Question

A garden is planned with a lawn area of 24m² and a path around the edge. The dimensions of the lawn and path are as shown in the diagram.

(i) Write down an expression for y in terms of x.
(ii) Find an expression for the overall area of the garden, A in terms of x.
(iii) Find the smallest possible overall area for the garden. ​

Answer 1

54 m²

Explanation:

i.

xy = 24

y = 24/x

ii.

A = (x + 2)(y + 3)

A = xy + 3x + 2y + 6

A = x(24/x) + 3x + 2(24/x) + 6

A = 24 + 3x + 48/x + 6

A = 3x + 48/x + 30

iii.

A = 3x + 48x^-1 + 30

dA/dx = 3 + (-1)(48)x^-2

dA/dx = 3 - 48/x²

3 - 48/x² = 0

48/x² = 3

x² = 16

x = 4 or x = -4

We discard the negative answer.

x = 4

y = 24/x

y = 24/4

y = 6

A = xy

A = (x + 2)(y + 3)

A = (4 + 2)(6 + 3)

A = 6(9)

A = 54