Final answer:
The volume V(t) of water in the tank at time t is determined by the formula 32⋅e10ln∣1+t∣−18t. This formula arises from a differential equation that models water flow into and out of the tank, with an initial condition V(0)=32 gallons.
Step-by-step explanation:
The given information states that the rate at which water flows into the tank is 10/1+t times the volume of water in the tank at time t, and water flows out of the tank at a rate of 18 gallons per hour.
To determine the formula for V(t), the volume of water in the tank at time t, we can set up a differential equation.
Let V(t) be the volume of water in the tank at time t. The rate of change of
V(t) is given by the difference between the inflow and outflow rates:
dV/dt = 10/ 1+t⋅V−18
Now, we can solve this differential equation with the initial condition
V(0)=32. The solution to this initial value problem will give us the formula for V(t).
dV/dt = 10/ 1+t⋅V−18
Separate variables and integrate:
∫ 1/V dV=∫( 10/1+t −18)dt
ln∣V∣=10ln∣1+t∣−18t+C
Now, apply the initial condition V(0)=32:
ln∣32∣=10ln∣1+0∣−18⋅0+C
ln∣32∣=C
So, the solution is:
ln∣V∣=10ln∣1+t∣−18t+ln∣32∣
Exponentiate both sides to solve for
V(t)=32e−18t (1+t)10
Therefore, the formula for V(t)=32e−18t (1+t)10, given the initial condition V(0)=32.