The product of 2 pos. numbers is 100 what is the smallest possible value of their sum?

Question

I know the answer but I'm not sure how to solve it using Calculus instead of common sense. I need to show work for this type of question.

Answer 1

Let's just go by examples:

x*y = 100

Lets replace x and y with some numbers:

1*100 = 100
2*50 = 100
4 * 25 = 100
8 * 12.5 = 100

So as X grows Z times in valid, Y will be divided by Z. However, this means that we have to find a balance between X and Y so that they have their smallest values. The smallest value for both X and Y that could make a product of 100 is 10. X*Y = 100
X = 10
Y = 10

So the result is X+Y = 20.


Good luck.

Answer 2

Let the two numbers you have given be a,b
and you said that a*b=100
there fore b=100/a
The sum of the two numbers is to be least implies that a+b is least=a+100/a=(a^2+100)/a to be least.
Let you think that f(a)=(a^2+100)/a where f(a) be a function of a.the value to be the least implies that differentiation of f=0 because if f is a curve you will get an extrimum value at differentiation of f=0
The differentiation of f(a)=(a(2a)-(a^2+100)*1)/a^2
=(a^2-100)a^2
=0 (As said above)
there fore a=+(or)-10 gives the answer for f to be least
So a=+(or)-10 is the least value for a+b to be minimum.
therefore b=+(or)-10
hence a+b will be minimum if a=b and a=+(or)-10
ans so the least valve of a+b=20 if a,b are positive and
largest value is -20 if a,b are negative

Answer 3

I find this interesting also, because what if we changed 100 with 102 and asked the same question. The answer would not be so easy to find and there must be an equation to determine an answer. I think if it was 102 the numbers would be 17 and 6 and their sum would be 23. But sorry I can't think of a formula that will determine this, besides trial and error.?

Answer 4

when a*b is fixed = 100,
(a + b) is least = sqrt(a*b) = sqrt(100) = 10, -- 10.
as Q requires on;y +ve integers, amswer is a = 10, b = 10
sp sum (a + b) = 10 + 10 = 20.

Answer 5

20

x*y=100
f(x,y) = x+y

your task is to minimize f
substitute x = 100/y into f(x,y)

f(x) = x+100/x

df/dx = 1 -100/x^2
for x = 10 ( or -10) you have that df/dx = 0
verify if this is an extreme and a minimum
verifying ....
yes it is

so x = 10 and thus y = 10 and the sum is 20.

Answer 6

the product of 2 postive numbers is 100

arrange it using equations (i have learnt calculus as yet)

let the two positive numbers be x and y respectively,

x+y=100
99.99 + 0.0011 = 100
so the value of y=0.0011 is the smallest possible

Answer 7

20 and i only have common sense as well. Sorry i couldn't help.