5. The rate (in liters per hour) at which oil leaked from a tank is recorded at

Question

5. The rate (in liters per hour) at which oil leaked from a tank is recorded at 3-hour intervals in the table below. Find upper and lower estimates for the amount of oil that leaked during the 9-hour period. t=0, 3, 6, 9 r(t)=3, 7, 10, 12 6. An object moves along the x-axis (where each unit represents 1 cm) and it's velocity after t seconds is v(t) cm per second. Tell what each of the following would mean relative to the object's motion. (a) S^4 (0) v(t)dt=-5 (b) S^4 (0) (absolute value v(t))dt=10 5. The rate (in liters per hour) at which oil leaked from a tank is recorded at 3-hour intervals in the table below. Find upper and lower estimates for the amount of oil that leaked during the 9-hour period. t=0, 3, 6, 9 r(t)=3, 7, 10, 12 6. An object moves along the x-axis (where each unit represents 1 cm) and it's velocity after t seconds is v(t) cm per second. Tell what each of the following would mean relative to the object's motion. (a) S^4 (0) v(t)dt=-5 (b) S^4 (0) (absolute value v(t))dt=10

Explanation / Answer

#5. A low estimate could be created by assuming linear increase between each pair of data points: thus, assume the average leakage in the first 3 hrs was 5 L/h, in the next 3 hrs 8.5 L/h, in the final 3 hrs 11 L/h. This gives a total of 73.5 liters leaked.
A higher estimate could be created by using the perfectly-fitting quadratic: leak-rate = -(1/2)(t/3)^2 + (9/2)(t/3) + 3, and then integrating that from t = 0 to 9.
Integral from t=0 to 9 of [3 + (3/2)t - (1/18)t^2] dt
= [3t + (3/4)t^2 - (1/54)t^3] to be evaluated at t = 9
= 27 + (243/4) - (27/2) = 74.25 liters leaked.

#6. It took me a couple of minutes to figure out your notation, but I guess your "S^4 (0)" means the integral from 0 to 4.
(a) If the integral from 0 to 4 of v(t) dt = -5, the particle's net displacement in the first four seconds was 5 cm to the left.
(b) If the integral from 0 to 4 of |v(t)| dt = 10, the total path-length in the first four seconds was 10 cm.

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