The equation d = 5sqrt(h - 11) relates the positive numbers d and h, where h gt 11. Which equation...

1. INFER the solving strategy

• Given: \(\mathrm{d = 5\sqrt{h - 11}}\), need to solve for h
• Key insight: To eliminate a square root, we need to square both sides, but first we should isolate the radical term

2. SIMPLIFY by isolating the radical

• Divide both sides by 5: \(\mathrm{\frac{d}{5} = \sqrt{h - 11}}\)
• This gets the radical by itself on one side

3. SIMPLIFY by eliminating the square root

• Square both sides: \(\mathrm{(\frac{d}{5})^2 = h - 11}\)
• Remember: When squaring a fraction, square both numerator and denominator
• \(\mathrm{(\frac{d}{5})^2 = \frac{d^2}{5^2} = \frac{d^2}{25}}\)

4. SIMPLIFY to solve for h

• We now have: \(\mathrm{\frac{d^2}{25} = h - 11}\)
• Add 11 to both sides: \(\mathrm{h = \frac{d^2}{25} + 11}\)

Answer: B) h = \(\mathrm{\frac{d^2}{25} + 11}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: When squaring \(\mathrm{\frac{d}{5}}\), students incorrectly apply exponent rules and get \(\mathrm{\frac{d^2}{5}}\) instead of \(\mathrm{\frac{d^2}{25}}\).

They forget that \(\mathrm{(\frac{a}{b})^2 = \frac{a^2}{b^2}}\), so they square only the numerator and leave the denominator as 5. This gives them \(\mathrm{h = \frac{d^2}{5} + 11}\), leading them to select Choice A (h = \(\mathrm{\frac{d^2}{5} + 11}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly get to \(\mathrm{\frac{d^2}{25} = h - 11}\) but make a sign error when isolating h.

Instead of adding 11 to both sides, they subtract 11 or get confused about which side the 11 should be on. This leads them to select Choice C (h = \(\mathrm{\frac{d^2}{25} - 11}\)).

The Bottom Line:

This problem tests precision in algebraic manipulation. The key challenge is remembering that squaring a fraction requires squaring both the numerator and denominator separately, and then carefully tracking signs when moving terms across the equation.