Subjects algebra

Phase Costs

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1. **State the problem:** We need to find how much more money is needed for Phase 3 to match the total cost of Phases 1 and 2 combined. 2. **Convert Phase 1 cost from base 4 to decimal:** Phase 1 cost is $(2203)_4$. Calculate decimal value: $$2 \times 4^3 + 2 \times 4^2 + 0 \times 4^1 + 3 \times 4^0 = 2 \times 64 + 2 \times 16 + 0 + 3 = 128 + 32 + 3 = 163$$ 3. **Convert Phase 2 cost from base 8 to decimal:** Phase 2 cost is $(124.2)_8$. Convert integer part $(124)_8$: $$1 \times 8^2 + 2 \times 8^1 + 4 \times 8^0 = 64 + 16 + 4 = 84$$ Convert fractional part $(0.2)_8$: $$2 \times 8^{-1} = 2 \times \frac{1}{8} = 0.25$$ Total Phase 2 cost in decimal: $$84 + 0.25 = 84.25$$ 4. **Calculate total cost of Phases 1 and 2:** $$163 + 84.25 = 247.25$$ 5. **Convert Phase 3 days from base 5 to decimal:** Phase 3 days is $(110)_5$. Calculate decimal value: $$1 \times 5^2 + 1 \times 5^1 + 0 \times 5^0 = 25 + 5 + 0 = 30$$ days 6. **Convert Phase 3 daily cost from base 6 to decimal:** Phase 3 daily cost is $(5.43)_6$. Convert integer part $(5)_6$: $$5 \times 6^0 = 5$$ Convert fractional part $(0.43)_6$: $$4 \times 6^{-1} + 3 \times 6^{-2} = 4 \times \frac{1}{6} + 3 \times \frac{1}{36} = \frac{4}{6} + \frac{3}{36} = 0.6667 + 0.0833 = 0.75$$ Total daily cost in decimal: $$5 + 0.75 = 5.75$$ 7. **Calculate total Phase 3 cost so far:** $$30 \text{ days} \times 5.75 = 172.5$$ 8. **Calculate how much more money is needed for Phase 3 to match Phases 1 and 2 combined:** $$247.25 - 172.5 = 74.75$$ **Final answer:** Phase 3 needs an additional $74.75$ dollars to match the total cost of Phases 1 and 2 combined.