When the ellipse b²x² + a²y² = a²b² is rotated about the x-axis, the volume V of the spheroid is 4 $\pi$ab²/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V. A) increase of 1% B) increase of 2% C) increase of 3% D) increase of 4% E) increase of 5%

Question

Quizplus · Accepted Answer

The volume V = (4/3)πab². Using differentials, dV ≈ (∂V/∂a)da + (∂V/∂b)db. Calculate partial derivatives: ∂V/∂a = (4/3)πb², ∂V/∂b = (8/3)πab. With da = 0.01a and db = 0.01b, dV ≈ (4/3)πb²(0.01a) + (8/3)πab(0.01b) = (4/3)πab²(0.01) + (8/3)πab²(0.01) = (12/3)πab²(0.01) = 0.04πab². The percentage change is (dV/V) * 100% = 0.03 * 100% = 3%.

When the Ellipse B2x2 + A2y2 = A2b2 Is Rotated π\piπ

When the Ellipse B²x² + A²y² = A²b² Is Rotated $\pi$