Among all pairs of numbers whose difference is 20, find a pair whose product is as small as possible. what is the minimum product?

the pair of numbers whose difference is 20 and whose product is as small as possible is

We want to find two numbers such that their difference is 20 and whose product is minimized.

The pair of numbers is 10 and -10.

Let's define our two numbers as A and B.

Such that A < B.

We want the difference to be equal to 20, then we have:

B - A  = 20.

The product between these two numbers is written as:

P = A*B

From the first equation, we can write:

B = 20 + A

Now we can replace that on the product equation to get:

P = A*(20 + A) = 20*A + A^2

Now we want to minimize this, notice that this is a quadratic polynomial, thus the minimum is at the vertex.

For a general quadratic polynomial.

y = a*x^2 + b*x + c

The vertex is at:

x = -b/2a

So in our case, P = 20*A + A^2

The vertex is at:

A = -20/(2*1) = -10

Then we have A = -10

To find the value of B we use the first equation:

B - A = 20

B = 20 + A = 20 + (-10) = 10

Then the pair of numbers is A = -10 and B = 10, and the product is:

P = 10*-10 = -100.

If you want to learn more, you can read:

link

-100

-10, 10

Step-by-step explanation:

Let the smaller number be x.

Then the larger number is x + 20.

The product is x(x + 20).

Now you can write the function

y = x(x + 20)

y = x^2 + 20x

Take the first derivative of y with respect to x.

y' = 2x + 20

Set the first derivative equal to zero to find the x value for the minimum value of the function.

2x + 20 = 0

2x = -20

x = -10

The minimum value of the function occurs at x = -10. -10 is one of the two numbers.

y = x^2 + 20x

For x = -10,

y = (-10)^2 + 20(-10)

y = 100 - 200

y = -100

The minimum value of the product is -100.

x = -10

x + 20 = -10 + 20 = 10

The numbers are -10 and 10.

If you have not learned derivatives yet, then plot the function

y = x^2 + 20x

Now look at the graph and find the minimum y value and the x value at which it occurs.

The minimum y value is -100. That is the minimum product you are looking for.

The x value of the minimum function value is x = -10.

Then x + 20 = -10 + 20 = 10.

The numbers are -10 and 10.

7

step-by-step explanation:

the answer would be 7

vwhat effect does the word head have on the meaning of this passage?

a. it demonstrates how intelligent he was.

b. it shows he was a favorite of the teachers.

c. it indicates he would be a good preacher.

d. it reveals how self-important he was.

how does the choice of the word head contribute to the passage’s tone?

a. it offers a sense of entitlement.

step-by-step explanation: