a large tank is filled with water when an outflow valve is opened at t = 0. water flows out at a rate in…
a large tank is filled with water when an outflow valve is opened at t = 0. water flows out at a rate in gal/min given by q(t)=0.5(196 - t²), for 0≤t≤14. a. find the amount of water q(t) that has flowed out of the tank after t minutes, given the initial condition q(0)=0. b. graph the function q for 0≤t≤14. c. how much water flows out of the tank in 14 min? a. q(t)=
Answer
Explanation:
Step1: Integrate the rate - function
We know that $Q(t)=\int Q^{\prime}(t)dt$. Given $Q^{\prime}(t)=0.5(196 - t^{2})=98-0.5t^{2}$, then $Q(t)=\int(98 - 0.5t^{2})dt$. Using the power - rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $Q(t)=98t-0.5\times\frac{t^{3}}{3}+C$.
Step2: Use the initial condition
Since $Q(0) = 0$, substituting $t = 0$ and $Q(0)=0$ into $Q(t)=98t-\frac{1}{6}t^{3}+C$, we get $0=98\times0-\frac{1}{6}\times0^{3}+C$, so $C = 0$. So, $Q(t)=98t-\frac{1}{6}t^{3}$.
Step3: Find the amount of water at $t = 14$
Substitute $t = 14$ into $Q(t)$: $Q(14)=98\times14-\frac{1}{6}\times14^{3}=98\times14-\frac{1}{6}\times2744=1372-\frac{2744}{6}=\frac{8232 - 2744}{6}=\frac{5488}{6}=\frac{2744}{3}\approx914.67$.
Answer:
a. $Q(t)=98t-\frac{1}{6}t^{3}$ c. $\frac{2744}{3}$ gallons