Expert: Scott A Wilson - 11/30/2009

Question
YOU HAVE 80 FEET OF FENCE YOU NEED TO MAKE A RECTANGLE PEN. ONE SIDE IS A WALL SO USE THE FENCE ON THREE OTHER SIDES AND MAKE IT AS BIG AS POSSIBLE WHAT ARE THE DEMINSIONS OF THE PEN?

2. YOU HAVE FIVE GALLONS OF SOLUTION. 80 PERCENT IS WATER AND 20 PERCENT IS ALCOHOL. HOW MUCH OF 100 PERCENT ALCOHOL DO WE NEED TO MAKE IT 50 PERCENT ALCOHOL?  

Answer
1. ON THAT PEN, THE TOTAL FENCE USED WOULD BE L + 2W, WHERE L IS FOR LENGTH AND W FOR WIDTH.
SINCE THIS IS 80, WE KNOW THAT L = 2W -80.  THE AREA OF THE PEN IS THEN W(2W-80).
MULTIPLY THAT OUT, TAKE THE DERIVATIVE, SET IT TO 0, SOLVE FOR W, AND NOTE THAT L IS 2W-80.
THAT WILL SOLVE FOR L AND W.

2. SINCE 20% IS ALCOHOL, THAT WOULD BE 1 GALLON.
THAT MEANS THE REMAINING 4 GALLONS IS WATER.
TO MAKE IT 50% ALCOHOL, THERE MUST BE AS MUCH WATER AS ALCOHOL.

TAKE THE GALLONS OF WATER - GALLONS OF ALCHOL = 4-1,
AND THAT WILL BE HOW MUCH ALCOHOL IS NEEDED.