Find two positive numbers that satisfy the given requirements: The product is 147 and the sum of the first number plus three times the second number is minimum
x y = 147 so x = 147/y
x + 3 y = a
147/y + 3 y = a
da/dy = 0 for max or min = 3 -147/y^2
y^2 = 147/3 = 49
y = 7
x = 147/7 = 21 so x = 21 and y = 7
Well, finding two positive numbers that satisfy these requirements would be quite a circus act! But fear not, I'm here to help you out. Let's call the first number x and the second number y.
Since the product of the two numbers is 147, we have xy = 147.
To minimize the sum of the first number (x) plus three times the second number (3y), we can use a little clown magic. Let's rewrite the sum as x + 3y = x + (3y/1).
Now it's time for the grand finish...we will use the fact that xy = 147 and solve for x in terms of y. So, x = 147/y.
Substituting this into our sum equation, we get (147/y) + 3y.
To minimize this expression, we can take its derivative with respect to y and set it equal to zero. But I won't bore you with all the mathematical shenanigans here. The minimized sum occurs when y equals approximately 4.16.
So, one possible pair of positive numbers that satisfy the requirements is x ≈ 35.39 and y ≈ 4.16.
To find the two positive numbers, let's assume the first number as "x" and the second number as "y".
Given that the product is 147, we have the equation:
x * y = 147
Now, we need to find the values of x and y that minimize the sum of the first number plus three times the second number. So, we have the expression:
f(x, y) = x + 3y
To minimize the expression, we can use the concept of derivatives. We take the partial derivatives with respect to x and y and set them equal to zero.
∂f/∂x = 1 + 0 = 0
∂f/∂y = 3 + 0 = 0
From the first derivative equation, we get:
1 = 0
Since this is not possible, it means that there is no minimum for the given expression. Hence, there is no specific value for x and y that will give us the minimum sum.
However, we can still find two positive numbers that satisfy the given requirements by trial and error.
Let's find possible values of x and y that satisfy the equation x * y = 147:
x = 1, y = 147
Sum: 1 + (3 * 147) = 442
x = 3, y = 49
Sum: 3 + (3 * 49) = 150
Therefore, two positive numbers that satisfy the given requirements are x = 3 and y = 49, with a sum of 150.
To find two positive numbers that satisfy the given requirements, we can use a basic mathematical approach. Let's denote the two positive numbers as x and y.
The product of the numbers is given as 147, so we can write the equation:
xy = 147
The sum of the first number (x) plus three times the second number (3y) is required to be at a minimum. Mathematically, we can express this as:
x + 3y = minimum
To minimize the sum x + 3y, we need to make x as small as possible while keeping y positive. Since y must be positive, the smallest possible value for x occurs when y is the largest possible factor of 147.
The factors of 147 are:
1, 3, 7, 21, 49, and 147
Starting with the largest factor, we can substitute each value for y and solve for x:
For y = 147:
x + 3(147) = minimum
x + 441 = minimum
For y = 49:
x + 3(49) = minimum
x + 147 = minimum
For y = 21:
x + 3(21) = minimum
x + 63 = minimum
For y = 7:
x + 3(7) = minimum
x + 21 = minimum
For y = 3:
x + 3(3) = minimum
x + 9 = minimum
For y = 1:
x + 3(1) = minimum
x + 3 = minimum
Comparing the different values of x for each y, we find that the smallest sum x + 3y occurs when:
x = 9 and y = 3
Therefore, the two positive numbers that satisfy the given requirements are x = 9 and y = 3.