Question

Matilda makes specialized chocolates. Her two best-selling chocolates, the chocolate heart and the chocolate flower, are made by pouring milk chocolate into a mold. A heart requires 3.5 ounces of milk chocolate and earns a profit of 20 cents, while a flower requires 1.5 ounces of milk chocolate and earns a profit of 30 cents. If she has 663 ounces of milk chocolate available and she wants to make at least twice as many hearts as flowers, what is the maximum profit she can make?

Maximum profit in dollars:

Answer 1

Let the numbers of the chocolate heart and the chocolate flower chocolates made by Matilda be x and y respectively.

Since Matilda wants to make at least twice as many hearts as flowers , hence x ≥ 2y or, y ≤ x/2 …(1)

A heart requires 3.5 ounces of milk chocolate, while a flower requires 1.5 ounces of milk chocolate. Since Matilda has 663 ounces of milk chocolate available, hence 3.5x+1.5y ≤ 663 , or, on multiplying both the sides by 2 ( to remove decimals), 7x+3y ≤ 1326 or, y ≤ -(7/3)x +442…(2).

The profit made by Matilda is P(x,y) = 0.20x + 0.30y…(3).

A graph of the lines y = x/2 ( in red) and y = -(7/3)x +442 ( in blue) is attached. The feasible region is on and below these 2 lines.

The 2 lines meet at the point (156,78) . Hence the conditions 1 and 2 are met when x = 156 and y = 78.

In such a case, the profit made by Matilda is 0.20*156+0.30*78 = 312+234= 546.

Thus, Matilda’s maximum profit is $ 546.