Calc 3.7b
- 1. 3.7B OPTIMIZATION Go for it!
- 2. Ex 4 p. 221 Finding minimum length. Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire? x 30-x y z 28’ 12’ W = y + z Primary, W is wire x 2 +12 2 = y 2 Secondary (30 – x) 2 + 28 2 = z 2 Secondary Solve for y, solve for z, replace so W is an equation in terms of x.
- 3. W = y + z, x 2 +12 2 = y 2 (30 – x) 2 + 28 2 = z 2 Let W’ = 0 to get critical #’s Square both sides Stake at 9 feet from short pole
- 4. You can confirm your math by graphing the Primary equation and find minimum of W. In each of the previous examples, the extreme value occurred at a critical number. This often happens, but remember that an extreme value can occur at an endpoint of an interval as well, as shown in the next example. Ex 5 p. 222 An endpoint maximum Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much for the wire to enclose the maximum area?
- 5. x x Area = x 2 Perimeter = 4x r Area = π r 2 Circum = 2 π r Total Area = x 2 + π r 2 Primary Wire is 4 ft, so 4 = 4x + 2 π r Secondary Solving for r, Feasible domain is 0 ≤ x ≤ 1, controlled by perimeter of square
- 6. 0 ≤ x ≤ 1 If we evaluate A(0) ≈ 1.273, A(.56) ≈ .56, and A(1) = 1, we can see that the greatest area occurs when x = 0, or in other words, all the wire is used to form the circle, none is used to form a square!
- 7. 3.7b: p. 223/ 23, 27, 33, 39, 41, 43, 54, 59