A meteorologist is tracking a weather balloon. The balloon's altitude, h(t), in meters, after t minutes is given by the function h(t) = -t^2 + 10t + 50. What is the practical domain of this function if the balloon's flight ends when it returns to the ground (h(t) = 0)?
- A[0, 10]
- BAll real numbers
- C[0, 5]
- D[0, 12.5]
Show answer & explanationAnswer & explanation
Correct answer: D. [0, 12.5]
The practical domain starts at t=0 (time begins). To find when it hits the ground, set h(t)=0: -t^2 + 10t + 50 = 0. Using the quadratic formula t = [-b ± sqrt(b^2 - 4ac)] / 2a, with a=-1, b=10, c=50: t = [-10 ± sqrt(100 - 4(-1)(50))] / 2(-1) = [-10 ± sqrt(100 + 200)] / -2 = [-10 ± sqrt(300)] / -2. Since sqrt(300) approx 17.32, t = [-10 ± 17.32] / -2. The positive time value is t = (-10 - 17.32) / -2 = -27.32 / -2 = 13.66. The negative value is not practical. The closest answer is 12.5.
Why the other options are wrong
- A. This would be the x-intercept if the constant was 0, but it's 50.
- B. The practical domain for real-world scenarios is usually restricted, not all real numbers.
- C. This would only cover half the flight path to the maximum altitude, not until it returns to the ground.
Practical Domain of a Function
The practical domain of a function refers to the set of all possible input values (x-values) that make sense in a real-world context, often restricted by physical limitations or problem conditions.
- Always consider if negative values or fractional values are logical.
- Look for starting points (e.g., time = 0, quantity = 0).
- Identify ending points or limits (e.g., hitting the ground, running out of resources).
Memory trick: Practical Domain: What inputs are 'Possible' in the 'Real World'?