Real-World Application: Cylinder Surface Area Functions
Cylinder surface area can be expressed as the sum of two functions: f(r) = 2πr² for the two circular bases and g(r) = 2πrh for the lateral surface — demonstrating function addition in a real-world context in enVision Algebra 1 Chapter 10 for Grade 11. For a cylinder with radius r = 3 inches and height h = 5 inches: f(3) = 18π and g(3) = 30π, giving total surface area (f+g)(3) = 48π ≈ 150.72 square inches. This application reinforces how combining functions models physical quantities, and shows how domain restrictions arise naturally — radius must be positive.
Key Concepts
A cylinder is a solid figure with two parallel circular bases of the same size. For a cylinder with radius $r$ and height $h$, the surface area can be expressed as the sum of two functions:.
Surface Area: $S = 2\pi r^2 + 2\pi rh$.
Common Questions
What are the two functions that make up cylinder surface area?
f(r) = 2πr² represents the area of the two circular bases combined, and g(r) = 2πrh represents the lateral (side) surface area. Total surface area S = f(r) + g(r) = 2πr² + 2πrh.
How do you find the surface area of a cylinder with r = 3 and h = 5?
f(3) = 2π(9) = 18π, g(3) = 2π(3)(5) = 30π. Total = (f+g)(3) = 48π ≈ 150.72 square inches.
Why is the domain of the surface area function restricted to r > 0?
A cylinder must have a positive radius to exist physically. A negative or zero radius has no geometric meaning, so the domain is r > 0.
How does this model illustrate function addition?
The total surface area is computed by adding two separate functions evaluated at the same radius r. This is the definition of (f+g)(r) = f(r) + g(r).
If height h doubles while radius stays constant, how does the surface area change?
Only the lateral term g(r) = 2πrh doubles. The base term f(r) = 2πr² is unchanged. So total surface area increases, but not by a full factor of 2 because the base area stays the same.