Sigma Notation

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Answer Key & Explanations

Sigma Notation · Grade 12 · Worksheet 3

1. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = 50te^(-0.2t) mg/L, 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 definite integral of C(t) from 0 to 10. Express this total drug exposure using sigma notation with 5 subintervals and right endpoints. Answer: ∑_{i=1}^5 [50(2i)e^(-0.2(2i)) * 2] Solution: C(t) = 50 t e^(-0.2 t) We want the definite integral from t = 0 to t = 10. They ask us to express the total drug exposure using sigma notation with 5 subintervals and right endpoints.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understanding the problem** We have C(t) = 50 t e^(-0.2 t) We want the definite integral from t = 0 to t = 10. They ask us to express the total drug exposure using **sigma notation** with **5 subintervals** and **right endpoints**. That means we are approximating the integral using a Riemann sum with n = 5, right endpoints. --- **Step 2: Determine Δt** Interval: [0, 10] Number of subintervals: n = 5 Δt = (10 - 0) / 5 = 10 / 5 = 2 So each subinterval has width 2. --- **Step 3: Right endpoints formula** For right endpoints: t_i = a + i * Δt Here a = 0, Δt = 2, so t_i = 0 + i * 2 = 2i We go from i = 1 to i = 5 because right endpoints: i=1 → t=2 i=2 → t=4 i=3 → t=6 i=4 → t=8 i=5 → t=10 --- **Step 4: Riemann sum in sigma notation** Riemann sum = Σ [ f(t_i) * Δt ] from i=1 to n. Here f(t) = C(t) = 50 t e^(-0.2 t) Δt = 2 So f(t_i) = 50 * (2i) * e^(-0.2 * (2i)) Thus the sum is: Σ_{i=1}^5 [ 50 * (2i) * e^(-0.2 * (2i)) * 2 ] --- **Step 5: Final check** Let's write it clearly: t_i = 2i C(t_i) = 50 * (2i) * e^(-0.2 * (2i)) Area of rectangle = C(t_i) * Δt = [50 * (2i) * e^(-0.2 * (2i))] * 2 So the sigma notation is: ∑_{i=1}^5 [50(2i)e^(-0.2(2i)) * 2] This matches the given correct answer. --- **Final answer:** ∑_{i=1}^5 [50(2i)e^(-0.2(2i)) * 2]

2. 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.

3. Zoe is analyzing the total profit from a series of weekly sales. The profit in week k is given by (2k + 8) hundred dollars, where k starts at 1 and ends at 3. Write the total profit for the 3 weeks using sigma notation, then evaluate the sum to find the total profit in hundreds of dollars. Answer: 36 Solution: Identify the series: Σ(k=1 to 3) (2k + 8). Split the sum: Σ(k=1 to 3) 2k + Σ(k=1 to 3) 8. Evaluate Σ(k=1 to 3) 2k = 2 * Σ(k=1 to 3) k = 2 * (3(3+1)/2) = 3(3+1).
   Full step-by-step solution

   Step 1: Identify the series: Σ(k=1 to 3) (2k + 8). Step 2: Split the sum: Σ(k=1 to 3) 2k + Σ(k=1 to 3) 8. Step 3: Evaluate Σ(k=1 to 3) 2k = 2 * Σ(k=1 to 3) k = 2 * (3(3+1)/2) = 3(3+1). Step 4: Evaluate Σ(k=1 to 3) 8 = 8 * 3. Step 5: Total = 83 + 3(3+1) = 36. The total profit is 36 hundred dollars.

4. Sophia is a biomedical engineer studying how a new drug disperses in a patient's bloodstream. The concentration C(t) in mg/L of the drug at time t hours is modeled by C(t) = 12t e^{-0.3t}. To estimate the total drug exposure (area under the curve) over the first 18 hours, she decides to use a Riemann sum with 6 equal subintervals and right endpoints. Write the sigma notation expression for this Riemann sum that approximates the total drug exposure. Answer: ∑_{i=1}^{6} [12(3i) e^{-0.3(3i)}] × 3 Solution: The interval is from t=0 to t=18, so the total width is 18 hours. With 6 equal subintervals, the width of each subinterval is Δt = (18-0)/6 = 3 hours.
   Full step-by-step solution

   Step 1: The interval is from t=0 to t=18, so the total width is 18 hours. Step 2: With 6 equal subintervals, the width of each subinterval is Δt = (18-0)/6 = 3 hours. Step 3: Using right endpoints, the t-values are: t₁ = 3, t₂ = 6, t₃ = 9, t₄ = 12, t₅ = 15, t₆ = 18. Step 4: In general, the right endpoint for the i-th subinterval is t_i = 0 + i×Δt = 3i, where i goes from 1 to 6. Step 5: The Riemann sum is ∑[i=1 to 6] C(t_i) × Δt. Step 6: Substitute C(t) = 12t e^{-0.3t}, t_i = 3i, and Δt = 3: ∑[i=1 to 6] [12(3i) e^{-0.3(3i)}] × 3. Step 7: The final sigma notation is ∑_{i=1}^{6} [12(3i) e^{-0.3(3i)}] × 3.

5. Noah is a biomedical engineer analyzing the decay of a radioactive isotope used in medical imaging. The rate of decay R(t) in milligrams per hour is modeled by the function R(t) = 6t e^{-0.1t} for t ≥ 0 hours. To estimate the total amount of isotope that decays during the first 6 hours, Noah wants to approximate the definite integral of R(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} [6(2i)e^{-0.1(2i)}] × 2 Solution: The interval is from t=0 to t=6, so the total width is 6 hours. With 3 subintervals, each subinterval has width Δt = (6-0)/3 = 2 hours. 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 hours. Step 2: With 3 subintervals, each subinterval has width Δt = (6-0)/3 = 2 hours. 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 to 3] R(t_i) × Δt. Step 6: Substitute R(t) = 6t e^{-0.1t} and t_i = 2i, Δt = 2: ∑[i=1 to 3] [6(2i) e^{-0.1(2i)}] × 2. Step 7: Simplify: ∑_{i=1}^{3} [12i e^{-0.2i}] × 2 or ∑_{i=1}^{3} 24i e^{-0.2i}. The final sigma notation is: ∑_{i=1}^{3} [6(2i)e^{-0.1(2i)}] × 2.

6. Σ(k=1 to 9) (3k - 4) = ? Answer: 99 Solution: Identify the number of terms: n = 9. Find the first term (k=1): 3(1) - 4 = 3 - 4 = -1. Find the last term (k=9): 3(9) - 4 = 27 - 4 = 23.
   Full step-by-step solution

   Step 1: Identify the number of terms: n = 9. Step 2: Find the first term (k=1): 3(1) - 4 = 3 - 4 = -1. Step 3: Find the last term (k=9): 3(9) - 4 = 27 - 4 = 23. Step 4: Use the sum formula for an arithmetic series: S = n/2 * (first term + last term). Step 5: S = 9/2 * (-1 + 23) = 9/2 * 22 = 9 * 11 = 99. The answer is 99.