The flow rate for water at a pumping
station on a given day is

is the flow rate in thousands of
gallons/hour and is in hours
Find the average flow rate during
the course of the day.
Flow Rate thousand gallons/hour

Ask by Sullivan Clark. in the United States

Mar 22,2025

Question

The flow rate for water at a pumping
station on a given day is

is the flow rate in thousands of
gallons/hour and is in hours
Find the average flow rate during
the course of the day.
Flow Rate thousand gallons/hour

Ask by Sullivan Clark. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

Let the average flow rate be given by  
$$
\text{Average} = \frac{1}{24-0}\int_{0}^{24} R(t)\,dt.
$$

Given  
$$
R(t) = -t^2 + 24t + 100,
$$  
we have  
$$
\text{Average} = \frac{1}{24}\int_{0}^{24} \left(-t^2 + 24t + 100\right) dt.
$$

First, evaluate the integral:
$$
\int \left(-t^2 + 24t + 100\right) dt = -\frac{t^3}{3} + 12t^2 + 100t + C.
$$

Now, evaluate from $$ t = 0 $$ to $$ t = 24 $$:
$$
\int_{0}^{24} \left(-t^2 + 24t + 100\right) dt = \left[-\frac{24^3}{3} + 12(24)^2 + 100(24)\right] - \left[0\right].
$$

Calculate each term:
- $$ 24^3 = 13824 $$ so $$-\frac{13824}{3} = -4608$$.
- $$ 12(24)^2 = 12 \cdot 576 = 6912 $$.
- $$ 100 \cdot 24 = 2400 $$.

Thus, the integral becomes:
$$
-4608 + 6912 + 2400 = 4608.
$$

Now, the average flow rate is:
$$
\text{Average} = \frac{4608}{24} = 192.
$$

So, the average flow rate during the day is  
$$
192 \text{ thousand gallons/hour}.
$$
The average flow rate during the day is 192 thousand gallons/hour.

The flow rate for water at a pumping
station on a given day is

is the flow rate in thousands of
gallons/hour and is in hours
Find the average flow rate during
the course of the day.
Flow Rate thousand gallons/hour

Upstudy AI Solution

Answer

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The flow rate for water at a pumping station on a given day is is the flow rate in thousands of gallons/hour and is in hours Find the average flow rate during the course of the day. Flow Rate thousand gallons/hour

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions

The flow rate for water at a pumping
station on a given day is

is the flow rate in thousands of
gallons/hour and is in hours
Find the average flow rate during
the course of the day.
Flow Rate thousand gallons/hour