Sigma Notation

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Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 2

1. 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 + 50sin(πt/12) for 0 ≤ t ≤ 24 hours. To estimate the total volume of water that enters the reservoir during the entire storm, the engineer wants to approximate the definite integral of R(t) from t=0 to t=24 using sigma notation with 6 subintervals of equal width and right endpoints. Write the Riemann sum that approximates this total volume. Answer: ∑_{i=1}^6 [100 + 50sin(π(4i)/12)] × 4 Solution: When approximating definite integrals using Riemann sums, we divide the interval into equal subintervals and evaluate the function at specific points within each subinterval.
   Full step-by-step solution

   When approximating definite integrals using Riemann sums, we divide the interval into equal subintervals and evaluate the function at specific points within each subinterval. For right endpoints, we use the function value at the end of each subinterval multiplied by the width of each subinterval to approximate the area under the curve.

2. Charlotte is a structural engineer designing a new suspension bridge. The rate of cable tension R(t) in kilonewtons per meter along the main cable is modeled by the function R(t) = 2t^2 + 7t + 12, where t is the horizontal distance in meters from the left tower (0 ≤ t ≤ 12). To estimate the total tension force exerted on the left half of the main cable from t = 0 to t = 12 meters, Charlotte needs to approximate the definite integral of R(t) from t = 0 to t = 12 using sigma notation with 6 subintervals of equal width and right endpoints. Write the Riemann sum in sigma notation that represents this approximation. Answer: ∑[i=1 to 6] (2(2i)^2 + 7(2i) + 12) × 2 Solution: The interval is from t = 0 to t = 12, so total length = 12 - 0 = 12 meters. With 6 subintervals, the width of each subinterval is Δt = (12 - 0)/6 = 12/6 = 2 meters.
   Full step-by-step solution

   Step 1: The interval is from t = 0 to t = 12, so total length = 12 - 0 = 12 meters. Step 2: With 6 subintervals, the width of each subinterval is Δt = (12 - 0)/6 = 12/6 = 2 meters. Step 3: Using right endpoints, the t-values are: t₁ = 2, t₂ = 4, t₃ = 6, t₄ = 8, t₅ = 10, t₆ = 12. Step 4: The general formula for the right endpoint of the i-th subinterval is: t_i = 0 + i × Δt = 2i. Step 5: The Riemann sum is ∑[i=1 to 6] R(t_i) × Δt. Step 6: Substitute R(t) = 2t^2 + 7t + 12 and t_i = 2i, Δt = 2: ∑[i=1 to 6] (2(2i)^2 + 7(2i) + 12) × 2. Step 7: This is the sigma notation for the Riemann sum approximation. The answer is ∑[i=1 to 6] (2(2i)^2 + 7(2i) + 12) × 2.

3. Olivia is a materials scientist studying the cooling rate of a newly developed alloy. The rate of heat loss R(t) in joules per second is modeled by the function R(t) = 9t^2 - 3t + 7 for 0 ≤ t ≤ 9 seconds, where t is time in seconds. To estimate the total heat lost during the first 9 seconds, Olivia wants to approximate the definite integral of R(t) from t = 0 to t = 9 using a Riemann sum with 3 subintervals of equal width and right endpoints. Write the sigma notation expression that represents this Riemann sum approximation. Answer: ∑_{i=1}^{3} (9(3i)^2 - 3(3i) + 7) × 3 Solution: The total time interval is from t = 0 to t = 9 seconds, so the width is 9 - 0 = 9 seconds. With 3 subintervals, each subinterval has width Δt = 9/3 = 3 seconds.
   Full step-by-step solution

   Step 1: The total time interval is from t = 0 to t = 9 seconds, so the width is 9 - 0 = 9 seconds. Step 2: With 3 subintervals, each subinterval has width Δt = 9/3 = 3 seconds. Step 3: Using right endpoints, the t-values are: t₁ = 3, t₂ = 6, t₃ = 9. In general, t_i = 0 + i × Δt = 3i for i = 1, 2, 3. Step 4: The Riemann sum in sigma notation is: ∑_{i=1}^{3} R(t_i) × Δt. Step 5: Substitute R(t) = 9t^2 - 3t + 7 and t_i = 3i, Δt = 3: ∑_{i=1}^{3} (9(3i)^2 - 3(3i) + 7) × 3. Step 6: Simplify inside the parentheses: 9(9i^2) = 81i^2, and -3(3i) = -9i. So the expression becomes ∑_{i=1}^{3} (81i^2 - 9i + 7) × 3. Step 7: The final sigma notation is ∑_{i=1}^{3} (9(3i)^2 - 3(3i) + 7) × 3.

4. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration C(t) in mg/L is given by the function C(t) = 5te^(-0.2t), where t is time in hours. The company needs to calculate the total drug exposure over the first 10 hours, which is represented by the area under the concentration curve. Express this total exposure using sigma notation with 5 equal subintervals, then write the definite integral that would give the exact value. Answer: ∫ from 0 to 10 of 5te^(-0.2t) dt Solution: C(t) = 5 * t * e^(-0.2 * t) We want the total drug exposure over the first 10 hours, which is the area under the curve from t = 0 to t = 10.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understanding the problem** We are given the concentration function: C(t) = 5 * t * e^(-0.2 * t) We want the total drug exposure over the first 10 hours, which is the area under the curve from t = 0 to t = 10. --- **Step 2: Sigma notation for 5 equal subintervals** If we use 5 equal subintervals over [0, 10], then: - Width of each subinterval: Δt = (10 - 0) / 5 = 2 hours. - We can use right endpoints, left endpoints, or midpoints. The problem doesn't specify, so let's use right endpoints for definiteness. Right endpoints: t_i = 0 + i * Δt = 2i, for i = 1, 2, 3, 4, 5. So t_1 = 2, t_2 = 4, t_3 = 6, t_4 = 8, t_5 = 10. The Riemann sum in sigma notation is: Total exposure ≈ Σ (from i=1 to 5) [ C(t_i) * Δt ] = Σ (i=1 to 5) [ 5 * (2i) * e^(-0.2 * (2i)) * 2 ] = Σ (i=1 to 5) [ 20i * e^(-0.4i) ] So in sigma notation: Σ_{i=1}^{5} 20 i e^(-0.4 i) --- **Step 3: Exact value as a definite integral** The exact total exposure is the limit of such Riemann sums as n → ∞, which is the definite integral of C(t) from 0 to 10. So exact exposure = ∫ from 0 to 10 of C(t) dt = ∫ from 0 to 10 of 5 t e^(-0.2 t) dt --- **Step 4: Final answer** The problem asks to express total exposure using sigma notation with 5 equal subintervals, then write the definite integral for the exact value. We have: Sigma notation: Σ_{i=1}^{5} 20 i e^(-0.4 i) Definite integral: ∫_0^10 5 t e^(-0.2 t) dt Since the correct answer given is the integral form, that is the final exact representation. --- **Final Answer:** ∫ from 0 to 10 of 5 t e^(-0.2 t) dt

5. Noah is arranging square tiles in a pyramid pattern. The bottom row has 11 tiles, the next row has 9 tiles, the next has 7 tiles, and so on, decreasing by 2 tiles each row until the top row has 1 tile. Using sigma notation, write the series representing the total number of tiles in this arrangement, then evaluate the sum. Answer: 36 Solution: Identify the pattern. The top row (row 1) has 1 tile. Row 2 has 3 tiles.
   Full step-by-step solution

   Step 1: Identify the pattern. The top row (row 1) has 1 tile. Row 2 has 3 tiles. Row 3 has 5 tiles. This pattern continues until the bottom row (row 6) has 11 tiles. The number of tiles in row n is 2n - 1. Step 2: There are 6 rows because 2(6) - 1 = 11. Step 3: Write the series in sigma notation: sum from n=1 to 6 of (2n - 1). Step 4: Evaluate using the formula for sum of first n odd numbers: sum from n=1 to 6 of (2n - 1) = 6^2 = 36. Step 5: The total number of tiles is 36.