For the positive integers x and y, x+y=21. What is the smallest possible value of xy?
a. 10
b. 20
c. 38
d. 54
e. 110
20 + 1
so that would be letter b?
yes b
To find the smallest possible value of xy when x+y=21, we need to determine the values of x and y that minimize the product xy.
Since the sum of x and y is fixed at 21, we can apply the concept of the arithmetic mean and the geometric mean.
The arithmetic mean of two numbers is equal to their sum divided by 2: (x+y)/2.
The geometric mean of two numbers is the square root of their product: √(xy).
According to the AM-GM inequality, the arithmetic mean is always greater than or equal to the geometric mean.
Mathematically, this can be written as: (x+y)/2 ≥ √(xy)
In this case, we have (x+y)/2 = 21/2 = 10.5.
Therefore, the smallest possible value of xy occurs when the equality holds: 10.5 = √(xy).
To find the value of xy, we need to square both sides of the equation: (10.5)^2 = (√(xy))^2.
Simplifying, we have 110.25 = xy.
Since xy must be an integer, the smallest possible value is 110 (Option e).
Hence, the correct answer is e. 110.