The the side of the cut-out square be x inches
base of box = 14-2x by 6-2x
(clearly x < 3 )
volume = V = x(14-2x)(6-2x)
= 84x - 40x^2 + 4x^3
dV/dx = 84 - 80x + 12x^2
= 0 for a max V
divide by 4
3x^2 - 20x + 21 = 0
x = (20 ± √148)/6
= 5.36 or 1.306 inches, but remember x < 3
so x = 1.31 correct to 2 decimals
check: pick a value slightly greater and slightly less than our answer
let x = 1.3 , V = 1.3(14-2.6)(6-2.6) = 50.388
let x = 1.31 , V = 1.31(14-2.62)(6-2.62) = 50.38836
let x = 1.4 , V = 1.4(14-2.8)(6-2.8) = 50.176
x = 1.31 yields the greatest volume
The demand function for a certain make of replacement cartridges for a water purifier is given by the following equation where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
Determine the consumers' surplus if the market price is set at $1/cartridge. (Round your answer to two decimal places.)
Find the point on the graph where the tangent line is horizontal.
I know you have to do the quotient rule to find the derivative of the function, but I do not know what to do after that.
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