find 3 positive real numbers whose sum is 500 and whose product is as large as possible.
this is the same as finding the volume of the largest brick with given sum of edges or given surface area.
As with the area of a rectangle, max is achieved for a square.
Max volume is achieved with a cube of side 500/3.
To find three positive real numbers whose sum is 500 and whose product is as large as possible, we can use the concept of optimization. We want to maximize the product of the three numbers given the constraint that their sum is 500.
Let's assume the three numbers are x, y, and z. We need to find the values of x, y, and z that maximize the product xyz.
To solve this problem, we can use the method of differentiation. We need to find the critical points where the derivative of the product function is zero.
1. Setup the equation: The product function is f(x, y, z) = xyz.
2. Apply the constraint: x + y + z = 500. Rearranging the equation gives z = 500 - x - y.
3. Rewrite f(x, y, z) in terms of two variables: f(x, y) = xy(500 - x - y).
4. Find the critical points: Take the partial derivatives of f(x, y) with respect to x and y and set them equal to zero:
df/dx = y(500 - 2x - y) = 0
df/dy = x(500 - x - 2y) = 0
5. Solve the equations simultaneously: Solve the system of equations to find the critical points. There may be one or more solutions.
6. Evaluate the function: Substitute the values of x, y, and z into the product function f(x, y, z) to find the maximum product.
Solving these equations may require advanced mathematical techniques such as calculus. Alternatively, you can use numerical optimization algorithms or graphing software to find the values of x, y, and z that maximize the product.
Note: While the above steps explain the general approach, solving the specific equations to find the precise values of x, y, and z requires further calculations that go beyond the scope of this explanation.
To find three positive real numbers whose sum is 500 and whose product is as large as possible, we can use the AM-GM inequality.
The AM-GM inequality states that for any set of positive real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM).
Let the three numbers be denoted as a, b, and c.
Since a + b + c = 500, let's solve for one variable in terms of the other two.
Let's express c in terms of a and b: c = 500 - a - b.
Now, let's find the product of a, b, and c.
P = a * b * c
= a * b * (500 - a - b)
= -a^2 * b - ab^2 + 500ab
To maximize the product, we need to find the critical points of P by taking the derivative of P with respect to both variables a and b and setting them equal to zero.
∂P/∂a = -2ab - b^2 + 500b = 0 (Equation 1)
∂P/∂b = -a^2 - 2ab + 500a = 0 (Equation 2)
We can now solve Equations 1 and 2 simultaneously to find the values of a, b, and c that maximize the product.
Solving Equation 1 for a:
-a^2 - b^2 + 500b = 0
a^2 = -b^2 + 500b
Substituting this into Equation 2:
-(-b^2 + 500b) - 2b(-b^2 + 500b) + 500(-b^2 + 500b) = 0
(b^2 - 500b) + 2b(b^2 - 500b) - 500(b^2 - 500b) = 0
b^3 - 1000b^2 + 2b^3 - 1000b^2 - 500b^2 + 250000b = 0
3b^3 - 2500b^2 + 250000b = 0
Factoring out 'b':
b(3b^2 - 2500b + 250000) = 0
Simplifying:
b(3b - 500)(b - 500) = 0
Now, let's consider each solution separately:
1. b = 0:
If b = 0, then from the equation a + b + c = 500, a + 0 + c = 500, and a + c = 500. However, since we're looking for positive real numbers, b cannot be 0. So, this solution is not valid.
2. 3b - 500 = 0:
If 3b - 500 = 0, then b = 500/3.
Substituting this into the equation a + b + c = 500:
a + 500/3 + c = 500.
To solve for a and c, we need additional information or constraints. Without any additional information, we cannot uniquely determine a and c.
3. b - 500 = 0:
If b - 500 = 0, then b = 500.
Substituting this into the equation a + b + c = 500:
a + 500 + c = 500.
This gives us a = -c, which means that a and c must be negative for b = 500. Since we're looking for positive real numbers, this solution is not valid.
Therefore, based on the given conditions, we cannot determine a unique set of three positive real numbers that would maximize the product.