In the space below, provide the smaller of the two positive integers that add to 40 and have the largest possible product
same as your fence really
a + b = 40 so b = 40 - a
P = a b = a(40-a) = 40 a - a^2
dP/da = 40 - 2 a
a = 20
then b = 20
so
20 and 20
a + b = 40 so b = 40 - a
P = a b = a(40-a) = 40 a - a^2
dP/da = 40 - 2 a
a = 20
then b = 20
so
20 and 20
Oh, I guess math is serious business for me. But hey, let's put on our clown noses and solve it with a smile! So we are looking for two positive integers that add up to 40 while having the largest possible product.
To maximize the product, we want the two numbers to be as close to each other as possible. So let's split 40 into two equal parts:
40 divided by 2 equals... (Don't worry, I'll do the math. You relax and enjoy the show!)
Oh, there you have it! The smaller of the two positive integers that add to 40 and have the largest possible product is 20.
Thank you for joining this mathematical circus act!
To find the two positive integers that add to 40 and have the largest possible product, we can use the fact that the product of two numbers is maximum when the numbers are closest to each other. Therefore, the two numbers should be as close to each other as possible.
Let's assume one of the integers is x. Then, the other integer can be expressed as 40 - x.
To maximize the product, we need to find the value of x that makes (40 - x) as close to x as possible.
To do this, we can find the average of the two numbers:
[(40 - x) + x] / 2
Simplifying:
(40 - x + x) / 2
40 / 2
20
Therefore, the two numbers that add to 40 and have the largest possible product are 20 and 20.
The smaller of the two positive integers is 20.
To find the smaller of the two positive integers that add to 40 and have the largest possible product, we need to consider the two numbers that would result in the largest product.
Let's assume the two numbers are x and y, where x is the smaller number and y is the larger number. Since the sum of the two numbers is 40, we can set up the equation: x + y = 40.
To find the largest possible product, we need to determine the value of x that maximizes the product xy. We can do this by considering that if x is smaller, then y must be larger, which will result in a larger product. Therefore, to maximize the product xy, x should be as small as possible.
If x is 1, then y would be 39 to satisfy the equation x + y = 40. The product xy would be 1 * 39 = 39.
If x is 2, then y would be 38, resulting in a product of 2 * 38 = 76.
Continuing this pattern, if x is 3, then y would be 37, resulting in a product of 3 * 37 = 111.
We can see that as x increases, the product xy also increases. Therefore, the largest possible product will occur when x is the smallest possible positive integer, which is 1.
Therefore, the smaller of the two positive integers that add to 40 and have the largest possible product is 1.