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The amount of water in a tank, in liters, can be modeled by the function 
Q(t), where 
t is measured in hours. The instantaneous rate of change of the amount of water in the tank is 4 liters per hour when 
t=5. Selected values of 
Q(t) are shown in the table below. Use a linear approximation when 
t=5 to estimate the amount of water in the tank at time 
t=5.1. Use proper units.





t
0
5
7
10
15



Q(t)
31
51
41
26
11




Answer kempricet ofs

The amount of water in a tank, in liters, can be modeled by the function Q(t) Q(t) , where t t is measured in hours. The instantaneous rate of change of the amount of water in the tank is 44 liters per hour when t=5 t=5 . Selected values of Q(t) Q(t) are shown in the table below. Use a linear approximation when t=5 t=5 to estimate the amount of water in the tank at time t=5.1 t=5.1 . Use proper units.\newline\begin{tabular}{|c|c|c|c|c|c|}\newline\hlinet t & 00 & 55 & 77 & 1010 & 1515 \\\newline\hlineQ(t) Q(t) & 3131 & 5151 & 4141 & 2626 & 1111 \\\newline\hline\newline\end{tabular}\newlineAnswer kempricet ofs

Full solution

Q. The amount of water in a tank, in liters, can be modeled by the function Q(t) Q(t) , where t t is measured in hours. The instantaneous rate of change of the amount of water in the tank is 44 liters per hour when t=5 t=5 . Selected values of Q(t) Q(t) are shown in the table below. Use a linear approximation when t=5 t=5 to estimate the amount of water in the tank at time t=5.1 t=5.1 . Use proper units.\newline\begin{tabular}{|c|c|c|c|c|c|}\newline\hlinet t & 00 & 55 & 77 & 1010 & 1515 \\\newline\hlineQ(t) Q(t) & 3131 & 5151 & 4141 & 2626 & 1111 \\\newline\hline\newline\end{tabular}\newlineAnswer kempricet ofs
  1. Identify Rate of Change: Identify the rate of change at t=5t=5, which is given as 44 liters per hour. This means for every hour, the amount of water changes by 44 liters.
  2. Use Value at t=5t=5: Use the value of Q(t)Q(t) at t=5t=5 from the table, which is 5151 liters.
  3. Calculate Change: Since the rate of change is 44 liters per hour, and we need to find the amount at t=5.1t=5.1, we calculate the change over 0.10.1 hours. The change in water amount =4= 4 liters/hour ×0.1\times 0.1 hours =0.4= 0.4 liters.
  4. Estimate Amount at t=5.1t=5.1: Add this change to the amount at t=5t=5 to estimate the amount at t=5.1t=5.1. So, Q(5.1)51 liters+0.4 liters=51.4 liters.Q(5.1) \approx 51 \text{ liters} + 0.4 \text{ liters} = 51.4 \text{ liters}.

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