# Thread: Optimization

1. ## Optimization

A fence is to be made to enclose three sides of a rectangular area beside a river. The river forms the fourth side of the enclosure. This fence costs $30/metre. The enclosure is divided in half by another fence at right angles to the river. This fence costs$20/metre. The area to be enclosed is 500m^2. Find the dimensions for minimum cost of fencing.

Please provide full solution. Thanks.

2. Originally Posted by meli3000
A fence is to be made to enclose three sides of a rectangular area beside a river. The river forms the fourth side of the enclosure. This fence costs $30/metre. The enclosure is divided in half by another fence at right angles to the river. This fence costs$20/metre. The area to be enclosed is 500m^2. Find the dimensions for minimum cost of fencing.

Please provide full solution. Thanks.
$\displaystyle C = 30x + 30w + 30x + 20x = 80x + 30w$ .... (A)

$\displaystyle A = wx \Rightarrow 500 = wx \Rightarrow w = \frac{500}{x}$ .... (B)

Substitute equation (B) into equation (A) and then use calculus to find the value of $\displaystyle x$ that gives a minimum of $\displaystyle C$.

Substitute your answer for $\displaystyle x$ into equation (B) to get the value of $\displaystyle w$.