The rale al which water is going into a sink is given by the function r(t) = gal
ID: 2860065 • Letter: T
Question
The rale al which water is going into a sink is given by the function r(t) = gallons per minute and t is time in minutes. How fast is the water amount changing at t = 7 minutes? What docs your answer tell you about the water in the sink? how much has the amount of water changed in the sink between t 2 and t - 7 minutes? How much water is in the sink at t = 4 minutes? Suppose there was a gallon of water initially in the sink. Derive a formula for the amount of water in the sink at any time t minutes.Explanation / Answer
solution:
a) rate of water going to sink
r(t) from t (6,8) is
r(t)=t-8
now rate of change of water
at t=7
hence at t=7
rate of change of water=t-8=7-8=-1
hence the water is coming out of the sink at a rate 1ga/min
b)
amount of water in sink from t=2 to t=7
at t=2 rate is =2
now amount of water at this time will be integral of the rate
= 2t
hence at t=2
amount
=4
and at t=7
rate= t-8
hence amount
= (t2-8t)
at t=7
=49-56=-7
now change of amount of water from t=2 to t=7
4-(-7)=11 gallons
c) at t=4
rate=0
hence amount is integration of the rate
and integration of 0 is constant
hence constant will be anything which should be provided in the question but as it is not given we will assume it to be k
d) now we have given that A(0)=1
and
dA/dt= 2 0<t<3
0 3<t<6
t-8 6<t<8
now amount will be integration of this function
A=2t+c for 0<t<3
=c1 3<t<6
= t2-8t+c2 6<t<8
now A(0)=1
hence 2(0)+c=1
c=1
hence function is
= 2t+1 0<t<3
c 3<t<6
t2-8t 6<t<8