Find the positive constant k such that the area bounded between the parabola y = x^2 and the line y = k*x is equal to 36.
Answer and explanation
Correct answer: C. 6
- Step 1: Intersection points: x^2 = k*x gives x = 0 and x = k (since k > 0).
- Step 2: The enclosed area is Area = int_0^k (kx - x^2) dx = [kx^2 / 2 - x^3 / 3]_0^k.
- Step 3: Substitute limits: k^3 / 2 - k^3 / 3 = k^3 / 6.
- Step 4: Set equal to 36: k^3 / 6 = 36, so k^3 = 216.
- Step 5: Cube root: k = 216^(1/3) = 6.
Answer
6.
Why not the others: 4 gives area 64/6 = 10.67; 8 gives area 512/6 = 85.33; 3 gives area 27/6 = 4.5.
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