Sigma Notation

lessonbunny.com

Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 3

1. Matiu is constructing a visual pattern using triangular tiles. The first layer has 1 tile, the second layer has 4 tiles, the third layer has 9 tiles, and this pattern continues for 12 layers. Each layer forms a perfect square arrangement. Using sigma notation, write the series representing the total number of tiles in the pattern, then evaluate the sum. Answer: 650 Solution: Identify the pattern. Layer 1 has 1 tile (1 squared). Layer 2 has 4 tiles (2 squared).
   Full step-by-step solution

   Step 1: Identify the pattern. Layer 1 has 1 tile (1 squared). Layer 2 has 4 tiles (2 squared). Layer 3 has 9 tiles (3 squared). This continues for 12 layers, so layer k has k squared tiles. Step 2: Write the series in sigma notation. Total tiles = sum from k=1 to 12 of k^2. Step 3: Use the formula for the sum of the first n squares: sum from k=1 to n of k^2 = n(n+1)(2n+1)/6. Step 4: Substitute n=12: sum = 12(12+1)(2*12+1)/6 = 12(13)(25)/6. Step 5: Simplify step by step. First, 12/6 = 2. So we have 2 * 13 * 25. Step 6: Multiply: 2 * 13 = 26. Then 26 * 25 = 650. Step 7: The total number of tiles is 650.

2. Mere is an environmental scientist modeling the rate at which a new species of plant absorbs carbon dioxide from the atmosphere. The absorption rate A(t) in kilograms per day is given by the function A(t) = 2t² + 4t for t ≥ 0 days, where t is time in days. To estimate the total carbon absorbed during the first 6 days, Mere wants to approximate the definite integral of A(t) from t=0 to t=6 using sigma notation with 3 subintervals of equal width and right endpoints. Write the Riemann sum in sigma notation that represents this approximation. Answer: ∑_{i=1}^{3} (2(2i)² + 4(2i)) × 2 Solution: The interval is from t=0 to t=6, so the total width is 6 days. With 3 subintervals, each subinterval has width Δt = (6-0)/3 = 2 days. Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6.
   Full step-by-step solution

   Step 1: The interval is from t=0 to t=6, so the total width is 6 days. Step 2: With 3 subintervals, each subinterval has width Δt = (6-0)/3 = 2 days. Step 3: Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6. Step 4: The general form for right endpoints is: t_i = 0 + i × Δt = 2i. Step 5: The Riemann sum using sigma notation is: ∑_{i=1}^{3} A(t_i) × Δt. Step 6: Substitute A(t) = 2t² + 4t and t_i = 2i, Δt = 2: ∑_{i=1}^{3} (2(2i)² + 4(2i)) × 2. Step 7: The final sigma notation is: ∑_{i=1}^{3} (2(2i)² + 4(2i)) × 2.

3. Charlotte is a materials engineer designing a heat shield for a spacecraft. During re-entry, the temperature T(t) in degrees Celsius on the shield's surface t seconds after re-entry begins is modeled by the function T(t) = 2000 - 12t^2 for 0 ≤ t ≤ 10 seconds. To calculate the total thermal energy absorbed over the first 10 seconds, she needs to approximate the definite integral of T(t) from t=0 to t=10. Using sigma notation with 5 subintervals of equal width and right endpoints, write the Riemann sum that approximates this total thermal energy. Answer: ∑_{i=1}^{5} (2000 - 12(2i)^2) × 2 Solution: The interval is from t=0 to t=10, so the total width is 10 seconds. With 5 subintervals, each subinterval has width Δt = (10-0)/5 = 2 seconds.
   Full step-by-step solution

   Step 1: The interval is from t=0 to t=10, so the total width is 10 seconds. Step 2: With 5 subintervals, each subinterval has width Δt = (10-0)/5 = 2 seconds. Step 3: Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6, t₄ = 8, t₅ = 10. Step 4: The general form for right endpoints is: t_i = 0 + i × Δt = 2i. Step 5: The Riemann sum using sigma notation is: ∑[i=1 to 5] T(t_i) × Δt Step 6: Substituting T(t) = 2000 - 12t² and t_i = 2i, Δt = 2: ∑[i=1 to 5] (2000 - 12(2i)²) × 2 Step 7: The final sigma notation is: ∑_{i=1}^{5} (2000 - 12(2i)²) × 2

4. An environmental engineer is modeling the rate of water flow into a reservoir during a storm. The flow rate R(t) in cubic meters per hour is given by the function R(t) = 100 + 20t - t² for 0 ≤ t ≤ 10 hours, where t is time in hours. To estimate the total volume of water that entered the reservoir during the storm, she needs to calculate the definite integral of R(t) from t=0 to t=10. Express this definite integral using sigma notation with 5 subintervals of equal width, using right endpoints for the Riemann sum approximation. Answer: ∑[i=1 to 5] (100 + 20(2i) - (2i)²) × 2 Solution: The interval is from t=0 to t=10, so the total width is 10 hours. With 5 subintervals, each subinterval has width Δt = (10-0)/5 = 2 hours. Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6, t₄ = 8, t₅ = 10.
   Full step-by-step solution

   Step 1: The interval is from t=0 to t=10, so the total width is 10 hours. Step 2: With 5 subintervals, each subinterval has width Δt = (10-0)/5 = 2 hours. Step 3: Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6, t₄ = 8, t₅ = 10. Step 4: The general form for right endpoints is: t_i = 0 + i × Δt = 2i. Step 5: The Riemann sum using sigma notation is: ∑[i=1 to 5] R(t_i) × Δt Step 6: Substituting R(t) = 100 + 20t - t² and t_i = 2i, Δt = 2: ∑[i=1 to 5] (100 + 20(2i) - (2i)²) × 2 Step 7: The final sigma notation is: ∑[i=1 to 5] (100 + 20(2i) - (2i)²) × 2

5. An environmental engineer is modeling the rate of carbon absorption by a forest over time. The absorption rate A(t) in tons per year is given by the function A(t) = 8t^2 * e^(-0.1t) for t ≥ 0 years. To estimate the total carbon absorbed during the first 20 years, the engineer wants to express the definite integral from t=0 to t=20 using sigma notation with 4 equal subintervals and right endpoints. Write the Riemann sum in sigma notation that approximates this total carbon absorption. Answer: ∑_{i=1}^4 [8*(5i)^2 * e^(-0.1*(5i)) * 5] Solution: Identify the interval [0,20] and number of subintervals n=4 Calculate the width of each subinterval: Δt = (20-0)/4 = 5 The right endpoints are: t_i = 0 + i*5 = 5i for i=1,2,3,4 The function is A(t) = 8t^2 * e^(-0.1t) The Riemann sum using right endpoints is: ∑_{i=1}^4 A(t_i) * Δt Substitute t_i…
   Full step-by-step solution

   Step 1: Identify the interval [0,20] and number of subintervals n=4 Step 2: Calculate the width of each subinterval: Δt = (20-0)/4 = 5 Step 3: The right endpoints are: t_i = 0 + i*5 = 5i for i=1,2,3,4 Step 4: The function is A(t) = 8t^2 * e^(-0.1t) Step 5: The Riemann sum using right endpoints is: ∑_{i=1}^4 A(t_i) * Δt Step 6: Substitute t_i = 5i and Δt = 5: ∑_{i=1}^4 [8*(5i)^2 * e^(-0.1*(5i)) * 5] The Riemann sum in sigma notation is ∑_{i=1}^4 [8*(5i)^2 * e^(-0.1*(5i)) * 5]