Question

A car's cooling system has a capacity of 20 quarts. Initially, the system contains a mixture of 3 quarts of antifreeze and 17 quarts of water. Water runs into the system at the rate of 1 gal min , then the homogeneous mixture runs out at the same rate. In quarts, how much antifreeze is in the system at the end of 5 minutes? (Round your answer to two decimal places.)

Answer 1

Answer:

1.10 quarts

Step-by-step explanation:

We are given that

Capacity of system=20 quarts=$\frac{20}{4}=5 gal$

1 gallon=4 quarts

Initially mixture contains

Antifreeze=3 quarts=$\frac{3}{4}=0.75 gal$

Water=17 quarts=$\frac{17}{4}=4.25 gal$

Water runs in to the system=1gal/min

We have to find the antifreeze in the system at the end of 5  minutes.

Concentration inflow=0

Let Concentration outflow=x

Rate of system=Concentration inflow(water inflow rate)-Concentration outflow(water outflow rate)

$\frac{dx}{dt}=0\times 1-\frac{1}{5} x=-\frac{x}{5}$

$\frac{dx}{x}=-\frac{dt}{5}$

Taking integration on both sides

$\int\frac{dx}{x}=-\frac{1}{5}\int dt$

$ln x=-\frac{1}{5}t+C$

$x=3 and t=0$

$ln 3=C$

$ln x-ln 3=-\frac{1}{5}t$

$ln\frac{x}{3}=-\frac{1}{5}t$

Using formula

$ln m-ln n=ln\frac{m}{n}$

$\frac{x}{3}=e^{-0.2t}$

$x=3e^{-0.2t}$

Substitute t=5

$x=3e^{-0.2\times 5}=3e^{-1}=1.10$

Hence, in the end of 5 minute ,the antifreeze in the system=1.10 quarts