The function f is defined by \(\mathrm{f(t) = 5 \cdot 2^t}\). What is the value of t when \(\mathrm{f(t)}\) is...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(t) = 5 \cdot 2^t}\)
  • Condition: \(\mathrm{f(t) = 40}\)
  • Find: value of t

• This tells us we need to set up an equation where f(t) equals 40

2. TRANSLATE the condition into an equation

• Since \(\mathrm{f(t) = 40}\) and \(\mathrm{f(t) = 5 \cdot 2^t}\):
  \(\mathrm{5 \cdot 2^t = 40}\)

3. SIMPLIFY to isolate the exponential term

• Divide both sides by 5:
  \(\mathrm{2^t = 8}\)

• Now we have a simpler exponential equation to solve

4. INFER the value of t using powers of 2

• We need to find what power of 2 equals 8
• Checking powers of 2:
  • \(\mathrm{2^1 = 2}\)
  • \(\mathrm{2^2 = 4}\)
  • \(\mathrm{2^3 = 8}\) ✓

• Therefore: \(\mathrm{t = 3}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students may incorrectly divide 40 by 5, getting \(\mathrm{2^t = 10}\) instead of \(\mathrm{2^t = 8}\).

This computational error leads to trying to solve \(\mathrm{2^t = 10}\), which doesn't have a nice integer solution among the choices. Students may then guess or try to force one of the answer choices, possibly selecting Choice D (4) since \(\mathrm{2^4 = 16}\) is "close" to 10.

Second Most Common Error:

Missing conceptual knowledge about powers of 2: Students may correctly get to \(\mathrm{2^t = 8}\) but not immediately recognize that \(\mathrm{8 = 2^3}\).

Instead, they might try substituting answer choices:

• Testing \(\mathrm{t = 1}\): \(\mathrm{2^1 = 2 \neq 8}\)
• Testing \(\mathrm{t = 2}\): \(\mathrm{2^2 = 4 \neq 8}\)
• But then make an error or give up before reaching \(\mathrm{t = 3}\)

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically work through exponential equations and recall basic powers of 2. Success depends on careful algebraic manipulation and familiarity with small exponential values.