Answer:
Part a) 10 gallons
Part b) 4.59 hours
Step-by-step explanation:
The picture of the question in the attached figure
Part a) How much water left the tank during the 6 hours shown?
To find out how much water left the tank during the 6 hours, determine the area of the graph in that interval
so
Calculate the area of the rectangle plus the area of the triangle
[tex](4)(2)+\frac{1}{2}(6-4)(2)[/tex]
[tex]8+2=10\ gal[/tex]
Part b) How many hours did it take for 9 gallons to leave the tank?
we know that
In the interval [0,4]
8 gallons of water leaves the tank in 4 hours (remember that the area in that interval is equal to 8 gallons)
so
Find out in the interval (4,6] how many hours did it take for 1 gallon to leave the tank
First determine the equation of the line in the interval (4,6)
we have the points (4,2) and (6,0)
The slope is equal to
[tex]m=(6-2)/(0-4)=-1[/tex]
The equation of the line is
[tex]y=-(x-6)\\y=-x+6[/tex]
Determine the area of graph in the interval (4,x) (Is the area of trapezoid)
so
[tex]A=\frac{1}{2}(2+(-x+6))((x-4)[/tex]
[tex]A=\frac{1}{2}(8-x)((x-4)[/tex]
[tex]A=\frac{1}{2}(-x^2+12x-32)\\\\A=-\frac{1}{2}x^{2}+6x-16[/tex]
For A=1 gal
[tex]1=-\frac{1}{2}x^{2}+6x-16[/tex]
[tex]-\frac{1}{2}x^{2}+6x-17=0[/tex]
solve the quadratic equation by graphing
The solution is x=4.59 hours
see the attached figure N 2
