Polar Graphing

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Answer Key & Explanations

Polar Graphing · Grade 12 · Worksheet 3

1. Ben is an astronomer studying the orbit of a comet. The comet's path relative to the sun at the origin is approximated by the polar equation r = 6 / (1 + 3 cos θ). What type of conic section does this represent, and what is the eccentricity? Answer: 3 Solution: The polar equation r = 6 / (1 + 3 cos θ) is in the standard form for a conic section with one focus at the origin: r = ed / (1 + e cos θ), where e is the eccentricity and d is the directrix.
   Full step-by-step solution

   Step 1: The polar equation r = 6 / (1 + 3 cos θ) is in the standard form for a conic section with one focus at the origin: r = ed / (1 + e cos θ), where e is the eccentricity and d is the directrix. Step 2: Comparing, we see that the eccentricity e = 3. Step 3: Since e = 3, and 3 is between 0 and 1 (if 3 < 1) or equal to 1 (if 3 == 1) or greater than 1 (if 3 > 1), we classify: - If e < 1, it's an ellipse. - If e = 1, it's a parabola. - If e > 1, it's a hyperbola. Step 4: Therefore, the eccentricity is 3. The answer is 3.

2. ∫(x²e^(x³)) dx from 0 to 1 = ? Answer: (e-1)/3 Solution: Let u = x³, then du = 3x² dx, so x² dx = du/3 When x = 0, u = 0³ = 0 When x = 1, u = 1³ = 1 Substitute: ∫(x²e^(x³)) dx from 0 to 1 = ∫(e^u)(du/3) from 0 to 1 = (1/3)∫(e^u) du from 0 to 1 = (1/3)[e^u] from 0 to 1 = (1/3)(e¹ - e⁰) = (1/3)(e - 1) = (e-1)/3 The answer is (e-1)/3.
   Full step-by-step solution

   Step 1: Let u = x³, then du = 3x² dx, so x² dx = du/3 Step 2: When x = 0, u = 0³ = 0 Step 3: When x = 1, u = 1³ = 1 Step 4: Substitute: ∫(x²e^(x³)) dx from 0 to 1 = ∫(e^u)(du/3) from 0 to 1 Step 5: = (1/3)∫(e^u) du from 0 to 1 Step 6: = (1/3)[e^u] from 0 to 1 Step 7: = (1/3)(e¹ - e⁰) Step 8: = (1/3)(e - 1) Step 9: = (e-1)/3 The answer is (e-1)/3.

3. Graph r = 5 + 3cos(θ) and identify the shape. Answer: limaçon with an inner loop Solution: Identify the general form. The equation r = 5 + 3cos(θ) is of the form r = a + bcos(θ) with a = 5 and b = 3. Determine the type of limaçon.
   Full step-by-step solution

   Step 1: Identify the general form. The equation r = 5 + 3cos(θ) is of the form r = a + bcos(θ) with a = 5 and b = 3. Step 2: Determine the type of limaçon. For a limaçon r = a + bcos(θ): - If a > b, the limaçon has a dimple (no inner loop). - If a = b, it is a cardioid. - If a < b, it has an inner loop. Here, a = 5 and b = 3, so a > b. This means the limaçon has a dimple, not an inner loop. Step 3: Check the ratio a/b = 5/3 ≈ 1.67. Since a/b > 1, the graph is a limaçon with a dimple (convex limaçon). Step 4: Key points for graphing: - At θ = 0: r = 5 + 3cos(0) = 5 + 3 = 8 - At θ = π/2: r = 5 + 3cos(π/2) = 5 + 0 = 5 - At θ = π: r = 5 + 3cos(π) = 5 - 3 = 2 - At θ = 3π/2: r = 5 + 3cos(3π/2) = 5 + 0 = 5 Step 5: The graph is symmetric about the polar axis (since cosine is even). Plotting these points and connecting them smoothly yields a limaçon with a dimple on the left side (near θ = π). The answer is a limaçon with a dimple.

4. Graph r = 8cos(2θ) and identify the shape. Answer: Rose with 4 petals Solution: The equation is r = 8cos(2θ). This is of the form r = a cos(nθ) with a = 8 and n = 2. For a polar rose, if n is even, the number of petals is 2n.
   Full step-by-step solution

   Step 1: The equation is r = 8cos(2θ). This is of the form r = a cos(nθ) with a = 8 and n = 2. Step 2: For a polar rose, if n is even, the number of petals is 2n. Here n = 2, so the number of petals is 2(2) = 4. Step 3: The maximum value of r is |a| = 8, which occurs when cos(2θ) = 1, i.e., when 2θ = 0, 2π, 4π, ... so θ = 0, π, 2π, ... Step 4: The petals are symmetric about the polar axis (horizontal axis) because of the cosine function. The graph is a rose with 4 petals. Step 5: To confirm, plot key points: at θ = 0, r = 8; at θ = π/4, r = 0; at θ = π/2, r = -8 (which places a point at (8, 3π/2) in polar coordinates); at θ = 3π/4, r = 0; at θ = π, r = 8; etc. This traces out 4 distinct petals. The answer is: Rose with 4 petals.

5. Graph r = 2 + 2cos θ and identify the shape. Answer: Cardioid Solution: The equation is in the form r = a + b cos θ, where a = 2 and b = 2. Since a = b, the graph is a cardioid. The cardioid is symmetric about the polar axis (since it uses cos θ).
   Full step-by-step solution

   Step 1: The equation is in the form r = a + b cos θ, where a = 2 and b = 2. Step 2: Since a = b, the graph is a cardioid. Step 3: The cardioid is symmetric about the polar axis (since it uses cos θ). Step 4: Key points: When θ = 0, r = 2 + 2(1) = 4. When θ = π/2, r = 2 + 2(0) = 2. When θ = π, r = 2 + 2(-1) = 0. When θ = 3π/2, r = 2 + 2(0) = 2. Step 5: Plotting these points and using symmetry gives a heart-shaped curve (cardioid) with a cusp at the origin. The answer is cardioid.

6. A polar graph is described by the equation r = 4 cos(2θ). This curve forms a rose with petals. How many petals does this rose have? Answer: 4 Solution: r = 4 cos(2θ) The equation is of the form r = a cos(nθ) with a = 4 and n = 2. - If n is even, the rose has 2n petals. - If n is odd, the rose has n petals.
   Full step-by-step solution

   Let's solve step by step. We have the polar equation: r = 4 cos(2θ) --- **Step 1: Understand the general shape** The equation is of the form r = a cos(nθ) with a = 4 and n = 2. For polar roses: - If n is even, the rose has 2n petals. - If n is odd, the rose has n petals. But we must check carefully because sometimes the number of petals can be n or 2n depending on symmetry and whether the equation uses sine or cosine. --- **Step 2: Determine the range of θ to trace the curve once** We know r = 4 cos(2θ). The cosine function cos(2θ) has period π in θ, so the function r(θ) repeats every π in θ. But in polar coordinates, (r, θ) and (−r, θ+π) represent the same point. So we must check if the curve repeats in less than 2π. --- **Step 3: Find where r = 0 to identify petal boundaries** Set 4 cos(2θ) = 0 → cos(2θ) = 0 2θ = π/2 + kπ, where k is an integer. θ = π/4 + kπ/2. So zeros occur at: θ = π/4, 3π/4, 5π/4, 7π/4 within 0 ≤ θ < 2π. That’s 4 zeros in one full cycle of θ from 0 to 2π. --- **Step 4: Check symmetry and number of petals** Between consecutive zeros, r has a constant sign and traces one petal. Let’s check: For n even in r = a cos(nθ), the number of petals is 2n. Here n = 2 → 2n = 4 petals. Let’s verify by plotting mentally: From θ = 0 to π/4: cos(2θ) decreases from 1 to 0, r positive → 1 petal in sector 1. From θ = π/4 to 3π/4: cos(2θ) negative → r negative, but negative r in polar means we plot in opposite direction, which actually produces the overlapping petals in the other quadrants. In fact, because n is even, the petals are symmetric and none overlap in the final figure; we get 4 distinct petals. --- **Step 5: Quick check with known rule** For r = a cos(nθ): - If n is even → 2n petals. - If n is odd → n petals. Here n = 2 (even) → 2n = 4 petals. --- **Final Answer:** 4

7. Graph r = 12cos(3θ) and identify the shape. Determine the number of petals and the length of each petal. Answer: Rose curve with 3 petals, each petal length 12 Solution: Identify the equation form. r = 12cos(3θ) is of the form r = a cos(nθ) with a = 12 and n = 3. Since n = 3 is odd, the rose curve has exactly n petals, so 3 petals.
   Full step-by-step solution

   Step 1: Identify the equation form. r = 12cos(3θ) is of the form r = a cos(nθ) with a = 12 and n = 3. Step 2: Since n = 3 is odd, the rose curve has exactly n petals, so 3 petals. Step 3: The length of each petal is given by the maximum value of |r|. The maximum of |cos(3θ)| is 1, so the maximum |r| = 12 * 1 = 12. Thus each petal has length 12. Step 4: The graph is symmetric about the polar axis (since cosine is an even function). The petals occur at angles where cos(3θ) = ±1, i.e., 3θ = 0, π, 2π, ... so θ = 0, π/3, 2π/3, etc. The first petal lies along the positive polar axis (θ = 0). Final answer: Rose curve with 3 petals, each petal length 12.

8. Liam is designing a roller coaster that follows a polar curve. The track's shape is given by the equation r = 3 + 2sin(θ), where r is in meters. To ensure safety, he needs to find the maximum distance from the center of the polar coordinate system that the track reaches. What is this maximum distance in meters? Answer: 5 Solution: r = 3 + 2 sin(θ) We need the maximum distance r from the origin. Since r depends on sin(θ), the maximum r occurs when sin(θ) is maximum.
   Full step-by-step solution

   We are given the polar equation: r = 3 + 2 sin(θ) Step 1: Understand the problem We need the maximum distance r from the origin. Since r depends on sin(θ), the maximum r occurs when sin(θ) is maximum. Step 2: Find the range of sin(θ) We know: -1 ≤ sin(θ) ≤ 1 Step 3: Substitute the maximum value of sin(θ) into r If sin(θ) = 1, then: r = 3 + 2*(1) = 3 + 2 = 5 Step 4: Check if this is indeed the maximum If sin(θ) = -1, then: r = 3 + 2*(-1) = 3 - 2 = 1 So r ranges from 1 to 5. Step 5: Conclusion The maximum distance from the center is 5 meters. Final answer: 5

9. 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.