Digital SATMath: Advanced MathMedium
A civil engineer is designing a curved ramp. The height of the ramp, h(x), in meters, at a horizontal distance x from the start, is given by the function h(x) = sqrt(16 - x^2). What is the domain of this function in the context of the real world?
- Ax ≥ 0
- B-4 ≤ x ≤ 4
- Cx ≤ 4
- D0 ≤ x ≤ 4
Show answer & explanationAnswer & explanation
Correct answer: D. 0 ≤ x ≤ 4
For the function h(x) = sqrt(16 - x^2) to be defined in real numbers, the expression under the square root must be non-negative: 16 - x^2 ≥ 0. This inequality can be rewritten as x^2 ≤ 16, which means -4 ≤ x ≤ 4. However, in the context of a 'horizontal distance from the start' of a ramp, x must also be non-negative (x ≥ 0). Combining these conditions, the domain is 0 ≤ x ≤ 4.
Why the other options are wrong
- A. This only considers the non-negative aspect of distance but ignores the square root restriction.
- B. This is the mathematical domain for the square root, but it does not account for x being a 'horizontal distance from the start' (x ≥ 0).
- C. This only considers one side of the square root restriction and ignores the non-negative distance for x.
Domain of Radical Functions (Real World)
For a square root function, the expression under the radical must be non-negative. In real-world contexts, additional restrictions (like distance being non-negative) must also be applied.
- Set the radicand ≥ 0 to find mathematical domain.
- Consider physical constraints of the problem (e.g., time, distance, quantity).
- The final domain is the intersection of mathematical and real-world restrictions.
Memory trick: Radicand non-negative, distance non-negative, combine!