Q: On a chicken farm, the poultry is given a
healthy diet to gain weight. The chickens have
to consume a minimum of 15 units of
Substance A and another 15 units of Substance
B. In the market there are only two classes of
compounds: Type X, with a composition of
one unit of A to five units of B, and another
type, Y, with a composition of five units of A
to one of B. The price of Type X is $10 and
Type Y, $30. What are the quantities of each
type of compound that have to be purchased to
cover the needs of the diet with a minimal cost?
Solve by using transition from graphically to
algebraic method.
quationer
Nothing now
To solve this problem using a transition from graphical to algebraic method, we can start by representing the problem on a graph. Let's use the x-axis to represent the number of Type X compounds and the y-axis to represent the number of Type Y compounds.
The first condition we need to satisfy is that the chickens have to consume a minimum of 15 units of Substance A. Since Type X has a composition of 1 unit of A to 5 units of B, the number of units of A provided by Type X compounds can be represented as 5x, where x is the number of Type X compounds. Similarly, the number of units of A provided by Type Y compounds can be represented as 5y, where y is the number of Type Y compounds. Therefore, the first condition can be represented as:
5x + 5y ≥ 15
Simplifying this equation, we get:
x + y ≥ 3
The second condition is that the chickens have to consume a minimum of 15 units of Substance B. Since Type X has a composition of 5 units of B to 1 unit of A, the number of units of B provided by Type X compounds can be represented as x/5. Similarly, the number of units of B provided by Type Y compounds can be represented as y/5. Therefore, the second condition can be represented as:
(x/5) + (y/5) ≥ 3
Simplifying this equation, we get:
x + y ≥ 15
Now, let's represent these conditions on a graph. Start by plotting the lines x + y = 3 and x + y = 15. These lines represent the boundary lines of the inequalities. Shade the region above both lines because we need x + y to be greater than or equal to 3 and 15. The shaded region represents all the possible combinations of Type X and Type Y compounds that satisfy the conditions.
Next, we need to plot the cost of the compounds. Type X costs $10 and Type Y costs $30. We want to minimize the cost, so we need to find the minimum point within the shaded region.
To find the minimum point, we can use the concept of vertices. The vertices are the points where the two boundary lines intersect. In this case, we have two vertices: (15, 0) and (0, 15). We can calculate the cost of each vertex by plugging the values into the cost equation.
For the vertex (15, 0):
Cost = (15 * $10) + (0 * $30) = $150
For the vertex (0, 15):
Cost = (0 * $10) + (15 * $30) = $450
Comparing the costs, we can see that the minimum cost is $150, which corresponds to the vertex (15, 0).
Therefore, to cover the needs of the diet with minimal cost, we need to purchase 15 units of Type X compounds and 0 units of Type Y compounds.
Please note that the graph and solution provided are based on the information provided in the question. If there are additional constraints or information, the solution may differ.
well, the cost function is c = 10x+30y
Now, what are the constraints?