Can someone help me out with this! Thank you in advance :)
Frank’s Fruit Stand sells fresh fruit as well as fruit dipped in chocolate and covered with chopped
almonds. Each dipped banana requires 6 grams of chocolate and 3 grams of almonds to coat. It takes 6 grams
of chocolate and 4 grams of almonds to coat a dipped apple and 1gram of chocolate and 1 gram of almonds
to coat a dipped strawberry. Frank has as much fruit as he could possibly need to make the dipped bananas,
apples and strawberries but he would like to clear out his inventory of chocolate and almonds. If Frank has
60 grams of chocolate and 42 grams of almonds left, what combinations of dipped bananas, apples and
strawberries could Frank make in order to use all of his chocolate and almonds.
what do we know?
if we let
x = # dipped bananas
y = # dipped apples
z = # dipped strawberries
so, to produce such a mix, we need
6x+6y+z = 60
3x+4y+z = 42
equating z, we wind up with
3x+2y = 18
So, let's see what we can do, using only integer values:
x y z
0 9 6
2 6 12
4 3 18
6 0 24
Thanks Steve!! You're a life saver ;)
Katy
To find out the combinations of dipped bananas, apples, and strawberries that Frank could make to use all of his chocolate and almonds, we can use a mathematical approach called linear programming.
Linear programming involves setting up a system of linear inequalities based on the constraints of the problem and then optimizing an objective function. In this case, our objective is to use all of Frank's chocolate and almonds, and our constraints are the grams of chocolate and almonds required for each type of dipped fruit.
Let's assign some variables:
Let X be the number of dipped bananas.
Let Y be the number of dipped apples.
Let Z be the number of dipped strawberries.
Based on the information given, we can set up the following inequalities:
6X + 6Y + 1Z ≤ 60 (constraint for chocolate)
3X + 4Y + 1Z ≤ 42 (constraint for almonds)
X, Y, Z ≥ 0 (non-negative constraint)
The first two inequalities represent the amount of chocolate and almonds used for each type of dipped fruit, respectively. The last inequality ensures that the number of each type of fruit cannot be negative.
To solve this linear programming problem, we can graph the feasible region using the inequalities and find the corner points of the region. The corner points represent possible combinations of dipped bananas, apples, and strawberries that satisfy all the constraints.
Once we have the corner points, we can evaluate the objective function (in this case, just the sum of X, Y, and Z) for each corner point to determine the optimal solution, which is a combination that uses all of Frank's chocolate and almonds.
However, solving this graphically can be complex and time-consuming. Alternatively, we can use software or online resources that can handle linear programming problems, such as linear programming solver websites or software packages like Microsoft Excel Solver or GNU Linear Programming Kit.
By plugging in the given constraints and objective function into one of these tools, we can quickly find the combinations of dipped bananas, apples, and strawberries that Frank could make to use all of his chocolate and almonds.