S8.3.7.3. whose sum is a minimum

In summary, the problem is to find two positive numbers whose product is 100 and whose sum is minimized. The sum function is $S = x + \frac{100}{x}$ and the derivative is $S'(x) = 1 - \frac{100}{x^2}$. The minimum value of the sum occurs at $x=10$.
  • #1
karush
Gold Member
MHB
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S8.3.7.3.
Find two positive numbers whose product is 100 and whose sum is a minimum
$x(100-x)=100x-x^2=100$

So far

Looks like it's 10+10=20Doing all my lockdown homework here
since I have no access to WiFi and a PC.
and just a tablet where overkeaf does not work
 
Last edited:
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  • #2
karush said:
S8.3.7.3.
Find two positive numbers whose product is 100 and whose sum is a minimum
$x(100-x)=100x-x^2=100$

So far

Looks like it's 10+10=20Doing all my lockdown homework here
since I have no access to WiFi and a PC.
and just a tablet where overkeaf does not work

The way you have defined the product implies that $\displaystyle x + y = 100 $, which it almost certainly doesn't.

You are told the product is 100, so $\displaystyle x\,y = 100 \implies y = \frac{100}{x} $.

The sum needs to be minimised, so your sum function is

$\displaystyle \begin{align*} S &= x + y \\ S &= x + \frac{100}{x} \end{align*} $

Now minimise the amount.
 
  • #3
$S'=\left(x + \dfrac{100}{x}\right)' = 1 - \dfrac{100}{x^2}$
So
$S'(0)=10$
 
  • #4
karush said:
$S'=\left(x + \dfrac{100}{x}\right)' = 1 - \dfrac{100}{x^2}$
So
$S'(0)=10$

No, $S'(0)$ is undefined ...

$S'(x) = 0$ at $x=10$
 

1. What does "S8.3.7.3. whose sum is a minimum" mean?

"S8.3.7.3. whose sum is a minimum" refers to a mathematical problem in which a set of numbers, represented by the letter S, must be added together in a specific way to achieve the smallest possible total.

2. How do you solve a problem involving "S8.3.7.3. whose sum is a minimum"?

To solve this type of problem, you must first identify the set of numbers represented by S. Then, use mathematical operations such as addition, subtraction, multiplication, and division to manipulate the numbers in a way that results in the smallest possible sum.

3. What is the significance of finding the minimum sum for "S8.3.7.3. whose sum is a minimum"?

Finding the minimum sum for this type of problem can be useful in various fields such as economics, engineering, and computer science. It allows for the optimization of resources and processes, leading to more efficient and effective solutions.

4. Are there any specific strategies or techniques for solving "S8.3.7.3. whose sum is a minimum" problems?

Yes, there are various strategies and techniques that can be used to solve these types of problems. Some common approaches include using algebraic equations, trial and error, and dynamic programming.

5. Can "S8.3.7.3. whose sum is a minimum" problems have multiple solutions?

Yes, it is possible for these types of problems to have multiple solutions. In some cases, there may be more than one way to manipulate the numbers in the set S to achieve the minimum sum. However, there may also be instances where there is only one unique solution.

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