Question

A valve in a full 6000 gallon water tank is slowly opening. Water flows out of the tank through the valve. The flow rate in gallons per hour is given by the function f(t)=300 t^2 where t is in minutes.
How much water flows out the tank in the first 7 minutes?
How many minutes does it take for the tank to be completely empty?,

Answer 1

The volume of the gallon is 6000 gallon
the rate of water flow per min is f(t)=300t^2
a] amount of water that flowed in the first 7 minutes=7/60 hours will be:
f(7)=300(7/60)^2
solving the above we get
f(7)=7/12 gallons

b] The time taken for the tank to be empty will be as follows:
amount of water that will flow out for the tank to be empty will be 6000 gallons
thus,
6000=300t^2
this can be simplified to
20=t^2
t^2-20=0
solving this we get:
t=-2√5 or 2√5
the time taken for the tank to be empty will be t=2√5 hours=5.4721 hours

Answer 2

Answer:

a) entire tank is empty hence 6000 gallons

b) 3.915 mins

Step-by-step explanation:

a)

$\int\limits^7_0 {f(t)} \, dt = \int\limits^7_0 {300t^2} \, dt \\\\= 100t^2\limits^7_0\\= 34300 > 6000 \\Hence, entire tank empt$

b) V (t) = 100t^3

6000 = 100t^3

$t = \sqrt[3]{60} = 3.915 mins$