The number of gallons of water in a large tank at time (t) minutes is given by the function w(t)=160,000-8,000t+t². find the average rate of change of the number of gallons of water in the tank from t=10 to t=10.5 minutes. interpret the sign of your answer in the context of this problem.

The number of gallons of water in the tank at t=10 is

... W(10) = 160,000 -10(8000 -10) = 80100

The number of gallons of water in the tank at t=10.5 is

... W(10.5) = 160,000 -10.5(8000 -10.5) = 76110.25

The rate of change over the interval is

... (W(10.5) - W(10))/(10.5 - 10) = (76110.25 - 80100)/(0.5) = -7979.5

The average rate of change in the number of gallons of water in the tank over the interval is -7979.5 gal/min.

The sign is negative, so the amount of water is decreasing.

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.