Question

determine the average rates of change over each of the following time intervals. 0 minutes to 4 minutes: gallons per minute 4 minutes to 8 minutes: gallons per minute 0 minutes to 24 minutes: gallons per minute complete the statement to explain the meaning of the rate of change in this situation. the amount of water remaining in the pool is decreasing at a rate of gallons per minute. what do the values above tell you about ( v(t) )? ( v(t) ) is linear.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = V(t)\) over the interval \([a,b]\) is \(\frac{V(b)-V(a)}{b - a}\).

Step2: Find \(V(0)\), \(V(4)\), \(V(8)\) and \(V(24)\)

From the graph: \(V(0)=48\), \(V(4) = 40\), \(V(8)=32\), \(V(24)=0\)

Step3: Calculate average rate of change from \(0\) to \(4\) minutes

\(\frac{V(4)-V(0)}{4 - 0}=\frac{40 - 48}{4}=\frac{- 8}{4}=-2\)

Step4: Calculate average rate of change from \(4\) to \(8\) minutes

\(\frac{V(8)-V(4)}{8 - 4}=\frac{32 - 40}{4}=\frac{-8}{4}=-2\)

Step5: Calculate average rate of change from \(0\) to \(24\) minutes

\(\frac{V(24)-V(0)}{24 - 0}=\frac{0 - 48}{24}=\frac{-48}{24}=-2\)

Answer:

0 minutes to 4 minutes: \(-2\) gallons per minute
4 minutes to 8 minutes: \(-2\) gallons per minute
0 minutes to 24 minutes: \(-2\) gallons per minute
The amount of water remaining in the pool is decreasing at a rate of \(-2\) gallons per minute.