To find two positive numbers that meet the criteria of their sum being 100 when the first number is added to twice the second number, as well as maximizing their product, we can use a bit of algebra and calculus.
Let’s denote the first number as x and the second number as y.
According to the information given, we have the following equations:
1. The sum condition: x + 2y = 100
2. We want to maximize the product: P = xy
From the first equation, we can solve for x:
x = 100 - 2y
Now, we can substitute this expression for x into the product equation:
P = (100 - 2y)y
This simplifies to:
P = 100y - 2y^2
To find the maximum product, we need to take the derivative of P with respect to y and set it to zero:
P' = 100 - 4y
Setting the derivative equal to zero gives us:
100 - 4y = 0
This leads to:
4y = 100
y = 25
Now that we have y, we can substitute this back to find x:
x = 100 - 2(25) = 100 - 50 = 50
Thus, the two positive numbers are:
- x = 50
- y = 25
Finally, let’s verify:
1. The sum: 50 + 2(25) = 50 + 50 = 100 (satisfied)
2. The product: P = 50 * 25 = 1250 (which is maximum)
Therefore, the two numbers you are looking for are 50 and 25.