The total cost, in dollars, to rent a surfboard consists of a $25 service fee and a $10 per hour...

1. TRANSLATE the cost information into mathematical expressions

• Given information:
  • Service fee: \(\$25\) (this is a one-time fixed cost)
  • Rental fee: \(\$10\) per hour
  • Time rented: \(\mathrm{t}\) hours
  • Maximum spending: \(\$75\)
• What this tells us: The total cost has two parts - a fixed part and a variable part.

2. INFER how to build the total cost expression

• The rental fee depends on time: \(\$10\) per hour × \(\mathrm{t}\) hours = \(\$10\mathrm{t}\)
• Total cost = Fixed cost + Variable cost = \(\$25 + \$10\mathrm{t}\)
• Since they want to spend "a maximum of \(\$75\)," this means the total cost must be \(\leq \$75\)

3. Set up the inequality

• Total cost \(\leq\) Maximum spending
• \(\$25 + \$10\mathrm{t} \leq \$75\)

Answer: D. \(25 + 10\mathrm{t} \leq 75\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which number goes with which variable, thinking the service fee is per hour.

They might reason: "It costs \(\$25\) per hour plus a \(\$10\) fee, so the cost is \(\$25\mathrm{t} + \$10\)." This leads them to select Choice B (\(10 + 25\mathrm{t} \leq 75\)).

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students only consider part of the total cost.

They might think: "The rental is \(\$10\) per hour, and they can spend at most \(\$75\) on rental, so \(\$10\mathrm{t} \leq \$75\)." They forget that the service fee is part of what they're spending. This leads them to select Choice A (\(10\mathrm{t} \leq 75\)).

The Bottom Line:

This problem tests whether students can carefully parse word problems to identify fixed vs. variable costs and correctly translate "maximum spending" into an inequality constraint.