kite glides horizontally at an altitude of 20 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 30 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 θ 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 θ as a function of x. (b) What is the value of x at the moment in question? Give the exact value. (c) What is the value of dθ/dt at the same moment? Give the exact value, paying attention to the sign.

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Transcript

00:01 So in this question we're told that a kite is gliding horizontally.

00:06 Here's our kite.

00:08 It's gliding horizontally at an altitude of 20 metres above the ground.

00:14 Let's see we're down on the ground here.

00:17 We've got the string of the kite.

00:21 And we've got this angle, theta, between the kite and the horizontal.

00:27 Now we want to find the rate at which the angle is decreasing.

00:32 So d theta by d t.

00:35 When 30 metres of string has been unspooled.

00:40 So let's say that the kite is at a position x.

00:44 We're going to have x squared plus 20 squared equals 30 squared.

00:52 And the kite's horizontal velocity, v equals 2 meters per second.

01:00 Now tan theta is equal to y over x, but our y is 20 meters.

01:18 So we can express theta as a function of x, that theta is the inverse tan of 20 over x.

01:34 Now, okay, so that's the first part.

01:38 So now what is x at the moment in question? we have x squared equals 30 squared minus 20 squared.

01:47 So 30 squared minus 20 squared is 500.

01:55 So when we take the square root of that, we get that x equals 10 root 5...

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