Sure, let's solve this problem step-by-step. We need to find two numbers [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that their difference is 90 ([tex]\( x - y = 90 \)[/tex]), and their product [tex]\( x \cdot y \)[/tex] is as small as possible.
1. Express one number in terms of the other:
Since [tex]\( x - y = 90 \)[/tex], we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = x - 90
\][/tex]
2. Formulate the product equation:
The product [tex]\( P \)[/tex] of the two numbers can be written as:
[tex]\[
P = x \cdot y = x \cdot (x - 90)
\][/tex]
3. Simplify the product equation:
So, we have:
[tex]\[
P = x^2 - 90x
\][/tex]
4. Find the minimum value of the product:
To minimize the quadratic function [tex]\( P = x^2 - 90x \)[/tex], we need to find the vertex of this parabola. The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
In our equation [tex]\( a = 1 \)[/tex] and [tex]\( b = -90 \)[/tex], so:
[tex]\[
x = -\frac{-90}{2 \cdot 1} = \frac{90}{2} = 45
\][/tex]
5. Calculate the value of [tex]\( y \)[/tex]:
Given [tex]\( x = 45 \)[/tex], we substitute it back into the equation [tex]\( y = x - 90 \)[/tex]:
[tex]\[
y = 45 - 90 = -45
\][/tex]
So, the two numbers whose difference is 90 and whose product is as small as possible are:
[tex]\[
45.0, -45.0
\][/tex]
