A gardener wants to build a rectangular garden of area 50 m² against an existing wall as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 24 - 2023 - Paper 1

Given the dimensions of the concrete path and garden, the area of the concrete path (A) can be expressed as:

$A = 2y + x - 2$

Substituting the expression for y:

$A = 2\left(\frac{50}{x - 2} + 1\right) + x - 2$

This simplifies to:

$A = 100/(x - 2) + x$

To find the minimum, we calculate the derivative (A') of the area with respect to x, set it to zero, and solve:

$A' = 100\left(-\frac{1}{(x - 2)^2}\right) + 1$

Setting it equal to zero:

$0 = -\frac{100}{(x - 2)^2} + 1\n$

Solving this leads to:

\Rightarrow x = 12$$ To confirm this gives a minimum area, we check the second derivative at this point.