1. **Problem statement:** We have two rental car cost options based on miles driven. Option A increases cost linearly from $20 at 0 miles to about $55 at 155 miles. Option B is a flat rate of $50 regardless of miles. 2. **Given:** - Option A cost at 0 miles: $20 - Option A cost at 155 miles: $55 - Option B cost: $50 flat 3. **Find:** (a) Which option costs more at 75 miles and by how much? (b) At what miles do costs equal? Which option is cheaper below that? 4. **Step for (a):** - Find the cost of Option A at 75 miles. - Since Option A is linear, find slope $m$: $$m=\frac{55-20}{155-0}=\frac{35}{155}=\frac{7}{31}$$ - Equation for Option A cost $y$ at miles $x$: $$y=mx+20=\frac{7}{31}x+20$$ - Calculate cost at $x=75$: $$y=\frac{7}{31}\times 75 + 20=\frac{525}{31}+20\approx 16.94 + 20=36.94$$ 5. **Compare costs at 75 miles:** - Option A: $36.94$ - Option B: $50$ - Option B costs more. - Difference: $$50 - 36.94 = 13.06$$ 6. **Step for (b):** - Find $x$ where Option A cost equals Option B cost: $$\frac{7}{31}x + 20 = 50$$ - Solve for $x$: $$\frac{7}{31}x = 30$$ $$x = 30 \times \frac{31}{7} = \frac{930}{7} \approx 132.86$$ 7. **Interpretation:** - At about 132.86 miles, both options cost the same. - For miles less than 132.86, Option A costs less. - For miles greater than 132.86, Option B costs less. **Final answers:** (a) Option B costs more at 75 miles by approximately 13.06. (b) Costs are equal at about 132.86 miles. For less than this, Option A is cheaper.