Unit 3, Lesson 1
Minimizing Surface Area
Integrated Math 3
1.1
Warm-up
The Least Material
Here are four circular cylinders that have the same volume.
- Which cylinder needs the least material to build? Explain your reasoning.
- What information would be useful to know to determine which cylinder takes the least material to build?
Student Lesson Summary
Some relationships cannot be described by polynomial functions. For example, let’s think about the relationship between the radius
We know these formulas are true for all cylinders with radius
Since we are only interested in cylinders with a volume of 330 cm3, we can use the volume formula to rewrite the surface area formula as:
Can you see how? We can use the volume formula rearranged as
We now have an equation giving
Graph of a rational function with a point (3 point 46 comma 264 point 312), origin O. Horizontal axis labeled r, scale 0 to 8 by 2’s. Vertical axis labeled S, scale 0 to 600, by 200’s.
In this situation, the height of a cylinder with fixed volume varies inversely with the square of the radius,
rational function
A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.
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