A scientist is studying the relationship between the dosage of a drug (d, in mg) and the reduction in symptoms (R, in percentage points). The relationship is modeled by the equation R = 0.5d² + 10d. If a patient experiences a 30% reduction in symptoms, what was the approximate dosage of the drug, to the nearest whole number?
- A8 mg
- B2 mg
- C5 mg
- D3 mg
Show answer & explanationAnswer & explanation
Correct answer: D. 3 mg
We are given R = 30 and the equation R = 0.5d² + 10d. Substitute R=30: 30 = 0.5d² + 10d. Rearrange into standard quadratic form (ax² + bx + c = 0): 0.5d² + 10d - 30 = 0. To simplify, multiply by 2: d² + 20d - 60 = 0. Use the quadratic formula: d = [-b ± sqrt(b² - 4ac)] / (2a). Here, a=1, b=20, c=-60. d = [-20 ± sqrt(20² - 4*1*(-60))] / (2*1) = [-20 ± sqrt(400 + 240)] / 2 = [-20 ± sqrt(640)] / 2. sqrt(640) ≈ 25.298. d = [-20 ± 25.298] / 2. We take the positive root for dosage: d = (-20 + 25.298) / 2 = 5.298 / 2 ≈ 2.649. To the nearest whole number, the dosage is 3 mg.
Why the other options are wrong
- A. This is significantly too high and likely results from substantial calculation errors.
- B. This is too low; it might be an incorrect use of the quadratic formula or rounding error.
- C. This is too high; it could result from calculation errors or choosing the incorrect root.
Solving Quadratic Equations (Applications)
Solving a quadratic equation in a real-world context involves setting the function equal to a given value and then using methods like factoring, completing the square, or the quadratic formula to find the unknown variable.
- Always set the equation to zero (ax² + bx + c = 0) before solving.
- The quadratic formula is d = [-b ± sqrt(b² - 4ac)] / (2a).
- In real-world problems, only positive and realistic solutions are typically valid.
Memory trick: Set the output, then solve for the input using the quadratic formula.