Question

Calculus I problem. Please assist.

Answer 1

The lower and upper estimates for the total amount of oil leaked from the tank over the interval [0, 10] are both 66.2 liters.

To estimate the total amount of oil leaked over the interval [0, 10], you can use the trapezoidal rule with the given values of the rate r(t) at two-hour time intervals.

The formula for the trapezoidal rule is:

`\[ \text{Estimate} = (h)/(2) \left[ f(a) + 2f(a+h) + 2f(a+2h) + \ldots + 2f(b-h) + f(b) \right] \]`

where h is the width of each subinterval, a is the starting point (0 in this case), b is the ending point (10 in this case), and f(x) is the rate function.

In this scenario, since you are using five equal subintervals:

`\[ h = (10 - 0)/(5) = 2 \]`

Now, apply the trapezoidal rule:

`\[ \text{Estimate} = (2)/(2) \left[ 8.9 + 2(7.9) + 2(6.9) + 2(6.2) + 2(5.6) + 5.1 \right] \]`

`\[ \text{Estimate} = 1 \left[ 8.9 + 15.8 + 13.8 + 12.4 + 11.2 + 5.1 \right] \]`

`\[ \text{Estimate} = 66.2 \]`

Therefore, the lower and upper estimates for the total amount of oil leaked from the tank over the interval [0, 10] are both 66.2 liters.