Question

The Banana Apple company sells its Golden, Macintosh, and Red Ida apples in mixes Box A costs 3€, Box B 3€ and Box C 4€. Box A contains 4 Golden, 2 Macintosh and 4 Red Ida; Box B contains 6 Golde, 2 Macintosh, and 2 Red Ida; and Box C contains no Golden, 6 Macintosh and 4 Red Ida apples. At the end of the season, the company has altogether 2800 Golden, 2200 Macintosh, and 2300 Red Ida apples left. Determine the number of each kind of boxes that the company should make to maximize their profit.

Answer 1

To solve this problem, we need to determine how many boxes of each type the company should make to maximize their profit. We can set up a system of equations to represent the number of apples in each box and the total number of apples the company has.

Let's call the number of boxes of type A x, the number of boxes of type B y, and the number of boxes of type C z. We can represent the number of apples in each box using the following equations:

x + y + z = 2800

4x + 6y + 0z = 2200

2x + 2y + 6z = 2300

To maximize the profit, the company should make as many boxes as possible while still having enough apples to fill all the boxes. We can solve this system of equations using substitution to find the values of x, y, and z that maximize the profit.

First, we can solve the first equation for z in terms of x and y:

z = 2800 - x - y

Next, we can substitute this expression for z into the second and third equations to eliminate z:

4x + 6y + 0(2800 - x - y) = 2200

4x + 6y = 2200

2x + 2y + 6(2800 - x - y) = 2300

2x + 2y + 17600 - 6x - 6y = 2300

2x + 2y = -15300

We can then solve this system of equations using substitution. From the second equation, we have:

4x + 6y = 2200

2x + 2y = -15300

We can solve the second equation for y in terms of x:

y = -15300/2 - x/2

We can substitute this expression for y into the first equation to solve for x:

4x + 6(-15300/2 - x/2) = 2200

4x - 45300 - 3x = 2200

x = (2200 + 45300) / (4 - 3)

x = 13600/1

x = 13600

We can then use this value of x to find the value of y:

y = -15300/2 - 13600/2

y = -13650

Finally, we can use these values of x and y to find the value of z:

z = 2800 - 13600 - (-13650)

z = 1650

Therefore, the company should make 13600 boxes of type A, -13650 boxes of type B, and 1650 boxes of type C to maximize their profit.

Explanation:

Answer 2

To maximize profit, the Banana Apple company should determine the number of each kind of apple box that they should make. To do this, they need to know the number of each type of apple that they have, as well as the price of each box.

Based on the information provided, the company has 2800 Golden apples, 2200 Macintosh apples, and 2300 Red Ida apples. This means that they can make 700 Box A's (2800 / 4 = 700), 1100 Box B's (2200 / 2 = 1100), and 575 Box C's (2300 / 4 = 575).

The price of each box is also given: Box A costs 3€, Box B costs 3€, and Box C costs 4€. This means that if the company makes 700 Box A's, they will earn 2100€ (700 x 3 = 2100€), if they make 1100 Box B's, they will earn 3300€ (1100 x 3 = 3300€), and if they make 575 Box C's, they will earn 2300€ (575 x 4 = 2300€).

To maximize profit, the company should make as many of the most profitable box as possible. In this case, the most profitable box is Box B, which earns 3€ per box. The company should make 1100 Box B's, which will earn them a total of 3300€.