The function h is defined by the equation \(\mathrm{h(t) = 500 - 2(t - 10)^2}\). What is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(t) = 500 - 2(t - 10)^2}\)
  • Need to find: \(\mathrm{h(3)}\)
• This means substitute \(\mathrm{t = 3}\) into the function expression

2. SIMPLIFY through systematic substitution

• Start with the function: \(\mathrm{h(t) = 500 - 2(t - 10)^2}\)
• Substitute \(\mathrm{t = 3}\): \(\mathrm{h(3) = 500 - 2(3 - 10)^2}\)

3. SIMPLIFY using order of operations

• Work inside parentheses first: \(\mathrm{(3 - 10) = -7}\)
• So we have: \(\mathrm{h(3) = 500 - 2(-7)^2}\)

4. SIMPLIFY the exponent

• Calculate \(\mathrm{(-7)^2}\): \(\mathrm{(-7)^2 = 49}\) (remember: negative squared equals positive)
• Now we have: \(\mathrm{h(3) = 500 - 2(49)}\)

5. SIMPLIFY the final calculation

• Multiply: \(\mathrm{2 \times 49 = 98}\)
• Subtract: \(\mathrm{500 - 98 = 402}\)

Answer: C. 402

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign error when squaring the negative number

Students often think that \(\mathrm{(-7)^2 = -49}\), forgetting that when you square any number (positive or negative), the result is always positive. With this error:

• \(\mathrm{h(3) = 500 - 2(-49) = 500 - (-98) = 500 + 98 = 598}\)

This may lead them to select Choice D (598).

Second Most Common Error:

Poor SIMPLIFY execution: Order of operations confusion

Some students might calculate \(\mathrm{2(3 - 10)^2}\) as \(\mathrm{2(3) - 10^2}\), getting \(\mathrm{6 - 100 = -94}\), then \(\mathrm{500 - (-94) = 594}\). Or they might make other arithmetic errors in the multi-step calculation process.

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

The Bottom Line:

This problem tests careful execution of order of operations and proper handling of negative numbers in exponents. The key insight is remembering that squaring always produces a positive result, regardless of whether the base is positive or negative.