Question

Limit of the radius of a cylinder A right circular cylinder with a height of \(10 \mathrm{cm}\) and a surface area of \(S \mathrm{cm}^{2}\) has a radius given by $$r(S)=\frac{1}{2}(\sqrt{100+\frac{2 S}{\pi}}-10)$$ Find \(\lim _{S \rightarrow 0^{+}} r(S)\) and interpret your result.

Short answer

Answer: The limit of the function r(S) as S approaches 0 from the positive side is 0.

Step-by-step solution

Recall the concepts of the limit

To find the limit of a function as its variable tends to a particular value (in this case, \(S\rightarrow 0^{+}\)), we try to check what happens to the function as the variable gets closer and closer to that value.

Analyze the given function

We have the function $$ r(S) =\frac{1}{2}\left(\sqrt{100+\frac{2 S}{\pi}}-10\right) $$ We need to find the limit of this function as \(S\) approaches 0 from the positive side. Mathematically, it is expressed as: $$ \lim _{S \rightarrow 0^{+}} r(S) $$

Substitute the value of S and calculate the limit

Now, we will replace \(S\) with zero in the function r(S) and simplify the expression: $$ \begin{aligned} \lim _{S \rightarrow 0^{+}} r(S) &=\lim _{S \rightarrow 0^{+}} \frac{1}{2}\left(\sqrt{100+\frac{2 S}{\pi}}-10\right) \\ &= \frac{1}{2}\left(\sqrt{100+\frac{2 (0)}{\pi}}-10\right) \\ &= \frac{1}{2}\left(\sqrt{100}-10\right) \\ &= \frac{1}{2}(10 - 10) \\ &= 0 \end{aligned} $$ So, the limit of the function r(S) as \(S\) approaches 0 from the positive side is 0.

Interpret the result

The result we just found is $$ \lim _{S \rightarrow 0^{+}} r(S) = 0 $$ This means that as the surface area of the right circular cylinder approaches 0, its radius also approaches 0. In other words, the cylinder becomes like a thin disk or a flat sheet of negligible thickness when the surface area is very close to zero.

Key concepts

Surface Area

The surface area of a geometrical shape refers to the total area covered by its surface. For a right circular cylinder, the surface area is calculated by adding the areas of the two circular bases and the rectangular side. The formula for the surface area of a cylinder is given by \[S = 2\pi r (r + h)\]where \( r \) is the radius and \( h \) is the height of the cylinder.

• The term \( 2\pi r^2 \) represents the area of the two circular bases.
• The term \( 2\pi rh \) accounts for the side of the cylinder, which is essentially a rectangle wrapped around the circular bases.

Understanding this formula helps in visualizing how a cylinder's surface is composed and how changing either the radius or height affects the total surface area. When solving problems involving limits, knowing how surface area relates to other variables in the problem is essential. As seen, the exercise explores how the surface area influences the radius of a cylinder.

Cylinder Radius

The cylinder radius is the distance from the center of a circular base to its edge. It is a key parameter in defining the cylinder's size and determining other properties, such as volume and surface area. In the context of limits, examining how the cylinder radius reacts as other parameters (like surface area) change can provide insight into the cylinder's behavior.

For the given function of radius:\[r(S) = \frac{1}{2}(\sqrt{100+\frac{2S}{\pi}}-10)\]we analyze how the radius changes when \( S \), the surface area, approaches zero. By substituting \( S = 0 \), we determine that the effective radius tends toward zero. This means:

• As surface area reduces, the cylinder's circular bases shrink.
• Ultimately, this results in a reduction in overall size, converging to a flat disk.

This interpretation connects the mathematical calculation to a physical understanding of the changing geometry of the cylinder.

Mathematical Functions

Mathematical functions are expressions that relate a set of inputs to a set of possible outputs, with each input being related to exactly one output. In calculus, functions often describe how quantities change over different variables, such as distance, time or, in our exercise, surface area.

The function \( r(S) \) in our exercise describes how the radius of the cylinder changes as the surface area \( S \) varies. Limits are used to analyze the behavior of these functions as variables approach specific values. We calculated:\[\lim _{S \rightarrow 0^{+}} r(S) = 0\]which means the radius becomes zero as the surface area approaches zero. This is a clear example of using limits to make predictions about the behavior of functions:

• They help evaluate what happens to a function as its input nears a particular point.
• Provide insight into continuity and the behavior of functions in constrained scenarios.

Understanding the function and its limit enables us to predict physical outcomes in scenarios involving shrinking or expanding geometries.