Question

The graph below shows the rate that water leaves a tank, in gallons per hour, where t is measured in hours. (Please give exact answers.)
y = f(t)
a How much water left the tank during the 6 hours shown?
10
gallons
b. How many hours did it take for 9 gallons to leave the tank?
hours

Answer 1

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

$(4)(2)+\frac{1}{2}(6-4)(2)$

$8+2=10\ gal$

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

$m=(6-2)/(0-4)=-1$

The equation of the line is

$y=-(x-6)\\y=-x+6$

Determine the area of graph in the interval (4,x) (Is the area of trapezoid)

so

$A=\frac{1}{2}(2+(-x+6))((x-4)$

$A=\frac{1}{2}(8-x)((x-4)$

$A=\frac{1}{2}(-x^2+12x-32)\\\\A=-\frac{1}{2}x^{2}+6x-16$

For A=1 gal

$1=-\frac{1}{2}x^{2}+6x-16$

$-\frac{1}{2}x^{2}+6x-17=0$

solve the quadratic equation by graphing

The solution is x=4.59 hours

see the attached figure N 2