Question

Queen Chloe is planning to build a castle inside of a rectangular moat. A river runs horizontally along the bottom of her land, and that river will form the bottom of the moat, but the other three sides must be dug by a ditch digging company. For the two vertical sections of the moat, the company will charge 1 gold piece per meter. But because the moat digging becomes easier further from the river, the company offers a discount to dig further from the river on the horizontal section. For the horizontal section, the price per meter is 1 gold piece divided by the vertical distance from the river.

If Queen Chloe has 12 gold pieces, what are the dimensions of the moat that will give her the maximum area for her castle? How do you know that these dimensions maximize the area?

Answer 1

let vertical side of castle is of length x meter and horizontal side y meter we will find area in terms of a single variable and then maximize it by following method

To maximize or minimize a function f(x) 1st we need to find its critical points

For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0

Then check second derivative of f(x) i.e. f ''(x) is positive or negative at critical numbers If f ''(x) >0 then that gives minimum value of f(x) and if f ''(x) <0 then that give maximum value of f(x)