1. **Problem 8:** A truck rental agency offers two plans: - Plan A: $85$ per week plus $0.30$ per mile driven. - Plan B: $120$ per week plus $0.25$ per mile driven. (a) Write the cost model for Plan A. (b) Write the cost model for Plan B. (c) Find the number of miles where both plans cost the same. (d) Decide which plan is better for $350$ miles. 2. **Formulas and rules:** - Cost for Plan A: $C_A = 85 + 0.30m$ where $m$ is miles. - Cost for Plan B: $C_B = 120 + 0.25m$. - To find when costs are equal, set $C_A = C_B$ and solve for $m$. 3. **Step-by-step solutions:** (a) Plan A cost equation: $$C_A = 85 + 0.30m$$ (b) Plan B cost equation: $$C_B = 120 + 0.25m$$ (c) Set $C_A = C_B$: $$85 + 0.30m = 120 + 0.25m$$ Subtract $0.25m$ from both sides: $$85 + 0.05m = 120$$ Subtract $85$ from both sides: $$0.05m = 35$$ Divide both sides by $0.05$: $$m = \frac{35}{0.05} = 700$$ So, at $700$ miles, both plans cost the same. (d) For $350$ miles, calculate costs: Plan A: $$C_A = 85 + 0.30 \times 350 = 85 + 105 = 190$$ Plan B: $$C_B = 120 + 0.25 \times 350 = 120 + 87.5 = 207.5$$ Since $190 < 207.5$, Plan A is cheaper for $350$ miles. --- 4. **Problem 9:** Mix a $25\%$ salt solution with a $60\%$ salt solution to get $20$ quarts of a $40\%$ salt solution. Let $x$ = quarts of $25\%$ solution. Then $20 - x$ = quarts of $60\%$ solution. 5. **Formula:** Salt from $25\%$ solution + Salt from $60\%$ solution = Salt in $40\%$ solution $$0.25x + 0.60(20 - x) = 0.40 \times 20$$ 6. **Solve:** $$0.25x + 12 - 0.60x = 8$$ Combine like terms: $$-0.35x + 12 = 8$$ Subtract $12$: $$-0.35x = -4$$ Divide by $-0.35$: $$x = \frac{-4}{-0.35} = \frac{4}{0.35} \approx 11.43$$ So, $11.43$ quarts of $25\%$ solution. Quarts of $60\%$ solution: $$20 - 11.43 = 8.57$$ **Answer:** Mix approximately $11.43$ quarts of $25\%$ solution and $8.57$ quarts of $60\%$ solution.