The cost of renting a large canopy tent for up to 10 days is $430 for the first day and...

1. TRANSLATE the problem information

• Given information:
  • First day rental cost: $430
  • Each additional day cost: $215
  • Need equation for total cost y after x days (where \(\mathrm{x \leq 10}\))

2. INFER the cost structure approach

• Key insight: Total cost = First day cost + (Additional days × Rate per additional day)
• Additional days = \(\mathrm{x - 1}\) (since first day is separate)
• This gives us: \(\mathrm{y = 430 + 215(x - 1)}\)

3. SIMPLIFY to standard form

• Expand: \(\mathrm{y = 430 + 215(x - 1)}\)
• Distribute: \(\mathrm{y = 430 + 215x - 215}\)
• Combine like terms: \(\mathrm{y = 215x + (430 - 215)}\)
• Final form: \(\mathrm{y = 215x + 215}\)

4. Verify with test case

• For \(\mathrm{x = 1}\): \(\mathrm{y = 215(1) + 215 = 430}\) ✓ (matches first day only)
• For \(\mathrm{x = 2}\): \(\mathrm{y = 215(2) + 215 = 645}\) ✓ (first day + one additional)

Answer: A. \(\mathrm{y = 215x + 215}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the cost structure, thinking the $430 applies to every day rather than just the first day.

This leads them to think: "Each day costs $430, plus an extra $215 fee" and set up \(\mathrm{y = 430x + 215}\), selecting Choice C (\(\mathrm{y = 430x + 215}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students reverse the daily costs, thinking the first day costs $215 and additional days cost $430 each.

This creates the equation \(\mathrm{y = 215 + 430(x - 1) = 430x - 215}\), leading them to select Choice B (\(\mathrm{y = 430x - 215}\)).

The Bottom Line:

The key challenge is carefully parsing the pricing structure to distinguish between the special first-day rate and the rate for additional days, then correctly translating this into algebraic form.