Hello! i'm having trouble on these questions.
1) Find two positive real numbers whose sum is 100 and whose product is a maximum
2) A farmer has 140 ft of fencing with which she will enclose a rectangular region adjacent to a barn 60 ft long. If she insists on using the
entire length of the barn as a
portion of one side of the rectangle, what is the maximum area that may be enclosed?
Also, 'm not looking for answers. I'm simply just trying to understand how to begin and how to go about solving it.
Any help would be great!!
thanks
We learn early in algebra to set up problems
initially in terms of a minimum number of unknowns. I must admit that I personally do not like that
standard approach because, in my opinion, it requires some analysis before even setting up the problem. I prefer to identify and name each unknown or variable as the first step in solving a problem and then look for ways to minimize the number of unknowns. Either method works, but your hybrid method will not.
Most problems in differential calculus require finding the value of one or more variables that maximize or minimize the value of another variable. So how does my method work in differential calculus problems. Let's take your first problem as an example.
Identification and naming step.
first positive real number = w
second positive real number = x
their product = y
I have just given names to the variables described in words in the word problem (just as tkhunny suggested in the very first post).
Now comes the translation step.
w > 0.
x > 0.
w + x = 100.
y = wx.
I just put into math symbolism the conditions specified in words in the problem.
Now comes the simplification step.
The problem as now formulated is to maximize y = wx.
I see two potential ways to solve this maximization problem in two independent variables. The easiest way is to reduce the number of independent variables to one and maximize y with respect to one variable. So let's do that.
w = 100 - x.
So the problem has been simplified to: Maximize y = (100 - x)x = 100x - x^2.
Now comes the solution step, which is a pure math problem of finding what value of x maximizes y and then calculating the corresponding value of w. What's your answer?
This process will work for just about any word problem. (1) Identify and name the relevant variables. (2) Translate the conditions specified in the problem (or implicitly assumed in the problem) into mathematical form. (3) Simplify if possible. (4) Solve the math problem.
So why don't you go as far as you can on problem 2 and show us your work up to where you get stuck.