Polar Graphing

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Answer Key & Explanations

Polar Graphing · Grade 12 · Worksheet 3

1. Graph r = 5 + 5cosθ and identify the shape. Answer: cardioid Solution: Recognize the equation r = 5 + 5cosθ is in the form r = a + bcosθ, where a = 5 and b = 5. Since a = b, the graph is a cardioid (a special limaçon with a cusp). The graph is symmetric about the polar axis (cosθ symmetry).
   Full step-by-step solution

   Step 1: Recognize the equation r = 5 + 5cosθ is in the form r = a + bcosθ, where a = 5 and b = 5. Step 2: Since a = b, the graph is a cardioid (a special limaçon with a cusp). Step 3: The graph is symmetric about the polar axis (cosθ symmetry). Step 4: Key points: when θ = 0, r = 5 + 5(1) = 10; when θ = π/2, r = 5 + 5(0) = 5; when θ = π, r = 5 + 5(-1) = 0 (cusp at the pole). Step 5: The shape is a heart-shaped curve (cardioid) opening to the right. The answer is cardioid.

2. An astronomer is tracking a comet's path through the solar system using polar coordinates. The comet's trajectory follows the polar equation r = 3/(1 + 2cosθ). Determine the eccentricity of this conic section and identify what type of orbit the comet has. Answer: eccentricity = 2, hyperbola Solution: Recall the standard polar form of a conic section. The standard polar equation for a conic section with one focus at the pole is: r = (e * d) / (1 + e * cos θ) where e is the eccentricity and d is a constant related to the directrix.
   Full step-by-step solution

   Step 1: Recall the standard polar form of a conic section. The standard polar equation for a conic section with one focus at the pole is: r = (e * d) / (1 + e * cos θ) where e is the eccentricity and d is a constant related to the directrix. Step 2: Compare the given equation to the standard form. Our given equation is: r = 3 / (1 + 2 cos θ) We can write it as: r = (2 * (3/2)) / (1 + 2 cos θ) But more directly, comparing r = 3 / (1 + 2 cos θ) with r = (e * d) / (1 + e cos θ), we see that: The numerator 3 corresponds to e * d. The coefficient of cos θ in the denominator is e. Step 3: Identify the eccentricity. From the denominator, the coefficient of cos θ is 2. Therefore, e = 2. Step 4: Identify the type of conic section. The type of conic is determined by the eccentricity e: - If e = 1, it's a parabola. - If e < 1, it's an ellipse. - If e > 1, it's a hyperbola. Since e = 2, which is greater than 1, the conic is a hyperbola. Step 5: Conclusion. The eccentricity is 2, and the orbit is a hyperbola. Final answer: eccentricity = 2, hyperbola

3. Graph r = 6 + 6cos(θ) and identify the shape. Answer: Cardioid Solution: Recognize the equation r = 6 + 6cos(θ) is of the form r = a + bcos(θ) with a = 6 and b = 6. Since a = b, the graph is a cardioid.
   Full step-by-step solution

   Step 1: Recognize the equation r = 6 + 6cos(θ) is of the form r = a + bcos(θ) with a = 6 and b = 6. Since a = b, the graph is a cardioid. Step 2: To confirm, evaluate r at key angles: θ = 0 gives r = 6 + 6(1) = 12; θ = π/2 gives r = 6 + 6(0) = 6; θ = π gives r = 6 + 6(-1) = 0; θ = 3π/2 gives r = 6 + 6(0) = 6. Step 3: Plot these points: (12, 0), (6, π/2), (0, π), (6, 3π/2). The curve starts at (12, 0), goes inward to the origin at θ = π, and returns to (12, 0) after 2π, forming a heart-shaped cardioid. The answer is cardioid.

4. Liam is designing a solar panel array that follows a polar equation r(θ) = 3 + 2cos(θ) to maximize sun exposure throughout the day. He needs to calculate the total area covered by the panels during one complete rotation. What is the area enclosed by this polar curve? Answer: 11π Solution: The area enclosed by a polar curve r(θ) from θ = α to θ = β is given by A = (1/2)∫[r(θ)]² dθ. For curves involving cosine terms, trigonometric identities like cos²θ = (1 + cos2θ)/2 can help simplify the integration.
   Full step-by-step solution

   The area enclosed by a polar curve r(θ) from θ = α to θ = β is given by A = (1/2)∫[r(θ)]² dθ. For curves involving cosine terms, trigonometric identities like cos²θ = (1 + cos2θ)/2 can help simplify the integration. This method is commonly used in engineering applications involving rotational symmetry.

5. A polar graph is described by the equation r = 2 + 4sin(θ). This limaçon has an inner loop. Find the exact area enclosed by the inner loop of this polar curve. Answer: 4π - 6√3 Solution: Find where the inner loop begins and ends by setting r = 0.
   Full step-by-step solution

   Step 1: Find where the inner loop begins and ends by setting r = 0. 0 = 2 + 4sin(θ) 4sin(θ) = -2 sin(θ) = -1/2 θ = 7π/6 and θ = 11π/6 Step 2: Use the polar area formula A = (1/2)∫[r(θ)]^2 dθ For the inner loop, r is negative between θ = 7π/6 and θ = 11π/6, but since we square r, we can integrate from 7π/6 to 11π/6. Step 3: Set up the integral: A = (1/2)∫[7π/6 to 11π/6] (2 + 4sin(θ))^2 dθ Step 4: Expand the integrand: (2 + 4sin(θ))^2 = 4 + 16sin(θ) + 16sin^2(θ) Step 5: Use the identity sin^2(θ) = (1 - cos(2θ))/2: 4 + 16sin(θ) + 16(1 - cos(2θ))/2 = 4 + 16sin(θ) + 8 - 8cos(2θ) = 12 + 16sin(θ) - 8cos(2θ) Step 6: Integrate term by term from 7π/6 to 11π/6: ∫(12)dθ = 12θ ∫(16sin(θ))dθ = -16cos(θ) ∫(-8cos(2θ))dθ = -4sin(2θ) Step 7: Evaluate at the bounds: At θ = 11π/6: 12(11π/6) - 16cos(11π/6) - 4sin(22π/6) = 22π - 16(√3/2) - 4sin(11π/3) = 22π - 8√3 - 4(-√3/2) = 22π - 8√3 + 2√3 = 22π - 6√3 At θ = 7π/6: 12(7π/6) - 16cos(7π/6) - 4sin(14π/6) = 14π - 16(-√3/2) - 4sin(7π/3) = 14π + 8√3 - 4(√3/2) = 14π + 8√3 - 2√3 = 14π + 6√3 Step 8: Subtract and multiply by 1/2: A = (1/2)[(22π - 6√3) - (14π + 6√3)] = (1/2)(8π - 12√3) = 4π - 6√3 The area enclosed by the inner loop is 4π - 6√3.

6. An engineer is designing a spiral ramp for a parking garage using polar coordinates. The ramp follows the equation r(θ) = 2θ, where θ is measured in radians and r is in meters. If the ramp makes exactly 3 complete rotations from the ground level, what is the total length of the ramp? Answer: 6π√(1+4π²) Solution: The arc length formula for polar curves is L = ∫√(r² + (dr/dθ)²)dθ from θ₁ to θ₂. For r(θ) = 2θ, we have dr/dθ = 2. Substitute into the formula: L = ∫√((2θ)² + (2)²)dθ = ∫√(4θ² + 4)dθ = ∫2√(θ² + 1)dθ.
   Full step-by-step solution

   Step 1: The arc length formula for polar curves is L = ∫√(r² + (dr/dθ)²)dθ from θ₁ to θ₂. Step 2: For r(θ) = 2θ, we have dr/dθ = 2. Step 3: Substitute into the formula: L = ∫√((2θ)² + (2)²)dθ = ∫√(4θ² + 4)dθ = ∫2√(θ² + 1)dθ. Step 4: For 3 complete rotations, θ goes from 0 to 6π. Step 5: Evaluate L = ∫[0 to 6π] 2√(θ² + 1)dθ. Step 6: Using the formula ∫√(θ² + 1)dθ = (θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)| + C. Step 7: Apply the limits: L = 2[(θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)|] from 0 to 6π. Step 8: At θ = 6π: (6π/2)√(36π² + 1) + (1/2)ln|6π + √(36π² + 1)| = 3π√(36π² + 1) + (1/2)ln|6π + √(36π² + 1)|. Step 9: At θ = 0: (0/2)√(0 + 1) + (1/2)ln|0 + √(0 + 1)| = 0 + (1/2)ln(1) = 0. Step 10: Since 36π² is much larger than 1, we can approximate √(36π² + 1) ≈ 6π. Step 11: Therefore L ≈ 2[3π(6π) + (1/2)ln|6π + 6π|] = 2[18π² + (1/2)ln(12π)] = 36π² + ln(12π). Step 12: For an exact answer: L = 6π√(1+4π²).

7. Sarah is tracking the orbit of a satellite using polar coordinates. The satellite's trajectory is modeled by the polar equation r = 13 + 5cos(θ). If the satellite reaches its closest point to Earth when θ = π, what is that minimum distance r from Earth's center? Answer: 8 Solution: The polar equation is r = 13 + 5cos(θ). At the closest point, θ = π, so cos(π) = -1. Substitute: r = 13 + 5 * (-1) = 13 - 5.
   Full step-by-step solution

   Step 1: The polar equation is r = 13 + 5cos(θ). Step 2: At the closest point, θ = π, so cos(π) = -1. Step 3: Substitute: r = 13 + 5 * (-1) = 13 - 5. Step 4: Calculate: 13 - 5 = 8. The minimum distance from Earth's center is 8 units.

8. Liam is designing a roller coaster that follows a spiral path. The track's position in polar coordinates is given by r(θ) = 3θ, where θ is measured in radians. If the roller coaster car starts at θ = 0 and travels to θ = 2π, what is the total distance traveled by the car along this spiral path? Answer: 3π√(1+4π²) + (3/2)ln(2π+√(1+4π²)) Solution: In polar coordinates, the arc length formula accounts for both the changing radius and the angular motion. For a curve defined by r = f(θ), the differential arc length element depends on both the current radius and how quickly the radius changes with respect to the angle.
   Full step-by-step solution

   In polar coordinates, the arc length formula accounts for both the changing radius and the angular motion. For a curve defined by r = f(θ), the differential arc length element depends on both the current radius and how quickly the radius changes with respect to the angle. This creates a Pythagorean-like relationship where the total displacement comes from both the radial and angular components of motion.