Question

Diana wants to enclose a rectangular flower plot that adjoins a brick retaining wall. If she has only 30m of decorative fencing, what should the dimensions of the flower plot be so that she has a maximum area?

Answer 1

The fence opposite the wall should be `15` m long and the other two sides should be `7 1/2` m each.

Explanation:

Let
`color(white)("XXX")y` be the length of the fence opposite the wall
`color(white)("XXX")x` be the length of the other two sides

We have
`color(white)("XXX")y+2x=30color(white)("xxx")rarrcolor(white)("xxx")y=30-2x`

Let the area of the garden relative to the variable `x` be `f(x)`
`color(white)("XXX")f(x)=x * y = x(30-2x)=30x-2x^2`

Maximum area will occur when the slope of the area function, `f(x)`, is equal to `0`;
i.e. when
`color(white)("XXX")f'(x)=0`

`f'(x)=30-4x`

If `f'(x)=30-4x=0color(white)("xxx")"then"color(white)("xxx")x=7.5`
and
since `y=30-2xcolor(white)("xxx")y=15`