# Rate of change of salt in the tank

A 100 gallon tank is filled with pure water. Water which has a concentration of 7g of salt per gallon flows into the tank at a rate of 10 gallons/min, and the mixture is stirred to a uniform concentration. Water also leaks from the tank at the same rate, 10 gallons/min.

Find a differential equation describing the rate of change of salt in the tank.

Hint: The concentration of salt in the tank is S(t)/100, where S(t) is the total amount of salt in the tank at time t, in grams. S′(t), the rate of change of salt in the tank over time, is equal to the [rate in] − [rate out].

My soultion was

$dS/dt = 7/10 − S(t)/10$ But that is incorrect. Can someone show me their workings to the solution so I know where I went wrong?

My workings were (10 gallons/mins)(7g/ gallons)-(10 gallons/mins)(s(t)/60)

• might want to convert grams to gallons ! – Saketh Malyala Apr 3 '17 at 3:49
• @SakethMalyala Gram is a unit of mass, gallon is a unit of volume. No need to convert. – bof Apr 3 '17 at 3:56
• In one minute, $10$ gallons of water flow into the tank, each gallon contains $7$ grams of salt, so that $7\times10$ (not $7/10$) grams of salt entering the tank each minute. – bof Apr 3 '17 at 3:59
• After $t$ minutes, the tank contains $S(t)$ grams of salt dissolved in $100$ gallons of water (this is a constant-volume problem), so the concentration of salt in the tank is $S(t)/100$ grams/gallon. But the rate of flow is $10$ gallons/minute, so salt is leaving the tank at the rate of $S(t)/10$ grams per minute. OK, you got that term right. Your only mistake is writing $7/10$ instead of $7\times10.$ – bof Apr 3 '17 at 4:02
• @ bof, I see where I went wrong now, so it should be dS/dt = 70−S(t)/10. Thank you – Michelle Apr 3 '17 at 4:04