100=1,478,300
1000=1,478,000
10,000=1,470,000
the average rate of change is just the slope from (0,f(0)) to (1,f(1))
find f(0) and f(1)
slope from (0,3) to (1,3) is rise/run=(3-3)/(0-1)=0/-1
the average rate of change from x=0 to x=1 for f(x) is 0
Assume:
Size of sides = x m
Depth of the pool = y m
Therefore, surface area = x^2+4xy =10 m^2
Then, y = (10-x^2)/(4x)
Now,
Volume (V) = x^2*y = x^2*y =x^2(10-x^2)/4x = (10x-x^3)/4 = 1/4(10x-x^3)
For maximum volume, first derivative of volume function is equal to zero.
That is,
dV/dx =0 = 1/4(10-3x^2)
Then,
1/4(10-3x^2) = 0
10-3x^2 = 0
3x^2=10
x= sqrt (10/3) = 1.826 m
And
y= (10-1.826^2)/(4*1.826) = 0.913 m
Therefore,
V= 1.826^2*0.913 = 3.044 m^3
If you want to check if your answer is correct, expanding it all out will help!
(x-6)(x+2)=0 <-- Use FOIL tactic
x²+2x-6x-12=0
x²-4x-12=0
As you can see, this is identical to the initial equation given. The x values would be 6 and -2, as seen from the factored equation.
Hope I helped :)
