Section 1.8 A Shrinking Circle Problem
Subsection 1.8.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 1.8.2 Introduction
Limits may seem trivial when you first learn them, but they are fundamental building blocks in calculus. They are used to explain terms like “infinitesimally small” and “infinitely large”. It may be interesting to know that they can also be used in applied problems. This activity will explore a geometry problem and will solve it using limits, instead of an elementary geometric approach.
Figure 1.1 below shows two circles:
-
\(C_1\text{,}\) centred at the point \((1,0)\) with radius \(1\) and equation\begin{equation*} (x-1)^2+y^2=1\text{.} \end{equation*}
-
\(C_2\text{,}\) centred at the origin with radius \(r\) and equation\begin{equation*} x^2+y^2=r^2\text{.} \end{equation*}
If we define \(P\) as the point \((0,r)\) at the top of the circle \(C_2\text{,}\) and \(Q\) as the upper point of intersection of the two circles, then we can construct the line \(PQ\) and see that it crosses the \(x\)-axis. Let \(R\) be the \(x\)-intercept of the line \(PQ\text{.}\)
Aside
Now, begin to shrink the radius of circle \(C_2\text{;}\) that is, let \(r\rightarrow 0^{+}\text{.}\) What happens to the point \(R\) as \(C_2\) shrinks?
Exercises 1.8.3 Exercises
2.
Find the point of intersection of \(C_1\) and \(C_2\) in quadrant I. These are the coordinates of \(Q\) and should be expressions of \(r\text{.}\)
Hint 1.
You can find the point of intersection with a single
solve() command, using both equations and solving for \(x\) and \(y\) at once. See Section 10.4 for an example.
3.
Now that you have the coordinates of two points, \(P = (0,r)\) and \(Q\) (from the previous exercise), you can construct the equation of the line \(PQ\text{.}\)
(a)
Find the slope of the line \(PQ\) using the slope equation
\begin{equation*}
m = \frac{\Delta y}{\Delta x}\text{.}
\end{equation*}
(b)
Using the fact that the \(y\)-intercept is known to be \((0,r)\text{,}\) assign the equation of the line
\begin{equation*}
L(x) = m x + r\text{.}
\end{equation*}
4.
Find the \(x\)-coordinate of point \(R\) by solving \(L(x)=0\) (the \(x\)-intercept of this line). This coordinate should also be an expression involving \(r\text{.}\)
5.
Finally, you can determine what happens as the red circle shrinks. Determine the limit of \(R\) as \(r \to 0^+\text{.}\)
