Lesson 16

Minimizing Surface Area

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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?

Solution

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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?

Solution

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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?
graph of y = S of r. vertical asymptote at x = 0. function decreasing to the left of asymptote. function decreases until y = 16 and then increases to the right of asymptote.

Solution

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Problem 4

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

 

polynomial function graphed. x intercepts = -3, -1, 1, 3. y intercept = 9. f of x increases as x increases in both the positive and negative direction.

Solution

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(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.

Solution

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(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:

\(x^3-3x^2+3x-1\)

Solution

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(From Unit 2, Lesson 15.)