Polar Graphing

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Answer Key & Explanations

Polar Graphing · Grade 12 · Worksheet 1

1. ∫(x³ - 2x² + 5) dx from -1 to 2 = ? Answer: 12.75 Solution: Find the antiderivative of x³ - 2x² + 5 Antiderivative = (1/4)x⁴ - (2/3)x³ + 5x Evaluate at upper bound x = 2 F(2) = (1/4)(2)⁴ - (2/3)(2)³ + 5(2) F(2) = (1/4)(16) - (2/3)(8) + 10 F(2) = 4 - 16/3 + 10 F(2) = 14 - 16/3 F(2) = 42/3 - 16/3 = 26/3 Evaluate at lower bound x = -1 F(-1) = (1/4)(-1)⁴ -…
   Full step-by-step solution

   Step 1: Find the antiderivative of x³ - 2x² + 5 Antiderivative = (1/4)x⁴ - (2/3)x³ + 5x Step 2: Evaluate at upper bound x = 2 F(2) = (1/4)(2)⁴ - (2/3)(2)³ + 5(2) F(2) = (1/4)(16) - (2/3)(8) + 10 F(2) = 4 - 16/3 + 10 F(2) = 14 - 16/3 F(2) = 42/3 - 16/3 = 26/3 Step 3: Evaluate at lower bound x = -1 F(-1) = (1/4)(-1)⁴ - (2/3)(-1)³ + 5(-1) F(-1) = (1/4)(1) - (2/3)(-1) - 5 F(-1) = 1/4 + 2/3 - 5 F(-1) = 3/12 + 8/12 - 60/12 F(-1) = (3 + 8 - 60)/12 = -49/12 Step 4: Subtract F(-1) from F(2) 26/3 - (-49/12) = 26/3 + 49/12 Convert to common denominator: 104/12 + 49/12 = 153/12 = 51/4 = 12.75 The answer is 12.75.

2. ∫(x² + 2x - 3) dx from 1 to 4 = ? Answer: 27 Solution: Find the antiderivative of x² + 2x - 3. The antiderivative of x² is (1/3)x³. The antiderivative of 2x is (2/2)x² = x².
   Full step-by-step solution

   Step 1: Find the antiderivative of x² + 2x - 3. The antiderivative of x² is (1/3)x³. The antiderivative of 2x is (2/2)x² = x². The antiderivative of -3 is -3x. So the antiderivative F(x) = (1/3)x³ + x² - 3x. Step 2: Evaluate F(4) - F(1). F(4) = (1/3)(4)³ + (4)² - 3(4) = (1/3)(64) + 16 - 12 = 64/3 + 4 = 64/3 + 12/3 = 76/3 F(1) = (1/3)(1)³ + (1)² - 3(1) = 1/3 + 1 - 3 = 1/3 - 2 = 1/3 - 6/3 = -5/3 Step 3: Calculate F(4) - F(1) = 76/3 - (-5/3) = 76/3 + 5/3 = 81/3 = 27 The answer is 27.

3. Liam is designing a roller coaster that follows a polar curve. The track's path is given by r(θ) = 4cos(3θ) where r is in meters. He needs to install safety barriers at all points where the track is exactly 2 meters from the center pole. Determine all angles θ in the interval [0, 2π) where this occurs. Answer: π/9, 2π/9, 4π/9, 5π/9, 7π/9, 8π/9, 10π/9, 11π/9, 13π/9, 14π/9, 16π/9, 17π/9 Solution: In polar coordinate systems, equations of the form r = a cos(nθ) create rose curves with n petals when n is odd.
   Full step-by-step solution

   In polar coordinate systems, equations of the form r = a cos(nθ) create rose curves with n petals when n is odd. To find where the curve is a specific distance from the pole, you set the polar equation equal to that distance and solve for θ. When solving trigonometric equations involving cos(nθ), you typically isolate the cosine term, then find the general solution before restricting to your interval of interest. The periodicity of trigonometric functions means multiple solutions often exist within any given interval.

4. Graph r = 2 + 2sin(θ) and identify the shape. Determine the maximum value of r. Answer: cardioid, 4 Solution: Recognize the equation r = 2 + 2sin(θ) is of the form r = a + b sin(θ) with a = 2 and b = 2. Since a = b, this is a cardioid. The cardioid is symmetric about the vertical axis (θ = π/2) because of the sine function.
   Full step-by-step solution

   Step 1: Recognize the equation r = 2 + 2sin(θ) is of the form r = a + b sin(θ) with a = 2 and b = 2. Since a = b, this is a cardioid. Step 2: The cardioid is symmetric about the vertical axis (θ = π/2) because of the sine function. Step 3: To find the maximum value of r, note that sin(θ) ranges from -1 to 1. The maximum occurs when sin(θ) = 1. Step 4: Substitute sin(θ) = 1 into the equation: r = 2 + 2(1) = 4. Step 5: The graph is a cardioid with maximum r = 4 at θ = π/2. The answer is cardioid, 4.

5. Graph r = 2 + 7sin(θ). Identify the shape and find the maximum value of r. Answer: cardioid, 9 Solution: Identify the form. The equation is r = 2 + 7sin(θ). This matches the form r = a + b sin(θ) with a = 2 and b = 7.
   Full step-by-step solution

   Step 1: Identify the form. The equation is r = 2 + 7sin(θ). This matches the form r = a + b sin(θ) with a = 2 and b = 7. Step 2: Determine the shape. Since b > a (7 > 2), the graph is a cardioid with a dimple (also called a limaçon with an inner loop). Specifically, because b > a, it is a cardioid that passes through the origin. Step 3: Find the maximum value of r. The maximum value of sin(θ) is 1. Substitute sin(θ) = 1 into the equation: r = 2 + 7(1) = 9. Step 4: Confirm the shape. The graph is symmetric about the vertical line θ = π/2 (since sine is used). The maximum r occurs at θ = π/2. The answer is cardioid, 9.

6. An engineer is designing a robotic arm that follows a spiral path in polar coordinates. The arm's position is given by r(θ) = 2θ, where θ is measured in radians and r is in meters. If the arm starts at θ = 0 and completes exactly 3 full rotations, what is the total distance traveled by the robotic arm? Answer: 12π√(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. Three full rotations means θ goes from 0 to 6π.
   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: Three full rotations means θ goes from 0 to 6π. Step 4: Substitute into the formula: L = ∫√((2θ)² + (2)²)dθ from 0 to 6π = ∫√(4θ² + 4)dθ from 0 to 6π. Step 5: Factor out 4: L = ∫√(4(θ² + 1))dθ from 0 to 6π = ∫2√(θ² + 1)dθ from 0 to 6π. Step 6: Use the formula ∫√(θ² + 1)dθ = (θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)| + C. Step 7: Evaluate from 0 to 6π: L = 2[(6π/2)√(36π² + 1) + (1/2)ln|6π + √(36π² + 1)| - (0 + 0)]. Step 8: Simplify: L = 2[3π√(36π² + 1) + (1/2)ln(6π + √(36π² + 1))]. Step 9: Since 36π² is much larger than 1, we can approximate √(36π² + 1) ≈ 6π. Step 10: This gives L ≈ 2[3π(6π) + (1/2)ln(6π + 6π)] = 2[18π² + (1/2)ln(12π)] = 36π² + ln(12π). Step 11: For an exact answer in simplified form: L = 12π√(1+4π²).

7. An engineer is designing a spiral antenna for a satellite communication system. The antenna's shape follows the polar equation r = 2θ, where θ is measured in radians. The engineer needs to calculate the total length of the spiral from θ = 0 to θ = 3π. Using the arc length formula for polar curves, what is the total length of this spiral antenna? Answer: 6π√(1+9π²) + (1/3)ln|3π+√(1+9π²)| Solution: Recall the arc length formula for polar curves: L = ∫√(r² + (dr/dθ)²) dθ from θ=a to θ=b. For r = 2θ, we have dr/dθ = 2.
   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 = 2θ, we have dr/dθ = 2. Step 3: Substitute into the formula: L = ∫√((2θ)² + (2)²) dθ from θ=0 to θ=3π = ∫√(4θ² + 4) dθ = ∫2√(θ² + 1) dθ from θ=0 to θ=3π. Step 4: Use the standard integral formula ∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2)ln|x + √(x² + a²)| + C. Step 5: Apply this formula with a=1: L = 2[(θ/2)√(θ² + 1) + (1/2)ln|θ + √(θ² + 1)|] evaluated from θ=0 to θ=3π. Step 6: Simplify: L = [θ√(θ² + 1) + ln|θ + √(θ² + 1)|] evaluated from θ=0 to θ=3π. Step 7: Evaluate at θ=3π: 3π√(9π² + 1) + ln|3π + √(9π² + 1)|. Step 8: Evaluate at θ=0: 0 + ln|0 + 1| = 0. Step 9: Subtract: L = 3π√(9π² + 1) + ln|3π + √(9π² + 1)|. The answer is 3π√(9π² + 1) + ln|3π + √(9π² + 1)|.

8. An engineer is designing a special antenna that follows a spiral path in polar coordinates. The antenna's shape is given by r(θ) = 2θ, where θ is measured in radians and r is in meters. If the antenna needs to extend from θ = 0 to θ = 3π, what is the total length of the antenna? Answer: 3π√(4π²+1) + (1/2)ln(3π+√(9π²+1)) 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 have 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 = ∫2√(θ² + 1)dθ from θ=0 to θ=3π Use the…
   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 have 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 = ∫2√(θ² + 1)dθ from θ=0 to θ=3π 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: Evaluate at θ=3π: (3π/2)√(9π² + 1) + ln(3π + √(9π² + 1)) Step 9: Evaluate at θ=0: (0/2)√(0 + 1) + ln(0 + √(0 + 1)) = 0 + ln(1) = 0 Step 10: Subtract: L = 3π√(9π² + 1) + ln(3π + √(9π² + 1)) The total length of the antenna is 3π√(9π² + 1) + ln(3π + √(9π² + 1)) meters.