By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box that will yield the maximum volume. (Round your answers to two decimal places.)

Answer 1

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

Answer 2

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.

-0.01x^2-0.3x+19
Determine the consumers' surplus if the market price is set at $1/cartridge. (Round your answer to two decimal places.)

Answer 3

Find the point on the graph where the tangent line is horizontal.
x/(x^2+25)

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.