Question

Find two numbers whose sum is 23 and whose product is a maximum.

Answer 1

Let x = the first numberLet y = the second number x + y = 23                    eq1       y = 23- x 
xy = Product                 eq2

Substitute eq1 into eq2.  Lets get eq2 in terms of x.  

Product = x(23 - x)Product = - x2 + 23x
 
we must find the vertex has coordinate (h, k). h = -b / 2a  where:a = -1b = 23
h= -23  / -2  =  11.5 

 x = 11.5   


Substitute this value of x into eq1 to find y. y = 23 - xy = 23 - 11.5y = 11.5

The two numbers that will give the largest product possible are 11.5 and 11.5.

Answer 2

The unknown numbers are 11.5 and 11.5

Let the unknown numbers be x and y

If the sum of the numbers is 23, hence;

x +  y = 23

x = 23 - y ............. 1

If the product is at maximum, hence;

xy = maximum ......... 2

Substitute equation 1 into 2

(23 - y )y = maximum

23y - y² = maximum

maximum = -y² + 23y

A(y) =  -y² + 23y

Since the product A(y) is at maximum, hence dA(y)/dy = 0 as shown:

dA(y)/dy= -2y + 23 = 0

-2y + 23 = 0

2y = 23

y = 23/2

y = 11.5

Since x + y = 23

x = 23 - y

x = 23 - 11.5

x = 11.5

Hence the unknown numbers are 11.5 and 11.5

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