The equation \(\mathrm{E(t) = 5(1.8)^t}\) gives the estimated number of employees at a restaurant, where t is the number of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{E(t) = 5(1.8)^t}\) gives estimated number of employees
  • \(\mathrm{t}\) = number of years since the restaurant opened
  • Need to interpret what the number 5 represents

2. INFER the mathematical approach

• Key insight: In exponential functions of the form \(\mathrm{E(t) = a(b)^t}\), the coefficient "a" represents the initial value
• Strategy: Find \(\mathrm{E(0)}\) to see what happens when \(\mathrm{t = 0}\) (when the restaurant opened)

3. Calculate the initial value

• When \(\mathrm{t = 0}\): \(\mathrm{E(0) = 5(1.8)^0}\)
• Since any number to the power of 0 equals 1: \(\mathrm{E(0) = 5(1) = 5}\)
• This means when the restaurant opened, there were 5 estimated employees

4. INFER the correct interpretation

• The number 5 represents the estimated number of employees when \(\mathrm{t = 0}\)
• Since \(\mathrm{t = 0}\) corresponds to when the restaurant opened, 5 is the initial number of employees

Answer: A. The estimated number of employees when the restaurant opened

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the different components of exponential functions

Many students don't clearly understand what each part of \(\mathrm{E(t) = a(b)^t}\) represents. They might think the coefficient 5 is related to the growth rate rather than the initial value. Some students focus on the base 1.8 and incorrectly associate the coefficient 5 with yearly increases or percentages.

This may lead them to select Choice B (The increase in the estimated number of employees each year) or Choice D (The percent increase in the estimated number of employees each year).

Second Most Common Error:

Poor TRANSLATE reasoning: Students don't connect \(\mathrm{t = 0}\) with "when the restaurant opened"

Students may struggle to understand that "years since the restaurant opened" means \(\mathrm{t = 0}\) represents the opening day. Without this connection, they can't evaluate what \(\mathrm{E(0)}\) means in context.

This leads to confusion and random guessing among the answer choices.

The Bottom Line:

Success on this problem requires understanding both the mathematical structure of exponential functions and the real-world interpretation of the variables. Students need to recognize that finding the initial value (when \(\mathrm{t = 0}\)) reveals what the coefficient represents in context.