The function h is defined by the equation \(\mathrm{h(t) = 60(1/2)^{(t/4)} + 15}\). What is the value of \(\mathrm{h(8)}\)?15304575

1. TRANSLATE the problem information

• Given: \(\mathrm{h(t) = 60(1/2)^{(t/4)} + 15}\)
• Find: \(\mathrm{h(8)}\)
• This means substitute 8 for every \(\mathrm{t}\) in the function definition

2. SIMPLIFY by direct substitution

• Replace \(\mathrm{t}\) with 8: \(\mathrm{h(8) = 60(1/2)^{(8/4)} + 15}\)
• First, evaluate inside the exponent: \(\mathrm{8/4 = 2}\)
• This gives us: \(\mathrm{h(8) = 60(1/2)^2 + 15}\)

3. SIMPLIFY using order of operations

• Evaluate the exponent first: \(\mathrm{(1/2)^2 = (1/2) \times (1/2) = 1/4}\)
• Substitute back: \(\mathrm{h(8) = 60(1/4) + 15}\)
• Multiply: \(\mathrm{60 \times (1/4) = 60/4 = 15}\)
• Add: \(\mathrm{h(8) = 15 + 15 = 30}\)

Answer: B. 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students might incorrectly evaluate \(\mathrm{(1/2)^2}\) as \(\mathrm{1/2}\) instead of \(\mathrm{1/4}\), confusing powers with the base itself.

If they think \(\mathrm{(1/2)^2 = 1/2}\), then \(\mathrm{h(8) = 60(1/2) + 15 = 30 + 15 = 45}\).
This may lead them to select Choice C (45).

Second Most Common Error:

Poor SIMPLIFY execution: Students might ignore order of operations and multiply \(\mathrm{60 \times 1/2 = 30}\) before handling the exponent, or make arithmetic errors in the final steps.

Various calculation mistakes in the multi-step process could lead them to select Choice A (15) or Choice D (75), or cause confusion and guessing.

The Bottom Line:

This problem tests careful function evaluation combined with proper order of operations. The key challenge is methodically working through each step without rushing, especially when evaluating fractional exponents.