Two numbers have a sum of 72. What is their product if it is a maximum?
72-x = 72x
Well, if you're looking to maximize the product of two numbers with a given sum, you'll want those numbers to be as close together as possible. In this case, the numbers that will give you the maximum product are 36 and 36.
So, the product of these two numbers would be... *drum roll*... *crash cymbals*... 1296!
Why did the mathematician bring a ladder to the party? Because he heard the drinks were on the house! 🍻🤓
To find the product of two numbers with a maximum value when their sum is 72, we can use the concept of maximizing the product of two numbers.
Let's assume the two numbers are x and y. We need to find the values of x and y such that their product, xy, is maximized.
According to the given information, we know that x + y = 72.
To maximize the product xy, we need to consider the case where x and y are as close to each other as possible. This is because when two numbers are close to each other, their product tends to be larger compared to when they are farther apart.
In this case, if we consider x = y, then the sum would be 2x = 72, which implies x = 36.
So, the two numbers are both equal to 36.
To find their product, we can multiply x and y:
xy = 36 * 36 = 1296.
Therefore, if the sum of two numbers is 72, their product is maximized at 1296 when both numbers are equal to 36.
To find the product of two numbers when their sum is given, we can use the concept of optimization. Let's denote the two numbers as x and y.
Given that the sum of the two numbers is 72, we can form an equation:
x + y = 72
To find the product of x and y, we need to maximize it. Since we don't have any constraints or limitations mentioned in the question, we can assume that x and y can be any real numbers.
To maximize the product, we can use calculus by finding the derivative of the product with respect to one of the variables (x or y), setting it equal to zero, and solving for x or y.
Let's solve for x in terms of y:
From the equation x + y = 72, we can solve for x:
x = 72 - y
Now, we can express the product of x and y:
P = xy
Substituting the value of x:
P = (72 - y)y
Expanding and simplifying:
P = 72y - y^2
To maximize P, we can differentiate it with respect to y:
dP/dy = 72 - 2y
Setting the derivative equal to zero:
72 - 2y = 0
Solving for y:
2y = 72
y = 36
Substituting this value back into the equation x + y = 72:
x + (36) = 72
x = 36
So, the two numbers are 36 and 36, and their product is:
Product = 36 * 36 = 1296
Therefore, when the sum of the two numbers is 72, the maximum product they can have is 1296.
let the two number be x and 72-x
let P be their product
P = x(72 - x) = -x^2 + 72x
the x of the vertex is -72/-2 = 36
then P = 36(72-36) = l6^2 = 1296