1. Problem 33: Equalizing Cost United Products Co. manufactures calculators at two plants: Exton and Whyton. - Exton fixed cost: 7000 - Exton variable cost per calculator: 7.50 - Whyton fixed cost: 8800 - Whyton variable cost per calculator: 6.00 - Total calculators to produce: 1500 We want to find how many calculators, $x$, to produce at Exton so that total costs at both plants are equal. 2. Define variables: Let $x$ = number of calculators made at Exton. Then $1500 - x$ = number made at Whyton. 3. Write cost equations: Exton cost: $$7000 + 7.5x$$ Whyton cost: $$8800 + 6(1500 - x)$$ 4. Set costs equal: $$7000 + 7.5x = 8800 + 6(1500 - x)$$ 5. Simplify right side: $$7000 + 7.5x = 8800 + 9000 - 6x$$ $$7000 + 7.5x = 17800 - 6x$$ 6. Bring variables to one side and constants to the other: $$7.5x + 6x = 17800 - 7000$$ $$13.5x = 10800$$ 7. Solve for $x$: $$x = \frac{10800}{13.5} = 800$$ 8. Calculate calculators at Whyton: $$1500 - 800 = 700$$ 9. Verify costs: Exton: $$7000 + 7.5 \times 800 = 7000 + 6000 = 13000$$ Whyton: $$8800 + 6 \times 700 = 8800 + 4200 = 13000$$ Costs are equal at 13000. --- 1. Problem 34: Coffee Blending A wholesaler blends three coffees priced at 2.20, 2.30, and 2.60 per pound to make 100 lb worth 2.40 per pound. The two higher priced coffees (2.30 and 2.60) are used in equal amounts. 2. Define variables: Let $x$ = pounds of coffee at 2.20 Let $y$ = pounds of coffee at 2.30 Let $y$ = pounds of coffee at 2.60 (same as above) 3. Total weight: $$x + y + y = 100 \Rightarrow x + 2y = 100$$ 4. Total value equation: $$2.20x + 2.30y + 2.60y = 2.40 \times 100 = 240$$ Simplify: $$2.20x + (2.30 + 2.60) y = 240$$ $$2.20x + 4.90y = 240$$ 5. From total weight, express $x$: $$x = 100 - 2y$$ 6. Substitute into value equation: $$2.20(100 - 2y) + 4.90y = 240$$ $$220 - 4.4y + 4.90y = 240$$ $$220 + 0.5y = 240$$ 7. Solve for $y$: $$0.5y = 20 \Rightarrow y = 40$$ 8. Find $x$: $$x = 100 - 2(40) = 100 - 80 = 20$$ 9. Final amounts: - 20 lb at 2.20 - 40 lb at 2.30 - 40 lb at 2.60 --- 1. Problem 35: Commissions A company pays salespeople a percentage of the first 100000 in sales plus a different percentage on sales above 100000. Given: - Salesperson A: sales 175000, earned 8500 - Salesperson B: sales 280000, earned 14800 Let $p$ = percentage on first 100000 sales (as decimal) Let $q$ = percentage on sales above 100000 2. Write equations: For A: $$8500 = 100000p + (175000 - 100000)q = 100000p + 75000q$$ For B: $$14800 = 100000p + (280000 - 100000)q = 100000p + 180000q$$ 3. Subtract first from second: $$14800 - 8500 = (100000p + 180000q) - (100000p + 75000q)$$ $$6300 = 105000q$$ 4. Solve for $q$: $$q = \frac{6300}{105000} = 0.06$$ 5. Substitute $q$ into first equation: $$8500 = 100000p + 75000(0.06)$$ $$8500 = 100000p + 4500$$ 6. Solve for $p$: $$100000p = 8500 - 4500 = 4000$$ $$p = \frac{4000}{100000} = 0.04$$ 7. Convert to percentages: - $p = 4\%$ - $q = 6\%$ --- "slug": "cost equalization", "subject": "algebra", "desmos": {"latex": "y=7000+7.5x", "features": {"intercepts": true, "extrema": true}}, "q_count": 3