To solve the problem of finding two positive numbers where the sum of the first number and twice the second number equals 56, while maximizing their product, we can approach it using algebra and calculus.
Let us define two positive numbers:
- x = the first number
- y = the second number
From the problem statement, we can create the following equation based on the given condition:
- Equation 1: x + 2y = 56
We need to maximize the product of these two numbers, represented by:
- Equation 2: P = x * y
Now, we can express x in terms of y using Equation 1:
- x = 56 – 2y
Next, we substitute this expression for x into Equation 2 to express the product solely in terms of y:
- P = (56 – 2y) * y
- P = 56y – 2y²
This is a quadratic equation in terms of y, and it will have a maximum point because the coefficient of y² is negative. The maximum value occurs at the vertex of the parabola, which can be found using the formula:
- y = -b/(2a>, where P = ay² + by + c. Here, a = -2 and b = 56.
Now, let’s calculate:
- y = -56/(2 * -2) = 56/4 = 14
Now that we have y, we can find x using Equation 1:
- x = 56 – 2 * 14 = 56 – 28 = 28
Thus, the two positive numbers that we are looking for are:
- x = 28
- y = 14
To conclude, the numbers 28 and 14 satisfy the conditions that their sum meets the requirement (28 + 2 * 14 = 56) and their product is maximized. Additionally, the maximum product is:
- P = 28 * 14 = 392
This provides not only the required values but also the highest product possible within the constraints given.