It takes one valve 5 hours to fill the tank
it takes one valve 6 hours to fill the tank
it takes another valve 10 hours to drain the tank
if all 3 valves are open at the same time how long does it take to fill the tank?

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

(5x-1)^2= (5x)^2-2(5x)×1+(1)^2=25x^2-10x+1

a ≈ 172.05

step-by-step explanation:

the pictures.