Answer:
Part a) 10 gallons
Part b) 4.59 hours
Step-by-step explanation:
<u><em>The picture of the question in the attached figure</em></u>
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
![(4)(2)+\frac{1}{2}(6-4)(2)](https://tex.z-dn.net/?f=%284%29%282%29%2B%5Cfrac%7B1%7D%7B2%7D%286-4%29%282%29)
![8+2=10\ gal](https://tex.z-dn.net/?f=8%2B2%3D10%5C%20gal)
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
![m=(6-2)/(0-4)=-1](https://tex.z-dn.net/?f=m%3D%286-2%29%2F%280-4%29%3D-1)
The equation of the line is
![y=-(x-6)\\y=-x+6](https://tex.z-dn.net/?f=y%3D-%28x-6%29%5C%5Cy%3D-x%2B6)
Determine the area of graph in the interval (4,x) (Is the area of trapezoid)
so
![A=\frac{1}{2}(2+(-x+6))((x-4)](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%282%2B%28-x%2B6%29%29%28%28x-4%29)
![A=\frac{1}{2}(8-x)((x-4)](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%288-x%29%28%28x-4%29)
![A=\frac{1}{2}(-x^2+12x-32)\\\\A=-\frac{1}{2}x^{2}+6x-16](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%28-x%5E2%2B12x-32%29%5C%5C%5C%5CA%3D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B2%7D%2B6x-16)
For A=1 gal
![1=-\frac{1}{2}x^{2}+6x-16](https://tex.z-dn.net/?f=1%3D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B2%7D%2B6x-16)
![-\frac{1}{2}x^{2}+6x-17=0](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B2%7D%2B6x-17%3D0)
solve the quadratic equation by graphing
The solution is x=4.59 hours
see the attached figure N 2