Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum?

Question

This is a business calculas question. 

Answer 1

Let one number = x

Let the other number = y

 

x + y = 9

 

xy²  is a maximum

 

 

Since we need positive numbers, we assign values of x and y respectively that will give us a sum of 9.

 

x: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 

y: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

 

 

Next, we square the values of y.

 

y²: 81, 64, 49, 36, 25, 16, 9, 4, 1, 0

 

 

Next, we multiply x by y² to see which product is the highest value.

 

x  : 0  , 1  , 2  , 3 , 4  , 5  , 6, 7, 8, 9

y² : 81, 64, 49, 36, 25, 16, 9, 4, 1, 0

 

 

We can eliminate x = 0, x = 8 , and x = 9 , since the product of xy² is of lowest value.

 

1 x 64 = 64

2 x 49 = 92

3 x 36 = 108

4 x 25 = 100

5 x 16 = 80

6 x 9 = 54

7 x 4 = 28

 

The highest product value here is 108. 

 

The two numbers in the solution are 3 and 6.