**Problem Statement:**Find two positive integers such that the sum of the first number and four times the second number is 1000, and the product of the two numbers is as large as possible. Show all of your work.---To solve this problem, we need to:1. **Define Variables:** - Let \( x \) be the first number. - Let \( y \) be the second number.2. **Set Up the Equation:** - The equation given by the problem is: \[ x + 4y = 1000 \] - We need to maximize the product \( P = xy \).3. **Express \( x \) in Terms of \( y \):** - From the equation \( x + 4y = 1000 \), solve for \( x \): \[ x = 1000 - 4y \]4. **Substitute into the Product Expression:** - Substitute \( x = 1000 - 4y \) into \( P = xy \): \[ P = (1000 - 4y)y = 1000y - 4y^2 \]5. **Find the Maximum Value of the Product:** - The expression \( P = 1000y - 4y^2 \) is a quadratic equation, which is concave down (since the coefficient of \( y^2 \) is negative). - The maximum value occurs at the vertex. For a quadratic equation \( ay^2 + by + c \), the vertex is at \( y = -\frac{b}{2a} \).6. **Calculate the Vertex:** - Identify coefficients: \( a = -4 \), \( b = 1000 \). - Compute: \[ y = -\frac{1000}{2(-4)} = \frac{1000}{8} = 125 \]7. **Determine \( x \):** - Substitute \( y = 125 \) back into \( x = 1000 - 4y \): \[ x = 1000 - 4(125) = 1000 - 500 = 500 \]8. **Solution:** - The two numbers are \( x = 500 \) and

Name: **Problem Statement:**Find two positive integers such that the sum of
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: **Problem Statement:**Find two positive integers such that the sum of the first number and four times the second number is 1000, and the product of the two numbers is as large as possible. Show all of your work.---To solve this problem, we need to:1