Question

Suppose sewage flows through an intake pipe in a sewage treatment plant at the rate of S(t) thousand gallons per hour, where t is time in minutes. The chart below gives values of S for particular values of t:

t S(t)
0 1.2
2 2.1
6 2.3
8 2.4
10 2.7
12 2.9
14 3.1
16 4.5

Using a Riemann sum with right endpoints and 8 subdivisions, estimate the amount of sewage that flows through the intake pipe on the interval 0 ≤ t ≤ 16 You get approximately:
a. 43.8 thousand gallons
b. 43.8 gallons
c. 730 gallons
d. 365 gallons
e. 620 gallons

Answer 1

To estimate the total amount of sewage flow using a Riemann sum with right endpoints over 8 subdivisions, we calculate the width of each subdivision and multiply the flow rates from the right endpoints by this width. After converting to gallons per minute, the estimated total is approximately 667.6 gallons, which does not match the provided options, suggesting a possible mistake.

Step-by-step explanation:

The student's question asks for an estimation of the amount of sewage that flows through an intake pipe over a certain period using a Riemann sum with right endpoints. To estimate the total amount using a Riemann sum, we consider the flow rates at right endpoints of the 8 subdivisions intervals of the time range from t = 0 to t = 16 minutes. First, we find the width of each subdivision by dividing the total range of t by the number of subdivisions (16 minutes / 8 subdivisions = 2 minutes per subdivision).

We then use the provided values for S(t) at right endpoints for each subdivision to estimate the Riemann sum. Here is the calculation:

• 
  
• From t=2 to t=4 (2 minutes), the flow rate is 2.1 thousand gallons per hour.
• 
  
• From t=4 to t=6 (2 minutes), the flow rate is 2.3 thousand gallons per hour.
• 
  
• From t=6 to t=8 (2 minutes), the flow rate is 2.4 thousand gallons per hour.
• 
  
• From t=8 to t=10 (2 minutes), the flow rate is 2.7 thousand gallons per hour.
• 
  
• From t=10 to t=12 (2 minutes), the flow rate is 2.9 thousand gallons per hour.
• 
  
• From t=12 to t=14 (2 minutes), the flow rate is 3.1 thousand gallons per hour.
• 
  
• From t=14 to t=16 (2 minutes), the flow rate is 4.5 thousand gallons per hour.

Converting the flow rates from gallons per hour to gallons per minute by dividing by 60 (since there are 60 minutes in an hour), we get:

• 
  
• 2.1 thousand gallons per hour = 0.035 thousand gallons per minute
• 
  
• 2.3 thousand gallons per hour = 0.0383 thousand gallons per minute
• 
  
• 2.4 thousand gallons per hour = 0.04 thousand gallons per minute
• 
  
• 2.7 thousand gallons per hour = 0.045 thousand gallons per minute
• 
  
• 2.9 thousand gallons per hour = 0.0483 thousand gallons per minute
• 
  
• 3.1 thousand gallons per hour = 0.0517 thousand gallons per minute
• 
  
• 4.5 thousand gallons per hour = 0.075 thousand gallons per minute

Now, multiplying each flow rate by the duration of its interval (2 minutes each): (0.035 + 0.0383 + 0.04 + 0.045 + 0.0483 + 0.0517 + 0.075) * 2 = 0.6676 thousand gallons over the 16 minute interval.

Finally, multiplying by 1,000 to convert thousand gallons to gallons gives us approximately 667.6 gallons of sewage flow. It seems there may be a discrepancy with the provided options, as none of them matches this calculation. Therefore, it is important to check the calculations and values provided for accuracy.