What Is the Greatest Length for the Pathway of the Swing?

[flash_659]

Experiments are being carried out on a new ‘high-tech’ swing in a playground, the motion of which follows the model of

y=(e^(-ct))*cos(at)

Where y is the distance in meters from the equilibrium point of the swing, t is the time in minutes from midday on Sunday, and a, c are real constants

a=pi/6, c=0.04

so: y=(e^(0.04t))*cos((pi/6)t)

You are then asked to graph this on a CAS calculator in the domain of [0,20] of t.
Which i had no problem with.

The question then asks:

1. Find the greatest distance the swing travels in one direction, to the nearest centimeter.

I assumed this was the greatest y value in the given domain as y is the distance from the equilibrium point. So I moved onto Q2:

2. Find the greatest length for the pathway of the swing, traveling in one direction, to the nearest centimeter (note: the pathway of the swing is the Arc of the swing and can be found using CAS)

I am unsure of what the second question is asking, it seems exactly like the first question to me. Or is it asking for the diameter of the arc, would that be the integral (area under that part of the graph)?

 

[HallsofIvy]

I suspect you haven't given use the entire question. If this is a swing then it is moving in two dimensions. Is y measured horizontally or vertically? Is there a formula for the other direction?

 

[flash_659]

I just checked over it and that's the entire question, I should've noted that e is eulers number if you haven't noted that. As it is a swing, I assume that it is moving horizontally as it swings from side to side.

 

[Char. Limit]

That sounds like an arc length problem. The arc length of a curve f(x) from a point a to a point b is equal to:

$$\int_a^b \sqrt{1+\left(\frac{df}{dx}\right)^2} dx$$