Scale: \( 3 \times 100 = \mathbf{300\text{ kits}} \)
Answer: 300 kits/quarter
PART B: MAXIMUM PROFIT
\( P(3) = -2(9) + 12(3) + 14 = 32 \)
Scale: \( 32 \times \$10,000 = \mathbf{\$320,000} \)
Answer: $320,000 quarterly profit
Optimal Profit Answer Key • Page 1 of 3 (Questions 1–5) Continue to Page 2 for Solutions 6–10 →
Optimal Profit Answer Key: Commercial Goods & Tech Products (Part 2 of 3)
Teacher Key | Page 2
6 Titan Drone Technologies • \( P = -5x^2 + 40x - 35 \)
\(P\) in $1,000s | \(x\) in 100 drones
PART A: PRODUCTION LEVEL
\( x = -\frac{40}{2(-5)} = \frac{-40}{-10} = 4 \)
Scale: \( 4 \times 100 = \mathbf{400\text{ drones}} \)
Answer: 400 drones/week
PART B: MAXIMUM PROFIT
\( P(4) = -5(16) + 40(4) - 35 = 45 \)
Scale: \( 45 \times \$1,000 = \mathbf{\$45,000} \)
Answer: $45,000 weekly profit
7 Zenith Audio Systems • \( P = -2x^2 + 20x - 18 \)
\(P\) in $100s | \(x\) in 1,000 headphones
PART A: PRODUCTION LEVEL
\( x = -\frac{20}{2(-2)} = \frac{-20}{-4} = 5 \)
Scale: \( 5 \times 1,000 = \mathbf{5,000\text{ headphones}} \)
Answer: 5,000 headphones/month
PART B: MAXIMUM PROFIT
\( P(5) = -2(25) + 20(5) - 18 = 32 \)
Scale: \( 32 \times \$100 = \mathbf{\$3,200} \)
Answer: $3,200 monthly profit
8 Lumina Lighting Co. • \( P = -x^2 + 8x + 9 \)
\(P\) in $100s | \(x\) in 100 fixtures
PART A: PRODUCTION LEVEL
\( x = -\frac{8}{2(-1)} = \frac{-8}{-2} = 4 \)
Scale: \( 4 \times 100 = \mathbf{400\text{ fixtures}} \)
Answer: 400 fixtures/day
PART B: MAXIMUM PROFIT
\( P(4) = -(16) + 8(4) + 9 = 25 \)
Scale: \( 25 \times \$100 = \mathbf{\$2,500} \)
Answer: $2,500 daily profit
9 Kinetix Fitness Equipment • \( P = -3x^2 + 18x - 15 \)
\(P\) in $1,000s | \(x\) in 100 rowers
PART A: PRODUCTION LEVEL
\( x = -\frac{18}{2(-3)} = \frac{-18}{-6} = 3 \)
Scale: \( 3 \times 100 = \mathbf{300\text{ rowers}} \)
Answer: 300 rowers/week
PART B: MAXIMUM PROFIT
\( P(3) = -3(9) + 18(3) - 15 = 12 \)
Scale: \( 12 \times \$1,000 = \mathbf{\$12,000} \)
Answer: $12,000 weekly profit
10 EcoPure Water Filters • \( P = -4x^2 + 32x + 16 \)
\(P\) in $1,000s | \(x\) in 1,000 systems
PART A: PRODUCTION LEVEL
\( x = -\frac{32}{2(-4)} = \frac{-32}{-8} = 4 \)
Scale: \( 4 \times 1,000 = \mathbf{4,000\text{ systems}} \)
Answer: 4,000 systems/month
PART B: MAXIMUM PROFIT
\( P(4) = -4(16) + 32(4) + 16 = 80 \)
Scale: \( 80 \times \$1,000 = \mathbf{\$80,000} \)
Answer: $80,000 monthly profit
Optimal Profit Answer Key • Page 2 of 3 (Questions 6–10) Continue to Page 3 for Solutions 11–15 →
Optimal Profit Answer Key: Industrial & Consumer Applications (Part 3 of 3)
Teacher Key | Page 3
11 Harbor Marine Engines • \( P = -2x^2 + 16x - 24 \)
\(P\) in $10,000s | \(x\) in 100 motors
PART A: PRODUCTION LEVEL
\( x = -\frac{16}{2(-2)} = \frac{-16}{-4} = 4 \)
Scale: \( 4 \times 100 = \mathbf{400\text{ motors}} \)
Answer: 400 motors/month
PART B: MAXIMUM PROFIT
\( P(4) = -2(16) + 16(4) - 24 = 8 \)
Scale: \( 8 \times \$10,000 = \mathbf{\$80,000} \)
Answer: $80,000 monthly profit
12 Velox Electric Scooters • \( P = -5x^2 + 30x + 20 \)
\(P\) in $100s | \(x\) in 100 scooters
PART A: PRODUCTION LEVEL
\( x = -\frac{30}{2(-5)} = \frac{-30}{-10} = 3 \)
Scale: \( 3 \times 100 = \mathbf{300\text{ scooters}} \)
Answer: 300 scooters/week
PART B: MAXIMUM PROFIT
\( P(3) = -5(9) + 30(3) + 20 = 65 \)
Scale: \( 65 \times \$100 = \mathbf{\$6,500} \)
Answer: $6,500 weekly profit
13 Pinnacle Outdoor Gear • \( P = -x^2 + 12x - 20 \)
\(P\) in $1,000s | \(x\) in 1,000 packs
PART A: PRODUCTION LEVEL
\( x = -\frac{12}{2(-1)} = \frac{-12}{-2} = 6 \)
Scale: \( 6 \times 1,000 = \mathbf{6,000\text{ backpacks}} \)
Answer: 6,000 backpacks/season
PART B: MAXIMUM PROFIT
\( P(6) = -(36) + 12(6) - 20 = 16 \)
Scale: \( 16 \times \$1,000 = \mathbf{\$16,000} \)
Answer: $16,000 seasonal profit
14 Terra Ceramics • \( P = -3x^2 + 24x - 21 \)
\(P\) in $100s | \(x\) in 100 sets
PART A: PRODUCTION LEVEL
\( x = -\frac{24}{2(-3)} = \frac{-24}{-6} = 4 \)
Scale: \( 4 \times 100 = \mathbf{400\text{ sets}} \)
Answer: 400 sets/month
PART B: MAXIMUM PROFIT
\( P(4) = -3(16) + 24(4) - 21 = 27 \)
Scale: \( 27 \times \$100 = \mathbf{\$2,700} \)
Answer: $2,700 monthly profit
15 AeroShield Composites • \( P = -4x^2 + 40x - 36 \)
\(P\) in $1,000s | \(x\) in 100 panels
PART A: PRODUCTION LEVEL
\( x = -\frac{40}{2(-4)} = \frac{-40}{-8} = 5 \)
Scale: \( 5 \times 100 = \mathbf{500\text{ panels}} \)
Answer: 500 panels/week
PART B: MAXIMUM PROFIT
\( P(5) = -4(25) + 40(5) - 36 = 64 \)
Scale: \( 64 \times \$1,000 = \mathbf{\$64,000} \)
Answer: $64,000 weekly profit
Optimal Profit Answer Key • Page 3 of 3 (Questions 11–15) All solutions verified with vertex formula \(x = -b/(2a)\).
