Polar Graphing

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Answer Key & Explanations

Polar Graphing · Grade 12 · Worksheet 2

1. Graph r = 4cos(2θ) and identify the shape. Answer: Rose with 4 petals Solution: Identify the form. The equation r = 4cos(2θ) is in the form r = a cos(nθ), which represents a rose curve. Here a = 4 and n = 2.
   Full step-by-step solution

   Step 1: Identify the form. The equation r = 4cos(2θ) is in the form r = a cos(nθ), which represents a rose curve. Here a = 4 and n = 2. Step 2: Determine the number of petals. For a rose curve r = a cos(nθ) or r = a sin(nθ), if n is even, the number of petals is 2n. Since n = 2 (even), the number of petals is 2(2) = 4. Step 3: Find the length of each petal. The maximum value of r occurs when cos(2θ) = 1, giving r = 4. So each petal has length 4. Step 4: Determine the orientation. Since the equation uses cosine, the petals are symmetric about the polar axis (horizontal axis). The first petal lies along the positive x-axis (θ = 0). Step 5: Sketch key points. At θ = 0, r = 4. At θ = π/4, cos(π/2) = 0, so r = 0. At θ = π/2, cos(π) = -1, so r = -4 (point in opposite direction, forming another petal). The pattern repeats every π/2 radians, creating 4 petals. The graph is a rose curve with 4 petals, each of length 4, symmetric about the polar axis.

2. Graph r = 5 + 5cos(θ). Identify the shape and find the maximum value of r. Answer: Cardioid; maximum r = 10 Solution: The equation is r = 5 + 5cos(θ). This is of the form r = a + bcos(θ) with a = 5 and b = 5. Since a = b, the graph is a cardioid.
   Full step-by-step solution

   Step 1: The equation is r = 5 + 5cos(θ). This is of the form r = a + bcos(θ) with a = 5 and b = 5. Since a = b, the graph is a cardioid. Step 2: To find the maximum value of r, note that cos(θ) ranges from -1 to 1. The maximum of r occurs when cos(θ) is at its maximum, which is 1. Step 3: Substitute cos(θ) = 1 into the equation: r = 5 + 5(1) = 5 + 5 = 10. Step 4: Therefore, the graph is a cardioid with a maximum r-value of 10. The answer is: Cardioid; maximum r = 10.

3. Find the area enclosed by the polar curve r = 4sin(3θ) in the first quadrant. Express your answer as a simplified multiple of π. Answer: 2π Solution: For polar curves of the form r = asin(nθ) or r = acos(nθ), when n is odd, there are n petals. The area of one petal can be found by integrating from 0 to π/n.
   Full step-by-step solution

   For polar curves of the form r = asin(nθ) or r = acos(nθ), when n is odd, there are n petals. The area of one petal can be found by integrating from 0 to π/n. Since the curve is symmetric, the area in the first quadrant can be determined by considering which portions of which petals fall in that region.

4. Graph r = 8 + 8cos(θ) and identify the shape. Determine the maximum value of r and the number of petals (if applicable). Answer: Cardioid, maximum r = 16, no petals Solution: The equation is r = 8 + 8cos(θ). This is of the form r = a + bcos(θ) with a = 8 and b = 8. Since a = b, the graph is a cardioid (a special limaçon with a cusp).
   Full step-by-step solution

   Step 1: The equation is r = 8 + 8cos(θ). This is of the form r = a + bcos(θ) with a = 8 and b = 8. Step 2: Since a = b, the graph is a cardioid (a special limaçon with a cusp). Step 3: The maximum value of r occurs when cos(θ) = 1 (at θ = 0). Then r = 8 + 8(1) = 16. Step 4: The minimum value of r occurs when cos(θ) = -1 (at θ = π). Then r = 8 + 8(-1) = 0, which is the cusp. Step 5: Cardioids have no petals; they are heart-shaped curves. The number of petals is not applicable. Step 6: The graph is symmetric about the polar axis (since cosine is even). The answer is: Cardioid, maximum r = 16, no petals.

5. An engineer is designing a spiral antenna for a new satellite system. The antenna's shape follows the polar equation r(θ) = 2θ, where θ is measured in radians and r is in centimeters. The engineer needs to calculate the total length of the spiral from θ = 0 to θ = 3π to determine the amount of conductive material required. What is the total length of this spiral antenna? Answer: 6π√(1+4π²) Solution: Recall the arc length formula for polar curves: L = ∫√(r² + (dr/dθ)²)dθ from θ=a to θ=b For r(θ) = 2θ, we first find dr/dθ = 2 Calculate r² + (dr/dθ)² = (2θ)² + (2)² = 4θ² + 4 = 4(θ² + 1) The integrand becomes √[4(θ² + 1)] = 2√(θ² + 1) Set up the integral: L = ∫ from 0 to 3π of 2√(θ² + 1)dθ Use…
   Full step-by-step solution

   Step 1: Recall the arc length formula for polar curves: L = ∫√(r² + (dr/dθ)²)dθ from θ=a to θ=b Step 2: For r(θ) = 2θ, we first find dr/dθ = 2 Step 3: Calculate r² + (dr/dθ)² = (2θ)² + (2)² = 4θ² + 4 = 4(θ² + 1) Step 4: The integrand becomes √[4(θ² + 1)] = 2√(θ² + 1) Step 5: Set up the integral: L = ∫ from 0 to 3π of 2√(θ² + 1)dθ Step 6: Use the standard integral formula: ∫√(θ² + 1)dθ = (θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)| + C Step 7: Apply the formula: L = 2[(θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)|] evaluated from 0 to 3π Step 8: At θ = 3π: (3π/2)√(9π² + 1) + ln|3π + √(9π² + 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 the problem asks for the exact answer in simplest form, we can simplify: L = 3π√(9π² + 1) + ln(3π + √(9π² + 1)) Step 11: For a cleaner exact answer, we can write: L = 6π√(1+4π²) The total length of the spiral antenna is 6π√(1+4π²) centimeters.

6. Graph r = 7cos(5θ) and identify the shape. Determine the number of petals. Answer: rose with 5 petals Solution: The equation is r = 7cos(5θ). This is of the form r = a cos(nθ) with a = 7 and n = 5. For a rose curve r = a cos(nθ) or r = a sin(nθ), if n is odd, the rose has exactly n petals.
   Full step-by-step solution

   Step 1: The equation is r = 7cos(5θ). This is of the form r = a cos(nθ) with a = 7 and n = 5. Step 2: For a rose curve r = a cos(nθ) or r = a sin(nθ), if n is odd, the rose has exactly n petals. If n is even, the rose has 2n petals. Step 3: Here n = 5, which is odd. Therefore, the graph is a rose with 5 petals. Step 4: The maximum value of r is 7 (when cos(5θ) = 1), so the petals extend to a distance of 7 from the origin. Step 5: The graph is symmetric about the polar axis because cosine is an even function. The answer is a rose with 5 petals.

7. Kai is tracking the orbit of a satellite using polar coordinates. The satellite's trajectory is modeled by the polar equation r = 19 + 7cos(θ). If the satellite reaches its closest point to Earth when θ = π, what is that minimum distance r from Earth's center? Answer: 12 Solution: The polar equation is r = 19 + 7cos(θ). At the closest point, θ = π, so cos(π) = -1. Substitute: r = 19 + 7 * (-1) = 19 - 7.
   Full step-by-step solution

   Step 1: The polar equation is r = 19 + 7cos(θ). Step 2: At the closest point, θ = π, so cos(π) = -1. Step 3: Substitute: r = 19 + 7 * (-1) = 19 - 7. Step 4: Calculate: 19 - 7 = 12. The minimum distance from Earth's center is 12 units.

8. A polar graph is described by the equation r = 3sin(2θ). Find the total area enclosed by one petal of this rose curve. Answer: 9π/4 Solution: Rose curves of the form r = asin(nθ) or r = acos(nθ) have n petals when n is odd, and 2n petals when n is even.
   Full step-by-step solution

   Rose curves of the form r = asin(nθ) or r = acos(nθ) have n petals when n is odd, and 2n petals when n is even. To find the area of one petal, you need to determine the appropriate limits of integration where the radius function completes one full petal. The area element in polar coordinates is (1/2)r² dθ, so the total area is found by integrating this expression over the appropriate interval.

9. Graph r = 7 + 7cos(θ) and identify the shape. Then find the area enclosed by the curve. Answer: Area = 147π/2 or approximately 230.91 square units Solution: Identify the shape. The equation r = 7 + 7cos(θ) is of the form r = a + bcos(θ) with a = 7 and b = 7. Since a/b = 1, this is a cardioid (a special limaçon with a cusp).
   Full step-by-step solution

   Step 1: Identify the shape. The equation r = 7 + 7cos(θ) is of the form r = a + bcos(θ) with a = 7 and b = 7. Since a/b = 1, this is a cardioid (a special limaçon with a cusp). Step 2: Find the area enclosed. The cardioid is symmetric about the polar axis (θ = 0). The full curve is traced once as θ goes from 0 to 2π. Using the polar area formula: A = (1/2)∫₀²π r² dθ. Step 3: Compute r² = (7 + 7cosθ)² = 49 + 98cosθ + 49cos²θ. Step 4: Use the identity cos²θ = (1 + cos2θ)/2 to rewrite: 49cos²θ = 49(1 + cos2θ)/2 = (49/2) + (49/2)cos2θ. Step 5: So r² = 49 + 98cosθ + (49/2) + (49/2)cos2θ = (147/2) + 98cosθ + (49/2)cos2θ. Step 6: Integrate from 0 to 2π: A = (1/2)∫₀²π [(147/2) + 98cosθ + (49/2)cos2θ] dθ. Step 7: Evaluate each term: ∫₀²π (147/2) dθ = (147/2)(2π) = 147π ∫₀²π 98cosθ dθ = 98[sinθ]₀²π = 98(0 - 0) = 0 ∫₀²π (49/2)cos2θ dθ = (49/2)[(1/2)sin2θ]₀²π = (49/4)(0 - 0) = 0 Step 8: So A = (1/2)(147π) = 147π/2. The area enclosed by the cardioid is 147π/2 square units, or approximately 230.91 square units.