find two positive real numbers x and y such that their product is 800 and x+2y is as small as possible
xy=800, so x == 800/y
x+2y = 800/y+2y
for minimum value, you need
-800/y^2 + 2 = 0
y=20
This is just like the fencing problems where minimum fencing for 3 sides of a rectangle is desired. The amount is evenly divided between the x's and the y's
Why did the scarecrow win an award?
Because he was outstanding in his field!
Now, let's find those positive real numbers, x and y.
To find the smallest value of x + 2y, we can use the concept of AM-GM inequality. According to this inequality, the arithmetic mean (AM) of two positive numbers is always greater than or equal to their geometric mean (GM).
Let's assume x and 2y as our two positive numbers. So, applying the AM-GM inequality, we have:
(x + 2y)/2 ≥ √(x * 2y)
Simplifying this, we get:
x + 2y ≥ 2√(2xy)
Now, we know that the product of x and y is 800. Substituting this value, we get:
x + 2y ≥ 2√(2 * 800)
x + 2y ≥ 2√(1600)
x + 2y ≥ 2 * 40
x + 2y ≥ 80
So, the smallest value of x + 2y is 80.
To find the exact values of x and y, we need to solve the equations:
xy = 800 and x + 2y = 80
One possible solution is x = 40 and y = 20, as this satisfies both equations and gives us the minimum value for x + 2y.
So, the two positive real numbers are x = 40 and y = 20, with their product being 800 and x + 2y equaling 80.
To find the positive real numbers x and y such that their product is 800 and x + 2y is as small as possible, we can use the method of Lagrange multipliers.
Let's define the objective function as f(x, y) = x + 2y and the constraint function as g(x, y) = xy - 800.
To find the minimum of f(x, y) subject to the constraint g(x, y) = 0, we need to find the partial derivatives:
∂f/∂x = 1
∂f/∂y = 2
∂g/∂x = y
∂g/∂y = x
Using Lagrange multipliers, we set up the following system of equations:
1 = λy
2 = λx
xy = 800
Solving this system of equations, we get the following solutions:
x = 16√5
y = 50/√5
Therefore, the two positive real numbers x and y that satisfy the given conditions and make x + 2y as small as possible are approximately x = 16√5 and y = 50/√5.
To find two positive real numbers x and y such that their product is 800 and x + 2y is as small as possible, we can use a strategy called optimization.
Step 1: Define the problem:
We want to find the values of x and y that satisfy the given conditions:
- x and y are positive real numbers.
- The product of x and y is 800: xy = 800.
- The sum x + 2y is as small as possible.
Step 2: Express one variable in terms of the other:
From the equation xy = 800, we can solve for one of the variables and express it in terms of the other. Let's solve for y:
y = 800 / x.
Step 3: Substitute the expression into the objective function:
Now, we substitute the expression for y in terms of x into the objective function (x + 2y) to get a single-variable function. Let's call this function f(x):
f(x) = x + 2 * (800 / x) = x + 1600 / x.
Step 4: Find the minimum value:
To find the minimum value of f(x), we can take the derivative of f(x) with respect to x, set it equal to zero, and solve for x.
f'(x) = 1 - 1600 / x^2
Setting f'(x) = 0, we have:
1 - 1600 / x^2 = 0
Simplifying the equation, we get:
x^2 = 1600
x = √1600
x = 40
Step 5: Solve for the values of the other variable:
Now that we have found x = 40, we can substitute it back into the equation xy = 800 to solve for y:
y = 800 / 40 = 20
Therefore, the two positive real numbers x and y that satisfy the given conditions, with their product being 800 and x + 2y being as small as possible, are x = 40 and y = 20.