Tiêu đề MSTE 14 Solutions.pdf ✅

14 Beyond Cartesian Solutions ▣ 1. Convert r = 3 cos θ into cartesian coordinates. [SOLUTION] r = 3 cos θ r 2 = 3r cos θ Since x = r cos θ , r 2 = x 2 + y 2 , x 2 + y 2 = 3x x 2 + y 2 − 3x = 0 ▣ 2. Convert x 2 + y 2 − 2x + 4y = 0 into polar form. [SOLUTION] Since x = r cos θ , y = r sin θ , r 2 = x 2 + y 2 , r 2 − 2r cos θ + 4r sin θ = 0 r − 2 cos θ + 4 sin θ = 0 r = 2 cos θ − 4 sin θ ▣ 3. Find the length of the sub-tangent and subnormal of the curve y 2 = 8x at the point (2,4). [SOLUTION] Length of subtangent ST = y y′ Solve for the derivative y 2 = 8x 2yy ′ = 8 y ′ = 4 y At (2,4), y ′ = 4 4 = 1 ST = 4 1
ST = 4 Length of subnormal SN = yy ′ SN = (4)(1) SN = 4 ▣ 4. Find the length of the latus rectum of the curve with the equation r = 2 1+cos θ . [SOLUTION] r = 2 1 + cos θ r(1 + cos θ) = 2 r + r cos θ = 2 r = 2 − r cos θ r 2 = (2 − r cos θ) 2 x 2 + y 2 = (2 − x) 2 x 2 + y 2 = x 2 − 4x + 4 y 2 = −4x + 4 y 2 = −4(x − 1) The length of the latus rectum is 4 . ▣ 5. Compute the area bounded by the curve r 2 (4 sin2 θ + 9 cos2 θ) = 36. [SOLUTION] r 2 (4 sin2 θ + 9 cos2 θ) = 36 4(r sin θ) 2 + 9(r cos θ) 2 = 36 4y 2 + 9x 2 = 36 x 2 9 + y 2 4 = 1 Therefore, a 2 = 9 → a = 3, b 2 = 4 → b = 2. The area of the ellipse is A = πab A = π(2)(3) = 6π ▣ 6. In the previous question, compute for the total length of the curve. [SOLUTION] There are few approximate formulas for the perimeter of an ellipse. • A. Quadratic Mean Approximation
P ≈ 2π√ a 2 + b 2 2 P ≈ 2π√ 2 2 + 3 2 2 = 16.019 units • B. Ramanujan Approximation 1 P ≈ π [3(a + b) − √(3a + b)(a + 3b)] P ≈ π [3(2 + 3) − √(3 × 3 + 2)(2 + 3 × 2)] = 17.6531 units • C. Ramanujan Approximation 2 P ≈ π(a + b) (1 + 3h 10 + √4 − 3h ) Where h = ( a − b a + b ) 2 h = ( 3 − 2 3 + 2 ) 2 = 1 25 P ≈ π(3 + 2) [ 1 + 3 ( 1 25) 10 + √4 − 3 ( 1 25) ] = 15.8654 units • D. Exact perimeter Formula (Elliptic Integral) P = 4a ∫ √1 − (e sin θ) 2 π 2 0 dθ Where e is the eccentricity. e = c a = √a 2 − b 2 a e = √3 2 − 2 2 3 = √5 3 P = 4(3) ∫ √1 − ( √5 3 sin θ) 2 dθ π 2 0 P = 15.856443958929 units ▣ 7. From question 5, determine the eccentricity of the curve.