91Ó°ÊÓ

A cylinder is a 3-dimensional shape with two parallel circular bases and a curved surface connecting these bases. The volume of a cylinder, which is the space inside it, is determined by its radius and height. The formula for the volume (V) is given by \( V = \pi r^2 h \). This formula is important because it connects the cylinder's radius (r) and height (h) to the volume. In exercises like this one, where the volume is fixed (V = 10 \text{ cm}^3) as in this problem, we can rearrange the formula to find the height in terms of the radius: \( h = \frac{V}{\pi r^2} \). This step is crucial for exploring how the surface area changes as the radius changes. So, when you see \( h = \frac{V}{\pi r^2} \), it simply means the height decreases as the radius increases, keeping the volume constant.

In mathematical terms, the behavior of a function as the variable approaches certain values can tell us a lot about the nature of the function. Here, we are interested in the limit behavior of the surface area \( S(r) = \frac{2V}{r} + 2\pi r^2 \) as the radius ( r ) becomes very large, or approaches infinity. When \( r \to \infty \), the term \( \frac{2V}{r} \) goes to 0 because the denominator \( r \) becomes very large, and therefore, the whole fraction becomes negligible. On the other hand, the term \( 2\pi r^2 \) will increase indefinitely. Thus, the surface area \( S \) will also increase without bound as \( r \to \infty \). This means, practically, as you make the radius very large, the contribution of the circular sides to the surface area dominates and continues to grow, implying a very large surface area, eventually approaching infinity.

Sketching graphs helps in visualizing mathematical relationships or functions, and it is especially useful in understanding how \( S \) changes with \( r \). With the surface area formula \( S(r) = \frac{20}{r} + 2\pi r^2 \) (where \( V = 10 \, \text{cm}^3 \)), we need to understand what happens at the extremes. Start by considering the limit behavior: as \( r \to 0, \frac{20}{r} \) becomes very large because you're dividing by a very small number, which drives \( S \) to positive infinity. Conversely, as \( r \to \infty \), \( 2\pi r^2 \) dominates making \( S \to \infty \). Hence, the shape of the graph will be such that there's a steep increase at small \( r \), it dips down, and then increases indefinitely, showing a minimum on the positive curve somewhere in between. To successfully sketch the graph:

• Note where it trends to infinity - both as \( r \to 0 \) and \( r \to \infty \).
• Identify the trough or minimum, indicating a certain optimal radius minimizing the surface area.
• Draw a smooth curve connecting these trends, visualizing the transition of \( S \) values over increasing \( r \).