Question

Joseph would like to make sixteen pounds of a coffee blend at a cost of $6 per pound. He blends Ground Chicory at $5 a pound with Jamaican Blue Mountain at $9 per pound. How I pounds of each type of coffee should he use?

Answer 1

Explanation:

Let x be the number of pounds of Ground Chicory and y be the number of pounds of Jamaican Blue Mountain coffee. Since he wants to make a blend of 16 pounds, we have:

x + y = 16

The cost of the coffee blend should be $6 per pound, so the total cost of the blend is:

6(16) = $96

The cost of x pounds of Ground Chicory at $5 per pound and y pounds of Jamaican Blue Mountain coffee at $9 per pound is:

5x + 9y

We want to minimize the cost of the blend, subject to the constraint that the total weight is 16 pounds. This gives us the following optimization problem:

Minimize 5x + 9y

Subject to: x + y = 16

To solve this problem, we can use the method of Lagrange multipliers. The Lagrangian function is:

L(x, y, λ) = 5x + 9y + λ(x + y - 16)

Taking partial derivatives and setting them to zero, we get:

∂L/∂x = 5 + λ = 0

∂L/∂y = 9 + λ = 0

∂L/∂λ = x + y - 16 = 0

Solving these equations, we get:

λ = -5

x = 7

y = 9

Therefore, Joseph should use 7 pounds of Ground Chicory and 9 pounds of Jamaican Blue Mountain coffee to make a 16-pound blend that costs $6 per pound.