Math SL Blackford 1. Constraint: ⋅ = 192 Optimizing: + 3 =

Calculus/Math SL Blackford 1. Constraint: 𝑥 ⋅ 𝑦 = 192 Optimizing: 𝑥 + 3𝑦 = 𝑀

192 𝑦 192 𝑀= + 3𝑦 𝑦 192 𝑀′ = − 2 + 3 𝑦 −192 + 3𝑦 2 = ; 𝑥 = ±8 𝑦2 The numbers that minimize the equation are 8 and 24. 𝑥=

2. Constraint: 𝑥 + 2𝑦 = 100 Optimizing: 𝑃 = 𝑥 ⋅ 𝑦 𝑥 = 100 − 2𝑦 𝑃 = (100 − 2𝑦)(𝑦) = 100𝑦 − 2𝑦 2 ′ 𝑃 = 100 − 4𝑦; 𝑦 = 25 The numbers that maximize the product are 25 and 50 3. Constraint: 𝑥 + 2𝑦 = 120 Optimizing: 𝐴 = 𝑥 ⋅ 𝑦 𝑥 = 120 − 2𝑦 𝐴 = (120 − 2𝑦)(𝑦) = 120𝑦 − 2𝑦 2 𝐴′ = 120 − 4𝑦; 𝑦 = 30 The dimensions that maximize the area are 60 feet parallel to the wall and a width of 30 feet. 4. Constraint: 2𝑥 + 2𝑦 = 80 Optimizing: 𝐴 = 𝑥 ⋅ 𝑦 𝑦 = 40 − 𝑥 𝐴 = 𝑥(40 − 𝑥) = 40𝑥 − 𝑥 2 𝐴′ = 40 − 2𝑥; 𝑥 = 20 The dimensions that maximize the area are 20 feet by 20 feet.

Calculus/Math SL Blackford 5. Constraint: 2𝑥 + 3𝑦 = 102 Optimizing: 𝐴 = 𝑥 ⋅ 𝑦

3 𝑥 = − 𝑦 + 51 2 3 𝐴 = (− 𝑦 + 51) (𝑦) 2 3 2 = − 𝑦 + 51𝑦 2 𝐴′ = −3𝑦 + 51; 𝑦 = 17 𝟓𝟏

The maximum area is (𝟏𝟕) ( 𝟐 ) = 𝟒𝟑𝟑. 𝟓 𝒎𝟐