Suppose you have 48 feet of fencing to enclose a rectangular dog pen. the function a = 24x-x2, where x = width, gives you the area of the dog pen in square feet. what width gives you the maximum area? what is the maximum area?

A = 24x - x^2

Finding the derivative:-

A' = 24 - 2x = 0  ( for minm/maxm)

2x = 24

x = 12 feet

A" is -2 so this is a maximum

Width for maximum area = 12 feet

length of  the dog pen =  (48 - 2(12) / 2 = 12 feet

so Maximum area = 12^2 = 144 ft^2

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: