A kite maintains a constant altitude of 50 feet and is...

Name: Problem 13, Chapter 2: Applied Calculus For Business, Economics, and Finance
Uploaded: 2021-02-25T01:56:35-08:00
Duration: 4 min 33 s
Channel: Jacquelyn Trost
Description: A kite maintains a constant altitude of 50 feet and is moving horizontally at the rate of 2 feet per second. Assume that there is no slack in the kite-string, at what rate is the string being paid out when its length is 130 feet?

A kite maintains a constant altitude of 50 feet and is moving horizontally at the rate of 2 feet per second. Assume that there is no slack in the kite-string, at what rate is the string being paid out when its length is 130 feet?

Key Concepts

Related Rates

Related rates problems involve determining the rate at which one quantity changes in relation to another by using the relationship between the quantities. In such problems, variables that depend on time are linked together through geometric or physical equations, allowing the calculation of an unknown rate using known changes of other related quantities.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. In problems involving a kite or similar objects, the theorem helps to establish a relationship between the horizontal distance, the altitude (which is constant in some cases), and the length of the string (or hypotenuse), forming the basis of the equation to be differentiated.

Differentiation with Respect to Time

Differentiation with respect to time is used to translate the static relationship between variables into a dynamic context. By applying implicit differentiation to the geometric equation derived from the Pythagorean theorem, one can obtain a relationship between the rates of change of different quantities, thus facilitating the computation of the rate at which a particular length or distance is changing over time.

Transcript

00:01 So if this problem is looking at a kite, let's see if we can draw what's happening with our kite.

00:06 The more we can picture it, the better we can visualize how all of our variables relate to each other.

00:12 So we have the ground down here.

00:16 And way up here at the top, we have a kite.

00:19 Okay.

00:19 Our kite is attached to a string and the kite's moving along in this direction.

00:25 So you can see it's the kite's pulling away.

00:27 This is not a great picture.

00:29 It doesn't have to be.

00:30 You don't have to be a great artist, but draw something.

00:32 Sketch it so you can see where our variables relate.

00:36 Okay, let's see what we know.

00:38 The kite maintains a constant altitude of 50 feet.

00:42 Because that is an unchanging number, i can put that on our picture.

00:46 That height right there is 50.

00:47 And in that little red line just to show, we are 50 feet high.

00:52 And it's moving horizontally.

00:54 Let's call that distance x here that's moved over from where the string is on the ground.

01:00 It's moving at two feet per second.

01:03 So that's telling me how big or how fast x is moving.

01:07 X is changing at two feet per second.

01:10 And x is getting bigger because the kite is moving toward the right.

01:13 So every second, x is getting two feet larger, that distance.

01:18 So we know that.

01:20 There's no slack in the kite string.

01:22 So that's a good straight line there.

01:24 And the question we were being asked is, at what rate is the string being played out? so let's call this l for length.

01:33 The question is how fast is l changing when the length is 130 feet? now unlike the height of the kite, i'm not putting this number on our picture because it's a variable.

01:49 It's a changing number.

01:50 At this particular snapshot in time, it's 130 feet, but it's changing every second...

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