# Lesson 16

Minimizing Surface Area

• Let’s investigate surface areas of different cylinders.

### Problem 1

There are many cylinders with a volume of $$144\pi$$ cubic inches. The height $$h(r)$$ in inches of one of these cylinders is a function of its radius $$r$$ in inches where $$h(r)=\frac{144}{r^2}$$.

1. What is the height of one of these cylinders if its radius is 2 inches?
2. What is the height of one of these cylinders if its radius is 3 inches?
3. What is the height of one of these cylinders if its radius is 6 inches?

### Problem 2

The surface area $$S(r)$$ in square units of a cylinder with a volume of 18 cubic units is a function of its radius $$r$$ in units where $$S(r)=2\pi r^2+\frac{36}{r}$$. What is the surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units?

### Problem 3

Han finds an expression for $$S(r)$$ that gives the surface area in square inches of any cylindrical can with a specific fixed volume, in terms of its radius $$r$$ in inches. This is the graph Han gets if he allows $$r$$ to take on any value between -1 and 5.

1. What would be a more appropriate domain for Han to use instead?
2. What is the approximate minimum surface area for the can?

### Problem 4

The graph of a polynomial function $$f$$ is shown. Is the degree of the polynomial even or odd? Explain your reasoning.

(From Unit 2, Lesson 8.)

### Problem 5

The polynomial function $$p(x)=x^4+4x^3-7x^2-22x+24$$ has known factors of $$(x+4)$$ and $$(x-1)$$.

1. Rewrite $$p(x)$$ as the product of linear factors.
2. Draw a rough sketch of the graph of the function.
(From Unit 2, Lesson 12.)

### Problem 6

Which polynomial has $$(x+1)$$ as a factor?

A:

$$x^3+2x^2-19x-20$$

B:

$$x^3-21x+20$$

C:

$$x^3+8x+11x-20$$

D:

$$x^3-3x^2+3x-1$$

(From Unit 2, Lesson 15.)