Questions: Find two positive numbers satisfying the given requirements. The sum is 36 and the product is a maximum. smaller value larger value

Transcript text: Find two positive numbers satisfying the given requirements. The sum is 36 and the product is a maximum. smaller value $\square$ larger value $\square$

Solution

Solution Steps

To find two positive numbers whose sum is 36 and whose product is maximized, we can use the concept of symmetry in quadratic functions. The product of two numbers $ x $ and $ y $ is maximized when the numbers are equal, given a constant sum. Therefore, we set $ x = y $ and solve for $ x $ using the equation $ x + y = 36 $.

Step 1: Define the Variables

Let $ x $ and $ y $ be the two positive numbers. According to the problem, we have the following equations: \[ x + y = 36 \] \[ P = x \cdot y \] where $ P $ is the product that we want to maximize.

Step 2: Express One Variable in Terms of the Other

From the sum equation, we can express $ y $ in terms of $ x $: \[ y = 36 - x \]

Step 3: Substitute and Maximize the Product

Substituting $ y $ into the product equation gives: \[ P = x(36 - x) = 36x - x^2 \] This is a quadratic function in the standard form $ P = -x^2 + 36x $.

Step 4: Find the Vertex

The maximum value of a quadratic function $ ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $. Here, $ a = -1 $ and $ b = 36 $: \[ x = -\frac{36}{2 \cdot -1} = 18 \] Thus, $ y $ is also: \[ y = 36 - x = 36 - 18 = 18 \]