Sigma Notation

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Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 2

1. Isabella, a biomedical engineer, is modeling the rate at which a new drug is metabolized in the body. The rate of change of the drug concentration in the bloodstream, R(t) in mg/L per hour, is given by R(t) = 30t^2 * e^(-0.15t) for t ≥ 0 hours. To estimate the total change in drug concentration over the first 12 hours, Isabella needs to approximate the definite integral of R(t) from t = 0 to t = 12. She decides to use a Riemann sum with 6 subintervals of equal width, using right endpoints. Write the sigma notation expression that represents this Riemann sum approximation. Answer: ∑_{i=1}^{6} [30(2i)^2 * e^(-0.15(2i))] * 2 Solution: The total interval is from t = 0 to t = 12, so the length is 12 hours. With 6 subintervals, the width of each subinterval is Δt = (12 - 0)/6 = 2 hours.
   Full step-by-step solution

   Step 1: The total interval is from t = 0 to t = 12, so the length is 12 hours. With 6 subintervals, the width of each subinterval is Δt = (12 - 0)/6 = 2 hours. Step 2: For right endpoints, the i-th endpoint is t_i = 0 + i * Δt = 2i, where i goes from 1 to 6. Step 3: The rate function is R(t) = 30t^2 * e^(-0.15t). At t = 2i, this becomes R(2i) = 30(2i)^2 * e^(-0.15(2i)). Step 4: The Riemann sum is the sum of the area of each rectangle: area = height * width = R(t_i) * Δt. So the sum is ∑_{i=1}^{6} R(2i) * 2. Step 5: Substituting R(2i) gives the final expression: ∑_{i=1}^{6} [30(2i)^2 * e^(-0.15(2i))] * 2. Therefore, the sigma notation expression is ∑_{i=1}^{6} [30(2i)^2 * e^(-0.15(2i))] * 2.

2. A triangular arrangement of stones is built by Kaia. The first row (top) has 1 stone, the second row has 3 stones, the third row has 5 stones, and this pattern continues. Each stone is a cube with side length 1 unit. The stones in each row are placed side by side to form a continuous line. Write, using sigma notation, the total volume of all stones in this triangular arrangement when there are 13 rows, and then evaluate the sum. Answer: 169 Solution: Identify the pattern. Row 1 has 1 stone, row 2 has 3 stones, row 3 has 5 stones. The number of stones in row n is 2n - 1.
   Full step-by-step solution

   Step 1: Identify the pattern. Row 1 has 1 stone, row 2 has 3 stones, row 3 has 5 stones. The number of stones in row n is 2n - 1. Step 2: The total number of stones in 13 rows is the sum of the odd numbers from 1 to (2*13 - 1) = 25. In sigma notation: sum from n=1 to 13 of (2n - 1). Step 3: Evaluate the sum. sum from n=1 to 13 of (2n - 1) = 2 * sum from n=1 to 13 of n - sum from n=1 to 13 of 1. Step 4: sum from n=1 to 13 of n = 13 * 14 / 2 = 91. So 2 * 91 = 182. Step 5: sum from n=1 to 13 of 1 = 13. Step 6: 182 - 13 = 169. Step 7: Since each stone has volume 1 cubic unit, the total volume is 169 cubic units. The answer is 169.

3. A hexagonal pattern is constructed by arranging circles in concentric hexagonal rings. The innermost ring (ring 1) has 1 circle. The second ring (ring 2) has 6 circles. The third ring (ring 3) has 12 circles, and the fourth ring (ring 4) has 18 circles. This pattern continues, where each subsequent ring adds 6 more circles than the previous ring. If the pattern has 11 rings total, write the total number of circles in the entire pattern using sigma notation, then evaluate the sum. Answer: 331 Solution: Identify the pattern. Ring 1 has 1 circle. Ring 2 has 6 circles.
   Full step-by-step solution

   Step 1: Identify the pattern. Ring 1 has 1 circle. Ring 2 has 6 circles. Ring 3 has 12 circles. Ring 4 has 18 circles. For ring n where n >= 2, the number of circles is 6(n-1). So ring 2: 6(2-1)=6, ring 3: 6(3-1)=12, ring 4: 6(4-1)=18. For ring 1, it's just 1. Step 2: Write the total number of circles using sigma notation. Total = 1 + sum from n=2 to 11 of 6(n-1). Alternatively, we can write it as sum from k=1 to 10 of 6k, plus the 1 from ring 1. So total = 1 + sum from k=1 to 10 of 6k. Step 3: Evaluate sum from k=1 to 10 of 6k. This is 6 times sum from k=1 to 10 of k. Sum from k=1 to 10 of k = 10*11/2 = 55. So 6 * 55 = 330. Step 4: Add the 1 from ring 1: 330 + 1 = 331. Step 5: The total number of circles is 331.

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 R(t) = 100 + 20t - t² for 0 ≤ t ≤ 10 hours. To estimate the total volume of water that entered the reservoir during the storm, the engineer wants to express the definite integral of R(t) from t=0 to t=10 using sigma notation with 5 equal subintervals and right endpoints. Write the Riemann sum in sigma notation. Answer: ∑[i=1 to 5] [100 + 20(2i) - (2i)²] × 2 Solution: The interval from t=0 to t=10 is divided into 5 equal subintervals, so each subinterval has width Δt = (10-0)/5 = 2 hours. Using right endpoints, the evaluation points are t_i = 0 + i×2 = 2i for i=1,2,3,4,5.
   Full step-by-step solution

   Step 1: The interval from t=0 to t=10 is divided into 5 equal subintervals, so each subinterval has width Δt = (10-0)/5 = 2 hours. Step 2: Using right endpoints, the evaluation points are t_i = 0 + i×2 = 2i for i=1,2,3,4,5. Step 3: The function value at each right endpoint is R(2i) = 100 + 20(2i) - (2i)². Step 4: The Riemann sum is the sum of R(2i)×Δt for i=1 to 5. Step 5: Therefore, the sigma notation is ∑[i=1 to 5] [100 + 20(2i) - (2i)²] × 2.

5. An environmental scientist 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) = 8t²e^(-0.1t) for t ≥ 0 years. To estimate the total carbon absorbed during the first 15 years, the scientist wants to approximate the definite integral from t=0 to t=15 using sigma notation with 6 equal subintervals and right endpoints. Write the Riemann sum that approximates this total carbon absorption. Answer: ∑_{i=1}^6 [8(2.5i)²e^(-0.1(2.5i)) × 2.5] Solution: The time interval is from t=0 to t=15, which has width 15 years. With 6 equal subintervals, each subinterval has width Δt = 15/6 = 2.5 years. Using right endpoints, the evaluation points are t_i = 2.5i for i=1,2,3,4,5,6.
   Full step-by-step solution

   Step 1: The time interval is from t=0 to t=15, which has width 15 years. Step 2: With 6 equal subintervals, each subinterval has width Δt = 15/6 = 2.5 years. Step 3: Using right endpoints, the evaluation points are t_i = 2.5i for i=1,2,3,4,5,6. Step 4: The function values at these points are A(t_i) = 8(2.5i)²e^(-0.1(2.5i)). Step 5: The Riemann sum is the sum of A(t_i) × Δt for i=1 to 6. Step 6: This gives ∑_{i=1}^6 [8(2.5i)²e^(-0.1(2.5i)) × 2.5]. The Riemann sum is ∑_{i=1}^6 [8(2.5i)²e^(-0.1(2.5i)) × 2.5].