The rate of U.S. per capita sales of bottled water for the period 2000-2010 could be approximated by s(t) = −0.18t2 + 3t + 15 gallons per year (0 ≤ t ≤ 10),

where t is time in years since the start of 2000. Use a Riemann sum with n = 5 to estimate the total U.S. per capita sales of bottled water from the start of 2001 to the start of 2006.

I posted this earlier but would appreciate someone checking my answer. Calculating the Left Riemann sum, I get 110.1 - is this correct?

To calculate the Left Riemann sum, we divide the interval [1, 6] into 5 subintervals of equal width. The width of each subinterval is (6-1)/5 = 1, and the partition points are t₁=1, t₂=2, t₃=3, t₄=4, and t₅=5.

The Left Riemann sum is calculated by evaluating the function at the left endpoint of each subinterval and summing the products with the width of the subinterval.

So, the Left Riemann sum for this problem is:

LRS = [s(1) x 1] + [s(2) x 1] + [s(3) x 1] + [s(4) x 1] + [s(5) x 1]

Let's calculate it step by step.

s(1) = -0.18(1)² + 3(1) + 15 = 14.82
s(2) = -0.18(2)² + 3(2) + 15 = 14.04
s(3) = -0.18(3)² + 3(3) + 15 = 14.06
s(4) = -0.18(4)² + 3(4) + 15 = 16.32
s(5) = -0.18(5)² + 3(5) + 15 = 19.2

LRS = (14.82 x 1) + (14.04 x 1) + (14.06 x 1) + (16.32 x 1) + (19.2 x 1)
= 14.82 + 14.04 + 14.06 + 16.32 + 19.2
= 78.44

Thus, the Left Riemann sum for this problem is approximately 78.44. Therefore, the total U.S. per capita sales of bottled water from the start of 2001 to the start of 2006 is estimated to be 78.44 gallons per year.

To estimate the total U.S. per capita sales of bottled water from the start of 2001 to the start of 2006 using a left Riemann sum with n = 5, we can follow these steps:

1. Calculate the width of each subinterval:
The total time period is from the start of 2001 to the start of 2006, which is 5 years. Since we are using five subintervals, each subinterval's width would be 5/5 = 1 year.

2. Calculate the left endpoints of each subinterval:
The left endpoints for the five subintervals are:
- Year 2001 (t = 1)
- Year 2002 (t = 2)
- Year 2003 (t = 3)
- Year 2004 (t = 4)
- Year 2005 (t = 5)

3. Evaluate the function at each left endpoint:
Substituting the left endpoints into the function s(t) = −0.18t^2 + 3t + 15, we get:
- s(1) = -0.18(1)^2 + 3(1) + 15 = 17.82 gallons per year
- s(2) = -0.18(2)^2 + 3(2) + 15 = 16.64 gallons per year
- s(3) = -0.18(3)^2 + 3(3) + 15 = 14.09 gallons per year
- s(4) = -0.18(4)^2 + 3(4) + 15 = 10.48 gallons per year
- s(5) = -0.18(5)^2 + 3(5) + 15 = 5.25 gallons per year

4. Calculate the sum of the function values multiplied by the width:
Using the left Riemann sum formula, we can calculate the sum as follows:
(left endpoint 1 * width) + (left endpoint 2 * width) + (left endpoint 3 * width) + (left endpoint 4 * width) + (left endpoint 5 * width) =
(17.82 * 1) + (16.64 * 1) + (14.09 * 1) + (10.48 * 1) + (5.25 * 1) = 64.28 gallons per year.

Therefore, the estimated total U.S. per capita sales of bottled water from the start of 2001 to the start of 2006 using a left Riemann sum with n = 5 is 64.28 gallons per year.

Based on the calculations, your answer of 110.1 gallons per year seems to be incorrect. Please double-check the calculation steps to ensure accuracy.

Yes, good work!