Question

s(t) = 0.29t2 − t + 25 gallons per year (0 ≤ t ≤ 7), where t is time in years since the start of 2007. Use a Riemann sum with n = 5 to estimate the total country's per capita sales of bottled water from the start of 2008 to the start of 2012. (Use a left Riemann sum to find your answer. Round your answer to the nearest gallon.) HINT [See Example 2.]

Answer 1

Answer: The answer is 300 gallons.

Step-by-step explanation: Riemann sum is a method of calculating the total area under a curve on a graph, which is also known as Integral.

To calculate that area, we divide it into a number of rectangles with one point touching the curve. The curve has a closed interval [a,b] that can be subdivided into n subintervals, each having a width of Δ
`x_(k)` =
`(b-a)/(n)`

If a function is defined on the closed interval [a,b] and
`c_(k)` is any point in [
`x_(k-1)`,
`x_(k)`], then a Riemann Sum is defined as ∑f(
`c_(k)`)Δ
`x_(k)`.

For this question:

Δ
`x_(k)` =
`(7-0)/(5)` = 1.4

Now, we have to find s(t) for each valor on the interval:

s(t) = 0.29
`t^(2)` - t +25

s(0) = 25

s(1) = 24.29

s(2) = 24.16

s(3) = 24.61

s(4) = 25.64

s(5) = 27.25

s(6) = 29.44

s(7) = 32.21

Now, using the formula:

∑f(
`c_(k)`)Δ
`x_(k)` = 1.4(25+24.29+24.16+24.61+25.64+29.44+32.21)

∑f(
`c_(k)`)Δ
`x_(k)` = 1.4(212.6)

∑f(
`c_(k)`)Δ
`x_(k)` ≅ 300

With Riemann Sum, it is estimated the total country's per capita sales of bottled water is 300 gallons.