Find two rational numbers such that the difference is 3 times the first and twice the second is 34.7 and whose product is a minimum. Also, find the minimum product.
To solve this problem, we can use algebraic representation and optimization techniques. Let's assume the two rational numbers as "x" and "y".
Given:
1. The difference between the two numbers is 3 times the first number: x - y = 3x.
2. Twice the second number is 34.7: 2y = 34.7.
To find the values of x and y, we can solve these two equations. Let's first solve the second equation for y:
2y = 34.7
y = 34.7 / 2
y = 17.35
Now, substitute the value of y in the first equation:
x - y = 3x
x - 17.35 = 3x
-17.35 = 2x
x = -17.35 / 2
x = -8.675
Thus, the two rational numbers are x = -8.675 and y = 17.35.
Now, let's find the minimum product of these two numbers. The product of x and y is:
Product = x * y
Product = (-8.675) * (17.35)
To find the minimum product, we can simply evaluate this expression:
Product = -8.675 * 17.35
Product ≈ -150.182
Therefore, the minimum product of the two rational numbers is approximately -150.182.