(CALC I) A rancher has to enclose 3 adjacent rectangular corrals (no spaces inbetween) along a straight river. The total area of the enclosures must be 90,000 square meters. What dimensions will use the least amount of fencing? How much fencing material does he need to purchase?

Question

anonymous · Answer

The total area, that includes the 3 adjacent rectangular corrals, measures$$150 \sqrt{2} \text{ meters} \text{ wide} \text{ by } 300 \sqrt{2} \text{ meters} \text{ long.}$$The total fence length to enclose all three corrals is:$$1200 \sqrt{2} \text{ meters} $$Set the derivative of$$\frac{180000}{w}+4 w $$to zero$$4-\frac{180000}{w^2}=0 $$and solve for w.$$w=150 \sqrt{2} $$

anonymous · Answer

Wow, thanks!! my teacher kept showing a much longer way and this makes more sense actually.

anonymous · Answer

The solution method was OK, however, the 180,000 should have been 90000 as stated in the problem narrative.  I don't understand how I make that mistake.

Using 90000, "w" works out to be 150 feet, not 150 * radical 2.
The four pens are four adjacent squares measuring 150 feet on a side.

The total area in feet is 4*150^2 = 90000 sq ft

You can take the medal away if you wish.