Among all pairs of numbers with the sum of 228 find the pair whose product is maximum

Question

Among all pairs of numbers with the sum of 228, find the pair whose product is maximum

Answer 1

the rectangle with the greatest area is always a square

all you have to do is divide 228 by 2 to get the answer

228/2=114, 114*114=12,996

if you use 113*115, you get12,995

112*116=12,992

if you need work, use the following...

x+y=228

y=228-x

xy=x(228-x)

x(228-x)=0

228x-x²=0 is a parabola that opens downward and whose vertex (h,k) is the maximum point

if you expand the vertex form of a parabola you get...

y=a(x-h)²+k

y=ax²-2ahx+ah²+k

using -b/2a you get -(-2ah)/2a=2ah/2a=h, the x coordinate of the vertex

using -b/2a with 228x-x²=0

-b/2a=-228/-2=114 for h and 114*114=12,996 for k

Answer 2

Problem statement:  "Among all pairs of numbers with the sum of 228, find the pair whose product is maximum"

Keywords:  maximum, product, all ... numbers, sum
  sum:  addition
  product:  multiplication
  all ... numbers:  numbers greater than zero (which, by the way, is neither positive nor negative) will produce a product greater than 0; negative numbers produce a negative product when multiplied by a positive number
  maximum:  the greatest value; a function has, at most, one value of the dependent variable for each allowable value of the input variables.

 

Let's find the pair of positive numbers.
 
The allowable input values (that is, the range) of this function is all positive numbers whose sum is 228.  Right away, that limits the values. Since they must be "positive," that means each number must be greater than 0;  since the sum must be 228 that means each value must be individually less than 228.  This concept will be used often in later math, so understanding it now is vey important.
 
Now, of the various numbers that are greater than 0 and less than 228, the products (that is, the output or domain of this function) goes from just greater than 0 to some unknown value (that the problem wants us to find) and then back down to just greater than 0.  The fact that the products do this is also an important concept, so, if you need to calculate them, find:
  1 * 228 = ??
  2 * 227 = ??
  3 * 226 = ??
 . . .
  226 * 3 = ??
  227 * 2 = ??
  228 * 1 = ??
 
[Note: the products of the above expressions are duplicates, increasing and then decreasing.  Since the problem stated "positive numbers," we must allow fractions even though I did not illustrate that.]
 
How to find the answer [algebra method]:
Look again at the list of products above (with answers increasing, then decreasing].  A important omitted answers are:
...
110      118      12980
111      117      12987
112      116      12992
113      115      12995
114      114      12996
115      113      12995
116      112      12992
117      111      12987
118      110      12980
...
Yep, the product increases to a maximum, then it decreases.  Now, you must decide whether only integers are allowed and whether there is a maximum between two of the values listed.  If the problem assumes that "numbers" means integers, then 114 and 114 are them.  Now, we would certainly look between two pairs of values if their products were the same, but because this is only algebra, not calculus, we can "keep it simple."
 
So, put a note in your math diary/journal (you will need this for later math):  "The maximum product of two positive numbers with a constant sum occurs when the numbers are equal (each half of the sum)."
 
Example of how to apply this rule:  "If the sides of a square add up to x, what is the length of a side that gives the square the maximum area possible?"
 
[Note:  if you take calculus, you will learn how to find formulas for the maximum and the minimum of complicated functions -- this is very handy, for example, when you must minimize the time a project takes, minimize the cost you must pay, maximize the profit you are going to earn, maximize the grade you are going to earn, ...]