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.

To estimate the total U.S. per capita sales of bottled water from the start of 2001 to the start of 2006 using a Riemann sum, we need to divide the time interval into smaller subintervals, calculate the value of the function at certain points within each subinterval, and then multiply these values by the width of each subinterval.

Here's how we can proceed:

1. Divide the time interval [1, 6] (start of 2001 to start of 2006) into 5 subintervals of equal width. The width of each subinterval is Δt = (6 - 1) / 5 = 1.

2. Choose a point within each subinterval. Since we are using the right endpoint Riemann sum, we will choose the right endpoint of each subinterval as our points. The chosen points are t = 1, 2, 3, 4, and 5.

3. Evaluate the function s(t) = -0.18t^2 + 3t + 15 at each chosen point within each subinterval.
s(1) = -0.18(1)^2 + 3(1) + 15 = 17.82 gallons per year
s(2) = -0.18(2)^2 + 3(2) + 15 = 18.12 gallons per year
s(3) = -0.18(3)^2 + 3(3) + 15 = 16.62 gallons per year
s(4) = -0.18(4)^2 + 3(4) + 15 = 12.12 gallons per year
s(5) = -0.18(5)^2 + 3(5) + 15 = 4.62 gallons per year

4. Multiply each function value by the width of the subinterval and sum them up:
Δt[ s(1) + s(2) + s(3) + s(4) + s(5) ] = 1[ 17.82 + 18.12 + 16.62 + 12.12 + 4.62 ] = 1[ 69.3 ] = 69.3 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 is approximately 69.3 gallons per year.