On the right side of the equation below, we seek an expression for how long it would require the ant to take a short lenfth h of soil and then carry it a distance x to the mouth of the tunnel. T(x+h)-T(x) = Notice that T(x + h) – T(x) ‡ T(h), for T(h) represents the time it takes to extend the tunnel a distance h from the mouth of the tunnel (at x = : 0), while T(x + h) — T(x) includes this time plus the time it takes to carry the soil to the mouth of the tunnel. Below are a few possibilities for the right side of the above equation. Defend or reject each and offer your reasons. One idea to consider in trying to reject a model consider some extreme cases and see
On the right side of the equation below, we seek an expression for how long it would require the ant to take a short lenfth h of soil and then carry it a distance x to the mouth of the tunnel. T(x+h)-T(x) = Notice that T(x + h) – T(x) ‡ T(h), for T(h) represents the time it takes to extend the tunnel a distance h from the mouth of the tunnel (at x = : 0), while T(x + h) — T(x) includes this time plus the time it takes to carry the soil to the mouth of the tunnel. Below are a few possibilities for the right side of the above equation. Defend or reject each and offer your reasons. One idea to consider in trying to reject a model consider some extreme cases and see
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. On the right side of the equation below, we seek an expression for how long it would require the ant
to take a short lenfth h of soil and then carry it a distance x to the mouth of the tunnel.
T(x+h)-T(x)
Notice that T(x + h) − T(x) ‡ T(h), for T(h) represents the time it takes to extend the tunnel a
distance h from the mouth of the tunnel (at x = 0), while T(x + h) - T(x) includes this time plus
the time it takes to carry the soil to the mouth of the tunnel.
=
Below are a few possibilities for the right side of the above equation. Defend or reject each and offer
your reasons. One idea to consider in trying to reject a model consider some extreme cases and see
if the model makes sense, e.g., h = 0 or x = 0, or either h or x very large. You might have other
reasons.
(a) T(x+h) — T(x) = x+h
(b) T(x+h) – T(x) = x - h
(c) T(x + h) − T(x) = xh
(d) T(x+h) - T(x) = x * h
(e) T(x+h) − T(x) = hª
(f) T(x+h)-T(x) = c
Transcribed Image Text:Ants are extremely industrious and tireless tunnel builders. How long does it take an ant to build a single
tunnel? To address the issue we need to narrow the scope of the problem, simplify, and identify some terms
and parameters.
To begin, let's define a few crucial variables. Consider a single ant digging a tunnel into a hillside as
illustrated in Figure 1. Let x denote the current length, in feet, of the tunnel that the ant is digging. Let
T(x) be the time, in hours, it has taken the ant to build the tunnel of length x. This is a rare case in this
class where the independent variable is not time!
Tunnel
Figure 1: Sketch of ant building a tunnel. Here x is the current length of the tunnel and T(x) is the time
it has taken the ant to build the tunnel of length x.
E+
h
X-
Instead of defining T(x) directly, we'll instead consider Figure 2, a depiction of the essential aspects of
the situation. When an ant digs a tunnel, the ant must extend the tunnel incrementally, by removing soil
between coordinates x and x + h, carrying this soil back to the tunnel entrance, and then returning to
remove more soil. Let us consider how long it takes the ant to complete the extension of the tunnel from
length x to length x + h.
Tunnel
Removed
Soil
Figure 2: A useful diagram for modeling the time it takes to extend a small section of the tunnel from
distance x to x + h.
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