Find two positive numbers that satisfy the requirements: "The product is 147 and the sum of the first number plus three times the second number is a minimum."
To find the two positive numbers that satisfy the given requirements, we can use the method of optimization by finding the minimum value of a certain function.
Let's represent the two positive numbers as x and y, where x is the first number and y is the second number.
According to the requirements, we have two conditions:
1. The product of the two numbers is 147, which can be expressed as:
x * y = 147
2. The sum of the first number (x) and three times the second number (3y) should be a minimum. Mathematically, this can be expressed as:
f(x,y) = x + 3y
To find the minimum value of the function f(x,y), we can use the method of derivative.
First, let's solve the first equation to express one variable in terms of the other. We can solve for x:
x = 147 / y
Then substitute this expression for x in the function f(x,y):
f(y) = (147 / y) + 3y
Now, we can take the derivative of f(y) with respect to y to find the critical points:
f'(y) = (-147 / y^2) + 3
Setting f'(y) equal to zero and solving for y, we get:
(-147 / y^2) + 3 = 0
(-147 + 3y^2) / y^2 = 0
-147 + 3y^2 = 0
3y^2 = 147
y^2 = 49
y = ±√49
y = ±7
Since we're looking for positive numbers, we take y = 7.
Now substitute this value of y into the first equation to find x:
x = 147 / y
x = 147 / 7
x = 21
Therefore, the two positive numbers that satisfy the given requirements are x = 21 and y = 7.