Expert: Socrates - 5/21/2008

Question
Hi Socrates,
I am working on this practice problem
Water usage rate for a city in a given 24-hour time period is represented in the table below (NOTE: t = 0 corresponds to midnight). Time is measured in hours, and the rate of water usage is measured in millions of gallons per hour.
t    R(t)
0    0.5
2    0.6
4    0.9
6    1.3
8    1.7
10   1.5
12   1.3
14   1.4
16   1.5
18   1.4
20   1.1
22   0.7
24    0.5

 
 a) Make an estimate of the average water usage over the 24-hour period.
 b) Find the average rate of change of R(t) over the 24-hour period.
  c) Estimate R'(12). Explain the physical meaning of this value. Be sure to use proper units in your answer.

could you tell me how to proceed - Thanks!

Answer
a) 27.8 million gallons

From t=0 to t=2  , the average rate of water use is (.5+.6)/2 =.55 million gallons per hour.
For the two hours, this means approximately 2 x .55 = 1.1 million gallons will be used


From t=2 to t=4  , the average rate of water use is (.6+.9)/2 = .75
For the two hours, this means approximately 2 x .75 = 1.5 million
gallons will be used

From t=4 to t=8  , the average rate of water use is (.9+ 1.3)/2 = 1.1
For the two hours, this means approximately 2 x 1.1 = 2.2 million gallons will be used.


Keep going like this until you get the last estimate for amount of water used from t=23 to t=24.  Then add up the estimates.


1.1 + 1.5 + 2.2 +.......= 27.8 million gallons used in 24 hours



b) 0

Average rate of change over the 24 hours will be

(R(24) - R(0))/(24 - 0) = (.5 - .5 )/(24 - 0) = 0/24 = 0


c) -.025 million gallons per hour per hour. The rate of water usage is decreasing at approximately .025 million gallons per hour per hour at t=12

You really need to see a graph for this part.
Use the slope of the line joining (10,R(10)) and (14,R(14))to estimate the slope of the tangent line when t=12 . The slope of the tangent line to the graph when t=12 will be R'(12).

(10,R(10)) = (10,1.5)

(14,R(14)) = (14,1.4)

Slope = (1.4 - 1.5)/ (14 - 10) = - .1/4 = -1/40 = -.025

Estimate for R'(12) = -.025 , this estimate says the rate of water use is decreasing by about .025 million gallons per hour per hour.