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April 15/16, 2009
MPM2D1
Name: ______________________
Maximum & Minimum – Day 2 – “Numbers”
Example 4
The sum of two numbers is 60. Find the numbers if their product is a maximum.
Our 2 numbers are n & 60-n. So,
our formula is:
p  n(60  n) , where n is one
We need to write formula with one variable to
represent the problem.
number and p is the product.
p  n(60  n)
p  n 2  60n
Put the formula into standard form.
Now we are ready to COMPLETE THE SQUARE!
p  (n 2  60n)
p  (n 2  60n  900  900)
p  (n 2  60n  900)  900
Factor out the a value.
Calculate and include the magic number!
 60  22 = 900
Boot the negative out of the brackets (remember
to multiply by the negative in front of the brackets).
Rewrite the perfect square portion in its factored
p  (n  30)  900
form.
Simplify if necessary.
Translate the information to make sense of the
Vertex: (30, 900)
question.
At the maximum, the value of n is 30. If one number is 30, the other number is 60-n, which
evaluates to 30 as well. So, the two numbers are 30 and 30.
2
Example 5
The sum of two numbers is 28. Find the numbers if the sum of their squares is a minimum.
Our 2 numbers are n & 28-n. So, our
formula is:
minimum= n 2  (28  n) 2 , where n is
one number and 28-n is the other
number.
minimum= n 2  784  56n  n 2
= 2n 2  56n  784
We need to write formula with one variable to
represent the problem.
Put the formula into standard form.
Now we are ready to COMPLETE THE SQUARE!
minimum= 2(n 2  28n)  784
minimum= 2(n 2  28n  196  196)  784
minimum= 2(n  28n  196)  392  784
2
Factor out the a value.
Calculate and include the magic number!
 28  22 = 196
Boot the negative out of the brackets
(remember to multiply by the negative in front of
the brackets).
April 15/16, 2009
MPM2D1
Name: ______________________
minimum= 2(n  14) 2  392
Vertex: (14, 392)
Therefore, the two numbers are 14 and 14.
Rewrite the perfect square portion in its
factored form.
Simplify if necessary.
Translate the information to make sense of
the question.
Example 6
Two numbers have a difference of 16. Find the numbers if the result of adding their sum and
their product is a minimum.
sum
Our 2 numbers are n & 16+n. So, our
formula is:
minimum= [n  (16  n)]  [n(16  n)] ,
where n is one number and 16+n is
the other number.
minimum= [n  (16  n)]  [n(16  n)]
2n  16  n 2  16n
product
We need to write formula with one variable to
represent the problem.
Put the formula into standard form.
n  18n  16
2
Now we are ready to COMPLETE THE SQUARE!
minimum= n 2  18n  16
minimum= n 2  18n  81  81  16
minimum= (n  9) 2  65
Vertex: (-9, -65)
Therefore, our two numbers are -9 and 7.
Factor out the a value. (BUT THERE IS NONE)
Calculate and include the magic number!
18  22 = 81
Nothing to boot out of brackets!
Rewrite the perfect square portion in its
factored form.
Simplify if necessary.
Translate the information to make sense of
the question.
Example 7
The sum of a number and three times another number is 18. Find the numbers if their product
is a maximum.
Assignment:
1. Page 272 #18, 20, 22 (Day1 type questions) AND
2. The sum of two natural numbers is 12. If their product is a maximum, find the numbers.
3. Two numbers have a difference of 20. Find the numbers if the sum of their squares is a
minimum.