Question

Determine two positive values such that the sum of the first number and nine times the second number is 1419, and whose product is a maximum.

a) 100, 150
b) 120, 130
c) 130, 120
d) 150, 100

Answer 1

To find two positive values such that the sum of the first number and nine times the second number is 1419 and their product is a maximum, set up the equations x + 9y = 1419 and xy = maximum product. To maximize the product, find the vertex of the parabola formed by the second equation.

Step-by-step explanation:

To find two positive values such that the sum of the first number and nine times the second number is 1419 and their product is a maximum, we can set up the equations:

x + 9y = 1419

xy = maximum product

Simplifying the first equation, we get x = 1419 - 9y. Substituting this into the second equation, we have (1419 - 9y)y = maximum product

To maximize the product, we can find the vertex of the parabola formed by the second equation. Using the formula for the x-coordinate of the vertex, x = -b/2a, we have x = -(-9)/2(1) = 9/2 = 4.5. Substituting this back into the first equation, we get y = (1419 - 9(4.5))/9 = 157. Therefore, the two positive values are x = 4.5 and y = 157.

Answer 2

The pair of numbers with the maximum product is (c) 130, 120. Their product is 15,600, exceeding the other options.

Step-by-step explanation:

Let x be the first number and y be the second number. We have two equations:

x + 9y = 1419 (sum constraint)

P = xy (product to maximize)

We want to maximize P subject to the constraint in equation 1. Using

Lagrange multipliers, we set up the following equation:

L(x, y, λ) = xy - λ(x + 9y - 1419)

Taking partial derivatives and setting them equal to zero:

∂L/∂x = y - λ = 0

∂L/∂y = x - 9λ = 0

∂L/∂λ = -x - 9y + 1419 = 0

Solving this system of equations, we get x = 130 and y = 120. Checking the other options, we find their products to be:

(a) 100 * 150 = 15,000

(b) 120 * 130 = 15,600 (matches our maximum)

(d) 150 * 100 = 15,000

Therefore, (c) 130, 120 has the highest product of 15,600, making it the optimal solution.