If the space below, provide the smaller of the two positive integers that add 30 and have the largest possible product.
Numerical Answers Expected!
common sense would tell us that the 2 numbers would have to be the same, each equal to 15
let's see:
let the smaller be x
larger = 30-x
product = x(30-x) = -x^2 + 30x
d(product)/dx = -2x + 30
= 0 for a max of product
2x = 30
x = 15
one number is 15, so is the other, their product = 225
I was right.
Ah, the quest for the largest product! To find the smaller positive integer that adds up to 30 and maximizes the product, we can use the trusty formula for a rectangle's area: length times width, or in our case, the two integers multiplied together. So, let's math it up:
Let's call our two positive integers "x" and "y." We know that x + y = 30.
To maximize the product, we want the two integers to be as close as possible. So, let's take a stab at it:
x = 15
y = 15
Now, let's check if that's correct by calculating the product:
Product = x * y
Product = 15 * 15
Product = 225
Voila! The smaller positive integer that adds up to 30 and has the largest possible product is 15.
To find the smaller of the two positive integers that add up to 30 and have the largest possible product, we can use the concept of optimization. Let's call the two positive integers x and y.
Since x + y = 30, we can express one of the integers in terms of the other:
x = 30 - y
Now, we can express the product of the two numbers in terms of y:
Product = x * y = (30 - y) * y
To find the value of y that maximizes the product, we can take the derivative of the product function with respect to y and set it equal to zero:
d(Product) / dy = 30 - 2y = 0
Solving this equation gives us:
30 - 2y = 0
2y = 30
y = 15
Therefore, y = 15 is the smaller of the two positive integers that add up to 30 and have the largest possible product.
To find the two positive integers that add up to 30 and have the largest possible product, let's call the numbers x and y. We want to maximize the product, so we need to find the pair of numbers that gives us the largest possible combination.
Since the two numbers add up to 30, we can set up the equation:
x + y = 30
To maximize the product, we need to consider the values that x and y can take. Since we are looking for positive integers, let's start with the largest possible value for one of the numbers and work our way down.
If we set x to be 29, then y would be 1, giving us a product of 29 * 1 = 29.
If we set x to be 28, then y would be 2, giving us a product of 28 * 2 = 56.
We can see that as we decrease the value of x, the value of y increases, resulting in a larger product. The largest product occurs when x = 15 and y = 15, resulting in a product of 15 * 15 = 225.
Therefore, the smaller of the two positive integers that add up to 30 and have the largest possible product is 15.